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介数中心性

2020-12-11T17:52:02Z

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[[File:Graph betweenness.svg|thumb|upright=1.3|An [[undirected graph]] colored based on the betweenness centrality of each vertex from least (red) to greatest (blue).]]

An [[undirected graph colored based on the betweenness centrality of each vertex from least (red) to greatest (blue).]]

根据每个顶点的介数中心性从最小(红色)到最大(蓝色)着色的无向图。

In [[graph theory]], '''betweenness centrality''' is a measure of [[centrality]] in a [[Graph (discrete mathematics)|graph]] based on [[Shortest path problem|shortest paths]]. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices such that either the number of edges that the path passes through (for unweighted graphs) or the sum of the weights of the edges (for weighted graphs) is minimized. The betweenness centrality for each [[Vertex (graph theory)|vertex]] is the number of these shortest paths that pass through the vertex.

In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices such that either the number of edges that the path passes through (for unweighted graphs) or the sum of the weights of the edges (for weighted graphs) is minimized. The betweenness centrality for each vertex is the number of these shortest paths that pass through the vertex.

在图论中，''' 介数中心性Betweenness centrality'''是基于最短路径的图中心性的一种度量。对于连通图中的每一对节点，在节点之间至少存在一条最短路径，使得路径通过的边数(对于未加权图)或者边权重的和(对于加权图)最小。节点的中心性是经过该节点的最短路径的数量。

Betweenness centrality was devised as a general measure of centrality:{{Sfnp|Freeman|1977|p=39}} it applies to a wide range of problems in network theory, including problems related to social [[network theory|networks]], biology, transport and scientific cooperation. Although earlier authors have intuitively described centrality as based on betweenness, {{Harvp|Freeman|1977}} gave the first formal definition of betweenness centrality.

Betweenness centrality was devised as a general measure of centrality: it applies to a wide range of problems in network theory, including problems related to social networks, biology, transport and scientific cooperation. Although earlier authors have intuitively described centrality as based on betweenness, gave the first formal definition of betweenness centrality.

''' 介数中心性Betweenness centrality'''的提出为中心性的衡量提供了一般的标准: 它适用于网络理论中广泛的问题，包括与社会网络、生物学、交通和科学合作有关的问题。尽管早期的学者直观地将中心性描述为基于介数，但是给出了''' 介数中心性Betweenness centrality'''的第一个正式定义。
--[[用户:小趣木木|小趣木木]]（[[用户讨论:小趣木木|讨论]]）'' 介数中心性Betweenness centrality'''被设计为中心性的一般衡量标准: 这里不能照搬机器翻译重新翻译

Betweenness centrality finds wide application in [[network theory]]; it represents the degree to which nodes stand between each other. For example, in a [[telecommunications network]], a node with higher betweenness centrality would have more control over the network, because more information will pass through that node.

Betweenness centrality finds wide application in network theory; it represents the degree to which nodes stand between each other. For example, in a telecommunications network, a node with higher betweenness centrality would have more control over the network, because more information will pass through that node.

''' 介数中心性Betweenness centrality'''在网络理论中有着广泛的应用，它代表了节点之间相互独立的程度。例如，在远程通信网络中，具有较高''' 介数中心性Betweenness centrality'''的节点将对网络有更多的控制，因为有更多的信息会通过该节点。

--[[用户:小趣木木|小趣木木]]（[[用户讨论:小趣木木|讨论]]）应该统一名词表达 node节点/顶点上文出现顶点，应该全文一致

== Definition定义 ==

The betweenness centrality of a node <math>v</math> is given by the expression:

The betweenness centrality of a node <math>v</math> is given by the expression:

一个节点 < math > v </math > 的''' 介数中心性Betweenness centrality'''通过以下表达式给出:

:<math>g(v)= \sum_{s \neq v \neq t}\frac{\sigma_{st}(v)}{\sigma_{st}}</math>

where <math>\sigma_{st}</math> is the total number of shortest paths from node <math>s</math> to node <math>t</math> and <math>\sigma_{st}(v)</math> is the number of those paths that pass through <math>v</math>.

where <math>\sigma_{st}</math> is the total number of shortest paths from node <math>s</math> to node <math>t</math> and <math>\sigma_{st}(v)</math> is the number of those paths that pass through <math>v</math>.

其中 < math > sigma { st } </math > 是从节点 < math > s </math > 到节点 < math > t </math >的最短路径总数， < math > sigma { st }(v) </math >是其中通过节点<math>v</math>的路径数量。

Note that the betweenness centrality of a node scales with the number of pairs of nodes as suggested by the summation indices. Therefore, the calculation may be rescaled by dividing through by the number of pairs of nodes not including <math>v</math>, so that <math>g \in [0,1]</math>. The division is done by <math>(N-1)(N-2)</math> for directed graphs and <math>(N-1)(N-2)/2</math> for undirected graphs, where <math>N</math> is the number of nodes in the giant component. Note that this scales for the highest possible value, where one node is crossed by every single shortest path. This is often not the case, and a normalization can be performed without a loss of precision

Note that the betweenness centrality of a node scales with the number of pairs of nodes as suggested by the summation indices. Therefore, the calculation may be rescaled by dividing through by the number of pairs of nodes not including <math>v</math>, so that <math>g \in [0,1]</math>. The division is done by <math>(N-1)(N-2)</math> for directed graphs and <math>(N-1)(N-2)/2</math> for undirected graphs, where <math>N</math> is the number of nodes in the giant component. Note that this scales for the highest possible value, where one node is crossed by every single shortest path. This is often not the case, and a normalization can be performed without a loss of precision

注意，如公式中求和指标所示，一个节点的''' 介数中心性Betweenness centrality'''与节点对的数量成比例缩放。因此，计算可以通过除以不包括 < math > v </math > 的节点对数来重新标定，使得<math>g \in [0,1]</math>。有向图是通过除以 < math > (N-1)(N-2) </math > 来实现的，而无向图是通过除以 < math > (N-1)(N-2)/2 </math >来实现的，其中 < math > n </math > 是巨组元中的节点数。请注意，当每一条最短路径都通过一个节点时，这个节点达到可以缩放的最大可能值。事实往往并非如此，可以在不损失精度的情况下执行标准化

--[[用户:小趣木木|小趣木木]]（[[用户讨论:小趣木木|讨论]]）有向图的除法是通过 < math > (N-1)(N-2) </math > 来完成的，而 < math > (N-1)(N-2)/2 </math > 的除法是通过无向图来完成的，两句可以采用相同的句式

:<math>\mbox{normal}(g(v)) = \frac{g(v) - \min(g)}{\max(g) - \min(g)}</math>

which results in:

which results in:

结果是:

:<math>\max(normal) = 1</math>

:<math>\min(normal) = 0</math>

Note that this will always be a scaling from a smaller range into a larger range, so no precision is lost.

Note that this will always be a scaling from a smaller range into a larger range, so no precision is lost.

请注意，这种方法将始终是从较小范围到较大范围的缩放，因此不会丢失精度。



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== Weighted networks 加权网络==

In a weighted network the links connecting the nodes are no longer treated as binary interactions, but are weighted in proportion to their capacity, influence, frequency, etc., which adds another dimension of heterogeneity within the network beyond the topological effects. A node's strength in a weighted network is given by the sum of the weights of its adjacent edges.

In a weighted network the links connecting the nodes are no longer treated as binary interactions, but are weighted in proportion to their capacity, influence, frequency, etc., which adds another dimension of heterogeneity within the network beyond the topological effects. A node's strength in a weighted network is given by the sum of the weights of its adjacent edges.

在加权网络中，连接节点的链路不再被视为二元的相互作用，而是根据其容量、影响力、频率等按比例加权，这在拓扑效应之外增加了网络内另一个异质性的维度。一个加权网络中的节点的强度是由其相邻边的权重之和来表示的。

:<math>s_{i} = \sum_{j=1}^{N} a_{ij}w_{ij}</math>

With <math>a_{ij}</math> and <math>w_{ij}</math> being adjacency and weight matrices between nodes <math>i</math> and <math>j</math>, respectively.

With <math>a_{ij}</math> and <math>w_{ij}</math> being adjacency and weight matrices between nodes <math>i</math> and <math>j</math>, respectively.

用 < math > a { ij } </math > 和 < math > w { ij } </math > 分别作为 < math > i </math > 和 < math > < j </math > 节点之间的邻接矩阵和权值矩阵。

Analogous to the power law distribution of degree found in scale free networks, the strength of a given node follows a power law distribution as well.

Analogous to the power law distribution of degree found in scale free networks, the strength of a given node follows a power law distribution as well.

类似于''' 无标度网络Scale free networks'''中度的''' 幂律分布Power law distribution'''，节点的强度也服从''' 幂律分布Power law distribution'''。

:<math>s(k) \approx k^\beta </math>

A study of the average value <math>s(b)</math> of the strength for vertices with betweenness <math>b</math> shows that the functional behavior can be approximated by a scaling form <ref name="Barrat">A. Barrat, M. Barthelemy, R. Pastor-Satorras, and A. Vespignani. The architecture of complex weighted networks. PNAS (2004) vol. 101 no. 11</ref>

A study of the average value <math>s(b)</math> of the strength for vertices with betweenness <math>b</math> shows that the functional behavior can be approximated by a scaling form

对具有介数<math>b</math>的节点强度的平均值<math>s(b)</math>的研究表明，功能行为可以用缩放形式来近似

:<math>s(b)\approx b^{\alpha} </math>

:

:

:

==''' Percolation centrality渗流中心性'''==

Percolation centrality is a version of weighted betweenness centrality, but it considers the 'state' of the source and target nodes of each shortest path in calculating this weight. Percolation of a ‘contagion’ occurs in complex networks in a number of scenarios. For example, viral or bacterial infection can spread over social networks of people, known as contact networks. The spread of disease can also be considered at a higher level of abstraction, by contemplating a network of towns or population centres, connected by road, rail or air links. Computer viruses can spread over computer networks. Rumours or news about business offers and deals can also spread via social networks of people. In all of these scenarios, a ‘contagion’ spreads over the links of a complex network, altering the ‘states’ of the nodes as it spreads, either recoverably or otherwise. For example, in an epidemiological scenario, individuals go from ‘susceptible’ to ‘infected’ state as the infection spreads. The states the individual nodes can take in the above examples could be binary (such as received/not received a piece of news), discrete (susceptible/infected/recovered), or even continuous (such as the proportion of infected people in a town), as the contagion spreads. The common feature in all these scenarios is that the spread of contagion results in the change of node states in networks. Percolation centrality (PC) was proposed with this in mind, which specifically measures the importance of nodes in terms of aiding the percolation through the network. This measure was proposed by Piraveenan et al.<ref name="piraveenan2013">{{cite journal |last1 = Piraveenan |first1 = Mahendra | year=2013| title = Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks | journal = PLOS ONE | volume=8 | issue=1 | doi=10.1371/journal.pone.0053095 | pages=e53095 | pmid=23349699 | pmc=3551907| bibcode=2013PLoSO...853095P }}</ref>

Percolation centrality is a version of weighted betweenness centrality, but it considers the 'state' of the source and target nodes of each shortest path in calculating this weight. Percolation of a ‘contagion’ occurs in complex networks in a number of scenarios. For example, viral or bacterial infection can spread over social networks of people, known as contact networks. The spread of disease can also be considered at a higher level of abstraction, by contemplating a network of towns or population centres, connected by road, rail or air links. Computer viruses can spread over computer networks. Rumours or news about business offers and deals can also spread via social networks of people. In all of these scenarios, a ‘contagion’ spreads over the links of a complex network, altering the ‘states’ of the nodes as it spreads, either recoverably or otherwise. For example, in an epidemiological scenario, individuals go from ‘susceptible’ to ‘infected’ state as the infection spreads. The states the individual nodes can take in the above examples could be binary (such as received/not received a piece of news), discrete (susceptible/infected/recovered), or even continuous (such as the proportion of infected people in a town), as the contagion spreads. The common feature in all these scenarios is that the spread of contagion results in the change of node states in networks. Percolation centrality (PC) was proposed with this in mind, which specifically measures the importance of nodes in terms of aiding the percolation through the network. This measure was proposed by Piraveenan et al.

''' Percolation centrality渗流中心性'''是加权的''' 介数中心性Betweenness centrality'''的一种形式，但它在计算该权重时考虑了每条最短路径的源节点和目标节点的“状态”。在许多情况下，复杂网络中都会出现“传染”的渗流现象。例如，病毒或细菌感染可以通过人们的社交网络传播，也就是所谓的接触网络。还可以在更高的抽象层次上考虑疾病的传播问题，设想通过公路、铁路或空中连接起来的城镇或人口中心网络。计算机病毒可以通过计算机网络传播。关于商业活动和交易的谣言或新闻也可以通过人们的社交网络传播。在所有这些情况下，一种“传染病”在一个复杂网络的链接上传播，随着它的传播，无论是可恢复的还是不可恢复的，都会改变节点的“状态”。例如，在流行病学情况下，随着传染病的扩散，个体从“易感”状态转变为“感染”状态。在上面的例子中，随着传染病的传播，每个节点的状态可以是二元的（如接收/没有接收到一条信息）、离散的（易感/受感染/康复），甚至是连续的（如一个城镇中受感染的人群比例）。这些情景的共同特点是，传染病的传播导致网络中节点状态的改变。''' Percolation centrality渗流中心性'''(PC)就是基于这个思想而提出的，它衡量了节点在帮助网络渗流方面的重要性。这个测度是由 Piraveenan 等人提出的。<ref name="piraveenan2013">{{cite journal |last1 = Piraveenan |first1 = Mahendra | year=2013| title = Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks | journal = PLOS ONE | volume=8 | issue=1 | doi=10.1371/journal.pone.0053095 | pages=e53095 | pmid=23349699 | pmc=3551907| bibcode=2013PLoSO...853095P }}</ref>
--[[用户:小趣木木|小趣木木]]（[[用户讨论:小趣木木|讨论]]）在所有这些情况下，一种“传染病”在一个复杂网络的链接上传播，随着它的传播，无论是可恢复的还是不可恢复的，都会改变节点的“状态”。例如，在流行病学方案中，随着感染扩散，个人从”易感”状态转变为”受感染”状态。在上面的例子中，每个节点可以采取的状态可以是二进制的(例如接收/没有接收到一条新闻)、离散的(易感/受感染/康复) ，甚至是连续的(例如一个城镇中受感染的人的比例) ，随着传染的扩散。这些情景的共同特点是，传染的扩散导致网络中节点状态的改变。该段还可以再精简优化一下。

Percolation centrality is defined for a given node, at a given time, as the proportion of ‘percolated paths’ that go through that node. A ‘percolated path’ is a shortest path between a pair of nodes, where the source node is percolated (e.g., infected). The target node can be percolated or non-percolated, or in a partially percolated state.

Percolation centrality is defined for a given node, at a given time, as the proportion of ‘percolated paths’ that go through that node. A ‘percolated path’ is a shortest path between a pair of nodes, where the source node is percolated (e.g., infected). The target node can be percolated or non-percolated, or in a partially percolated state.

''' Percolation centrality渗流中心性'''定义了在给定时间内，通过一个给定节点的渗流路径的比例。“渗流路径”是一对节点之间的最短路径，其中源节点被渗流的（例如，被感染）。目标节点可以是渗流的或非渗流的，或处于部分渗流状态。

:<math>PC^t(v)= \frac{1}{N-2}\sum_{s \neq v \neq r}\frac{\sigma_{sr}(v)}{\sigma_{sr}}\frac{{x^t}_s}{{\sum {[{x^t}_i}]}-{x^t}_v}</math>

where <math>\sigma_{sr}</math> is total number of shortest paths from node <math>s</math> to node <math>r</math> and <math>\sigma_{sr}(v)</math> is the number of those paths that pass through <math>v</math>. The percolation state of the node <math>i</math> at time <math>t</math> is denoted by <math>{x^t}_i</math> and two special cases are when <math>{x^t}_i=0</math> which indicates a non-percolated state at time <math>t</math> whereas when <math>{x^t}_i=1</math> which indicates a fully percolated state at time <math>t</math>. The values in between indicate partially percolated states ( e.g., in a network of townships, this would be the percentage of people infected in that town).

where <math>\sigma_{sr}</math> is total number of shortest paths from node <math>s</math> to node <math>r</math> and <math>\sigma_{sr}(v)</math> is the number of those paths that pass through <math>v</math>. The percolation state of the node <math>i</math> at time <math>t</math> is denoted by <math>{x^t}_i</math> and two special cases are when <math>{x^t}_i=0</math> which indicates a non-percolated state at time <math>t</math> whereas when <math>{x^t}_i=1</math> which indicates a fully percolated state at time <math>t</math>. The values in between indicate partially percolated states ( e.g., in a network of townships, this would be the percentage of people infected in that town).

其中， < math >\sigma_{sr}</math > 是从节点 < math > s </math > 到节点 < math > r </math > 的最短路径的数量总数， < math > \sigma_{sr}(v) </math > 是其中通过节点 < math > v </math > 的路径总数。在时间 < math > t </math > 时，节点的渗流状态用 < math > {x^t}_i </math > 表示，两个特殊情况是当 < math > {x^t}_i=0</math >，表示在时间<math>t</math>时是非渗流状态，而当 <math>{x^t}_i=1</math>时，表示在时间<math>t</math>时是完全渗流状态。两者之间的值表示部分渗流状态（例如，在一个城镇网络中，这是该城镇感染者的百分比）。

The attached weights to the percolation paths depend on the percolation levels assigned to the source nodes, based on the premise that the higher the percolation level of a source node is, the more important are the paths that originate from that node. Nodes which lie on shortest paths originating from highly percolated nodes are therefore potentially more important to the percolation. The definition of PC may also be extended to include target node weights as well. Percolation centrality calculations run in [[Big O notation|<math>O(NM)</math>]] time with an efficient implementation adopted from Brandes' fast algorithm and if the calculation needs to consider target nodes weights, the worst case time is [[Big O notation|<math>O(N^3)</math>]].

The attached weights to the percolation paths depend on the percolation levels assigned to the source nodes, based on the premise that the higher the percolation level of a source node is, the more important are the paths that originate from that node. Nodes which lie on shortest paths originating from highly percolated nodes are therefore potentially more important to the percolation. The definition of PC may also be extended to include target node weights as well. Percolation centrality calculations run in <math>O(NM)</math> time with an efficient implementation adopted from Brandes' fast algorithm and if the calculation needs to consider target nodes weights, the worst case time is <math>O(N^3)</math>.

渗流路径的权重取决于分配给源节点的渗流水平，前提是源节点的渗流水平越高，源节点的路径就越重要。因此，位于源自高渗流节点的最短路径上的节点可能对渗流更为重要。PC 的定义也可以扩展到包括目标节点的权值。当采用Brandes的快速算法计算''' Percolation centrality渗滤中心性'''时，计算时间复杂度为 < math > O(NM) </math >，如果计算需要考虑目标节点的权值，最高的计算时间复杂度为 < math > O(N^3) </math > 。

== Algorithms 算法==

Calculating the betweenness and closeness centralities of all the [[Vertex (graph theory)|vertices]] in a graph involves calculating the shortest paths between all pairs of vertices on a graph, which takes [[big theta|<math>\Theta(|V|^3)</math>]] time with the [[Floyd–Warshall algorithm]], modified to not only find one but count all shortest paths between two nodes. On a sparse graph, [[Johnson's algorithm]] or Brandes' algorithm may be more efficient, both taking [[Big O notation|<math>O(|V|^2 \log |V| + |V| |E|)</math>]] time. On unweighted graphs, calculating betweenness centrality takes [[Big O notation|<math>O(|V| |E|)</math>]] time using Brandes' algorithm.{{Sfnp|Brandes|2001|p=1}}

Calculating the betweenness and closeness centralities of all the vertices in a graph involves calculating the shortest paths between all pairs of vertices on a graph, which takes ^3)</math> time with the Floyd–Warshall algorithm, modified to not only find one but count all shortest paths between two nodes. On a sparse graph, Johnson's algorithm or Brandes' algorithm may be more efficient, both taking )</math> time. On unweighted graphs, calculating betweenness centrality takes )</math> time using Brandes' algorithm.

计算一个图中所有顶点的介数和''' 紧密中心性Closeness centralities'''涉及到计算图中所有节点对之间的最短路径，使用 Floyd-Warshall 算法时，这需要[[big theta|<math>\Theta(|V|^3)</math>]]的时间复杂度，改进后的算法不仅可以找到一条路径，还可以计算两个节点之间的所有最短路径。在稀疏图上，Johnson算法或Brandes算法可能更有效率，两者的时间复杂度为[[Big O notation|<math>O(|V|^2 \log |V| + |V| |E|)</math>]]。在未加权图上，使用 Brandes 算法计算''' 介数中心性Betweenness centrality'''需要的时间复杂度为[[Big O notation|<math>O(|V| |E|)</math>]]。{{Sfnp|Brandes|2001|p=1}}

In calculating betweenness and closeness centralities of all vertices in a graph, it is assumed that graphs are undirected and connected with the allowance of loops and multiple edges. When specifically dealing with network graphs, often graphs are without loops or multiple edges to maintain simple relationships (where edges represent connections between two people or vertices). In this case, using Brandes' algorithm will divide final centrality scores by 2 to account for each shortest path being counted twice.{{Sfnp|Brandes|2001|p=9}}

In calculating betweenness and closeness centralities of all vertices in a graph, it is assumed that graphs are undirected and connected with the allowance of loops and multiple edges. When specifically dealing with network graphs, often graphs are without loops or multiple edges to maintain simple relationships (where edges represent connections between two people or vertices). In this case, using Brandes' algorithm will divide final centrality scores by 2 to account for each shortest path being counted twice.

在计算一个图的所有节点的介数和''' 紧密中心性Closeness centralities'''时，假定图是无向的，并且图在允许自环和多边时是连通的。当专门处理网络图时，图通常没有自环或多边来维持简单的关系（其中的边表示两个人或节点之间的联系）。在这种情况下，使用 Brandes 算法将最终的中心性除以2，因为每条最短路径被计算两次。

Another algorithm generalizes the Freeman's betweenness computed on geodesics and Newman's betweenness computed on all paths, by introducing a hyper-parameter controlling the trade-off between exploration and exploitation. The time complexity is the number of edges times the number of nodes in the graph.{{Sfnp|Mantrach|2010}}

Another algorithm generalizes the Freeman's betweenness computed on geodesics and Newman's betweenness computed on all paths, by introducing a hyper-parameter controlling the trade-off between exploration and exploitation. The time complexity is the number of edges times the number of nodes in the graph.

另一个算法通过引入一个超参数来控制勘探和开发之间的平衡，将计算测地线的弗里曼（Freeman）介数和所有路径上计算纽曼（Newman）介数的结果进行了推广。时间复杂度是图中的边数乘以节点数。{{Sfnp|Mantrach|2010}}

The concept of centrality was extended to a group level as well.<ref name=group>Puzis, R., Yagil, D., Elovici, Y., Braha, D. (2009)[http://necsi.edu/affiliates/braha/Internet_Research_Anonimity.pdf Collaborative attack on Internet users’ anonymity] {{Webarchive|url=https://web.archive.org/web/20131207133417/http://necsi.edu/affiliates/braha/Internet_Research_Anonimity.pdf |date=2013-12-07 }}, ''Internet Research'' '''19'''(1)</ref> Group betweenness centrality shows the proportion of geodesics connecting pairs of non-group members that pass through a group of nodes. Brandes' algorithm for computing the betweenness centrality of all vertices was modified to compute the group betweenness centrality of one group of nodes with the same asymptotic running time.<ref name="group"/>

The concept of centrality was extended to a group level as well. Group betweenness centrality shows the proportion of geodesics connecting pairs of non-group members that pass through a group of nodes. Brandes' algorithm for computing the betweenness centrality of all vertices was modified to compute the group betweenness centrality of one group of nodes with the same asymptotic running time.

中心性的概念也扩展到了群体层次。<ref name=group>Puzis, R., Yagil, D., Elovici, Y., Braha, D. (2009)[http://necsi.edu/affiliates/braha/Internet_Research_Anonimity.pdf Collaborative attack on Internet users’ anonymity] {{Webarchive|url=https://web.archive.org/web/20131207133417/http://necsi.edu/affiliates/braha/Internet_Research_Anonimity.pdf |date=2013-12-07 }}, ''Internet Research'' '''19'''(1)</ref>群体介数中心性反映了通过一组节点连接非组成员节点对的测地线所占的比例。修改了计算所有节点之间的介数中心性的Brandes算法，以计算具有相同渐近运行时间的一组节点之间的群体介数中心性。

== Related concepts相关概念 ==

''Betweenness centrality'' is related to a network's [[Connectivity (graph theory)|connectivity]], in so much as high betweenness vertices have the potential to disconnect graphs if removed (see [[Cut (graph theory)|cut set]]) .

Betweenness centrality is related to a network's connectivity, in so much as high betweenness vertices have the potential to disconnect graphs if removed (see cut set) .

''' 介数中心性Betweenness centrality'''与网络的连通性有关，如果高介数顶点被移除，就有可能断开图的连接（参见割集）。

== See also另请参阅==

* [[Centrality]]

*[[中心性]]

== Notes 备注==

{{Reflist}}

== References 参考文献==

{{Refbegin|40em}}

*{{cite journal | last1=Brandes | first1=Ulrik | year=2001 | journal=Journal of Mathematical Sociology | volume=25 | issue=2 | pages=163–177 | title=A faster algorithm for betweenness centrality | url=https://algo.uni-konstanz.de/publications/b-fabc-01.pdf | ref=harv | doi=10.1080/0022250x.2001.9990249 | citeseerx=10.1.1.11.2024 | access-date=2018-11-18 | archive-url=https://web.archive.org/web/20171013152036/http://algo.uni-konstanz.de/publications/b-fabc-01.pdf | archive-date=2017-10-13 | url-status=dead }}

*{{cite journal

|last1 = Freeman

1 = Freeman

|first1 = Linton

1 = Linton

|authorlink = Linton Freeman

| year=1977

| title = A set of measures of centrality based on betweenness

| journal = Sociometry

| volume=40

|issue = 1

| pages=35–41

|doi=10.2307/3033543|ref=harv|jstor = 3033543

}}

*{{cite journal

|title=Universal Behavior of Load Distribution in Scale-Free Networks

|last1=Goh

|first1=K. I.

|last2=Kahng

|first2=B.

|last3=Kim

|first3=D.|journal=Physical Review Letters

|volume=87

|issue=27

|pages=278701

|date=2001

|doi=10.1103/physrevlett.87.278701|ref=harv

|bibcode=2001PhRvL..87A8701G

|arxiv=cond-mat/0106565

}}

*{{cite journal |last1=Mantrach |first1=Amin | title = The sum-over-paths covariance kernel: A novel covariance measure between nodes of a directed graph|journal= IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=32|issue=6|pages=1112–1126|year=2010 |display-authors=etal|ref=harv |doi=10.1109/tpami.2009.78}}

*{{cite journal|last1=Moxley|first1=Robert L.|last2=Moxley|first2=Nancy F.|title=Determining Point-Centrality in Uncontrived Social Networks|journal=Sociometry|date=1974|volume=37|issue=1|pages=122–130|doi=10.2307/2786472|ref=harv|jstor=2786472}}

*{{cite book

|last1=Newman

|first1= M. E. J.

|year=2010

|title=Networks: An Introduction

|place=Oxford, UK

|publisher=Oxford University Press

|isbn=978-0199206650

|ref=harv

}}

*{{cite journal|last1=Dolev|first1=Shlomi|last2=Elovici|first2=Yuval|last3=Puzis|first3=Rami|title=Routing betweenness centrality|journal=J. ACM|date=2010|volume=57|issue=4|pages=25:1–25:27|doi=10.1145/1734213.1734219}}

{{Refend}}

[[Category:Network theory]]

[[Category:Graph invariants]]

<noinclude>

This page was moved from [[wikipedia:en:Betweenness centrality]]. Its edit history can be viewed at [[介数中心性/edithistory]]</noinclude>

[[Category:待整理页面]]

沙堆模型

2020-11-15T02:34:04Z

Zcy：

此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。此词条由Zcy初步审校。
{{short description|Cellular automaton}}
[[File:Sandpile identity 300x205.png|upright=1.25|thumb|The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.]]

[[File:Sandpile identity 300x205.png|upright=1.25|thumb|沙堆在矩形网格上的标识。黄色像素对应三颗沙粒的顶点，淡紫色代表两颗沙粒，绿色表示一颗沙粒，黑色表示零颗沙粒。]]

The '''Abelian sandpile model''', also known as the '''Bak–Tang–Wiesenfeld model''', was the first discovered example of a [[dynamical system]] displaying [[self-organized criticality]]. It was introduced by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] in a 1987 paper.<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>

''' 阿贝尔沙堆模型Abelian sandpile model'''，也被称为 Bak-Tang-Wiesenfeld 模型，是第一个发现的动力系统展现自组织临界性的例子。它是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld 在1987年的一篇论文<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>中提出的。

The model is a [[cellular automaton]]. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.

这个模型是一种''' 细胞自动机模型Cellular automaton'''。在最初的公式中，有限网格上的每个位置都有一个与沙堆的坡度相对应的关联值。当“沙粒”(或“碎片”)被随机放置在沙堆上时，放置位置的斜坡就会堆积起来，直到倾斜程度超过一个特定的阈值，这个位置倒塌，沙子会转移到邻近的位置，增加它们的斜坡。Bak，Tang和 Wiesenfeld考虑了在网格上连续随机放置沙粒的过程; 每次这样在特定位置放置沙粒有可能不会产生影响，也有可能会引起级联反应，影响到周围的其他位置。

The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs).<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref> It is closely related to the [[Chip-firing game#Biggs's Variant|dollar game]], a variant of the [[chip-firing game]] introduced by Biggs.<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

该模型已经在无限栅格、其他(非方形)栅格和任意图(包括有向多重图)上进行了研究。<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref>它与美元游戏密切相关，美元游戏是Biggs引入的一种chip-firing游戏的变体。<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

==Definition (rectangular grids)定义（矩形网格）==
The sandpile model is a [[cellular automaton]] originally defined on a <math>N\times M</math> rectangular grid (''checkerboard'') <math>\Gamma\subset\mathbb{Z}^2</math> of the [[Square lattice|standard square lattice]] <math>\mathbb{Z}^2</math>.

沙堆模型是一个'''元胞自动机cellular automaton '''，它最初定义在<math> N\times M </math>矩形网格(棋盘格)上，其顶点在标准的正方形格子<math> \mathbb{Z}^2 </math>上<math> \Gamma\subset\mathbb{Z}^2 </math>。

To each vertex (''side'', ''field'') <math>(x,y)\in\Gamma</math> of the grid, we associate a value (''grains of sand'', ''slope'', ''particles'') <math>z_0(x,y)\in\{0,1,2,3\}</math>, with <math>z_0\in\{0,1,2,3\}^\Gamma</math> referred to as the (initial) configuration of the sandpile.

对于栅格中的每个顶点<math> (x,y)\in\Gamma </math>，我们关联一个值(沙粒、坡度、颗粒) <math> z_0(x,y)\in\{0,1,2,3\} </math>，这样所有顶点的初始状态<math> z_0\in\{0,1,2,3\}^\Gamma </math>被称为沙堆的(初始)构型。

The dynamics of the automaton at iteration <math>i\in\mathbb{N}</math> are then defined as follows:

自动机的动力学过程在第<math> i\in\mathbb{N} </math>次迭代时的定义如下:

# Choose a random vertex <math>(x_i,y_i)\in\Gamma</math> according to some probability distribution (usually uniform).
# Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set <math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> and <math>z_i(x,y)=z_{i-1}(x,y)</math> for all <math>(x,y)\neq(x_i,y_i)</math>.
# If all vertices are ''stable'', i.e. <math>z_i(x,y)<4</math> for all <math>(x,y)\in\Gamma</math>, also the configuration <math>z_i</math> is said to be stable. In this case, continue with the next iteration.
# If at least one vertex is ''unstable'', i.e. <math>z_i(x_u,y_u)\geq 4</math> for some <math>(x_u,y_u)\in\Gamma</math>, the whole configuration <math>z_i</math> is said to be unstable. In this case, choose any unstable vertex <math>(x_u,y_u)\in\Gamma</math> at random. ''Topple'' this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>, and <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
# Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of <math>z_i</math> eventually become stable and continue with the next iteration.

#根据某种概率分布(通常为均匀分布)随机选择一个顶点。根据一些概率分布(通常是均匀的)选择一个随机顶点<math> (x_i,y_i)\in\Gamma </math>。
#向这个顶点添加一粒沙子，同时让其他顶点的沙粒数保持不变，也就是对于所有的<math>(x,y)\neq(x_i,y_i)</math>，设定 <math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> 和 <math>z_i(x,y)=z_{i-1}(x,y)</math>。
#如果所有的顶点都是稳定的，即如果对于<math>(x,y)\in\Gamma</math>，<math>z_i(x,y)<4</math>，那么构型<math>z_i</math>被认为是稳定的。在这种情况下，继续下一轮迭代。
#如果至少有一个顶点是不稳定的，即对于一些<math>(x_u,y_u)\in\Gamma</math>，<math>z_i(x_u,y_u)\geq 4</math>，<math>z_i</math>被认为是不稳定的。在这种情况下，随机选择任意不稳定顶点<math> (x_u,y_u)\in\Gamma</math>。将该顶点的沙粒数减少4个，清空这个顶点，并将其每个(最多4个)直接邻居的沙粒数增加1个。即： <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>，如果 <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. ， <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math>。如果一个在边界的顶点产生崩塌，这将导致沙粒的净损失（两粒在网格的角落，一粒在其他地方）。
#由于沙粒的重新分布，一个顶点的崩塌会使其他顶点不稳定。这样，重复崩塌的过程，直到<math>z_i</math>状态下的所有顶点最终稳定下来，继续下一轮迭代。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]） (two grains at the corner of the grid, one grain otherwise)翻译存疑

The toppling of several vertices during one iteration is referred to as an ''avalanche''. Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final stable configuration does not depend on the chosen order; this is one sense in which the sandpile is [[Abelian group|''abelian'']]. Similarly, the number of times each vertex topples during each iteration is also independent of the choice of toppling order.

在一次迭代中多个顶点的崩塌被称为雪崩。每一次雪崩最终都会停止，也就是说，经过有限数量的顶点崩塌，会达到某种稳定的配置，这样自动机就得到了很好的定义。此外，尽管顶点崩塌的顺序常常有许多可能的选择，但最终的稳定状态并不依赖于所选择的顺序; 这是沙堆模型具有的可交换性质。类似地，在每次迭代过程中，每个顶点的崩塌次数也与崩塌顺序的选择是无关的。

==Definition (undirected finite multigraphs)定义（无向有限多图）==
To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph <math>G=(V,E)</math>, a special vertex <math>s\in V</math> called the ''sink'' is specified that is not allowed to topple. A ''configuration'' (state) of the model is then a function <math>z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math> counting the non-negative number of grains on each non-sink vertex. A non-sink vertex <math>v\in V\setminus\{s\}</math> with
:<math>z(v)\geq \deg(v)</math>

is unstable; it can be toppled, which sends one of its grains to each of its (non-sink) neighbors:
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math> for all <math>u\sim v</math>, <math>u\neq s</math>.

为了将''' 沙堆模型'''从标准方格的矩形网格推广到任意无向有限多重图 <math> G=(V,E)</math> ，在 <math> V</math> 中指定了一个不允许崩塌的特殊沉没顶点<math> s</math>。模型的构型(状态)服从函数<math> z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math>，计算每个非沉没顶点上的非负沙粒数。非沉没顶点<math> v\in V\setminus\{s\} </math>当满足<math> z(v)\geq \deg(v) </math>时是不稳定的，它会产生崩塌，向给它的每个(非沉没)邻居分发一颗沙粒：
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math>对于所有的<math>u\sim v</math>, <math>u\neq s</math>.

The cellular automaton then progresses as before, i.e. by adding, in each iteration, one particle to a randomly chosen non-sink vertex and toppling until all vertices are stable.

元胞自动机像之前一样进行，即在每次迭代中，向随机选择的非沉没顶点添加一个沙粒，不断进行崩塌过程，直到所有顶点都稳定。

The definition of the sandpile model given above for finite rectangular grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math> can then be seen as a special case of this definition: consider the graph <math>G=(V,E)</math> which is obtained from <math>\Gamma</math> by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of <math>\Gamma</math> such that the [[Degree (graph theory)|degree]] of every non-sink vertex of <math>G</math> is four. In this manner, also sandpile models on non-rectangular grids of the standard square lattice (or of any other lattice) can be defined: Intersect some bounded subset <math>S</math> of <math>\mathbb{R}^2</math> with <math>\mathbb{Z}^2</math>. [[Edge contraction|Contract every edge]] of <math>\mathbb{Z}^2</math> whose two endpoints are not in <math>S\cap\mathbb{Z}^2</math>. The single remaining vertex outside of <math>S\cap\mathbb{Z}^2</math> then constitutes the sink of the resulting sandpile graph.

上面给出的沙堆模型的定义，是在标准正方形网格<math>\mathbb{Z}^2</math>上的有限矩形网格<math>\Gamma\subset\mathbb{Z}^2</math>上，它可以看作是下面定义的一个特例：考虑图<math>G=(V,E)</math>，从<math>\Gamma</math>添加一个沉没顶点，并添加从沉没顶点到每个边界顶点的边，使得<math>G</math>的每个非沉没顶点的度数为4。以这种方式，也可以定义标准正方形网格(或任何其他类型网格)的非矩形格上的沙堆模型: 将<math>\mathbb{R}^2</math>的一些有界子集<math>S</math>与<math>\mathbb{R}^2</math>相交。收缩<math>\mathbb{Z}^2</math>的每条边，其两个端点不在<math>S\cap\mathbb{Z}^2</math>中。<math>S\cap\mathbb{Z}^2</math>之外的一个单独剩余顶点构成了最终沙堆图的沉没顶点。

==Transient and recurrent configurations瞬态和循环构型==
In the dynamics of the sandpile automaton defined above, some stable configurations (<math>0\leq z(v)<4</math> for all <math>v\in G\setminus\{s\}</math>) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as ''recurrent configurations'', while the latter are referred to as ''transient configurations''. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the ''minimally stable configuration'' <math>z_m</math>, where every vertex carries <math>z_m(v)=deg(v)-1</math> grains of sand, is reachable from any other stable configuration (add <math>deg(v)-z(v)-1\geq 0</math> grains to every vertex). Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing.

在上面定义的沙堆自动机的动力学过程中，一些稳定状态的构型（对于所有<math>v\in G\setminus\{s\}</math>，<math>0\leq z(v)<4</math>）经常无限次出现，而另一些则只能出现有限次（如果真的发生的话）。前者被称为“循环构型”，而后者被称为“瞬态构型”。因此，周期性构形由所有稳定的非负构形组成，这些构形可以从任何其他稳定构形中，通过反复向顶点添加沙粒，产生崩塌而得到。很容易看出，“最小稳定构型”<math>zum</math>，其中每个顶点放置<math>z_m(v)=deg(v)-1</math>颗沙粒，可从任何其他稳定构型得到（通过向每个顶点添加<math>deg(v)-z(v)-1\geq 0</math>颗沙粒）。因此，也就是说，周期性构型可以从最小稳定构型开始，通过添加沙粒，再稳定化得到。

Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together. To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.

并非所有非负稳定构型都是循环的。例如，在一个至少包含由两个连通的非沉没顶点的图结构的沙堆模型中，如果这两个顶点没有放置沙粒，那么这个稳定结构是非循环的。为了证明这一点，首先要注意的是，沙粒的增加只能增加两个顶点放置的沙粒的总数。为了达到两个顶点都不放置沙粒的构型，从一个不是这种情况的构型出发，必然涉及到两个顶点中至少有一个崩塌的步骤。考虑这些步骤中的最后一个，在这个步骤中，两个顶点中的一个必须最后崩塌。由于崩塌会将对每个相邻的顶点转移一颗沙粒，这意味着两个顶点共同放置的沙粒总数不能低于一颗，因而得证。

==Sandpile group沙堆群==
Given a configuration <math>z</math>, <math>z(v)\in\mathbb{N}_0</math> for all <math>v\in G\setminus\{s\}</math>, toppling unstable non-sink vertices on a finite connected graph until no unstable non-sink vertex remains leads to a unique ''stable'' configuration <math>z^\circ</math>, which is called the ''stabilization'' of <math>z</math>. Given two stable configurations <math>z</math> and <math>w</math>, we can define the operation <math>z*w \to (z+w)^\circ</math>, corresponding to the vertex-wise addition of grains followed by the stabilization of the resulting sandpile.

给定一个构型<math>z</math>，<math>z(v)\in\mathbb{N}_0</math>对于所有<math>v\in G\setminus\{s\}</math>，在有限连通图上使不稳定的非沉没顶点崩塌，直到没有不稳定的非汇顶点保留，这将导致唯一的“稳定”配置math>z^\circ</math>，这就是<math>z</math>的“稳定化”。给定两个稳定构型<math>z</math>和<math>w</math>，我们可以定义运算<math>z*w \to (z+w)^\circ</math>，对应于沙粒的顶点方向相加，然后稳定得到的沙堆。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）corresponding to the vertex-wise addition of grains翻译存疑

Given an arbitrary but fixed ordering of the non-sink vertices, multiple toppling operations, which can e.g. occur during the stabilization of an unstable configuration, can be efficiently encoded by using the [[Laplacian matrix|graph Laplacian]] <math>\Delta=D-A</math>, where <math>D</math> is the [[degree matrix]] and <math>A</math> is the [[adjacency matrix]] of the graph.

给定一个任意但固定的非沉没顶点的顺序，进行多个崩塌操作，可以通过使用[[拉普拉斯矩阵|图拉普拉斯]]<math>\Delta=D-A</math>高效地编码多个崩塌操作（例如，在不稳定构型的稳定过程中可能发生），其中<math>D</math>是图的[[度矩阵]]，<math>A</math>是图的[[邻接矩阵]]。

Deleting the row and column of <math>\Delta</math> corresponding with the sink yields the ''reduced graph Laplacian'' <math>\Delta'</math>. Then, when starting with a configuration <math>z</math> and toppling each vertex <math>v</math> a total of <math>\mathbf{x}(v)\in\mathbb{N}_0</math> times yields the configuration <math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>, where <math>\boldsymbol{\cdot}</math> is the contraction product. Furthermore, if <math>\mathbf{x}</math> corresponds to the number of times each vertex is toppled during the stabilization of a given configuration <math>z</math>, then
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

删除与沉没顶点相对应的 <math>\Delta</math>的行和列，得到简化图拉普拉斯矩阵 <math>\Delta'</math>。然后，当以一个构型<math>z</math> 开始，并将每个顶点<math>v</math>进行总共<math>\mathbf{x}(v)\in\mathbb{N}_0</math>次的崩塌操作时，产生<math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>构型，其中<math>\boldsymbol{\cdot}</math>是收缩积。此外，如果 <math>\mathbf{x}</math> 对应于在给定构型<math>z</math>的稳定过程中每个顶点产生崩塌的次数，则
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

In this case, <math>\mathbf{x}</math> is referred to as the ''toppling'' or ''odometer function'' (of the stabilization of <math>z</math>).

在这种情况下，<math>\mathbf{x}</math>被称为崩塌或<math>z</math>的稳定过程的里程计函数。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）odometer function翻译存疑

Under the operation <math>*</math>, the set of recurrent configurations forms an [[abelian group]] isomorphic to the cokernel of the reduced graph Laplacian <math>\Delta'</math>, i.e. to <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>, whereby <math>n</math> denotes the number of vertices (including the sink). More generally, the set of stable configurations (transient and recurrent) forms a [[commutative monoid]] under the operation <math>*</math>. The minimal [[Semigroup#Subsemigroups and ideals|ideal]] of this monoid is then isomorphic to the group of recurrent configurations.

在<math>*</math>运算下，递归构形的集合构成一个与约化图拉普拉斯矩阵<math>\Delta'</math>的核同构的阿贝尔群。对于 <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，其中<math>n</math> 表示顶点数（包括沉没顶点）。更一般地说，稳定构型集（瞬态和循环）在<math>*</math>.运算下形成''' 交换幺半群Commutative monoid'''。这个幺半群的最小理想同构于循环构型群。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）ideal翻译存疑

The group formed by the recurrent configurations, as well as the group <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> to which the former is isomorphic, is most commonly referred to as the ''sandpile group''. Other common names for the same group are ''critical group'', ''Jacobian group'' or (less often) ''Picard group''. Note, however, that some authors only denote the group formed by the recurrent configurations as the sandpile group, while reserving the name Jacobian group or critical group for the (isomorphic) group defined by <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> (or for related isomorphic definitions). Finally, some authors use the name Picard group to refer to the direct product of the sandpile group and <math>\mathbb{Z}</math>, which naturally appears in a cellular automaton closely related to the sandpile model, referred to as the chip firing or dollar game.

由循环构形形成的群，以及与之同构的群<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，通常称为''' 沙堆群Sandpile group'''。相同群的其它常用名称是“临界群”、“Jacobian群”或(不常见的)“Picard群”。然而，要注意的是，有些作者只把循环构型形成的组称为沙堆组，而为由<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>（或相关的同构定义）定义的同构群保留Jacobian群或临界群的称呼。最后，一些作者使用Picard群来指代沙堆群和<math>\mathbb{Z}</math>的直接产物，后者出现在与沙堆模型密切相关的元胞自动机中，被称为''' 碎片点火或美元游戏Chip firing or Dollar game'''。

Given the isomorphisms stated above, the order of the sandpile group is the determinant of <math>\Delta'</math>, which by the [[Kirchhoff's theorem|matrix tree theorem]] is the number of spanning trees of the graph.

给定上述同构，沙堆群的顺序是<math>\Delta'</math>的行列式，根据矩阵树定理，它是图的生成树数目。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）the order of the sandpile group is the determinant of <math>\Delta'</math>的翻译存疑，order是否翻译成顺序

==Self-organized criticality自组织临界性==
{{main|Self-organized criticality}}
The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its [[critical state]], at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the [[phase transition]]s between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is [[self-organization|self-organized]].

模型最初源于这样一个事实，即在晶格上的模拟中，它被吸引到了它的[[临界状态]]，此时系统的关联长度和关联时间趋于无穷大，不需要对系统参数进行任何微调。这与早期临界现象的例子形成了对比，例如固体和液体之间，或液体和气体之间的[[相变]]，其中临界点只能通过精确调节（例如，温度）来达到。因此，在沙堆模型中，我们可以说临界性是自组织的。

Once the sandpile model reaches its critical state there is no correlation between the system's response to a [[wiktionary:perturbation|perturbation]] and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays [[1/f noise|1/''&fnof;'' noise]], a feature common to many complex systems in nature.

一旦沙堆模型达到其临界状态，系统对扰动的响应和扰动细节之间就没有关联。一般来说，这意味着再往沙堆形成的斜坡上撒一粒沙子可能不会导致任何事情发生，或者可能导致整个沙堆形成的斜坡在大规模滑坡中崩塌。该模型还显示了[[1/f noise|1/''&fnof;'' noise]]，这是自然界中许多复杂系统的共同特征。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）[[1/f noise|1/''&fnof;'' noise]]是什么

This model only displays critical behaviour in two or more dimensions. The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.

此模型仅在两个或多个维度中显示临界现象。沙堆模型可以用一维来表示; 然而，一维沙堆模型不是演化到临界状态，而是达到最小稳定状态，其中每个格点都趋向临界坡度。

For two dimensions, it has been hypothesized that the associated [[conformal field theory]] is consists of [[symplectic fermion]]s with a [[central charge]] ''c'' = −2.<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

对于二维，相关共形场理论被认为是中心电荷为''c'' = −2的symplectic fermion辛费米子组成的；<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

==Properties属性==
===Least action principle最小作用原理===
The stabilization of chip configurations obeys a form of ''[[principle of least action|least action principle]]'': each vertex topples no more than necessary in the course of the stabilization.<ref name=Fey2010>
{{cite journal
| author = Fey, A. |author2=Levine, L.|author3=Peres, Y.
| year=2010
| issn=0022-4715
| journal=Journal of Statistical Physics
| volume=138
| number=1–3
| doi=10.1007/s10955-009-9899-6
| title=Growth Rates and Explosions in Sandpiles
| pages=143–159|arxiv = 0901.3805 |bibcode = 2010JSP...138..143F |s2cid=7180488}}</ref>

芯片构型的稳定遵循一种“[[最小作用原理|最小作用原理]]”的形式：每个顶点在稳定过程中不超过必要的崩塌量。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）chip configurations的翻译存疑

This can be formalized as follows. Call a sequence of topples ''legal'' if it only topples unstable vertices, and ''stabilizing'' if it results in a stable configuration. The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible. Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex <math>v</math>, the number of times <math>v</math> topples is the same in all legal stabilizing sequences. According to the least action principle, a '''minimal stabilizing''' sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence. In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence.

这可以形式化如下。如果一个崩塌序列只推倒不稳定的顶点，则称其为“合法的”，使用一组合法的顶点序列，如果它的结果是一个稳定的构型，则称其为“稳定的”。稳定沙堆的标准方法是找到一个最大的合法崩塌序列，也就是说，让崩塌序列尽可能地长。这种序列具有明显的稳定性，沙堆的可交换性质是所有这些序列都等价于倾斜序列的置换，也就是说，对于任何顶点<math>v</math>，在所有合法的稳定序列中<math>v</math>的崩塌次数都是相同的。根据最小作用原理，最小稳定序列等价于合法的（且稳定的）崩塌序列的置换。特别地，由最小稳定序列产生的构型与由最大合法序列产生的构型是相同的。

More formally, if <math>\mathbf{u}</math> is a vector such that <math>\mathbf{u}(v)</math> is the number of times the vertex <math>v</math> topples during the stabilization (via the toppling of unstable vertices) of a chip configuration <math>z</math>, and <math>\mathbf{n}</math> is an integral vector (not necessarily non-negative) such that <math>z-\mathbf{n}\Delta'</math> is stable, then <math>\mathbf{u}(v) \leq \mathbf{n}(v)</math> for all vertices <math>v</math>.

更正式地说，如果<math>\mathbf{u}</math>是一个向量， <math>\mathbf{u}（v）</math>是碎片构型<math>z</math>在稳定过程中（通过不稳定顶点的崩塌）顶点<math>v</math>崩塌的次数，并且<math>\mathbf{n}</math>是一个积分向量（不一定是非负的），使得<math>z-\mathbf{n}\Delta'</math>是稳定的，那么对于所有顶点<math>v</math>，<math>\mathbf{u}(v) \leq \mathbf{n}(v)</math>。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）<math>\mathbf{n}</math> is an integral vector (not necessarily non-negative)这一句话的翻译存疑。

=== Scaling limits缩放限制===
[[File:Scaling sandpile identity.gif|thumb|Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.]]

[[File:Scaling sandpile identity.gif|thumb|沙堆标识在方形网格不断增加的动画。黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。]]

The animation shows the recurrent configuration corresponding to the identity of the sandpile group on different <math>N\times N</math> square grids of increasing sizes <math>N\geq 1</math>, whereby the configurations are rescaled to always have the same physical dimension. Visually, the identities on larger grids seem to become more and more detailed and to "converge to a continuous image". Mathematically, this suggests the existence of scaling-limits of the sandpile identity on square grids based on the notion of weak-* convergence (or some other, generalized notion of convergence). Indeed, existence of scaling limits of recurrent sandpile configurations has been proved by Wesley Pegden and Charles Smart
<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复构型，从而重新缩放构型，使其最终具有相同的物理尺度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）whereby the configurations are rescaled to always have the same physical dimension翻译存疑

== Generalizations and related models归纳与相关模型==
=== Sandpile models on infinite grids有向图上的沙堆模型===
[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|30 million grains dropped to a site of the infinite square grid, then toppled according to the rules of the sandpile model. White color denotes sites with 0 grains, green is for 1, purple is for 2, gold is for 3. The bounding box is 3967×3967.]]

[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|3000万粒沙粒落在无限方形网格的一个位置上，然后按照沙堆模型的规则产生崩塌。白色表示0颗沙粒位置，绿色表示1颗，紫色表示2颗，金色表示3颗。图中框的大小是3967×3967。]]

There exist several generalizations of the sandpile model to infinite grids. A challenge in such generalizations is that, in general, it is not guaranteed anymore that every avalanche will eventually stop. Several of the generalization thus only consider the stabilization of configurations for which this can be guaranteed.

沙堆模型可以推广到无限网格中。这种归纳法的一个挑战是，一般来说，不能保证每次雪崩最终都会停止。因此，一些一般化方法只考虑了构型的稳定性，因为这一点是能保证的。

A rather popular model on the (infinite) square lattice with sites <math>(x,y)\in\mathbb{Z}^2</math> is defined as follows:

在（无限）方格上有一个相当流行的模型，其位置<math>(x,y)\in\mathbb{Z}^2</math>定义如下：

Begin with some nonnegative configuration of values <math>z(x,y)\in \mathbf{Z}</math> which is finite, meaning

从\mathbf{z}</math>中有限值<math>z(x,y)\in \mathbf{Z}</math>的一些非负配置开始，这意味着

:<math>\sum_{x,y}z(x,y)<\infty.</math>

Any site <math>(x,y)</math> with

任何位置<math>(x,y)</math>有
:<math>z(x,y)\geq 4</math>
is ''unstable'' and can ''topple'' (or ''fire''), sending one of its chips to each of its four neighbors:

就是“不稳定的”，并且会产生崩塌，将它位置上的沙粒分发给它的四个邻居：
:<math>z(x,y) \rightarrow z(x,y) - 4,</math>
:<math>z( x \pm 1, y) \rightarrow z( x \pm 1, y) + 1,</math>
:<math>z(x, y \pm 1) \rightarrow z( x, y \pm 1 ) + 1.</math>

Since the initial configuration is finite, the [[Action (philosophy)|process]] is guaranteed to terminate, with the grains scattering outward.

由于初始构型是有限的，这一过程必然会随着沙粒向外散布终止。

A popular special case of this model is given when the initial configuration is zero for all vertices except the origin. If the origin carries a huge number of grains of sand, the configuration after relaxation forms fractal patterns (see figure). When letting the initial number of grains at the origin go to infinity, the rescaled stabilized configurations were shown to converge to a unique limit.<ref name="Pegden2013" /><ref name="Levine2016" />

给出了该模型的一个常见特例，即除原点外的所有顶点的初始配置都为零。如果原点放置大量沙粒，松弛后的构型形成分形图案(见图)。当初始颗粒数趋于无穷时，重缩放的稳定构型收敛到唯一极限。<ref name="Pegden2013" /><ref name="Levine2016" />

=== Sandpile models on directed graphs有向图上的沙堆模型===
The sandpile model can be generalized to arbitrary directed multigraphs. The rules are that any vertex <math>v</math> with

沙堆模型可以推广到任意有向多重图。规则是任何顶点<math>v</math>有
:<math>z(v)\geq \deg^{+}(v)</math>
is unstable; toppling again sends chips to each of its neighbors, one along each outgoing edge:

则是不稳定的；崩塌将它的碎片沿着边的输出方向，分发给它的邻居：
:<math>z(v) \rightarrow z(v) - \deg^{+}(v) + \deg(v,v)</math>
and, for each <math>u\neq v</math>:

并且，对于每个<math>u\neq v</math>：
:<math>z(u) \rightarrow z(u) + \deg(v,u)</math>
where <math>\deg(v,u)</math> is the number of edges from <math>v</math> to <math>u</math>.

其中，<math>\deg(v,u)</math>是从<math>v</math>到<math>u</math>的边数。

In this case the Laplacian matrix is not symmetric. If we specify a sink <math>s</math> such that there is a path from every other vertex to <math>s</math>, then the stabilization operation on finite graphs is well-defined and the sandpile group can be written

在这种情况下，拉普拉斯矩阵是不对称的。如果我们指定一个沉没顶点<math>s</math>，使得每一个顶点都有一条到<math>s</math>的路径，那么有限图上的稳定操作是定义良好的，并且沙堆群可以像之前一样被写出来

:<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>
as before.

The order of the sandpile group is again the determinant of <math>\Delta'</math>, which by the general version of the [[Kirchhoff's theorem|matrix tree theorem]] is the number of oriented [[spanning tree]]s rooted at the sink.

沙堆群的顺序又是<math>\Delta'</math>的行列式，根据[[矩阵树定理]]的一般版本，它是根在沉没顶点的有向[[spanning tree]]生成树的数目。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）The order of the sandpile group翻译存疑。

=== The extended sandpile model扩展沙堆模型===
[[File:Harmonic Sandpile Dynamics.gif|thumb|Sandpile dynamics induced by the harmonic function H=x*y on a 255x255 square grid.]]

[[File:Harmonic Sandpile Dynamics.gif|thumb|由谐波函数H=x*y在255x255方格网格上引起的沙堆动力学。]]

To better understand the structure of the sandpile group for different finite convex grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math>, Lang and Shkolnikov introduced the ''extended sandpile model'' in 2019.<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref> The extended sandpile model is defined nearly exactly the same as the ''usual sandpile model'' (i.e. the original Bak–Tang–Wiesenfeld model <ref name="Bak1987" />), except that vertices at the boundary <math>\partial\Gamma</math> of the grid are now allowed to carry a non-negative real number of grains. In contrast, vertices in the interior of the grid are still only allowed to carry integer numbers of grains. The toppling rules remain unchanged, i.e. both interior and boundary vertices are assumed to become unstable and topple if the grain number reaches or exceeds four.

为了更好地理解不同有限凸网格<math>\Gamma\subset\mathbb{Z}^2</math>的沙堆群的结构，Lang和Shkolnikov在2019年提出了“扩展沙堆模型”。<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref>扩展沙堆模型的定义与“通常的沙堆模型”几乎完全相同（即原始的Bak–Tang–Wiesenfeld模型<ref name="Bak1987" />），除了网格边界<math>\partial\Gamma</math>的顶点现在允许放置非负实数的沙粒。相比之下，网格内部的顶点仍然只允许放置整数个粒子。崩塌规则保持不变，即假设当沙粒数达到或超过4时，内部顶点和边界顶点都变得不稳定并发生崩塌。

Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the ''extended sandpile group'', of which the usual sandpile group is a [[discrete subgroup]]. Different to the usual sandpile group, the extended sandpile group is however a continuous [[Lie group]]. Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid, the extended sandpile group furthermore has the [[topological group|topology]] of a [[torus]] of dimension <math>|\partial\Gamma|</math> and a volume given by the order of the usual sandpile group.<ref name="Lang2019" />

扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid翻译存疑。a volume given by the order of the usual sandpile group.翻译存疑。

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

特别感兴趣的问题是循环构型如何通过恒等式，沿着这个环面的连续[[测地线]]动态变化的问题。这个问题引出了沙堆动力学的定义
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？
:<math>D_H(t)=(I-t\Delta H)^\circ</math> （扩展沙堆模型）
respectively
:<math>\tilde{D}_H(t)=(I+\lfloor-t\Delta H\rfloor)^\circ</math> （普通沙堆模型）

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />
For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the
smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid. Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" />

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在标度限制。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复配置映射到子网格上的重复配置。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）上面的整个长句翻译需要重新审校Informaly, this renormalization simply maps

=== The divisible sandpile 可分割的沙堆===
A strongly related model is the so called '''divisible sandpile model''', introduced by Levine and Peres in 2008,<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref> in which, instead of a discrete number of particles in each site <math>x</math>, there is a real number <math>s(x)</math> representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time <math>t</math>, it will be full for all later times.

Levine和Peres在2008年提出了一个与之密切相关的模型，即所谓的“可分割的沙堆模型”。<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref>与每个位置<math>x</math>上的沙粒数量为离散数不同，有一个实数<math>s(x)</math>代表位置的总质量。如果这个质量是负的，我们就可以把它理解为一个空洞。当一个位置上的质量大于1时，就会发生崩塌; 它将多余的部分均匀地分发给它的邻居，这就导致了如果一个位置在<math>t</math>的时刻质量是1，它在以后的所有时间质量都是1。

== References引用 ==
{{Reflist|30em}}

== Further reading ==
* {{cite book
| author = Per Bak
| year = 1996
| title = How Nature Works: The Science of Self-Organized Criticality
| publisher = Copernicus
| location = New York
| isbn = 978-0-387-94791-4
}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B | pmid=10035754}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1988
| title = Self-organized criticality
| journal = [[Physical Review A]]
| volume = 38
| issue = 1
| pages = 364–374
| doi = 10.1103/PhysRevA.38.364
|pmid=9900174 | bibcode=1988PhRvA..38..364B
}}
* {{cite journal | last1=Cori | first1=Robert | last2=Rossin | first2=Dominique | last3=Salvy | first3=Bruno | title=Polynomial ideals for sandpiles and their Gröbner bases | zbl=1002.68105 | journal=Theor. Comput. Sci. | volume=276 | issue=1–2 | year=2002 | doi=10.1016/S0304-3975(00)00397-2 | pages=1–15| url=https://hal.archives-ouvertes.fr/hal-00016378/file/sandpile.pdf }}
* {{cite book|first=Caroline|last=Klivans|authorlink=Caroline Klivans|title= The Mathematics of Chip-Firing|title-link= The Mathematics of Chip-Firing|publisher=CRC Press|year=2018}}
* {{cite book | author1-last=Perkinson | author1-first=David | author2-last=Perlman | author2-first=Jacob | author3-last=Wilmes | author3-first=John | chapter=Algebraic geometry of sandpiles | pages=211–256 | editor1-last=Amini | editor1-first=Omid | editor2-last=Baker | editor2-first=Matthew | editor3-last=Faber | editor3-first=Xander | title=Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 | zbl=1281.14002 | series=Contemporary Mathematics | volume=605 | doi=10.1090/conm/605/12117 
| location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-1-4704-1021-6 | year=2013 | citeseerx=10.1.1.760.283 | s2cid=7851577 }}

==External links==
*{{cite web|last1=Garcia-Puente|first1=Luis David|title=Sandpiles|url=https://www.youtube.com/watch?v=1MtEUErz7Gg|website=YouTube|publisher=[[Brady Haran]]|accessdate=15 January 2017|format=YouTube video}}

[[Category:Self-organization]]
[[Category:Phase transitions]]
[[Category:Dynamical systems]]
[[Category:Critical phenomena]]
[[Category:Nonlinear systems]]
[[Category:Cellular automaton rules]]

沙堆模型

2020-11-13T20:50:24Z

Zcy：

此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。此词条由Zcy初步审校。
{{short description|Cellular automaton}}
[[File:Sandpile identity 300x205.png|upright=1.25|thumb|The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.]]

[[File:Sandpile identity 300x205.png|upright=1.25|thumb|沙堆在矩形网格上的标识。黄色像素对应三颗沙粒的顶点，淡紫色代表两颗沙粒，绿色表示一颗沙粒，黑色表示零颗沙粒。]]

The '''Abelian sandpile model''', also known as the '''Bak–Tang–Wiesenfeld model''', was the first discovered example of a [[dynamical system]] displaying [[self-organized criticality]]. It was introduced by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] in a 1987 paper.<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>

''' 阿贝尔沙堆模型Abelian sandpile model'''，也被称为 Bak-Tang-Wiesenfeld 模型，是第一个发现的动力系统展示自组织临界性的例子。它是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld 在1987年的一篇论文<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>中提出的。

The model is a [[cellular automaton]]. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.

这个模型是一种''' 细胞自动机模型Cellular automaton'''。在最初的公式中，有限网格上的每个位置都有一个与沙堆的坡度相对应的关联值。当“沙粒”(或“碎片”)被随机放置在沙堆上时，放置位置的斜坡就会堆积起来，直到倾斜程度超过一个特定的阈值，这个位置倒塌，沙子会转移到邻近的位置，增加它们的斜坡。这个斜坡以“沙粒”(或“碎片”)随机放置的方式逐渐形成，直到斜坡超过一个特定的阈值，在这个阈值的时候，这个位置倒塌，将沙子转移到邻近的位置，增加它们的斜坡。Bak，Tang和 Wiesenfeld考虑了在网格上连续随机放置沙粒的过程; 每次这样在特定位置放置沙粒有可能没有产生影响，也有可能会引起级联反应，影响到周围的其他位置。

The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs).<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref> It is closely related to the [[Chip-firing game#Biggs's Variant|dollar game]], a variant of the [[chip-firing game]] introduced by Biggs.<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

该模型已经在无限栅格、其他(非方形)栅格和任意图(包括有向多重图)上进行了研究。<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref>它与美元游戏密切相关，美元游戏是Biggs引入的一种chip-firing游戏的变体。<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

==Definition (rectangular grids)定义（矩形网格）==
The sandpile model is a [[cellular automaton]] originally defined on a <math>N\times M</math> rectangular grid (''checkerboard'') <math>\Gamma\subset\mathbb{Z}^2</math> of the [[Square lattice|standard square lattice]] <math>\mathbb{Z}^2</math>.

沙堆模型是一个'''元胞自动机cellular automaton '''，它最初定义在<math> N\times M </math>矩形网格(棋盘格)上，其顶点在标准的正方形格子<math> \mathbb{Z}^2 </math>上<math> \Gamma\subset\mathbb{Z}^2 </math>。

To each vertex (''side'', ''field'') <math>(x,y)\in\Gamma</math> of the grid, we associate a value (''grains of sand'', ''slope'', ''particles'') <math>z_0(x,y)\in\{0,1,2,3\}</math>, with <math>z_0\in\{0,1,2,3\}^\Gamma</math> referred to as the (initial) configuration of the sandpile.

对于栅格中的每个顶点<math> (x,y)\in\Gamma </math>，我们关联一个值(沙粒、坡度、颗粒) <math> z_0(x,y)\in\{0,1,2,3\} </math>，这样所有顶点的初始状态<math> z_0\in\{0,1,2,3\}^\Gamma </math>被称为沙堆的(初始)构型。

The dynamics of the automaton at iteration <math>i\in\mathbb{N}</math> are then defined as follows:

自动机的动力学过程在第<math> i\in\mathbb{N} </math>次迭代时的定义如下:

# Choose a random vertex <math>(x_i,y_i)\in\Gamma</math> according to some probability distribution (usually uniform).
# Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set <math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> and <math>z_i(x,y)=z_{i-1}(x,y)</math> for all <math>(x,y)\neq(x_i,y_i)</math>.
# If all vertices are ''stable'', i.e. <math>z_i(x,y)<4</math> for all <math>(x,y)\in\Gamma</math>, also the configuration <math>z_i</math> is said to be stable. In this case, continue with the next iteration.
# If at least one vertex is ''unstable'', i.e. <math>z_i(x_u,y_u)\geq 4</math> for some <math>(x_u,y_u)\in\Gamma</math>, the whole configuration <math>z_i</math> is said to be unstable. In this case, choose any unstable vertex <math>(x_u,y_u)\in\Gamma</math> at random. ''Topple'' this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>, and <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
# Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of <math>z_i</math> eventually become stable and continue with the next iteration.

#根据某种概率分布(通常为均匀分布)随机选择一个顶点。根据一些概率分布(通常是均匀的)选择一个随机顶点<math> (x_i,y_i)\in\Gamma </math>。
#向这个顶点添加一粒沙子，同时让其他顶点的沙粒数保持不变，也就是对于所有的<math>(x,y)\neq(x_i,y_i)</math>，设定 <math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> 和 <math>z_i(x,y)=z_{i-1}(x,y)</math>。
#如果所有的顶点都是稳定的，即如果对于<math>(x,y)\in\Gamma</math>，<math>z_i(x,y)<4</math>，那么构型<math>z_i</math>被认为是稳定的。在这种情况下，继续下一轮迭代。
#如果至少有一个顶点是不稳定的，即对于一些<math>(x_u,y_u)\in\Gamma</math>，<math>z_i(x_u,y_u)\geq 4</math>，<math>z_i</math>被认为是不稳定的。在这种情况下，随机选择任意不稳定顶点<math> (x_u,y_u)\in\Gamma</math>。将该顶点的沙粒数减少4个，清空这个顶点，并将其每个(最多4个)直接邻居的沙粒数增加1个。即： <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>， <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> 如果 <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. 。如果一个在边界的顶点产生崩塌，这将导致沙粒的净损失（两粒在网格的角落，一粒在其他地方）。
#由于沙粒的重新分布，一个顶点的崩塌会使其他顶点不稳定。这样，重复崩塌的过程，直到<math>z_i</math>状态下的所有顶点最终稳定下来，继续下一轮迭代。

The toppling of several vertices during one iteration is referred to as an ''avalanche''. Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final stable configuration does not depend on the chosen order; this is one sense in which the sandpile is [[Abelian group|''abelian'']]. Similarly, the number of times each vertex topples during each iteration is also independent of the choice of toppling order.

在一次迭代中多个顶点的崩塌被称为雪崩。每一次雪崩最终都会停止，也就是说，经过有限数量的顶点崩塌，会达到某种稳定的配置，这样自动机就得到了很好的定义。此外，尽管顶点崩塌的顺序常常有许多可能的选择，但最终的稳定状态并不依赖于所选择的顺序; 这是沙堆模型具有的可交换性质。类似地，在每次迭代过程中，每个顶点的崩塌次数也与崩塌顺序的选择是无关。

==Definition (undirected finite multigraphs)定义（无向有限多图）==
To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph <math>G=(V,E)</math>, a special vertex <math>s\in V</math> called the ''sink'' is specified that is not allowed to topple. A ''configuration'' (state) of the model is then a function <math>z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math> counting the non-negative number of grains on each non-sink vertex. A non-sink vertex <math>v\in V\setminus\{s\}</math> with
:<math>z(v)\geq \deg(v)</math>

is unstable; it can be toppled, which sends one of its grains to each of its (non-sink) neighbors:
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math> for all <math>u\sim v</math>, <math>u\neq s</math>.

为了将''' 沙堆模型'''从标准方格的矩形网格推广到任意无向有限多重图 <math> G=(V,E)</math> ，在 <math> V</math> 中指定了一个不允许崩塌的特殊沉没顶点<math> s</math>。模型的配置(状态)服从函数<math> z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math>，计算每个非沉没顶点上的非负沙粒数。非沉没顶点<math> v\in V\setminus\{s\} </math>当满足<math> z(v)\geq \deg(v) </math>时是不稳定的，它会产生崩塌，将给它的每个(非沉没)邻居分发一颗沙粒：
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math>对于所有的<math>u\sim v</math>, <math>u\neq s</math>.

The cellular automaton then progresses as before, i.e. by adding, in each iteration, one particle to a randomly chosen non-sink vertex and toppling until all vertices are stable.

元胞自动机像之前一样进行，即在每次迭代中，向随机选择的非沉没顶点添加一个沙粒，不断进行崩塌过程，直到所有顶点都稳定。

The definition of the sandpile model given above for finite rectangular grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math> can then be seen as a special case of this definition: consider the graph <math>G=(V,E)</math> which is obtained from <math>\Gamma</math> by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of <math>\Gamma</math> such that the [[Degree (graph theory)|degree]] of every non-sink vertex of <math>G</math> is four. In this manner, also sandpile models on non-rectangular grids of the standard square lattice (or of any other lattice) can be defined: Intersect some bounded subset <math>S</math> of <math>\mathbb{R}^2</math> with <math>\mathbb{Z}^2</math>. [[Edge contraction|Contract every edge]] of <math>\mathbb{Z}^2</math> whose two endpoints are not in <math>S\cap\mathbb{Z}^2</math>. The single remaining vertex outside of <math>S\cap\mathbb{Z}^2</math> then constitutes the sink of the resulting sandpile graph.

上面给出的沙堆模型的定义，是在标准正方形网格<math>\mathbb{Z}^2</math>上的有限矩形网格<math>\Gamma\subset\mathbb{Z}^2</math>，它可以看作是下面定义的一个特例：考虑图<math>G=(V,E)</math>，从<math>\Gamma</math>添加一个沉没顶点，并添加从沉没顶点到每个边界顶点的边，使得<math>G</math>的每个非沉没顶点的度数为4。以这种方式，也可以定义标准正方形网格(或任何其他类型网格)的非矩形格上的沙堆模型: 将<math>\mathbb{R}^2</math>的一些有界子集<math>S</math>与<math>\mathbb{R}^2</math>相交。收缩<math>\mathbb{Z}^2</math>的每条边，其两个端点不在<math>S\cap\mathbb{Z}^2</math>中。<math>S\cap\mathbb{Z}^2</math>之外的一个单独剩余顶点构成了最终沙堆图的沉没顶点。

==Transient and recurrent configurations瞬态和循环构型==
In the dynamics of the sandpile automaton defined above, some stable configurations (<math>0\leq z(v)<4</math> for all <math>v\in G\setminus\{s\}</math>) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as ''recurrent configurations'', while the latter are referred to as ''transient configurations''. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the ''minimally stable configuration'' <math>z_m</math>, where every vertex carries <math>z_m(v)=deg(v)-1</math> grains of sand, is reachable from any other stable configuration (add <math>deg(v)-z(v)-1\geq 0</math> grains to every vertex). Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing.

在上面定义的沙堆自动机的动力学过程中，一些稳定状态的构型（对于所有<math>v\in G\setminus\{s\}</math>，<math>0\leq z(v)<4</math>）经常无限次出现，而另一些则只能出现有限次（如果真的发生的话）。前者被称为“循环构型”，而后者被称为“瞬态构型”。因此，周期性构形由所有稳定的非负构形组成，这些构形可以从任何其他稳定构形中，通过反复向顶点添加沙粒，产生崩塌而得到。很容易看出，“最小稳定配置”<math>zum</math>，其中每个顶点放置<math>z_m(v)=deg(v)-1</math>颗沙粒，可从任何其他稳定构型得到（通过向每个顶点添加<math>deg(v)-z(v)-1\geq 0</math>颗沙粒）。因此，也就是说，周期性构型可以从最小稳定构型开始，通过添加沙粒，再稳定化得到。

Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together. To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.

并非所有非负稳定构型都是循环的。例如，在一个至少包含由两个连通的非沉没顶点的图结构的沙堆模型中，如果这两个顶点没有放置沙粒，那么这个稳定结构是非循环的。为了证明这一点，首先要注意的是，沙粒的增加只能增加两个顶点放置的沙粒的总数。为了达到两个顶点都不放置沙粒的构型，从一个不是这种情况的构型出发，必然涉及到两个顶点中至少有一个崩塌的步骤。考虑这些步骤中的最后一个，在这个步骤中，两个顶点中的一个必须最后崩塌。由于崩塌会将对每个相邻的顶点转移一颗沙粒，这意味着两个顶点共同放置的沙粒总数不能低于一颗，因而得证。

==Sandpile group沙堆群==
Given a configuration <math>z</math>, <math>z(v)\in\mathbb{N}_0</math> for all <math>v\in G\setminus\{s\}</math>, toppling unstable non-sink vertices on a finite connected graph until no unstable non-sink vertex remains leads to a unique ''stable'' configuration <math>z^\circ</math>, which is called the ''stabilization'' of <math>z</math>. Given two stable configurations <math>z</math> and <math>w</math>, we can define the operation <math>z*w \to (z+w)^\circ</math>, corresponding to the vertex-wise addition of grains followed by the stabilization of the resulting sandpile.

给定一个构型<math>z</math>，<math>z(v)\in\mathbb{N}_0</math>对于所有<math>v\in G\setminus\{s\}</math>，在有限连通图上使不稳定的非沉没顶点崩塌，直到没有不稳定的非汇顶点保留，这将导致唯一的“稳定”配置math>z^\circ</math>，这就是<math>z</math>的“稳定化”。给定两个稳定构型<math>z</math>和<math>w</math>，我们可以定义运算<math>z*w \to (z+w)^\circ</math>，对应于沙粒的顶点方向相加，然后稳定得到的沙堆。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）corresponding to the vertex-wise addition of grains翻译存疑

Given an arbitrary but fixed ordering of the non-sink vertices, multiple toppling operations, which can e.g. occur during the stabilization of an unstable configuration, can be efficiently encoded by using the [[Laplacian matrix|graph Laplacian]] <math>\Delta=D-A</math>, where <math>D</math> is the [[degree matrix]] and <math>A</math> is the [[adjacency matrix]] of the graph.

给定一个任意但固定的非沉没顶点的顺序，进行多个崩塌操作，可以通过使用[[拉普拉斯矩阵|图拉普拉斯]]<math>\Delta=D-A</math>高效地编码多个崩塌操作（例如，在不稳定构型的稳定过程中可能发生），其中<math>D</math>是图的[[度矩阵]]，<math>A</math>是图的[[邻接矩阵]]。

Deleting the row and column of <math>\Delta</math> corresponding with the sink yields the ''reduced graph Laplacian'' <math>\Delta'</math>. Then, when starting with a configuration <math>z</math> and toppling each vertex <math>v</math> a total of <math>\mathbf{x}(v)\in\mathbb{N}_0</math> times yields the configuration <math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>, where <math>\boldsymbol{\cdot}</math> is the contraction product. Furthermore, if <math>\mathbf{x}</math> corresponds to the number of times each vertex is toppled during the stabilization of a given configuration <math>z</math>, then
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

删除与沉没顶点相对应的 <math>\Delta</math>的行和列，得到简化图拉普拉斯矩阵 <math>\Delta'</math>。然后，当以一个构型<math>z</math> 开始，并将每个顶点<math>v</math>进行总共<math>\mathbf{x}(v)\in\mathbb{N}_0</math>次的崩塌操作时，产生<math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>构型，其中<math>\boldsymbol{\cdot}</math>是收缩积。此外，如果 <math>\mathbf{x}</math> 对应于在给定构型<math>z</math>的稳定过程中每个顶点产生崩塌的次数，则
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

In this case, <math>\mathbf{x}</math> is referred to as the ''toppling'' or ''odometer function'' (of the stabilization of <math>z</math>).

在这种情况下，<math>\mathbf{x}</math>被称为崩塌或<math>z</math>的稳定过程的里程计函数。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）odometer function翻译存疑

Under the operation <math>*</math>, the set of recurrent configurations forms an [[abelian group]] isomorphic to the cokernel of the reduced graph Laplacian <math>\Delta'</math>, i.e. to <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>, whereby <math>n</math> denotes the number of vertices (including the sink). More generally, the set of stable configurations (transient and recurrent) forms a [[commutative monoid]] under the operation <math>*</math>. The minimal [[Semigroup#Subsemigroups and ideals|ideal]] of this monoid is then isomorphic to the group of recurrent configurations.

在<math>*</math>运算下，递归构形的集合构成一个与约化图拉普拉斯矩阵<math>\Delta'</math>的核同构的阿贝尔群。对于 <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，其中<math>n</math> 表示顶点数（包括沉没顶点）。更一般地说，稳定构型集（瞬态和循环）在<math>*</math>.运算下形成''' 交换幺半群Commutative monoid'''。这个幺半群的最小理想同构于循环构型群。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）ideal翻译存疑

The group formed by the recurrent configurations, as well as the group <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> to which the former is isomorphic, is most commonly referred to as the ''sandpile group''. Other common names for the same group are ''critical group'', ''Jacobian group'' or (less often) ''Picard group''. Note, however, that some authors only denote the group formed by the recurrent configurations as the sandpile group, while reserving the name Jacobian group or critical group for the (isomorphic) group defined by <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> (or for related isomorphic definitions). Finally, some authors use the name Picard group to refer to the direct product of the sandpile group and <math>\mathbb{Z}</math>, which naturally appears in a cellular automaton closely related to the sandpile model, referred to as the chip firing or dollar game.

由循环构形形成的群，以及与之同构的群<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，通常称为''' 沙堆群Sandpile group'''。相同群的其它常用名称是“临界群”、“Jacobian群”或(不常见的)“Picard群”。然而，要注意的是，有些作者只把循环构型形成的组称为沙堆组，而为由<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>（或相关的同构定义）定义的同构群保留Jacobian群或临界群的称呼。最后，一些作者使用Picard群来指代沙堆群和<math>\mathbb{Z}</math>的直接产物，后者出现在与沙堆模型密切相关的元胞自动机中，被称为''' 碎片点火或美元游戏Chip firing or Dollar game'''。

Given the isomorphisms stated above, the order of the sandpile group is the determinant of <math>\Delta'</math>, which by the [[Kirchhoff's theorem|matrix tree theorem]] is the number of spanning trees of the graph.

给定上述同构，沙堆群的顺序是<math>\Delta'</math>的行列式，根据矩阵树定理，它是图的生成树数目。

==Self-organized criticality自组织临界性==
{{main|Self-organized criticality}}
The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its [[critical state]], at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the [[phase transition]]s between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is [[self-organization|self-organized]].

模型最初源于这样一个事实，即在晶格上的模拟中，它被吸引到了它的[[临界状态]]，此时系统的关联长度和关联时间趋于无穷大，不需要对系统参数进行任何微调。这与早期临界现象的例子形成了对比，例如固体和液体之间，或液体和气体之间的[[相变]]，其中临界点只能通过精确调节（例如，温度）来达到。因此，在沙堆模型中，我们可以说临界性是自组织的。

Once the sandpile model reaches its critical state there is no correlation between the system's response to a [[wiktionary:perturbation|perturbation]] and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays [[1/f noise|1/''&fnof;'' noise]], a feature common to many complex systems in nature.

一旦沙堆模型达到其临界状态，系统对扰动的响应和扰动细节之间就没有关联。一般来说，这意味着再往沙堆形成的斜坡上撒一粒沙子可能不会导致任何事情发生，或者可能导致整个沙堆形成的斜坡在大规模滑坡中崩塌。该模型还显示了[[1/f noise|1/''&fnof;'' noise]]，这是自然界中许多复杂系统的共同特征。

This model only displays critical behaviour in two or more dimensions. The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.

此模型仅在两个或多个维度中显示关键行为。沙堆模型可以用一维来表示; 然而，一维沙堆模型不是演化到临界状态，而是达到最小稳定状态，其中每个格点都趋向临界坡度。

For two dimensions, it has been hypothesized that the associated [[conformal field theory]] is consists of [[symplectic fermion]]s with a [[central charge]] ''c'' = −2.<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

对于二维，相关共形场理论被认为是中心电荷为''c'' = −2的symplectic fermion辛费米子组成的；<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

==Properties属性==
===Least action principle最小作用原理===
The stabilization of chip configurations obeys a form of ''[[principle of least action|least action principle]]'': each vertex topples no more than necessary in the course of the stabilization.<ref name=Fey2010>
{{cite journal
| author = Fey, A. |author2=Levine, L.|author3=Peres, Y.
| year=2010
| issn=0022-4715
| journal=Journal of Statistical Physics
| volume=138
| number=1–3
| doi=10.1007/s10955-009-9899-6
| title=Growth Rates and Explosions in Sandpiles
| pages=143–159|arxiv = 0901.3805 |bibcode = 2010JSP...138..143F |s2cid=7180488}}</ref>

芯片构型的稳定遵循一种“[[最小作用原理|最小作用原理]]”的形式：每个顶点在稳定过程中不超过必要的崩塌量。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）chip configurations的翻译存疑

This can be formalized as follows. Call a sequence of topples ''legal'' if it only topples unstable vertices, and ''stabilizing'' if it results in a stable configuration. The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible. Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex <math>v</math>, the number of times <math>v</math> topples is the same in all legal stabilizing sequences. According to the least action principle, a '''minimal stabilizing''' sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence. In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence.

这可以形式化如下。如果一个崩塌序列只推倒不稳定的顶点，则称其为“合法的”，使用一组合法的顶点序列，如果它的结果是一个稳定的构型，则称其为“稳定的”。稳定沙堆的标准方法是找到一个最大的合法崩塌序列，也就是说，让崩塌序列尽可能地长。这种序列具有明显的稳定性，沙堆的可交换性质是所有这些序列都等价于倾斜序列的置换，也就是说，对于任何顶点<math>v</math>，在所有合法的稳定序列中<math>v</math>的崩塌次数都是相同的。根据最小作用原理，最小稳定序列等价于合法的（且稳定的）崩塌序列的置换。特别地，由最小稳定序列产生的构型与由最大合法序列产生的构型是相同的。

More formally, if <math>\mathbf{u}</math> is a vector such that <math>\mathbf{u}(v)</math> is the number of times the vertex <math>v</math> topples during the stabilization (via the toppling of unstable vertices) of a chip configuration <math>z</math>, and <math>\mathbf{n}</math> is an integral vector (not necessarily non-negative) such that <math>z-\mathbf{n}\Delta'</math> is stable, then <math>\mathbf{u}(v) \leq \mathbf{n}(v)</math> for all vertices <math>v</math>.

更正式地说，如果<math>\mathbf{u}</math>是一个向量， <math>\mathbf{u}（v）</math>是碎片构型<math>z</math>在稳定过程中（通过不稳定顶点的崩塌）顶点<math>v</math>崩塌的次数，并且<math>\mathbf{n}</math>是一个积分向量（不一定是非负的），使得<math>z-\mathbf{n}\Delta'</math>是稳定的，那么对于所有顶点<math>v</math>，<math>\mathbf{u}(v) \leq \mathbf{n}(v)</math>。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）<math>\mathbf{n}</math> is an integral vector (not necessarily non-negative)这一句话的翻译存疑。

=== Scaling limits缩放限制===
[[File:Scaling sandpile identity.gif|thumb|Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.]]

[[File:Scaling sandpile identity.gif|thumb|沙堆标识在方形网格不断增加的动画。黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。]]

The animation shows the recurrent configuration corresponding to the identity of the sandpile group on different <math>N\times N</math> square grids of increasing sizes <math>N\geq 1</math>, whereby the configurations are rescaled to always have the same physical dimension. Visually, the identities on larger grids seem to become more and more detailed and to "converge to a continuous image". Mathematically, this suggests the existence of scaling-limits of the sandpile identity on square grids based on the notion of weak-* convergence (or some other, generalized notion of convergence). Indeed, existence of scaling limits of recurrent sandpile configurations has been proved by Wesley Pegden and Charles Smart
<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复配置，从而重新缩放配置以始终具有相同的物理维度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

== Generalizations and related models归纳与相关模型==
=== Sandpile models on infinite grids有向图上的沙堆模型===
[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|30 million grains dropped to a site of the infinite square grid, then toppled according to the rules of the sandpile model. White color denotes sites with 0 grains, green is for 1, purple is for 2, gold is for 3. The bounding box is 3967×3967.]]

[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|3000万粒沙粒落在无限方形网格的一个位置上，然后按照沙堆模型的规则产生崩塌。白色表示0颗沙粒位置，绿色表示1颗，紫色表示2颗，金色表示3颗。图中框的大小是3967×3967。]]

There exist several generalizations of the sandpile model to infinite grids. A challenge in such generalizations is that, in general, it is not guaranteed anymore that every avalanche will eventually stop. Several of the generalization thus only consider the stabilization of configurations for which this can be guaranteed.

沙堆模型可以推广到无限网格中。这种归纳法的一个挑战是，一般来说，不再保证每次雪崩最终都会停止。因此，一些一般化方法只考虑了构型的稳定性，因为这一点是能保证的。

A rather popular model on the (infinite) square lattice with sites <math>(x,y)\in\mathbb{Z}^2</math> is defined as follows:

在（无限）方格上有一个相当流行的模型，其位置<math>(x,y)\in\mathbb{Z}^2</math>定义如下：

Begin with some nonnegative configuration of values <math>z(x,y)\in \mathbf{Z}</math> which is finite, meaning

从\mathbf{z}</math>中有限值<math>z(x,y)\in \mathbf{Z}</math>的一些非负配置开始，这意味着

:<math>\sum_{x,y}z(x,y)<\infty.</math>

Any site <math>(x,y)</math> with

任何位置<math>(x,y)</math>有
:<math>z(x,y)\geq 4</math>
is ''unstable'' and can ''topple'' (or ''fire''), sending one of its chips to each of its four neighbors:

就是“不稳定的”，并且会产生崩塌，将它位置上的沙粒分发给它的四个邻居：
:<math>z(x,y) \rightarrow z(x,y) - 4,</math>
:<math>z( x \pm 1, y) \rightarrow z( x \pm 1, y) + 1,</math>
:<math>z(x, y \pm 1) \rightarrow z( x, y \pm 1 ) + 1.</math>

Since the initial configuration is finite, the [[Action (philosophy)|process]] is guaranteed to terminate, with the grains scattering outward.

由于初始构型是有限的，这一过程必然会随着沙粒向外散布终止。

A popular special case of this model is given when the initial configuration is zero for all vertices except the origin. If the origin carries a huge number of grains of sand, the configuration after relaxation forms fractal patterns (see figure). When letting the initial number of grains at the origin go to infinity, the rescaled stabilized configurations were shown to converge to a unique limit.<ref name="Pegden2013" /><ref name="Levine2016" />

给出了该模型的一个常见特例，即除原点外的所有顶点的初始配置都为零。如果原点放置大量沙粒，松弛后的构型形成分形图案(见图)。当初始颗粒数趋于无穷时，重缩放的稳定构型收敛到唯一极限。<ref name="Pegden2013" /><ref name="Levine2016" />

=== Sandpile models on directed graphs有向图上的沙堆模型===
The sandpile model can be generalized to arbitrary directed multigraphs. The rules are that any vertex <math>v</math> with

沙堆模型可以推广到任意有向多重图。规则是任何顶点<math>v</math>有
:<math>z(v)\geq \deg^{+}(v)</math>
is unstable; toppling again sends chips to each of its neighbors, one along each outgoing edge:

则是不稳定的；崩塌将它的碎片沿着边的输出方向，分发给它的邻居：
:<math>z(v) \rightarrow z(v) - \deg^{+}(v) + \deg(v,v)</math>
and, for each <math>u\neq v</math>:

并且，对于每个<math>u\neq v</math>：
:<math>z(u) \rightarrow z(u) + \deg(v,u)</math>
where <math>\deg(v,u)</math> is the number of edges from <math>v</math> to <math>u</math>.

其中，<math>\deg(v,u)</math>是从<math>v</math>到<math>u</math>的边数。

In this case the Laplacian matrix is not symmetric. If we specify a sink <math>s</math> such that there is a path from every other vertex to <math>s</math>, then the stabilization operation on finite graphs is well-defined and the sandpile group can be written

在这种情况下，拉普拉斯矩阵是不对称的。如果我们指定一个沉没顶点<math>s</math>，使得每一个顶点都有一条到<math>s</math>的路径，那么有限图上的稳定操作是定义良好的，并且沙堆群可以像之前一样被写出来

:<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>
as before.

The order of the sandpile group is again the determinant of <math>\Delta'</math>, which by the general version of the [[Kirchhoff's theorem|matrix tree theorem]] is the number of oriented [[spanning tree]]s rooted at the sink.

沙堆群的顺序又是<math>\Delta'</math>的行列式，根据[[矩阵树定理]]的一般版本，它是根在沉没顶点的有向[[spanning tree]]生成树的数目。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）The order of the sandpile group翻译存疑。

=== The extended sandpile model扩展沙堆模型===
[[File:Harmonic Sandpile Dynamics.gif|thumb|Sandpile dynamics induced by the harmonic function H=x*y on a 255x255 square grid.]]

[[File:Harmonic Sandpile Dynamics.gif|thumb|由谐波函数H=x*y在255x255方格网格上引起的沙堆动力学。]]

To better understand the structure of the sandpile group for different finite convex grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math>, Lang and Shkolnikov introduced the ''extended sandpile model'' in 2019.<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref> The extended sandpile model is defined nearly exactly the same as the ''usual sandpile model'' (i.e. the original Bak–Tang–Wiesenfeld model <ref name="Bak1987" />), except that vertices at the boundary <math>\partial\Gamma</math> of the grid are now allowed to carry a non-negative real number of grains. In contrast, vertices in the interior of the grid are still only allowed to carry integer numbers of grains. The toppling rules remain unchanged, i.e. both interior and boundary vertices are assumed to become unstable and topple if the grain number reaches or exceeds four.

为了更好地理解不同有限凸网格<math>\Gamma\subset\mathbb{Z}^2</math>的沙堆群的结构，Lang和Shkolnikov在2019年提出了“扩展沙堆模型”。<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref>扩展沙堆模型的定义与“通常的沙堆模型”几乎完全相同（即原始的Bak–Tang–Wiesenfeld模型<ref name="Bak1987" />），除了网格边界<math>\partial\Gamma</math>的顶点现在允许放置非负实数的沙粒。相比之下，网格内部的顶点仍然只允许放置整数个粒子。崩塌规则保持不变，即假设当沙粒数达到或超过4时，内部顶点和边界顶点都变得不稳定并发生崩塌。

Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the ''extended sandpile group'', of which the usual sandpile group is a [[discrete subgroup]]. Different to the usual sandpile group, the extended sandpile group is however a continuous [[Lie group]]. Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid, the extended sandpile group furthermore has the [[topological group|topology]] of a [[torus]] of dimension <math>|\partial\Gamma|</math> and a volume given by the order of the usual sandpile group.<ref name="Lang2019" />

扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）a volume given by the order of the usual sandpile group.翻译存疑。

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

特别感兴趣的问题是循环构型如何通过恒等式，沿着这个环面的连续[[测地线]]动态变化的问题。这个问题引出了沙堆动力学的定义
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？
:<math>D_H(t)=(I-t\Delta H)^\circ</math> （扩展沙堆模型）
respectively
:<math>\tilde{D}_H(t)=(I+\lfloor-t\Delta H\rfloor)^\circ</math> （普通沙堆模型）

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />
For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the
smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid. Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" />

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在标度限制。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复配置映射到子网格上的重复配置。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）上面的长句翻译需要重新审校Informaly, this renormalization simply maps

=== The divisible sandpile 可分割的沙堆===
A strongly related model is the so called '''divisible sandpile model''', introduced by Levine and Peres in 2008,<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref> in which, instead of a discrete number of particles in each site <math>x</math>, there is a real number <math>s(x)</math> representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time <math>t</math>, it will be full for all later times.

Levine和Peres在2008年提出了一个与之密切相关的模型，即所谓的“可分割的沙堆模型”。<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref>与每个位置<math>x</math>上的沙粒数量为离散数不同，有一个实数<math>s(x)</math>代表位置的总质量。如果这个质量是负的，我们就可以把它理解为一个空洞。当一个位置上的质量大于1时，就会发生崩塌; 它将多余的部分均匀地分发给它的邻居，这就导致了如果一个位置在<math>t</math>的时刻是满的，它在以后的所有时间都是满的。

== References引用 ==
{{Reflist|30em}}

== Further reading ==
* {{cite book
| author = Per Bak
| year = 1996
| title = How Nature Works: The Science of Self-Organized Criticality
| publisher = Copernicus
| location = New York
| isbn = 978-0-387-94791-4
}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B | pmid=10035754}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1988
| title = Self-organized criticality
| journal = [[Physical Review A]]
| volume = 38
| issue = 1
| pages = 364–374
| doi = 10.1103/PhysRevA.38.364
|pmid=9900174 | bibcode=1988PhRvA..38..364B
}}
* {{cite journal | last1=Cori | first1=Robert | last2=Rossin | first2=Dominique | last3=Salvy | first3=Bruno | title=Polynomial ideals for sandpiles and their Gröbner bases | zbl=1002.68105 | journal=Theor. Comput. Sci. | volume=276 | issue=1–2 | year=2002 | doi=10.1016/S0304-3975(00)00397-2 | pages=1–15| url=https://hal.archives-ouvertes.fr/hal-00016378/file/sandpile.pdf }}
* {{cite book|first=Caroline|last=Klivans|authorlink=Caroline Klivans|title= The Mathematics of Chip-Firing|title-link= The Mathematics of Chip-Firing|publisher=CRC Press|year=2018}}
* {{cite book | author1-last=Perkinson | author1-first=David | author2-last=Perlman | author2-first=Jacob | author3-last=Wilmes | author3-first=John | chapter=Algebraic geometry of sandpiles | pages=211–256 | editor1-last=Amini | editor1-first=Omid | editor2-last=Baker | editor2-first=Matthew | editor3-last=Faber | editor3-first=Xander | title=Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 | zbl=1281.14002 | series=Contemporary Mathematics | volume=605 | doi=10.1090/conm/605/12117 
| location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-1-4704-1021-6 | year=2013 | citeseerx=10.1.1.760.283 | s2cid=7851577 }}

==External links==
*{{cite web|last1=Garcia-Puente|first1=Luis David|title=Sandpiles|url=https://www.youtube.com/watch?v=1MtEUErz7Gg|website=YouTube|publisher=[[Brady Haran]]|accessdate=15 January 2017|format=YouTube video}}

[[Category:Self-organization]]
[[Category:Phase transitions]]
[[Category:Dynamical systems]]
[[Category:Critical phenomena]]
[[Category:Nonlinear systems]]
[[Category:Cellular automaton rules]]

沙堆模型

2020-11-13T20:40:43Z

Zcy：

此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。此词条由Zcy初步审校。
{{short description|Cellular automaton}}
[[File:Sandpile identity 300x205.png|upright=1.25|thumb|The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.]]

[[File:Sandpile identity 300x205.png|upright=1.25|thumb|沙堆在矩形网格上的标识。黄色像素对应三颗沙粒的顶点，淡紫色代表两颗沙粒，绿色表示一颗沙粒，黑色表示零颗沙粒。]]

The '''Abelian sandpile model''', also known as the '''Bak–Tang–Wiesenfeld model''', was the first discovered example of a [[dynamical system]] displaying [[self-organized criticality]]. It was introduced by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] in a 1987 paper.<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>

''' 阿贝尔沙堆模型Abelian sandpile model'''，也被称为 Bak-Tang-Wiesenfeld 模型，是第一个发现的动力系统展示自组织临界性的例子。它是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld 在1987年的一篇论文<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>中提出的。

The model is a [[cellular automaton]]. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.

这个模型是一种''' 细胞自动机模型Cellular automaton'''。在最初的公式中，有限网格上的每个位置都有一个与沙堆的坡度相对应的关联值。当“沙粒”(或“碎片”)被随机放置在沙堆上时，放置位置的斜坡就会堆积起来，直到倾斜程度超过一个特定的阈值，这个位置倒塌，沙子会转移到邻近的位置，增加它们的斜坡。这个斜坡以“沙粒”(或“碎片”)随机放置的方式逐渐形成，直到斜坡超过一个特定的阈值，在这个阈值的时候，这个位置倒塌，将沙子转移到邻近的位置，增加它们的斜坡。Bak，Tang和 Wiesenfeld考虑了在网格上连续随机放置沙粒的过程; 每次这样在特定位置放置沙粒有可能没有产生影响，也有可能会引起级联反应，影响到周围的其他位置。

The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs).<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref> It is closely related to the [[Chip-firing game#Biggs's Variant|dollar game]], a variant of the [[chip-firing game]] introduced by Biggs.<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

该模型已经在无限栅格、其他(非方形)栅格和任意图(包括有向多重图)上进行了研究。<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref>它与美元游戏密切相关，美元游戏是Biggs引入的一种chip-firing游戏的变体。<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

==Definition (rectangular grids)定义（矩形网格）==
The sandpile model is a [[cellular automaton]] originally defined on a <math>N\times M</math> rectangular grid (''checkerboard'') <math>\Gamma\subset\mathbb{Z}^2</math> of the [[Square lattice|standard square lattice]] <math>\mathbb{Z}^2</math>.

沙堆模型是一个'''元胞自动机cellular automaton '''，它最初定义在<math> N\times M </math>矩形网格(棋盘格)上，其顶点在标准的正方形格子<math> \mathbb{Z}^2 </math>上<math> \Gamma\subset\mathbb{Z}^2 </math>。

To each vertex (''side'', ''field'') <math>(x,y)\in\Gamma</math> of the grid, we associate a value (''grains of sand'', ''slope'', ''particles'') <math>z_0(x,y)\in\{0,1,2,3\}</math>, with <math>z_0\in\{0,1,2,3\}^\Gamma</math> referred to as the (initial) configuration of the sandpile.

对于栅格中的每个顶点<math> (x,y)\in\Gamma </math>，我们关联一个值(沙粒、坡度、颗粒) <math> z_0(x,y)\in\{0,1,2,3\} </math>，这样所有顶点的初始状态<math> z_0\in\{0,1,2,3\}^\Gamma </math>被称为沙堆的(初始)构型。

The dynamics of the automaton at iteration <math>i\in\mathbb{N}</math> are then defined as follows:

自动机的动力学过程在第<math> i\in\mathbb{N} </math>次迭代时的定义如下:

# Choose a random vertex <math>(x_i,y_i)\in\Gamma</math> according to some probability distribution (usually uniform).
# Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set <math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> and <math>z_i(x,y)=z_{i-1}(x,y)</math> for all <math>(x,y)\neq(x_i,y_i)</math>.
# If all vertices are ''stable'', i.e. <math>z_i(x,y)<4</math> for all <math>(x,y)\in\Gamma</math>, also the configuration <math>z_i</math> is said to be stable. In this case, continue with the next iteration.
# If at least one vertex is ''unstable'', i.e. <math>z_i(x_u,y_u)\geq 4</math> for some <math>(x_u,y_u)\in\Gamma</math>, the whole configuration <math>z_i</math> is said to be unstable. In this case, choose any unstable vertex <math>(x_u,y_u)\in\Gamma</math> at random. ''Topple'' this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>, and <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
# Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of <math>z_i</math> eventually become stable and continue with the next iteration.

#根据某种概率分布(通常为均匀分布)随机选择一个顶点。根据一些概率分布(通常是均匀的)选择一个随机顶点<math> (x_i,y_i)\in\Gamma </math>。
#向这个顶点添加一粒沙子，同时让其他顶点的沙粒数保持不变，也就是设定<math> z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 </math>，对于所有的<math> (x,y)\neq(x_i,y_i) </math>，<math> z_i(x,y)=z_{i-1}(x,y) </math>。
#如果所有的顶点都是稳定的，即如果<math> (x,y)\in\Gamma </math>中<math> z_i(x,y)<4 </math>，那么<math> z_i </math>被认为是稳定的。在这种情况下，继续下一轮迭代。
#如果至少有一个顶点是不稳定的，即对于一些<math> (x_u,y_u)\in\Gamma</math>，<math>z_i(x_u,y_u)\geq 4</math>，<math>z_i</math>被认为是不稳定的。在这种情况下，随机选择任意不稳定顶点<math> (x_u,y_u)\in\Gamma</math>。将该顶点的沙粒数减少4个，清空这个顶点，并将其每个(最多4个)直接邻居的沙粒数增加1个。即： <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>， <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. 。如果一个在边界的顶点产生崩塌，这将导致沙粒的净损失（两粒在网格的角落，一粒在其他地方）。
#由于沙粒的重新分布，一个顶点的崩塌会使其他顶点不稳定。这样，重复崩塌的过程，直到<math>z_i</math>状态下的所有顶点最终稳定下来，继续下一轮迭代。

The toppling of several vertices during one iteration is referred to as an ''avalanche''. Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final stable configuration does not depend on the chosen order; this is one sense in which the sandpile is [[Abelian group|''abelian'']]. Similarly, the number of times each vertex topples during each iteration is also independent of the choice of toppling order.

在一次迭代中多个顶点的崩塌被称为雪崩。每一次雪崩最终都会停止，也就是说，经过有限数量的顶点崩塌，会达到某种稳定的配置，这样自动机就得到了很好的定义。此外，尽管顶点崩塌的顺序常常有许多可能的选择，但最终的稳定状态并不依赖于所选择的顺序; 这是沙堆模型具有的可交换性质。类似地，在每次迭代过程中，每个顶点的崩塌次数也与崩塌顺序的选择是无关。

==Definition (undirected finite multigraphs)定义（无向有限多图）==
To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph <math>G=(V,E)</math>, a special vertex <math>s\in V</math> called the ''sink'' is specified that is not allowed to topple. A ''configuration'' (state) of the model is then a function <math>z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math> counting the non-negative number of grains on each non-sink vertex. A non-sink vertex <math>v\in V\setminus\{s\}</math> with
:<math>z(v)\geq \deg(v)</math>

is unstable; it can be toppled, which sends one of its grains to each of its (non-sink) neighbors:
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math> for all <math>u\sim v</math>, <math>u\neq s</math>.

为了将''' 沙堆模型'''从标准方格的矩形网格推广到任意无向有限多重图 <math> G=(V,E)</math> ，在 <math> V</math> 中指定了一个不允许崩塌的特殊沉没顶点<math> s</math>。模型的配置(状态)服从函数<math> z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math>，计算每个非沉没顶点上的非负沙粒数。非沉没顶点<math> v\in V\setminus\{s\} </math>当满足<math> z(v)\geq \deg(v) </math>时是不稳定的，它会产生崩塌，将给它的每个(非沉没)邻居分发一颗沙粒：
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math>对于所有的<math>u\sim v</math>, <math>u\neq s</math>.

The cellular automaton then progresses as before, i.e. by adding, in each iteration, one particle to a randomly chosen non-sink vertex and toppling until all vertices are stable.

元胞自动机像之前一样进行，即在每次迭代中，向随机选择的非沉没顶点添加一个沙粒，不断进行崩塌过程，直到所有顶点都稳定。

The definition of the sandpile model given above for finite rectangular grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math> can then be seen as a special case of this definition: consider the graph <math>G=(V,E)</math> which is obtained from <math>\Gamma</math> by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of <math>\Gamma</math> such that the [[Degree (graph theory)|degree]] of every non-sink vertex of <math>G</math> is four. In this manner, also sandpile models on non-rectangular grids of the standard square lattice (or of any other lattice) can be defined: Intersect some bounded subset <math>S</math> of <math>\mathbb{R}^2</math> with <math>\mathbb{Z}^2</math>. [[Edge contraction|Contract every edge]] of <math>\mathbb{Z}^2</math> whose two endpoints are not in <math>S\cap\mathbb{Z}^2</math>. The single remaining vertex outside of <math>S\cap\mathbb{Z}^2</math> then constitutes the sink of the resulting sandpile graph.

上面给出的沙堆模型的定义，是在标准正方形网格<math>\mathbb{Z}^2</math>上的有限矩形网格<math>\Gamma\subset\mathbb{Z}^2</math>，它可以看作是下面定义的一个特例：考虑图<math>G=(V,E)</math>，从<math>\Gamma</math>添加一个沉没顶点，并添加从沉没顶点到每个边界顶点的边，使得<math>G</math>的每个非沉没顶点的度数为4。以这种方式，也可以定义标准正方形网格(或任何其他类型网格)的非矩形格上的沙堆模型: 将<math>\mathbb{R}^2</math>的一些有界子集<math>S</math>与<math>\mathbb{R}^2</math>相交。收缩<math>\mathbb{Z}^2</math>的每条边，其两个端点不在<math>S\cap\mathbb{Z}^2</math>中。<math>S\cap\mathbb{Z}^2</math>之外的一个单独剩余顶点构成了最终沙堆图的沉没顶点。

==Transient and recurrent configurations瞬态和循环构型==
In the dynamics of the sandpile automaton defined above, some stable configurations (<math>0\leq z(v)<4</math> for all <math>v\in G\setminus\{s\}</math>) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as ''recurrent configurations'', while the latter are referred to as ''transient configurations''. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the ''minimally stable configuration'' <math>z_m</math>, where every vertex carries <math>z_m(v)=deg(v)-1</math> grains of sand, is reachable from any other stable configuration (add <math>deg(v)-z(v)-1\geq 0</math> grains to every vertex). Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing.

在上面定义的沙堆自动机的动力学过程中，一些稳定状态的构型（对于所有<math>v\in G\setminus\{s\}</math>，<math>0\leq z(v)<4</math>）经常无限次出现，而另一些则只能出现有限次（如果真的发生的话）。前者被称为“循环构型”，而后者被称为“瞬态构型”。因此，周期性构形由所有稳定的非负构形组成，这些构形可以从任何其他稳定构形中，通过反复向顶点添加沙粒，产生崩塌而得到。很容易看出，“最小稳定配置”<math>zum</math>，其中每个顶点放置<math>z_m(v)=deg(v)-1</math>颗沙粒，可从任何其他稳定构型得到（通过向每个顶点添加<math>deg(v)-z(v)-1\geq 0</math>颗沙粒）。因此，也就是说，周期性构型可以从最小稳定构型开始，通过添加沙粒，再稳定化得到。

Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together. To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.

并非所有非负稳定构型都是循环的。例如，在一个至少包含由两个连通的非沉没顶点的图结构的沙堆模型中，如果这两个顶点没有放置沙粒，那么这个稳定结构是非循环的。为了证明这一点，首先要注意的是，沙粒的增加只能增加两个顶点放置的沙粒的总数。为了达到两个顶点都不放置沙粒的构型，从一个不是这种情况的构型出发，必然涉及到两个顶点中至少有一个崩塌的步骤。考虑这些步骤中的最后一个，在这个步骤中，两个顶点中的一个必须最后崩塌。由于崩塌会将对每个相邻的顶点转移一颗沙粒，这意味着两个顶点共同放置的沙粒总数不能低于一颗，因而得证。

==Sandpile group沙堆群==
Given a configuration <math>z</math>, <math>z(v)\in\mathbb{N}_0</math> for all <math>v\in G\setminus\{s\}</math>, toppling unstable non-sink vertices on a finite connected graph until no unstable non-sink vertex remains leads to a unique ''stable'' configuration <math>z^\circ</math>, which is called the ''stabilization'' of <math>z</math>. Given two stable configurations <math>z</math> and <math>w</math>, we can define the operation <math>z*w \to (z+w)^\circ</math>, corresponding to the vertex-wise addition of grains followed by the stabilization of the resulting sandpile.

给定一个构型<math>z</math>，<math>z(v)\in\mathbb{N}_0</math>对于所有<math>v\in G\setminus\{s\}</math>，在有限连通图上使不稳定的非沉没顶点崩塌，直到没有不稳定的非汇顶点保留，这将导致唯一的“稳定”配置math>z^\circ</math>，这就是<math>z</math>的“稳定化”。给定两个稳定构型<math>z</math>和<math>w</math>，我们可以定义运算<math>z*w \to (z+w)^\circ</math>，对应于沙粒的顶点方向相加，然后稳定得到的沙堆。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）corresponding to the vertex-wise addition of grains翻译存疑

Given an arbitrary but fixed ordering of the non-sink vertices, multiple toppling operations, which can e.g. occur during the stabilization of an unstable configuration, can be efficiently encoded by using the [[Laplacian matrix|graph Laplacian]] <math>\Delta=D-A</math>, where <math>D</math> is the [[degree matrix]] and <math>A</math> is the [[adjacency matrix]] of the graph.

给定一个任意但固定的非沉没顶点的顺序，进行多个崩塌操作，可以通过使用[[拉普拉斯矩阵|图拉普拉斯]]<math>\Delta=D-A</math>高效地编码多个崩塌操作（例如，在不稳定构型的稳定过程中可能发生），其中<math>D</math>是图的[[度矩阵]]，<math>A</math>是图的[[邻接矩阵]]。

Deleting the row and column of <math>\Delta</math> corresponding with the sink yields the ''reduced graph Laplacian'' <math>\Delta'</math>. Then, when starting with a configuration <math>z</math> and toppling each vertex <math>v</math> a total of <math>\mathbf{x}(v)\in\mathbb{N}_0</math> times yields the configuration <math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>, where <math>\boldsymbol{\cdot}</math> is the contraction product. Furthermore, if <math>\mathbf{x}</math> corresponds to the number of times each vertex is toppled during the stabilization of a given configuration <math>z</math>, then
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

删除与沉没顶点相对应的 <math>\Delta</math>的行和列，得到简化图拉普拉斯矩阵 <math>\Delta'</math>。然后，当以一个构型<math>z</math> 开始，并将每个顶点<math>v</math>进行总共<math>\mathbf{x}(v)\in\mathbb{N}_0</math>次的崩塌操作时，产生<math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>构型，其中<math>\boldsymbol{\cdot}</math>是收缩积。此外，如果 <math>\mathbf{x}</math> 对应于在给定构型<math>z</math>的稳定过程中每个顶点产生崩塌的次数，则
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

In this case, <math>\mathbf{x}</math> is referred to as the ''toppling'' or ''odometer function'' (of the stabilization of <math>z</math>).

在这种情况下，<math>\mathbf{x}</math>被称为崩塌或<math>z</math>的稳定过程的里程计函数。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）odometer function翻译存疑

Under the operation <math>*</math>, the set of recurrent configurations forms an [[abelian group]] isomorphic to the cokernel of the reduced graph Laplacian <math>\Delta'</math>, i.e. to <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>, whereby <math>n</math> denotes the number of vertices (including the sink). More generally, the set of stable configurations (transient and recurrent) forms a [[commutative monoid]] under the operation <math>*</math>. The minimal [[Semigroup#Subsemigroups and ideals|ideal]] of this monoid is then isomorphic to the group of recurrent configurations.

在<math>*</math>运算下，递归构形的集合构成一个与约化图拉普拉斯矩阵<math>\Delta'</math>的核同构的阿贝尔群。对于 <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，其中<math>n</math> 表示顶点数（包括沉没顶点）。更一般地说，稳定构型集（瞬态和循环）在<math>*</math>.运算下形成''' 交换幺半群Commutative monoid'''。这个幺半群的最小理想同构于循环构型群。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）ideal翻译存疑

The group formed by the recurrent configurations, as well as the group <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> to which the former is isomorphic, is most commonly referred to as the ''sandpile group''. Other common names for the same group are ''critical group'', ''Jacobian group'' or (less often) ''Picard group''. Note, however, that some authors only denote the group formed by the recurrent configurations as the sandpile group, while reserving the name Jacobian group or critical group for the (isomorphic) group defined by <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> (or for related isomorphic definitions). Finally, some authors use the name Picard group to refer to the direct product of the sandpile group and <math>\mathbb{Z}</math>, which naturally appears in a cellular automaton closely related to the sandpile model, referred to as the chip firing or dollar game.

由循环构形形成的群，以及与之同构的群<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，通常称为''' 沙堆群Sandpile group'''。相同群的其它常用名称是“临界群”、“Jacobian群”或(不常见的)“Picard群”。然而，要注意的是，有些作者只把循环构型形成的组称为沙堆组，而为由<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>（或相关的同构定义）定义的同构群保留Jacobian群或临界群的称呼。最后，一些作者使用Picard群来指代沙堆群和<math>\mathbb{Z}</math>的直接产物，后者出现在与沙堆模型密切相关的元胞自动机中，被称为''' 碎片点火或美元游戏Chip firing or Dollar game'''。

Given the isomorphisms stated above, the order of the sandpile group is the determinant of <math>\Delta'</math>, which by the [[Kirchhoff's theorem|matrix tree theorem]] is the number of spanning trees of the graph.

给定上述同构，沙堆群的顺序是<math>\Delta'</math>的行列式，根据矩阵树定理，它是图的生成树数目。

==Self-organized criticality自组织临界性==
{{main|Self-organized criticality}}
The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its [[critical state]], at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the [[phase transition]]s between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is [[self-organization|self-organized]].

模型最初源于这样一个事实，即在晶格上的模拟中，它被吸引到了它的[[临界状态]]，此时系统的关联长度和关联时间趋于无穷大，不需要对系统参数进行任何微调。这与早期临界现象的例子形成了对比，例如固体和液体之间，或液体和气体之间的[[相变]]，其中临界点只能通过精确调节（例如，温度）来达到。因此，在沙堆模型中，我们可以说临界性是自组织的。

Once the sandpile model reaches its critical state there is no correlation between the system's response to a [[wiktionary:perturbation|perturbation]] and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays [[1/f noise|1/''&fnof;'' noise]], a feature common to many complex systems in nature.

一旦沙堆模型达到其临界状态，系统对扰动的响应和扰动细节之间就没有关联。一般来说，这意味着再往沙堆形成的斜坡上撒一粒沙子可能不会导致任何事情发生，或者可能导致整个沙堆形成的斜坡在大规模滑坡中崩塌。该模型还显示了[[1/f noise|1/''&fnof;'' noise]]，这是自然界中许多复杂系统的共同特征。

This model only displays critical behaviour in two or more dimensions. The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.

此模型仅在两个或多个维度中显示关键行为。沙堆模型可以用一维来表示; 然而，一维沙堆模型不是演化到临界状态，而是达到最小稳定状态，其中每个格点都趋向临界坡度。

For two dimensions, it has been hypothesized that the associated [[conformal field theory]] is consists of [[symplectic fermion]]s with a [[central charge]] ''c'' = −2.<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

对于二维，相关共形场理论被认为是中心电荷为''c'' = −2的symplectic fermion辛费米子组成的；<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

==Properties属性==
===Least action principle最小作用原理===
The stabilization of chip configurations obeys a form of ''[[principle of least action|least action principle]]'': each vertex topples no more than necessary in the course of the stabilization.<ref name=Fey2010>
{{cite journal
| author = Fey, A. |author2=Levine, L.|author3=Peres, Y.
| year=2010
| issn=0022-4715
| journal=Journal of Statistical Physics
| volume=138
| number=1–3
| doi=10.1007/s10955-009-9899-6
| title=Growth Rates and Explosions in Sandpiles
| pages=143–159|arxiv = 0901.3805 |bibcode = 2010JSP...138..143F |s2cid=7180488}}</ref>

芯片构型的稳定遵循一种“[[最小作用原理|最小作用原理]]”的形式：每个顶点在稳定过程中不超过必要的崩塌量。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）chip configurations的翻译存疑

This can be formalized as follows. Call a sequence of topples ''legal'' if it only topples unstable vertices, and ''stabilizing'' if it results in a stable configuration. The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible. Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex <math>v</math>, the number of times <math>v</math> topples is the same in all legal stabilizing sequences. According to the least action principle, a '''minimal stabilizing''' sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence. In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence.

这可以形式化如下。如果一个崩塌序列只推倒不稳定的顶点，则称其为“合法的”，使用一组合法的顶点序列，如果它的结果是一个稳定的构型，则称其为“稳定的”。稳定沙堆的标准方法是找到一个最大的合法崩塌序列，也就是说，让崩塌序列尽可能地长。这种序列具有明显的稳定性，沙堆的可交换性质是所有这些序列都等价于倾斜序列的置换，也就是说，对于任何顶点<math>v</math>，在所有合法的稳定序列中<math>v</math>的崩塌次数都是相同的。根据最小作用原理，最小稳定序列等价于合法的（且稳定的）崩塌序列的置换。特别地，由最小稳定序列产生的构型与由最大合法序列产生的构型是相同的。

More formally, if <math>\mathbf{u}</math> is a vector such that <math>\mathbf{u}(v)</math> is the number of times the vertex <math>v</math> topples during the stabilization (via the toppling of unstable vertices) of a chip configuration <math>z</math>, and <math>\mathbf{n}</math> is an integral vector (not necessarily non-negative) such that <math>z-\mathbf{n}\Delta'</math> is stable, then <math>\mathbf{u}(v) \leq \mathbf{n}(v)</math> for all vertices <math>v</math>.

更正式地说，如果<math>\mathbf{u}</math>是一个向量， <math>\mathbf{u}（v）</math>是碎片构型<math>z</math>在稳定过程中（通过不稳定顶点的崩塌）顶点<math>v</math>崩塌的次数，并且<math>\mathbf{n}</math>是一个积分向量（不一定是非负的），使得<math>z-\mathbf{n}\Delta'</math>是稳定的，那么对于所有顶点<math>v</math>，<math>\mathbf{u}(v) \leq \mathbf{n}(v)</math>。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）<math>\mathbf{n}</math> is an integral vector (not necessarily non-negative)这一句话的翻译存疑。

=== Scaling limits缩放限制===
[[File:Scaling sandpile identity.gif|thumb|Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.]]

[[File:Scaling sandpile identity.gif|thumb|沙堆标识在方形网格不断增加的动画。黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。]]

The animation shows the recurrent configuration corresponding to the identity of the sandpile group on different <math>N\times N</math> square grids of increasing sizes <math>N\geq 1</math>, whereby the configurations are rescaled to always have the same physical dimension. Visually, the identities on larger grids seem to become more and more detailed and to "converge to a continuous image". Mathematically, this suggests the existence of scaling-limits of the sandpile identity on square grids based on the notion of weak-* convergence (or some other, generalized notion of convergence). Indeed, existence of scaling limits of recurrent sandpile configurations has been proved by Wesley Pegden and Charles Smart
<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复配置，从而重新缩放配置以始终具有相同的物理维度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

== Generalizations and related models归纳与相关模型==
=== Sandpile models on infinite grids有向图上的沙堆模型===
[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|30 million grains dropped to a site of the infinite square grid, then toppled according to the rules of the sandpile model. White color denotes sites with 0 grains, green is for 1, purple is for 2, gold is for 3. The bounding box is 3967×3967.]]

[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|3000万粒沙粒落在无限方形网格的一个位置上，然后按照沙堆模型的规则产生崩塌。白色表示0颗沙粒位置，绿色表示1颗，紫色表示2颗，金色表示3颗。图中框的大小是3967×3967。]]

There exist several generalizations of the sandpile model to infinite grids. A challenge in such generalizations is that, in general, it is not guaranteed anymore that every avalanche will eventually stop. Several of the generalization thus only consider the stabilization of configurations for which this can be guaranteed.

沙堆模型可以推广到无限网格中。这种归纳法的一个挑战是，一般来说，不再保证每次雪崩最终都会停止。因此，一些一般化方法只考虑了构型的稳定性，因为这一点是能保证的。

A rather popular model on the (infinite) square lattice with sites <math>(x,y)\in\mathbb{Z}^2</math> is defined as follows:

在（无限）方格上有一个相当流行的模型，其位置<math>(x,y)\in\mathbb{Z}^2</math>定义如下：

Begin with some nonnegative configuration of values <math>z(x,y)\in \mathbf{Z}</math> which is finite, meaning

从\mathbf{z}</math>中有限值<math>z(x,y)\in \mathbf{Z}</math>的一些非负配置开始，这意味着

:<math>\sum_{x,y}z(x,y)<\infty.</math>

Any site <math>(x,y)</math> with

任何位置<math>(x,y)</math>有
:<math>z(x,y)\geq 4</math>
is ''unstable'' and can ''topple'' (or ''fire''), sending one of its chips to each of its four neighbors:

就是“不稳定的”，并且会产生崩塌，将它位置上的沙粒分发给它的四个邻居：
:<math>z(x,y) \rightarrow z(x,y) - 4,</math>
:<math>z( x \pm 1, y) \rightarrow z( x \pm 1, y) + 1,</math>
:<math>z(x, y \pm 1) \rightarrow z( x, y \pm 1 ) + 1.</math>

Since the initial configuration is finite, the [[Action (philosophy)|process]] is guaranteed to terminate, with the grains scattering outward.

由于初始构型是有限的，这一过程必然会随着沙粒向外散布终止。

A popular special case of this model is given when the initial configuration is zero for all vertices except the origin. If the origin carries a huge number of grains of sand, the configuration after relaxation forms fractal patterns (see figure). When letting the initial number of grains at the origin go to infinity, the rescaled stabilized configurations were shown to converge to a unique limit.<ref name="Pegden2013" /><ref name="Levine2016" />

给出了该模型的一个常见特例，即除原点外的所有顶点的初始配置都为零。如果原点放置大量沙粒，松弛后的构型形成分形图案(见图)。当初始颗粒数趋于无穷时，重缩放的稳定构型收敛到唯一极限。<ref name="Pegden2013" /><ref name="Levine2016" />

=== Sandpile models on directed graphs有向图上的沙堆模型===
The sandpile model can be generalized to arbitrary directed multigraphs. The rules are that any vertex <math>v</math> with

沙堆模型可以推广到任意有向多重图。规则是任何顶点<math>v</math>有
:<math>z(v)\geq \deg^{+}(v)</math>
is unstable; toppling again sends chips to each of its neighbors, one along each outgoing edge:

则是不稳定的；崩塌将它的碎片沿着边的输出方向，分发给它的邻居：
:<math>z(v) \rightarrow z(v) - \deg^{+}(v) + \deg(v,v)</math>
and, for each <math>u\neq v</math>:

并且，对于每个<math>u\neq v</math>：
:<math>z(u) \rightarrow z(u) + \deg(v,u)</math>
where <math>\deg(v,u)</math> is the number of edges from <math>v</math> to <math>u</math>.

其中，<math>\deg(v,u)</math>是从<math>v</math>到<math>u</math>的边数。

In this case the Laplacian matrix is not symmetric. If we specify a sink <math>s</math> such that there is a path from every other vertex to <math>s</math>, then the stabilization operation on finite graphs is well-defined and the sandpile group can be written

在这种情况下，拉普拉斯矩阵是不对称的。如果我们指定一个沉没顶点<math>s</math>，使得每一个顶点都有一条到<math>s</math>的路径，那么有限图上的稳定操作是定义良好的，并且沙堆群可以像之前一样被写出来

:<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>
as before.

The order of the sandpile group is again the determinant of <math>\Delta'</math>, which by the general version of the [[Kirchhoff's theorem|matrix tree theorem]] is the number of oriented [[spanning tree]]s rooted at the sink.

沙堆群的顺序又是<math>\Delta'</math>的行列式，根据[[矩阵树定理]]的一般版本，它是根在沉没顶点的有向[[spanning tree]]生成树的数目。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）The order of the sandpile group翻译存疑。

=== The extended sandpile model扩展沙堆模型===
[[File:Harmonic Sandpile Dynamics.gif|thumb|Sandpile dynamics induced by the harmonic function H=x*y on a 255x255 square grid.]]

[[File:Harmonic Sandpile Dynamics.gif|thumb|由谐波函数H=x*y在255x255方格网格上引起的沙堆动力学。]]

To better understand the structure of the sandpile group for different finite convex grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math>, Lang and Shkolnikov introduced the ''extended sandpile model'' in 2019.<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref> The extended sandpile model is defined nearly exactly the same as the ''usual sandpile model'' (i.e. the original Bak–Tang–Wiesenfeld model <ref name="Bak1987" />), except that vertices at the boundary <math>\partial\Gamma</math> of the grid are now allowed to carry a non-negative real number of grains. In contrast, vertices in the interior of the grid are still only allowed to carry integer numbers of grains. The toppling rules remain unchanged, i.e. both interior and boundary vertices are assumed to become unstable and topple if the grain number reaches or exceeds four.

为了更好地理解不同有限凸网格<math>\Gamma\subset\mathbb{Z}^2</math>的沙堆群的结构，Lang和Shkolnikov在2019年提出了“扩展沙堆模型”。<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref>扩展沙堆模型的定义与“通常的沙堆模型”几乎完全相同（即原始的Bak–Tang–Wiesenfeld模型<ref name="Bak1987" />），除了网格边界<math>\partial\Gamma</math>的顶点现在允许放置非负实数的沙粒。相比之下，网格内部的顶点仍然只允许放置整数个粒子。崩塌规则保持不变，即假设当沙粒数达到或超过4时，内部顶点和边界顶点都变得不稳定并发生崩塌。

Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the ''extended sandpile group'', of which the usual sandpile group is a [[discrete subgroup]]. Different to the usual sandpile group, the extended sandpile group is however a continuous [[Lie group]]. Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid, the extended sandpile group furthermore has the [[topological group|topology]] of a [[torus]] of dimension <math>|\partial\Gamma|</math> and a volume given by the order of the usual sandpile group.<ref name="Lang2019" />

扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）a volume given by the order of the usual sandpile group.翻译存疑。

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

特别感兴趣的问题是循环构型如何通过恒等式，沿着这个环面的连续[[测地线]]动态变化的问题。这个问题引出了沙堆动力学的定义
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？
:<math>D_H(t)=(I-t\Delta H)^\circ</math> （扩展沙堆模型）
respectively
:<math>\tilde{D}_H(t)=(I+\lfloor-t\Delta H\rfloor)^\circ</math> （普通沙堆模型）

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />
For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the
smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid. Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" />

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在标度限制。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复配置映射到子网格上的重复配置。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）上面的长句翻译需要重新审校Informaly, this renormalization simply maps

=== The divisible sandpile 可分割的沙堆===
A strongly related model is the so called '''divisible sandpile model''', introduced by Levine and Peres in 2008,<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref> in which, instead of a discrete number of particles in each site <math>x</math>, there is a real number <math>s(x)</math> representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time <math>t</math>, it will be full for all later times.

Levine和Peres在2008年提出了一个与之密切相关的模型，即所谓的“可分割的沙堆模型”。<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref>与每个位置<math>x</math>上的沙粒数量为离散数不同，有一个实数<math>s(x)</math>代表位置的总质量。如果这个质量是负的，我们就可以把它理解为一个空洞。当一个位置上的质量大于1时，就会发生崩塌; 它将多余的部分均匀地分发给它的邻居，这就导致了如果一个位置在<math>t</math>的时刻是满的，它在以后的所有时间都是满的。

== References引用 ==
{{Reflist|30em}}

== Further reading ==
* {{cite book
| author = Per Bak
| year = 1996
| title = How Nature Works: The Science of Self-Organized Criticality
| publisher = Copernicus
| location = New York
| isbn = 978-0-387-94791-4
}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B | pmid=10035754}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1988
| title = Self-organized criticality
| journal = [[Physical Review A]]
| volume = 38
| issue = 1
| pages = 364–374
| doi = 10.1103/PhysRevA.38.364
|pmid=9900174 | bibcode=1988PhRvA..38..364B
}}
* {{cite journal | last1=Cori | first1=Robert | last2=Rossin | first2=Dominique | last3=Salvy | first3=Bruno | title=Polynomial ideals for sandpiles and their Gröbner bases | zbl=1002.68105 | journal=Theor. Comput. Sci. | volume=276 | issue=1–2 | year=2002 | doi=10.1016/S0304-3975(00)00397-2 | pages=1–15| url=https://hal.archives-ouvertes.fr/hal-00016378/file/sandpile.pdf }}
* {{cite book|first=Caroline|last=Klivans|authorlink=Caroline Klivans|title= The Mathematics of Chip-Firing|title-link= The Mathematics of Chip-Firing|publisher=CRC Press|year=2018}}
* {{cite book | author1-last=Perkinson | author1-first=David | author2-last=Perlman | author2-first=Jacob | author3-last=Wilmes | author3-first=John | chapter=Algebraic geometry of sandpiles | pages=211–256 | editor1-last=Amini | editor1-first=Omid | editor2-last=Baker | editor2-first=Matthew | editor3-last=Faber | editor3-first=Xander | title=Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 | zbl=1281.14002 | series=Contemporary Mathematics | volume=605 | doi=10.1090/conm/605/12117 
| location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-1-4704-1021-6 | year=2013 | citeseerx=10.1.1.760.283 | s2cid=7851577 }}

==External links==
*{{cite web|last1=Garcia-Puente|first1=Luis David|title=Sandpiles|url=https://www.youtube.com/watch?v=1MtEUErz7Gg|website=YouTube|publisher=[[Brady Haran]]|accessdate=15 January 2017|format=YouTube video}}

[[Category:Self-organization]]
[[Category:Phase transitions]]
[[Category:Dynamical systems]]
[[Category:Critical phenomena]]
[[Category:Nonlinear systems]]
[[Category:Cellular automaton rules]]

沙堆模型

2020-11-13T20:34:56Z

Zcy：

{{short description|Cellular automaton}}
[[File:Sandpile identity 300x205.png|upright=1.25|thumb|The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.]]

[[File:Sandpile identity 300x205.png|upright=1.25|thumb|沙堆在矩形网格上的标识。黄色像素对应三颗沙粒的顶点，淡紫色代表两颗沙粒，绿色表示一颗沙粒，黑色表示零颗沙粒。]]

The '''Abelian sandpile model''', also known as the '''Bak–Tang–Wiesenfeld model''', was the first discovered example of a [[dynamical system]] displaying [[self-organized criticality]]. It was introduced by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] in a 1987 paper.<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>

''' 阿贝尔沙堆模型Abelian sandpile model'''，也被称为 Bak-Tang-Wiesenfeld 模型，是第一个发现的动力系统展示自组织临界性的例子。它是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld 在1987年的一篇论文<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>中提出的。

The model is a [[cellular automaton]]. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.

这个模型是一种''' 细胞自动机模型Cellular automaton'''。在最初的公式中，有限网格上的每个位置都有一个与沙堆的坡度相对应的关联值。当“沙粒”(或“碎片”)被随机放置在沙堆上时，放置位置的斜坡就会堆积起来，直到倾斜程度超过一个特定的阈值，这个位置倒塌，沙子会转移到邻近的位置，增加它们的斜坡。这个斜坡以“沙粒”(或“碎片”)随机放置的方式逐渐形成，直到斜坡超过一个特定的阈值，在这个阈值的时候，这个位置倒塌，将沙子转移到邻近的位置，增加它们的斜坡。Bak，Tang和 Wiesenfeld考虑了在网格上连续随机放置沙粒的过程; 每次这样在特定位置放置沙粒有可能没有产生影响，也有可能会引起级联反应，影响到周围的其他位置。

The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs).<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref> It is closely related to the [[Chip-firing game#Biggs's Variant|dollar game]], a variant of the [[chip-firing game]] introduced by Biggs.<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

该模型已经在无限栅格、其他(非方形)栅格和任意图(包括有向多重图)上进行了研究。<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref>它与美元游戏密切相关，美元游戏是Biggs引入的一种chip-firing游戏的变体。<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

==Definition (rectangular grids)定义（矩形网格）==
The sandpile model is a [[cellular automaton]] originally defined on a <math>N\times M</math> rectangular grid (''checkerboard'') <math>\Gamma\subset\mathbb{Z}^2</math> of the [[Square lattice|standard square lattice]] <math>\mathbb{Z}^2</math>.

沙堆模型是一个'''元胞自动机cellular automaton '''，它最初定义在<math> N\times M </math>矩形网格(棋盘格)上，其顶点在标准的正方形格子<math> \mathbb{Z}^2 </math>上<math> \Gamma\subset\mathbb{Z}^2 </math>。

To each vertex (''side'', ''field'') <math>(x,y)\in\Gamma</math> of the grid, we associate a value (''grains of sand'', ''slope'', ''particles'') <math>z_0(x,y)\in\{0,1,2,3\}</math>, with <math>z_0\in\{0,1,2,3\}^\Gamma</math> referred to as the (initial) configuration of the sandpile.

对于栅格中的每个顶点<math> (x,y)\in\Gamma </math>，我们关联一个值(沙粒、坡度、颗粒) <math> z_0(x,y)\in\{0,1,2,3\} </math>，这样所有顶点的初始状态<math> z_0\in\{0,1,2,3\}^\Gamma </math>被称为沙堆的(初始)构型。

The dynamics of the automaton at iteration <math>i\in\mathbb{N}</math> are then defined as follows:

自动机的动力学过程在第<math> i\in\mathbb{N} </math>次迭代时的定义如下:

# Choose a random vertex <math>(x_i,y_i)\in\Gamma</math> according to some probability distribution (usually uniform).
# Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set <math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> and <math>z_i(x,y)=z_{i-1}(x,y)</math> for all <math>(x,y)\neq(x_i,y_i)</math>.
# If all vertices are ''stable'', i.e. <math>z_i(x,y)<4</math> for all <math>(x,y)\in\Gamma</math>, also the configuration <math>z_i</math> is said to be stable. In this case, continue with the next iteration.
# If at least one vertex is ''unstable'', i.e. <math>z_i(x_u,y_u)\geq 4</math> for some <math>(x_u,y_u)\in\Gamma</math>, the whole configuration <math>z_i</math> is said to be unstable. In this case, choose any unstable vertex <math>(x_u,y_u)\in\Gamma</math> at random. ''Topple'' this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>, and <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
# Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of <math>z_i</math> eventually become stable and continue with the next iteration.

#根据某种概率分布(通常为均匀分布)随机选择一个顶点。根据一些概率分布(通常是均匀的)选择一个随机顶点<math> (x_i,y_i)\in\Gamma </math>。
#向这个顶点添加一粒沙子，同时让其他顶点的沙粒数保持不变，也就是设定<math> z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 </math>，对于所有的<math> (x,y)\neq(x_i,y_i) </math>，<math> z_i(x,y)=z_{i-1}(x,y) </math>。
#如果所有的顶点都是稳定的，即如果<math> (x,y)\in\Gamma </math>中<math> z_i(x,y)<4 </math>，那么<math> z_i </math>被认为是稳定的。在这种情况下，继续下一轮迭代。
#如果至少有一个顶点是不稳定的，即对于一些<math> (x_u,y_u)\in\Gamma</math>，<math>z_i(x_u,y_u)\geq 4</math>，<math>z_i</math>被认为是不稳定的。在这种情况下，随机选择任意不稳定顶点<math> (x_u,y_u)\in\Gamma</math>。将该顶点的沙粒数减少4个，清空这个顶点，并将其每个(最多4个)直接邻居的沙粒数增加1个。即：<math> z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) – 4</math>，<math> z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1 if ( x_u \pm 1, y_u\pm 1)\in\Gamma. </math>。如果一个在边界的顶点产生崩塌，这将导致沙粒的净损失（两粒在网格的角落，一粒在其他地方）。
#由于沙粒的重新分布，一个顶点的崩塌会使其他顶点不稳定。这样，重复崩塌的过程，直到<math>z_i</math>状态下的所有顶点最终稳定下来，继续下一轮迭代。

The toppling of several vertices during one iteration is referred to as an ''avalanche''. Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final stable configuration does not depend on the chosen order; this is one sense in which the sandpile is [[Abelian group|''abelian'']]. Similarly, the number of times each vertex topples during each iteration is also independent of the choice of toppling order.

在一次迭代中多个顶点的崩塌被称为雪崩。每一次雪崩最终都会停止，也就是说，经过有限数量的顶点崩塌，会达到某种稳定的配置，这样自动机就得到了很好的定义。此外，尽管顶点崩塌的顺序常常有许多可能的选择，但最终的稳定状态并不依赖于所选择的顺序; 这是沙堆模型具有的可交换性质。类似地，在每次迭代过程中，每个顶点的崩塌次数也与崩塌顺序的选择是无关。

==Definition (undirected finite multigraphs)定义（无向有限多图）==
To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph <math>G=(V,E)</math>, a special vertex <math>s\in V</math> called the ''sink'' is specified that is not allowed to topple. A ''configuration'' (state) of the model is then a function <math>z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math> counting the non-negative number of grains on each non-sink vertex. A non-sink vertex <math>v\in V\setminus\{s\}</math> with
:<math>z(v)\geq \deg(v)</math>

is unstable; it can be toppled, which sends one of its grains to each of its (non-sink) neighbors:
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math> for all <math>u\sim v</math>, <math>u\neq s</math>.

为了将''' 沙堆模型'''从标准方格的矩形网格推广到任意无向有限多重图 <math> G=(V,E)</math> ，在 <math> V</math> 中指定了一个不允许崩塌的特殊沉没顶点<math> s</math>。模型的配置(状态)服从函数<math> z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math>，计算每个非沉没顶点上的非负沙粒数。非沉没顶点<math> v\in V\setminus\{s\} </math>当满足<math> z(v)\geq \deg(v) </math>时是不稳定的，它会产生崩塌，将给它的每个(非沉没)邻居分发一颗沙粒：
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math>对于所有的<math>u\sim v</math>, <math>u\neq s</math>.

The cellular automaton then progresses as before, i.e. by adding, in each iteration, one particle to a randomly chosen non-sink vertex and toppling until all vertices are stable.

元胞自动机像之前一样进行，即在每次迭代中，向随机选择的非沉没顶点添加一个沙粒，不断进行崩塌过程，直到所有顶点都稳定。

The definition of the sandpile model given above for finite rectangular grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math> can then be seen as a special case of this definition: consider the graph <math>G=(V,E)</math> which is obtained from <math>\Gamma</math> by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of <math>\Gamma</math> such that the [[Degree (graph theory)|degree]] of every non-sink vertex of <math>G</math> is four. In this manner, also sandpile models on non-rectangular grids of the standard square lattice (or of any other lattice) can be defined: Intersect some bounded subset <math>S</math> of <math>\mathbb{R}^2</math> with <math>\mathbb{Z}^2</math>. [[Edge contraction|Contract every edge]] of <math>\mathbb{Z}^2</math> whose two endpoints are not in <math>S\cap\mathbb{Z}^2</math>. The single remaining vertex outside of <math>S\cap\mathbb{Z}^2</math> then constitutes the sink of the resulting sandpile graph.

上面给出的沙堆模型的定义，是在标准正方形网格<math>\mathbb{Z}^2</math>上的有限矩形网格<math>\Gamma\subset\mathbb{Z}^2</math>，它可以看作是下面定义的一个特例：考虑图<math>G=(V,E)</math>，从<math>\Gamma</math>添加一个沉没顶点，并添加从沉没顶点到每个边界顶点的边，使得<math>G</math>的每个非沉没顶点的度数为4。以这种方式，也可以定义标准正方形网格(或任何其他类型网格)的非矩形格上的沙堆模型: 将<math>\mathbb{R}^2</math>的一些有界子集<math>S</math>与<math>\mathbb{R}^2</math>相交。收缩<math>\mathbb{Z}^2</math>的每条边，其两个端点不在<math>S\cap\mathbb{Z}^2</math>中。<math>S\cap\mathbb{Z}^2</math>之外的一个单独剩余顶点构成了最终沙堆图的沉没顶点。

==Transient and recurrent configurations瞬态和循环构型==
In the dynamics of the sandpile automaton defined above, some stable configurations (<math>0\leq z(v)<4</math> for all <math>v\in G\setminus\{s\}</math>) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as ''recurrent configurations'', while the latter are referred to as ''transient configurations''. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the ''minimally stable configuration'' <math>z_m</math>, where every vertex carries <math>z_m(v)=deg(v)-1</math> grains of sand, is reachable from any other stable configuration (add <math>deg(v)-z(v)-1\geq 0</math> grains to every vertex). Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing.

在上面定义的沙堆自动机的动力学过程中，一些稳定状态的构型（对于所有<math>v\in G\setminus\{s\}</math>，<math>0\leq z(v)<4</math>）经常无限次出现，而另一些则只能出现有限次（如果真的发生的话）。前者被称为“循环构型”，而后者被称为“瞬态构型”。因此，周期性构形由所有稳定的非负构形组成，这些构形可以从任何其他稳定构形中，通过反复向顶点添加沙粒，产生崩塌而得到。很容易看出，“最小稳定配置”<math>zum</math>，其中每个顶点放置<math>z_m(v)=deg(v)-1</math>颗沙粒，可从任何其他稳定构型得到（通过向每个顶点添加<math>deg(v)-z(v)-1\geq 0</math>颗沙粒）。因此，也就是说，周期性构型可以从最小稳定构型开始，通过添加沙粒，再稳定化得到。

Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together. To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.

并非所有非负稳定构型都是循环的。例如，在一个至少包含由两个连通的非沉没顶点的图结构的沙堆模型中，如果这两个顶点没有放置沙粒，那么这个稳定结构是非循环的。为了证明这一点，首先要注意的是，沙粒的增加只能增加两个顶点放置的沙粒的总数。为了达到两个顶点都不放置沙粒的构型，从一个不是这种情况的构型出发，必然涉及到两个顶点中至少有一个崩塌的步骤。考虑这些步骤中的最后一个，在这个步骤中，两个顶点中的一个必须最后崩塌。由于崩塌会将对每个相邻的顶点转移一颗沙粒，这意味着两个顶点共同放置的沙粒总数不能低于一颗，因而得证。

==Sandpile group沙堆群==
Given a configuration <math>z</math>, <math>z(v)\in\mathbb{N}_0</math> for all <math>v\in G\setminus\{s\}</math>, toppling unstable non-sink vertices on a finite connected graph until no unstable non-sink vertex remains leads to a unique ''stable'' configuration <math>z^\circ</math>, which is called the ''stabilization'' of <math>z</math>. Given two stable configurations <math>z</math> and <math>w</math>, we can define the operation <math>z*w \to (z+w)^\circ</math>, corresponding to the vertex-wise addition of grains followed by the stabilization of the resulting sandpile.

给定一个构型<math>z</math>，<math>z(v)\in\mathbb{N}_0</math>对于所有<math>v\in G\setminus\{s\}</math>，在有限连通图上使不稳定的非沉没顶点崩塌，直到没有不稳定的非汇顶点保留，这将导致唯一的“稳定”配置math>z^\circ</math>，这就是<math>z</math>的“稳定化”。给定两个稳定构型<math>z</math>和<math>w</math>，我们可以定义运算<math>z*w \to (z+w)^\circ</math>，对应于沙粒的顶点方向相加，然后稳定得到的沙堆。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）corresponding to the vertex-wise addition of grains翻译存疑

Given an arbitrary but fixed ordering of the non-sink vertices, multiple toppling operations, which can e.g. occur during the stabilization of an unstable configuration, can be efficiently encoded by using the [[Laplacian matrix|graph Laplacian]] <math>\Delta=D-A</math>, where <math>D</math> is the [[degree matrix]] and <math>A</math> is the [[adjacency matrix]] of the graph.

给定一个任意但固定的非沉没顶点的顺序，进行多个崩塌操作，可以通过使用[[拉普拉斯矩阵|图拉普拉斯]]<math>\Delta=D-A</math>高效地编码多个崩塌操作（例如，在不稳定构型的稳定过程中可能发生），其中<math>D</math>是图的[[度矩阵]]，<math>A</math>是图的[[邻接矩阵]]。

Deleting the row and column of <math>\Delta</math> corresponding with the sink yields the ''reduced graph Laplacian'' <math>\Delta'</math>. Then, when starting with a configuration <math>z</math> and toppling each vertex <math>v</math> a total of <math>\mathbf{x}(v)\in\mathbb{N}_0</math> times yields the configuration <math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>, where <math>\boldsymbol{\cdot}</math> is the contraction product. Furthermore, if <math>\mathbf{x}</math> corresponds to the number of times each vertex is toppled during the stabilization of a given configuration <math>z</math>, then
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

删除与沉没顶点相对应的 <math>\Delta</math>的行和列，得到简化图拉普拉斯矩阵 <math>\Delta'</math>。然后，当以一个构型<math>z</math> 开始，并将每个顶点<math>v</math>进行总共<math>\mathbf{x}(v)\in\mathbb{N}_0</math>次的崩塌操作时，产生<math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>构型，其中<math>\boldsymbol{\cdot}</math>是收缩积。此外，如果 <math>\mathbf{x}</math> 对应于在给定构型<math>z</math>的稳定过程中每个顶点产生崩塌的次数，则
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

In this case, <math>\mathbf{x}</math> is referred to as the ''toppling'' or ''odometer function'' (of the stabilization of <math>z</math>).

在这种情况下，<math>\mathbf{x}</math>被称为崩塌或<math>z</math>的稳定过程的里程计函数。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）odometer function翻译存疑

Under the operation <math>*</math>, the set of recurrent configurations forms an [[abelian group]] isomorphic to the cokernel of the reduced graph Laplacian <math>\Delta'</math>, i.e. to <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>, whereby <math>n</math> denotes the number of vertices (including the sink). More generally, the set of stable configurations (transient and recurrent) forms a [[commutative monoid]] under the operation <math>*</math>. The minimal [[Semigroup#Subsemigroups and ideals|ideal]] of this monoid is then isomorphic to the group of recurrent configurations.

在<math>*</math>运算下，递归构形的集合构成一个与约化图拉普拉斯矩阵<math>\Delta'</math>的核同构的阿贝尔群。对于 <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，其中<math>n</math> 表示顶点数（包括沉没顶点）。更一般地说，稳定构型集（瞬态和循环）在<math>*</math>.运算下形成''' 交换幺半群Commutative monoid'''。这个幺半群的最小理想同构于循环构型群。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）ideal翻译存疑

The group formed by the recurrent configurations, as well as the group <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> to which the former is isomorphic, is most commonly referred to as the ''sandpile group''. Other common names for the same group are ''critical group'', ''Jacobian group'' or (less often) ''Picard group''. Note, however, that some authors only denote the group formed by the recurrent configurations as the sandpile group, while reserving the name Jacobian group or critical group for the (isomorphic) group defined by <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> (or for related isomorphic definitions). Finally, some authors use the name Picard group to refer to the direct product of the sandpile group and <math>\mathbb{Z}</math>, which naturally appears in a cellular automaton closely related to the sandpile model, referred to as the chip firing or dollar game.

由循环构形形成的群，以及与之同构的群<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，通常称为''' 沙堆群Sandpile group'''。相同群的其它常用名称是“临界群”、“Jacobian群”或(不常见的)“Picard群”。然而，要注意的是，有些作者只把循环构型形成的组称为沙堆组，而为由<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>（或相关的同构定义）定义的同构群保留Jacobian群或临界群的称呼。最后，一些作者使用Picard群来指代沙堆群和<math>\mathbb{Z}</math>的直接产物，后者出现在与沙堆模型密切相关的元胞自动机中，被称为''' 碎片点火或美元游戏Chip firing or Dollar game'''。

Given the isomorphisms stated above, the order of the sandpile group is the determinant of <math>\Delta'</math>, which by the [[Kirchhoff's theorem|matrix tree theorem]] is the number of spanning trees of the graph.

给定上述同构，沙堆群的顺序是<math>\Delta'</math>的行列式，根据矩阵树定理，它是图的生成树数目。

==Self-organized criticality自组织临界性==
{{main|Self-organized criticality}}
The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its [[critical state]], at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the [[phase transition]]s between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is [[self-organization|self-organized]].

模型最初源于这样一个事实，即在晶格上的模拟中，它被吸引到了它的[[临界状态]]，此时系统的关联长度和关联时间趋于无穷大，不需要对系统参数进行任何微调。这与早期临界现象的例子形成了对比，例如固体和液体之间，或液体和气体之间的[[相变]]，其中临界点只能通过精确调节（例如，温度）来达到。因此，在沙堆模型中，我们可以说临界性是自组织的。

Once the sandpile model reaches its critical state there is no correlation between the system's response to a [[wiktionary:perturbation|perturbation]] and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays [[1/f noise|1/''&fnof;'' noise]], a feature common to many complex systems in nature.

一旦沙堆模型达到其临界状态，系统对扰动的响应和扰动细节之间就没有关联。一般来说，这意味着再往沙堆形成的斜坡上撒一粒沙子可能不会导致任何事情发生，或者可能导致整个沙堆形成的斜坡在大规模滑坡中崩塌。该模型还显示了[[1/f noise|1/''&fnof;'' noise]]，这是自然界中许多复杂系统的共同特征。

This model only displays critical behaviour in two or more dimensions. The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.

此模型仅在两个或多个维度中显示关键行为。沙堆模型可以用一维来表示; 然而，一维沙堆模型不是演化到临界状态，而是达到最小稳定状态，其中每个格点都趋向临界坡度。

For two dimensions, it has been hypothesized that the associated [[conformal field theory]] is consists of [[symplectic fermion]]s with a [[central charge]] ''c'' = −2.<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

对于二维，相关共形场理论被认为是中心电荷为''c'' = −2的symplectic fermion辛费米子组成的；<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

==Properties属性==
===Least action principle最小作用原理===
The stabilization of chip configurations obeys a form of ''[[principle of least action|least action principle]]'': each vertex topples no more than necessary in the course of the stabilization.<ref name=Fey2010>
{{cite journal
| author = Fey, A. |author2=Levine, L.|author3=Peres, Y.
| year=2010
| issn=0022-4715
| journal=Journal of Statistical Physics
| volume=138
| number=1–3
| doi=10.1007/s10955-009-9899-6
| title=Growth Rates and Explosions in Sandpiles
| pages=143–159|arxiv = 0901.3805 |bibcode = 2010JSP...138..143F |s2cid=7180488}}</ref>

芯片构型的稳定遵循一种“[[最小作用原理|最小作用原理]]”的形式：每个顶点在稳定过程中不超过必要的崩塌量。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）chip configurations的翻译存疑

This can be formalized as follows. Call a sequence of topples ''legal'' if it only topples unstable vertices, and ''stabilizing'' if it results in a stable configuration. The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible. Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex <math>v</math>, the number of times <math>v</math> topples is the same in all legal stabilizing sequences. According to the least action principle, a '''minimal stabilizing''' sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence. In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence.

这可以形式化如下。如果一个崩塌序列只推倒不稳定的顶点，则称其为“合法的”，使用一组合法的顶点序列，如果它的结果是一个稳定的构型，则称其为“稳定的”。稳定沙堆的标准方法是找到一个最大的合法崩塌序列，也就是说，让崩塌序列尽可能地长。这种序列具有明显的稳定性，沙堆的可交换性质是所有这些序列都等价于倾斜序列的置换，也就是说，对于任何顶点<math>v</math>，在所有合法的稳定序列中<math>v</math>的崩塌次数都是相同的。根据最小作用原理，最小稳定序列等价于合法的（且稳定的）崩塌序列的置换。特别地，由最小稳定序列产生的构型与由最大合法序列产生的构型是相同的。

More formally, if <math>\mathbf{u}</math> is a vector such that <math>\mathbf{u}(v)</math> is the number of times the vertex <math>v</math> topples during the stabilization (via the toppling of unstable vertices) of a chip configuration <math>z</math>, and <math>\mathbf{n}</math> is an integral vector (not necessarily non-negative) such that <math>z-\mathbf{n}\Delta'</math> is stable, then <math>\mathbf{u}(v) \leq \mathbf{n}(v)</math> for all vertices <math>v</math>.

更正式地说，如果<math>\mathbf{u}</math>是一个向量， <math>\mathbf{u}（v）</math>是碎片构型<math>z</math>在稳定过程中（通过不稳定顶点的崩塌）顶点<math>v</math>崩塌的次数，并且<math>\mathbf{n}</math>是一个积分向量（不一定是非负的），使得<math>z-\mathbf{n}\Delta'</math>是稳定的，那么对于所有顶点<math>v</math>，<math>\mathbf{u}(v) \leq \mathbf{n}(v)</math>。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）<math>\mathbf{n}</math> is an integral vector (not necessarily non-negative)这一句话的翻译存疑。

=== Scaling limits缩放限制===
[[File:Scaling sandpile identity.gif|thumb|Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.]]

[[File:Scaling sandpile identity.gif|thumb|沙堆标识在方形网格不断增加的动画。黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。]]

The animation shows the recurrent configuration corresponding to the identity of the sandpile group on different <math>N\times N</math> square grids of increasing sizes <math>N\geq 1</math>, whereby the configurations are rescaled to always have the same physical dimension. Visually, the identities on larger grids seem to become more and more detailed and to "converge to a continuous image". Mathematically, this suggests the existence of scaling-limits of the sandpile identity on square grids based on the notion of weak-* convergence (or some other, generalized notion of convergence). Indeed, existence of scaling limits of recurrent sandpile configurations has been proved by Wesley Pegden and Charles Smart
<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复配置，从而重新缩放配置以始终具有相同的物理维度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

== Generalizations and related models归纳与相关模型==
=== Sandpile models on infinite grids有向图上的沙堆模型===
[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|30 million grains dropped to a site of the infinite square grid, then toppled according to the rules of the sandpile model. White color denotes sites with 0 grains, green is for 1, purple is for 2, gold is for 3. The bounding box is 3967×3967.]]

[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|3000万粒沙粒落在无限方形网格的一个位置上，然后按照沙堆模型的规则产生崩塌。白色表示0颗沙粒位置，绿色表示1颗，紫色表示2颗，金色表示3颗。图中框的大小是3967×3967。]]

There exist several generalizations of the sandpile model to infinite grids. A challenge in such generalizations is that, in general, it is not guaranteed anymore that every avalanche will eventually stop. Several of the generalization thus only consider the stabilization of configurations for which this can be guaranteed.

沙堆模型可以推广到无限网格中。这种归纳法的一个挑战是，一般来说，不再保证每次雪崩最终都会停止。因此，一些一般化方法只考虑了构型的稳定性，因为这一点是能保证的。

A rather popular model on the (infinite) square lattice with sites <math>(x,y)\in\mathbb{Z}^2</math> is defined as follows:

在（无限）方格上有一个相当流行的模型，其位置<math>(x,y)\in\mathbb{Z}^2</math>定义如下：

Begin with some nonnegative configuration of values <math>z(x,y)\in \mathbf{Z}</math> which is finite, meaning

从\mathbf{z}</math>中有限值<math>z(x,y)\in \mathbf{Z}</math>的一些非负配置开始，这意味着

:<math>\sum_{x,y}z(x,y)<\infty.</math>

Any site <math>(x,y)</math> with

任何位置<math>(x,y)</math>有
:<math>z(x,y)\geq 4</math>
is ''unstable'' and can ''topple'' (or ''fire''), sending one of its chips to each of its four neighbors:

就是“不稳定的”，并且会产生崩塌，将它位置上的沙粒分发给它的四个邻居：
:<math>z(x,y) \rightarrow z(x,y) - 4,</math>
:<math>z( x \pm 1, y) \rightarrow z( x \pm 1, y) + 1,</math>
:<math>z(x, y \pm 1) \rightarrow z( x, y \pm 1 ) + 1.</math>

Since the initial configuration is finite, the [[Action (philosophy)|process]] is guaranteed to terminate, with the grains scattering outward.

由于初始构型是有限的，这一过程必然会随着沙粒向外散布终止。

A popular special case of this model is given when the initial configuration is zero for all vertices except the origin. If the origin carries a huge number of grains of sand, the configuration after relaxation forms fractal patterns (see figure). When letting the initial number of grains at the origin go to infinity, the rescaled stabilized configurations were shown to converge to a unique limit.<ref name="Pegden2013" /><ref name="Levine2016" />

给出了该模型的一个常见特例，即除原点外的所有顶点的初始配置都为零。如果原点放置大量沙粒，松弛后的构型形成分形图案(见图)。当初始颗粒数趋于无穷时，重缩放的稳定构型收敛到唯一极限。<ref name="Pegden2013" /><ref name="Levine2016" />

=== Sandpile models on directed graphs有向图上的沙堆模型===
The sandpile model can be generalized to arbitrary directed multigraphs. The rules are that any vertex <math>v</math> with

沙堆模型可以推广到任意有向多重图。规则是任何顶点<math>v</math>有
:<math>z(v)\geq \deg^{+}(v)</math>
is unstable; toppling again sends chips to each of its neighbors, one along each outgoing edge:

则是不稳定的；崩塌将它的碎片沿着边的输出方向，分发给它的邻居：
:<math>z(v) \rightarrow z(v) - \deg^{+}(v) + \deg(v,v)</math>
and, for each <math>u\neq v</math>:

并且，对于每个<math>u\neq v</math>：
:<math>z(u) \rightarrow z(u) + \deg(v,u)</math>
where <math>\deg(v,u)</math> is the number of edges from <math>v</math> to <math>u</math>.

其中，<math>\deg(v,u)</math>是从<math>v</math>到<math>u</math>的边数。

In this case the Laplacian matrix is not symmetric. If we specify a sink <math>s</math> such that there is a path from every other vertex to <math>s</math>, then the stabilization operation on finite graphs is well-defined and the sandpile group can be written

在这种情况下，拉普拉斯矩阵是不对称的。如果我们指定一个沉没顶点<math>s</math>，使得每一个顶点都有一条到<math>s</math>的路径，那么有限图上的稳定操作是定义良好的，并且沙堆群可以像之前一样被写出来

:<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>
as before.

The order of the sandpile group is again the determinant of <math>\Delta'</math>, which by the general version of the [[Kirchhoff's theorem|matrix tree theorem]] is the number of oriented [[spanning tree]]s rooted at the sink.

沙堆群的顺序又是<math>\Delta'</math>的行列式，根据[[矩阵树定理]]的一般版本，它是根在沉没顶点的有向[[spanning tree]]生成树的数目。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）The order of the sandpile group翻译存疑。

=== The extended sandpile model扩展沙堆模型===
[[File:Harmonic Sandpile Dynamics.gif|thumb|Sandpile dynamics induced by the harmonic function H=x*y on a 255x255 square grid.]]

[[File:Harmonic Sandpile Dynamics.gif|thumb|由谐波函数H=x*y在255x255方格网格上引起的沙堆动力学。]]

To better understand the structure of the sandpile group for different finite convex grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math>, Lang and Shkolnikov introduced the ''extended sandpile model'' in 2019.<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref> The extended sandpile model is defined nearly exactly the same as the ''usual sandpile model'' (i.e. the original Bak–Tang–Wiesenfeld model <ref name="Bak1987" />), except that vertices at the boundary <math>\partial\Gamma</math> of the grid are now allowed to carry a non-negative real number of grains. In contrast, vertices in the interior of the grid are still only allowed to carry integer numbers of grains. The toppling rules remain unchanged, i.e. both interior and boundary vertices are assumed to become unstable and topple if the grain number reaches or exceeds four.

为了更好地理解不同有限凸网格<math>\Gamma\subset\mathbb{Z}^2</math>的沙堆群的结构，Lang和Shkolnikov在2019年提出了“扩展沙堆模型”。<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref>扩展沙堆模型的定义与“通常的沙堆模型”几乎完全相同（即原始的Bak–Tang–Wiesenfeld模型<ref name="Bak1987" />），除了网格边界<math>\partial\Gamma</math>的顶点现在允许放置非负实数的沙粒。相比之下，网格内部的顶点仍然只允许放置整数个粒子。崩塌规则保持不变，即假设当沙粒数达到或超过4时，内部顶点和边界顶点都变得不稳定并发生崩塌。

Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the ''extended sandpile group'', of which the usual sandpile group is a [[discrete subgroup]]. Different to the usual sandpile group, the extended sandpile group is however a continuous [[Lie group]]. Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid, the extended sandpile group furthermore has the [[topological group|topology]] of a [[torus]] of dimension <math>|\partial\Gamma|</math> and a volume given by the order of the usual sandpile group.<ref name="Lang2019" />

扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）a volume given by the order of the usual sandpile group.翻译存疑。

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics

特别感兴趣的问题是循环构型如何通过恒等式，沿着这个环面的连续[[测地线]]动态变化的问题。这个问题引出了沙堆动力学的定义
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？
:<math>D_H(t)=(I-t\Delta H)^\circ</math> （扩展沙堆模型）
respectively
:<math>\tilde{D}_H(t)=(I+\lfloor-t\Delta H\rfloor)^\circ</math> （普通沙堆模型）

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />
For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the
smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid. Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" />

由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在标度限制。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复配置映射到子网格上的重复配置。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）上面的长句翻译需要重新审校Informaly, this renormalization simply maps

=== The divisible sandpile 可分割的沙堆===
A strongly related model is the so called '''divisible sandpile model''', introduced by Levine and Peres in 2008,<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref> in which, instead of a discrete number of particles in each site <math>x</math>, there is a real number <math>s(x)</math> representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time <math>t</math>, it will be full for all later times.

Levine和Peres在2008年提出了一个与之密切相关的模型，即所谓的“可分割的沙堆模型”。<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref>与每个位置<math>x</math>上的沙粒数量为离散数不同，有一个实数<math>s(x)</math>代表位置的总质量。如果这个质量是负的，我们就可以把它理解为一个空洞。当一个位置上的质量大于1时，就会发生崩塌; 它将多余的部分均匀地分发给它的邻居，这就导致了如果一个位置在<math>t</math>的时刻是满的，它在以后的所有时间都是满的。

== References引用 ==
{{Reflist|30em}}

== Further reading ==
* {{cite book
| author = Per Bak
| year = 1996
| title = How Nature Works: The Science of Self-Organized Criticality
| publisher = Copernicus
| location = New York
| isbn = 978-0-387-94791-4
}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B | pmid=10035754}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1988
| title = Self-organized criticality
| journal = [[Physical Review A]]
| volume = 38
| issue = 1
| pages = 364–374
| doi = 10.1103/PhysRevA.38.364
|pmid=9900174 | bibcode=1988PhRvA..38..364B
}}
* {{cite journal | last1=Cori | first1=Robert | last2=Rossin | first2=Dominique | last3=Salvy | first3=Bruno | title=Polynomial ideals for sandpiles and their Gröbner bases | zbl=1002.68105 | journal=Theor. Comput. Sci. | volume=276 | issue=1–2 | year=2002 | doi=10.1016/S0304-3975(00)00397-2 | pages=1–15| url=https://hal.archives-ouvertes.fr/hal-00016378/file/sandpile.pdf }}
* {{cite book|first=Caroline|last=Klivans|authorlink=Caroline Klivans|title= The Mathematics of Chip-Firing|title-link= The Mathematics of Chip-Firing|publisher=CRC Press|year=2018}}
* {{cite book | author1-last=Perkinson | author1-first=David | author2-last=Perlman | author2-first=Jacob | author3-last=Wilmes | author3-first=John | chapter=Algebraic geometry of sandpiles | pages=211–256 | editor1-last=Amini | editor1-first=Omid | editor2-last=Baker | editor2-first=Matthew | editor3-last=Faber | editor3-first=Xander | title=Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 | zbl=1281.14002 | series=Contemporary Mathematics | volume=605 | doi=10.1090/conm/605/12117 
| location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-1-4704-1021-6 | year=2013 | citeseerx=10.1.1.760.283 | s2cid=7851577 }}

==External links==
*{{cite web|last1=Garcia-Puente|first1=Luis David|title=Sandpiles|url=https://www.youtube.com/watch?v=1MtEUErz7Gg|website=YouTube|publisher=[[Brady Haran]]|accessdate=15 January 2017|format=YouTube video}}

[[Category:Self-organization]]
[[Category:Phase transitions]]
[[Category:Dynamical systems]]
[[Category:Critical phenomena]]
[[Category:Nonlinear systems]]
[[Category:Cellular automaton rules]]

沙堆模型

2020-11-13T20:23:03Z

Zcy：

此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。此词条由Zcy初步审校。
{{short description|Cellular automaton}}
[[File:Sandpile identity 300x205.png|upright=1.25|thumb|The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.]]
[[File:Sandpile identity 300x205.png|upright=1.25|thumb|沙堆在矩形网格上的标识。黄色像素对应三颗沙粒的顶点，淡紫色代表两颗沙粒，绿色表示一颗沙粒，黑色表示零颗沙粒。]]

The '''Abelian sandpile model''', also known as the '''Bak–Tang–Wiesenfeld model''', was the first discovered example of a [[dynamical system]] displaying [[self-organized criticality]]. It was introduced by [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] in a 1987 paper.<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>

''' 阿贝尔沙堆模型Abelian sandpile model'''，也被称为 Bak-Tang-Wiesenfeld 模型，是第一个发现的动力系统展示自组织临界性的例子。它是由 Per Bak，Chao Tang 和 Kurt Wiesenfeld 在1987年的一篇论文<ref name=Bak1987>
{{cite journal
| author = Bak, P. |author2=Tang, C. |author3-link=Kurt Wiesenfeld |author3=Wiesenfeld, K.
| year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B
| pmid=10035754
|author-link=Per Bak |author2-link=Chao Tang }}</ref>中提出的。

The model is a [[cellular automaton]]. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.
这个模型是一种''' 细胞自动机模型Cellular automaton'''。在最初的公式中，有限网格上的每个位置都有一个与沙堆的坡度相对应的关联值。当“沙粒”(或“碎片”)被随机放置在沙堆上时，放置位置的斜坡就会堆积起来，直到倾斜程度超过一个特定的阈值，这个位置倒塌，沙子会转移到邻近的位置，增加它们的斜坡。这个斜坡以“沙粒”(或“碎片”)随机放置的方式逐渐形成，直到斜坡超过一个特定的阈值，在这个阈值的时候，这个位置倒塌，将沙子转移到邻近的位置，增加它们的斜坡。Bak，Tang和 Wiesenfeld考虑了在网格上连续随机放置沙粒的过程; 每次这样在特定位置放置沙粒有可能没有产生影响，也有可能会引起级联反应，影响到周围的其他位置。

The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs).<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref> It is closely related to the [[Chip-firing game#Biggs's Variant|dollar game]], a variant of the [[chip-firing game]] introduced by Biggs.<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>
该模型已经在无限栅格、其他(非方形)栅格和任意图(包括有向多重图)上进行了研究。<ref name=Hol2008>{{cite book
| author = Holroyd, A. |author2=Levine, L. |author3=Mészáros, K. |author4=Peres, Y. |author5=Propp, J. |author6=Wilson, B.
| year = 2008
| title = Chip-Firing and Rotor-Routing on Directed Graphs
| journal = In and Out of Equilibrium 2
| volume = 60
| pages = 331–364
| doi = 10.1007/978-3-7643-8786-0_17
| bibcode=1987PhRvL..59..381B
|arxiv=0801.3306|isbn=978-3-7643-8785-3 |s2cid=7313023 }}</ref>它与美元游戏密切相关，美元游戏是Biggs引入的一种chip-firing游戏的变体。<ref>{{cite journal|last=Biggs|first=Norman L.|date=25 June 1997|title=Chip-Firing and the Critical Group of a Graph|url=ftp://ftp.math.ethz.ch/hg/EMIS/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf|journal=Journal of Algebraic Combinatorics|pages=25–45|accessdate=10 May 2014}}</ref>

==Definition (rectangular grids)定义（矩形网格）==
The sandpile model is a [[cellular automaton]] originally defined on a <math>N\times M</math> rectangular grid (''checkerboard'') <math>\Gamma\subset\mathbb{Z}^2</math> of the [[Square lattice|standard square lattice]] <math>\mathbb{Z}^2</math>.

沙堆模型是一个'''元胞自动机cellular automaton '''，它最初定义在<math> N\times M </math>矩形网格(棋盘格)上，其顶点在标准的正方形格子<math> \mathbb{Z}^2 </math>上<math> \Gamma\subset\mathbb{Z}^2 </math>。

To each vertex (''side'', ''field'') <math>(x,y)\in\Gamma</math> of the grid, we associate a value (''grains of sand'', ''slope'', ''particles'') <math>z_0(x,y)\in\{0,1,2,3\}</math>, with <math>z_0\in\{0,1,2,3\}^\Gamma</math> referred to as the (initial) configuration of the sandpile.

对于栅格中的每个顶点<math> (x,y)\in\Gamma </math>，我们关联一个值(沙粒、坡度、颗粒) <math> z_0(x,y)\in\{0,1,2,3\} </math>，这样所有顶点的初始状态<math> z_0\in\{0,1,2,3\}^\Gamma </math>被称为沙堆的(初始)构型。

The dynamics of the automaton at iteration <math>i\in\mathbb{N}</math> are then defined as follows:
自动机的动力学过程在第<math> i\in\mathbb{N} </math>次迭代时的定义如下:

# Choose a random vertex <math>(x_i,y_i)\in\Gamma</math> according to some probability distribution (usually uniform).
# Add one grain of sand to this vertex while letting the grain numbers for all other vertices unchanged, i.e. set <math>z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1</math> and <math>z_i(x,y)=z_{i-1}(x,y)</math> for all <math>(x,y)\neq(x_i,y_i)</math>.
# If all vertices are ''stable'', i.e. <math>z_i(x,y)<4</math> for all <math>(x,y)\in\Gamma</math>, also the configuration <math>z_i</math> is said to be stable. In this case, continue with the next iteration.
# If at least one vertex is ''unstable'', i.e. <math>z_i(x_u,y_u)\geq 4</math> for some <math>(x_u,y_u)\in\Gamma</math>, the whole configuration <math>z_i</math> is said to be unstable. In this case, choose any unstable vertex <math>(x_u,y_u)\in\Gamma</math> at random. ''Topple'' this vertex by reducing its grain number by four and by increasing the grain numbers of each of its (at maximum four) direct neighbors by one, i.e. set <math>z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) - 4,</math>, and <math>z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1</math> if <math>( x_u \pm 1, y_u\pm 1)\in\Gamma</math>. If a vertex at the boundary of the domain topples, this results in a net loss of grains (two grains at the corner of the grid, one grain otherwise).
# Due to the redistribution of grains, the toppling of one vertex can render other vertices unstable. Thus, repeat the toppling procedure until all vertices of <math>z_i</math> eventually become stable and continue with the next iteration.

#根据某种概率分布(通常为均匀分布)随机选择一个顶点。根据一些概率分布(通常是均匀的)选择一个随机顶点<math> (x_i,y_i)\in\Gamma </math>。
#向这个顶点添加一粒沙子，同时让其他顶点的沙粒数保持不变，也就是设定<math> z_i(x_i,y_i)=z_{i-1}(x_i,y_i)+1 </math>，对于所有的<math> (x,y)\neq(x_i,y_i) </math>，<math> z_i(x,y)=z_{i-1}(x,y) </math>。
#如果所有的顶点都是稳定的，即如果<math> (x,y)\in\Gamma </math>中<math> z_i(x,y)<4 </math>，那么<math> z_i </math>被认为是稳定的。在这种情况下，继续下一轮迭代。
#如果至少有一个顶点是不稳定的，即对于一些<math> (x_u,y_u)\in\Gamma</math>，<math>z_i(x_u,y_u)\geq 4</math>，<math>z_i</math>被认为是不稳定的。在这种情况下，随机选择任意不稳定顶点<math> (x_u,y_u)\in\Gamma</math>。将该顶点的沙粒数减少4个，清空这个顶点，并将其每个(最多4个)直接邻居的沙粒数增加1个。即：<math> z_i(x_u,y_u) \rightarrow z_i(x_u,y_u) – 4</math>，<math> z_i( x_u \pm 1, y_u \pm 1) \rightarrow z_i( x_u \pm 1, y_u\pm 1) + 1 if ( x_u \pm 1, y_u\pm 1)\in\Gamma. </math>。如果一个在边界的顶点产生崩塌，这将导致沙粒的净损失（两粒在网格的角落，一粒在其他地方）。
#由于沙粒的重新分布，一个顶点的崩塌会使其他顶点不稳定。这样，重复崩塌的过程，直到<math>z_i</math>状态下的所有顶点最终稳定下来，继续下一轮迭代。

The toppling of several vertices during one iteration is referred to as an ''avalanche''. Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final stable configuration does not depend on the chosen order; this is one sense in which the sandpile is [[Abelian group|''abelian'']]. Similarly, the number of times each vertex topples during each iteration is also independent of the choice of toppling order.

在一次迭代中多个顶点的崩塌被称为雪崩。每一次雪崩最终都会停止，也就是说，经过有限数量的顶点崩塌，会达到某种稳定的配置，这样自动机就得到了很好的定义。此外，尽管顶点崩塌的顺序常常有许多可能的选择，但最终的稳定状态并不依赖于所选择的顺序; 这是沙堆模型具有的可交换性质。类似地，在每次迭代过程中，每个顶点的崩塌次数也与崩塌顺序的选择是无关。

==Definition (undirected finite multigraphs)定义（无向有限多图）==
To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph <math>G=(V,E)</math>, a special vertex <math>s\in V</math> called the ''sink'' is specified that is not allowed to topple. A ''configuration'' (state) of the model is then a function <math>z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math> counting the non-negative number of grains on each non-sink vertex. A non-sink vertex <math>v\in V\setminus\{s\}</math> with
:<math>z(v)\geq \deg(v)</math>

is unstable; it can be toppled, which sends one of its grains to each of its (non-sink) neighbors:
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math> for all <math>u\sim v</math>, <math>u\neq s</math>.
为了将''' 沙堆模型'''从标准方格的矩形网格推广到任意无向有限多重图 <math> G=(V,E)</math> ，在 <math> V</math> 中指定了一个不允许崩塌的特殊沉没顶点<math> s</math>。模型的配置(状态)服从函数<math> z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math>，计算每个非沉没顶点上的非负沙粒数。非沉没顶点<math> v\in V\setminus\{s\} </math>当满足<math> z(v)\geq \deg(v) </math>时是不稳定的，它会产生崩塌，将给它的每个(非沉没)邻居分发一颗沙粒：
:<math>z(v) \to z(v) - \deg(v)</math>
:<math>z(u) \to z(u) + 1</math>对于所有的<math>u\sim v</math>, <math>u\neq s</math>.

The cellular automaton then progresses as before, i.e. by adding, in each iteration, one particle to a randomly chosen non-sink vertex and toppling until all vertices are stable.
元胞自动机像之前一样进行，即在每次迭代中，向随机选择的非沉没顶点添加一个沙粒，不断进行崩塌过程，直到所有顶点都稳定。

The definition of the sandpile model given above for finite rectangular grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math> can then be seen as a special case of this definition: consider the graph <math>G=(V,E)</math> which is obtained from <math>\Gamma</math> by adding an additional vertex, the sink, and by drawing additional edges from the sink to every boundary vertex of <math>\Gamma</math> such that the [[Degree (graph theory)|degree]] of every non-sink vertex of <math>G</math> is four. In this manner, also sandpile models on non-rectangular grids of the standard square lattice (or of any other lattice) can be defined: Intersect some bounded subset <math>S</math> of <math>\mathbb{R}^2</math> with <math>\mathbb{Z}^2</math>. [[Edge contraction|Contract every edge]] of <math>\mathbb{Z}^2</math> whose two endpoints are not in <math>S\cap\mathbb{Z}^2</math>. The single remaining vertex outside of <math>S\cap\mathbb{Z}^2</math> then constitutes the sink of the resulting sandpile graph.
上面给出的沙堆模型的定义，是在标准正方形网格\mathbb{Z}^2上的有限矩形网格\Gamma\subset\mathbb{Z}^2，它可以看作是下面定义的一个特例：考虑图G=(V,E)，从\Gamma添加一个沉没顶点，并添加从沉没顶点到每个边界顶点的边，使得G的每个非沉没顶点的度数为4。以这种方式，也可以定义标准正方形网格(或任何其他类型网格)的非矩形格上的沙堆模型: 将\mathbb{R}^2的一些有界子集S与\mathbb{Z}^2相交。收缩\mathbb{Z}^2的每条边，其两个端点不在S\cap\mathbb{Z}^2中。S\cap\mathbb{Z}^2之外的一个单独剩余顶点构成了最终沙堆图的沉没顶点。

==Transient and recurrent configurations瞬态和循环构型==
In the dynamics of the sandpile automaton defined above, some stable configurations (<math>0\leq z(v)<4</math> for all <math>v\in G\setminus\{s\}</math>) appear infinitely often, while others can only appear a finite number of times (if at all). The former are referred to as ''recurrent configurations'', while the latter are referred to as ''transient configurations''. The recurrent configurations thereby consist of all stable non-negative configurations which can be reached from any other stable configuration by repeatedly adding grains of sand to vertices and toppling. It is easy to see that the ''minimally stable configuration'' <math>z_m</math>, where every vertex carries <math>z_m(v)=deg(v)-1</math> grains of sand, is reachable from any other stable configuration (add <math>deg(v)-z(v)-1\geq 0</math> grains to every vertex). Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing.
在上面定义的沙堆自动机的动力学过程中，一些稳定状态的构型（对于所有<math>v\In G\setminus\{s\}</math>，<math>0\leq z（v）<4</math>）经常无限次出现，而另一些则只能出现有限次（如果真的发生的话）。前者被称为“循环构型”，而后者被称为“瞬态构型”。因此，周期性构形由所有稳定的非负构形组成，这些构形可以从任何其他稳定构形中，通过反复向顶点添加沙粒，产生崩塌而得到。很容易看出，“最小稳定配置”<math>zum</math>，其中每个顶点放置<math>zum（v）=deg（v）-1</math>颗沙粒，可从任何其他稳定构型得到（通过向每个顶点添加<math>deg（v）-z（v）-1\geq 0</math>颗沙粒）。因此，也就是说，周期性构型可以从最小稳定构型开始，通过添加沙粒，再稳定化得到。

Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together. To reach a configuration where both vertices carry zero particles from a configuration where this is not the case thus necessarily involves steps where at least one of the two vertices is toppled. Consider the last one of these steps. In this step, one of the two vertices has to topple last. Since toppling transfers a grain of sand to every neighboring vertex, this implies that the total number of grains carried by both vertices together cannot be lower than one, which concludes the proof.
并非所有非负稳定构型都是循环的。例如，在一个至少包含由两个连通的非沉没顶点的图结构的沙堆模型中，如果这两个顶点没有放置沙粒，那么这个稳定结构是非循环的。为了证明这一点，首先要注意的是，沙粒的增加只能增加两个顶点放置的沙粒的总数。为了达到两个顶点都不放置沙粒的构型，从一个不是这种情况的构型出发，必然涉及到两个顶点中至少有一个崩塌的步骤。考虑这些步骤中的最后一个，在这个步骤中，两个顶点中的一个必须最后崩塌。由于崩塌会将对每个相邻的顶点转移一颗沙粒，这意味着两个顶点共同放置的沙粒总数不能低于一颗，因而得证。

==Sandpile group沙堆群==
Given a configuration <math>z</math>, <math>z(v)\in\mathbb{N}_0</math> for all <math>v\in G\setminus\{s\}</math>, toppling unstable non-sink vertices on a finite connected graph until no unstable non-sink vertex remains leads to a unique ''stable'' configuration <math>z^\circ</math>, which is called the ''stabilization'' of <math>z</math>. Given two stable configurations <math>z</math> and <math>w</math>, we can define the operation <math>z*w \to (z+w)^\circ</math>, corresponding to the vertex-wise addition of grains followed by the stabilization of the resulting sandpile.
给定一个构型<math>z</math>，<math>z(v)\in\mathbb{N}_0</math>对于所有<math>v\in G\setminus\{s\}</math>，在有限连通图上使不稳定的非沉没顶点崩塌，直到没有不稳定的非汇顶点保留，这将导致唯一的“稳定”配置math>z^\circ</math>，这就是<math>z</math>的“稳定化”。给定两个稳定构型<math>z</math>和<math>w</math>，我们可以定义运算<math>z*w \to (z+w)^\circ</math>，对应于沙粒的顶点方向相加，然后稳定得到的沙堆。

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）corresponding to the vertex-wise addition of grains翻译存疑

Given an arbitrary but fixed ordering of the non-sink vertices, multiple toppling operations, which can e.g. occur during the stabilization of an unstable configuration, can be efficiently encoded by using the [[Laplacian matrix|graph Laplacian]] <math>\Delta=D-A</math>, where <math>D</math> is the [[degree matrix]] and <math>A</math> is the [[adjacency matrix]] of the graph.

给定一个任意但固定的非沉没顶点的顺序，进行多个崩塌操作，可以通过使用[[拉普拉斯矩阵|图拉普拉斯]]<math>\Delta=D-A</math>高效地编码多个崩塌操作（例如，在不稳定构型的稳定过程中可能发生），其中<math>D</math>是图的[[度矩阵]]，<math>A</math>是图的[[邻接矩阵]]。

Deleting the row and column of <math>\Delta</math> corresponding with the sink yields the ''reduced graph Laplacian'' <math>\Delta'</math>. Then, when starting with a configuration <math>z</math> and toppling each vertex <math>v</math> a total of <math>\mathbf{x}(v)\in\mathbb{N}_0</math> times yields the configuration <math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>, where <math>\boldsymbol{\cdot}</math> is the contraction product. Furthermore, if <math>\mathbf{x}</math> corresponds to the number of times each vertex is toppled during the stabilization of a given configuration <math>z</math>, then
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

删除与沉没顶点相对应的 <math>\Delta</math>的行和列，得到简化图拉普拉斯矩阵 <math>\Delta'</math>。然后，当以一个构型<math>z</math> 开始，并将每个顶点<math>v</math>进行总共<math>\mathbf{x}(v)\in\mathbb{N}_0</math>次的崩塌操作时，产生<math>z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>构型，其中<math>\boldsymbol{\cdot}</math>是收缩积。此外，如果 <math>\mathbf{x}</math> 对应于在给定构型<math>z</math>的稳定过程中每个顶点产生崩塌的次数，则
:<math>z^\circ=z-\Delta'\boldsymbol{\cdot}~\mathbf{x}</math>

In this case, <math>\mathbf{x}</math> is referred to as the ''toppling'' or ''odometer function'' (of the stabilization of <math>z</math>).
在这种情况下，<math>\mathbf{x}</math>被称为崩塌或<math>z</math>的稳定过程的里程计函数。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）odometer function翻译存疑

Under the operation <math>*</math>, the set of recurrent configurations forms an [[abelian group]] isomorphic to the cokernel of the reduced graph Laplacian <math>\Delta'</math>, i.e. to <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>, whereby <math>n</math> denotes the number of vertices (including the sink). More generally, the set of stable configurations (transient and recurrent) forms a [[commutative monoid]] under the operation <math>*</math>. The minimal [[Semigroup#Subsemigroups and ideals|ideal]] of this monoid is then isomorphic to the group of recurrent configurations.
在<math>*</math>运算下，递归构形的集合构成一个与约化图拉普拉斯矩阵<math>\Delta'</math>的核同构的阿贝尔群。对于 <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，其中<math>n</math> 表示顶点数（包括沉没顶点）。更一般地说，稳定构型集（瞬态和循环）在<math>*</math>.运算下形成''' 交换幺半群Commutative monoid'''。这个幺半群的最小理想同构于循环构型群。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）ideal翻译存疑

The group formed by the recurrent configurations, as well as the group <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> to which the former is isomorphic, is most commonly referred to as the ''sandpile group''. Other common names for the same group are ''critical group'', ''Jacobian group'' or (less often) ''Picard group''. Note, however, that some authors only denote the group formed by the recurrent configurations as the sandpile group, while reserving the name Jacobian group or critical group for the (isomorphic) group defined by <math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math> (or for related isomorphic definitions). Finally, some authors use the name Picard group to refer to the direct product of the sandpile group and <math>\mathbb{Z}</math>, which naturally appears in a cellular automaton closely related to the sandpile model, referred to as the chip firing or dollar game.

由循环构形形成的群，以及与之同构的群<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>，通常称为''' 沙堆群Sandpile group'''。相同群的其它常用名称是“临界群”、“Jacobian群”或(不常见的)“Picard群”。然而，要注意的是，有些作者只把循环构型形成的组称为沙堆组，而为由<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>（或相关的同构定义）定义的同构群保留Jacobian群或临界群的称呼。最后，一些作者使用Picard群来指代沙堆群和<math>\mathbb{Z}</math>的直接产物，后者出现在与沙堆模型密切相关的元胞自动机中，被称为''' 碎片点火或美元游戏Chip firing or Dollar game'''。

Given the isomorphisms stated above, the order of the sandpile group is the determinant of <math>\Delta'</math>, which by the [[Kirchhoff's theorem|matrix tree theorem]] is the number of spanning trees of the graph.

给定上述同构，沙堆群的顺序是<math>\Delta'</math>的行列式，根据矩阵树定理，它是图的生成树数目。

==Self-organized criticality自组织临界性==
{{main|Self-organized criticality}}
The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its [[critical state]], at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the [[phase transition]]s between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is [[self-organization|self-organized]].
模型最初源于这样一个事实，即在晶格上的模拟中，它被吸引到了它的[[临界状态]]，此时系统的关联长度和关联时间趋于无穷大，不需要对系统参数进行任何微调。这与早期临界现象的例子形成了对比，例如固体和液体之间，或液体和气体之间的[[相变]]，其中临界点只能通过精确调节（例如，温度）来达到。因此，在沙堆模型中，我们可以说临界性是自组织的。

Once the sandpile model reaches its critical state there is no correlation between the system's response to a [[wiktionary:perturbation|perturbation]] and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays [[1/f noise|1/''&fnof;'' noise]], a feature common to many complex systems in nature.
一旦沙堆模型达到其临界状态，系统对扰动的响应和扰动细节之间就没有关联。一般来说，这意味着再往沙堆形成的斜坡上撒一粒沙子可能不会导致任何事情发生，或者可能导致整个沙堆形成的斜坡在大规模滑坡中崩塌。该模型还显示了[[1/f noise|1/''&fnof;'' noise]]，这是自然界中许多复杂系统的共同特征。

This model only displays critical behaviour in two or more dimensions. The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.
此模型仅在两个或多个维度中显示关键行为。沙堆模型可以用一维来表示; 然而，一维沙堆模型不是演化到临界状态，而是达到最小稳定状态，其中每个格点都趋向临界坡度。

For two dimensions, it has been hypothesized that the associated [[conformal field theory]] is consists of [[symplectic fermion]]s with a [[central charge]] ''c'' = −2.<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>
对于二维，相关共形场理论被认为是中心电荷为''c'' = −2的symplectic fermion辛费米子组成的；<ref>{{cite journal |author=S. Moghimi-Araghi |author2=M. A. Rajabpour |author3=S. Rouhani |title=Abelian Sandpile Model: a Conformal Field Theory Point of View |arxiv=cond-mat/0410434 |year=2004 |doi=10.1016/j.nuclphysb.2005.04.002 |volume=718|issue=3|journal=Nuclear Physics B|pages=362–370|bibcode = 2005NuPhB.718..362M |s2cid=16233977 }}</ref>

==Properties属性==
===Least action principle最小作用原理===
The stabilization of chip configurations obeys a form of ''[[principle of least action|least action principle]]'': each vertex topples no more than necessary in the course of the stabilization.<ref name=Fey2010>
{{cite journal
| author = Fey, A. |author2=Levine, L.|author3=Peres, Y.
| year=2010
| issn=0022-4715
| journal=Journal of Statistical Physics
| volume=138
| number=1–3
| doi=10.1007/s10955-009-9899-6
| title=Growth Rates and Explosions in Sandpiles
| pages=143–159|arxiv = 0901.3805 |bibcode = 2010JSP...138..143F |s2cid=7180488}}</ref>

芯片结构的稳定遵循一种“[[最小作用原理|最小作用原理]]”的形式：每个顶点在稳定过程中不超过必要的崩塌量。

This can be formalized as follows. Call a sequence of topples ''legal'' if it only topples unstable vertices, and ''stabilizing'' if it results in a stable configuration. The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible. Such a sequence is obviously stabilizing, and the Abelian property of the sandpile is that all such sequences are equivalent up to permutation of the toppling order; that is, for any vertex <math>v</math>, the number of times <math>v</math> topples is the same in all legal stabilizing sequences. According to the least action principle, a '''minimal stabilizing''' sequence is also equivalent up to permutation of the toppling order to a legal (and still stabilizing) sequence. In particular, the configuration resulting from a minimal stabilizing sequence is the same as results from a maximal legal sequence.
这可以形式化如下。如果一个崩塌序列只推倒不稳定的顶点，则称其为“合法的”，使用一组合法的顶点序列，如果它的结果是一个稳定的构型，则称其为“稳定的”。稳定沙堆的标准方法是找到一个最大的合法崩塌序列，也就是说，让崩塌序列尽可能地长。这种序列具有明显的稳定性，沙堆的可交换性质是所有这些序列都等价于倾斜序列的置换，也就是说，对于任何顶点<math>v</math>，在所有合法的稳定序列中<math>v</math>的崩塌次数都是相同的。根据最小作用原理，最小稳定序列等价于合法的（且稳定的）崩塌序列的置换。特别地，由最小稳定序列产生的构型与由最大合法序列产生的构型是相同的。

More formally, if <math>\mathbf{u}</math> is a vector such that <math>\mathbf{u}(v)</math> is the number of times the vertex <math>v</math> topples during the stabilization (via the toppling of unstable vertices) of a chip configuration <math>z</math>, and <math>\mathbf{n}</math> is an integral vector (not necessarily non-negative) such that <math>z-\mathbf{n}\Delta'</math> is stable, then <math>\mathbf{u}(v) \leq \mathbf{n}(v)</math> for all vertices <math>v</math>.
更正式地说，如果<math>\mathbf{u}</math>是一个向量， <math>\mathbf{u}（v）</math>是碎片构型<math>z</math>在稳定过程中（通过不稳定顶点的崩塌）顶点<math>v</math>崩塌的次数，并且<math>\mathbf{n}</math>是一个积分向量（不一定是非负的），使得<math>z-\mathbf{n}\Delta'</math>是稳定的，那么对于所有顶点<math>v</math>，<math>\mathbf{u}(v) \leq \mathbf{n}(v)</math>。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）<math>\mathbf{n}</math> is an integral vector (not necessarily non-negative)这一句话的翻译存疑。
=== Scaling limits缩放限制===
[[File:Scaling sandpile identity.gif|thumb|Animation of the sandpile identity on square grids of increasing size. Black color denotes vertices with 0 grains, green is for 1, purple is for 2, and gold is for 3.]]
[[File:Scaling sandpile identity.gif|thumb|沙堆标识在方形网格不断增加的动画。黑色表示沙粒数为0的顶点，绿色表示沙粒数为1，紫色表示沙粒数为2，金色表示沙粒数为3。]]

The animation shows the recurrent configuration corresponding to the identity of the sandpile group on different <math>N\times N</math> square grids of increasing sizes <math>N\geq 1</math>, whereby the configurations are rescaled to always have the same physical dimension. Visually, the identities on larger grids seem to become more and more detailed and to "converge to a continuous image". Mathematically, this suggests the existence of scaling-limits of the sandpile identity on square grids based on the notion of weak-* convergence (or some other, generalized notion of convergence). Indeed, existence of scaling limits of recurrent sandpile configurations has been proved by Wesley Pegden and Charles Smart
<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>
动画显示了对应网格尺寸<math>N\geq 1</math>不断增大，不同大小的<math>N\times N</math>正方形网格上的沙堆群标识的重复配置，从而重新缩放配置以始终具有相同的物理维度。从视觉上看，更大网格上的标识似乎变得越来越详细，并且“收敛到一个连续的图像”。从数学上讲，这表明基于弱收敛的概念（或其他一些广义的收敛概念），正方形网格上沙堆恒等式存在标度极限。事实上，Wesley-Pegden和Charles-Smart已经证明了循环沙堆结构标度极限的存在性。<ref name=Pegden2016>{{cite arxiv |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Stability of patterns in the Abelian sandpile.|eprint=1708.09432 | date=2017 | ref=Pegden2017|class=math.AP }}</ref>
.<ref name=Pegden2013>{{cite journal |last1=Pegden |first1=Wesley |last2=Smart |first2=Charles |title=Convergence of the Abelian sandpile |journal=Duke Mathematical Journal |date=2013 |volume=162 |issue=4 |pages=627–642 |doi=10.1215/00127094-2079677 |ref=Pegden2013|arxiv=1105.0111 |s2cid=13027232 }}</ref> In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.<ref name=Levine2016>{{cite journal |last1=Levine |first1=Lionel |last2=Pegden |first2=Wesley |title=Apollonian structure in the Abelian sandpile |journal=Geometric and Functional Analysis |date=2016 |volume=26 |issue=1 |pages=306–336 |doi=10.1007/s00039-016-0358-7 |ref=Levine2016|hdl=1721.1/106972 |s2cid=119626417 |hdl-access=free }}</ref>

== Generalizations and related models归纳与相关模型==
=== Sandpile models on infinite grids有向图上的沙堆模型===
[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|30 million grains dropped to a site of the infinite square grid, then toppled according to the rules of the sandpile model. White color denotes sites with 0 grains, green is for 1, purple is for 2, gold is for 3. The bounding box is 3967×3967.]]
[[File:Sandpile on infinite grid, 3e7 grains.png|thumb|right|upright=1.25|3000万粒沙粒落在无限方形网格的一个位置上，然后按照沙堆模型的规则产生崩塌。白色表示0颗沙粒位置，绿色表示1颗，紫色表示2颗，金色表示3颗。图中框的大小是3967×3967。]]

There exist several generalizations of the sandpile model to infinite grids. A challenge in such generalizations is that, in general, it is not guaranteed anymore that every avalanche will eventually stop. Several of the generalization thus only consider the stabilization of configurations for which this can be guaranteed.

沙堆模型可以推广到无限网格中。这种归纳法的一个挑战是，一般来说，不再保证每次雪崩最终都会停止。因此，一些一般化方法只考虑了构型的稳定性，因为这一点是能保证的。

A rather popular model on the (infinite) square lattice with sites <math>(x,y)\in\mathbb{Z}^2</math> is defined as follows:
在（无限）方格上有一个相当流行的模型，其位置<math>(x,y)\in\mathbb{Z}^2</math>定义如下：

Begin with some nonnegative configuration of values <math>z(x,y)\in \mathbf{Z}</math> which is finite, meaning
从\mathbf{z}</math>中有限值<math>z(x,y)\in \mathbf{Z}</math>的一些非负配置开始，这意味着

:<math>\sum_{x,y}z(x,y)<\infty.</math>

Any site <math>(x,y)</math> with
任何位置<math>(x,y)</math>有
:<math>z(x,y)\geq 4</math>
is ''unstable'' and can ''topple'' (or ''fire''), sending one of its chips to each of its four neighbors:
就是“不稳定的”，并且会产生崩塌，将它位置上的沙粒分发给它的四个邻居：
:<math>z(x,y) \rightarrow z(x,y) - 4,</math>
:<math>z( x \pm 1, y) \rightarrow z( x \pm 1, y) + 1,</math>
:<math>z(x, y \pm 1) \rightarrow z( x, y \pm 1 ) + 1.</math>

Since the initial configuration is finite, the [[Action (philosophy)|process]] is guaranteed to terminate, with the grains scattering outward.
由于初始构型是有限的，这一过程必然会随着沙粒向外散布终止。

A popular special case of this model is given when the initial configuration is zero for all vertices except the origin. If the origin carries a huge number of grains of sand, the configuration after relaxation forms fractal patterns (see figure). When letting the initial number of grains at the origin go to infinity, the rescaled stabilized configurations were shown to converge to a unique limit.<ref name="Pegden2013" /><ref name="Levine2016" />
给出了该模型的一个常见特例，即除原点外的所有顶点的初始配置都为零。如果原点放置大量沙粒，松弛后的构型形成分形图案(见图)。当初始颗粒数趋于无穷时，重缩放的稳定构型收敛到唯一极限。<ref name="Pegden2013" /><ref name="Levine2016" />

=== Sandpile models on directed graphs有向图上的沙堆模型===
The sandpile model can be generalized to arbitrary directed multigraphs. The rules are that any vertex <math>v</math> with
沙堆模型可以推广到任意有向多重图。规则是任何顶点<math>v</math>有
:<math>z(v)\geq \deg^{+}(v)</math>
is unstable; toppling again sends chips to each of its neighbors, one along each outgoing edge:
则是不稳定的；崩塌将它的碎片沿着边的输出方向，分发给它的邻居：
:<math>z(v) \rightarrow z(v) - \deg^{+}(v) + \deg(v,v)</math>
and, for each <math>u\neq v</math>:
并且，对于每个<math>u\neq v</math>：
:<math>z(u) \rightarrow z(u) + \deg(v,u)</math>
where <math>\deg(v,u)</math> is the number of edges from <math>v</math> to <math>u</math>.
其中，<math>\deg(v,u)</math>是从<math>v</math>到<math>u</math>的边数。

In this case the Laplacian matrix is not symmetric. If we specify a sink <math>s</math> such that there is a path from every other vertex to <math>s</math>, then the stabilization operation on finite graphs is well-defined and the sandpile group can be written
在这种情况下，拉普拉斯矩阵是不对称的。如果我们指定一个沉没顶点<math>s</math>，使得每一个顶点都有一条到<math>s</math>的路径，那么有限图上的稳定操作是定义良好的，并且沙堆群可以像之前一样被写出来

:<math>\mathbf{Z}^{n-1}/\mathbf{Z}^{n-1}\Delta'</math>
as before.

The order of the sandpile group is again the determinant of <math>\Delta'</math>, which by the general version of the [[Kirchhoff's theorem|matrix tree theorem]] is the number of oriented [[spanning tree]]s rooted at the sink.
沙堆群的顺序又是<math>\Delta'</math>的行列式，根据[[矩阵树定理]]的一般版本，它是根在沉没顶点的有向[[spanning tree]]生成树的数目。
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）The order of the sandpile group翻译存疑。

=== The extended sandpile model扩展沙堆模型===
[[File:Harmonic Sandpile Dynamics.gif|thumb|Sandpile dynamics induced by the harmonic function H=x*y on a 255x255 square grid.]]
[[File:Harmonic Sandpile Dynamics.gif|thumb|由谐波函数H=x*y在255x255方格网格上引起的沙堆动力学。]]

To better understand the structure of the sandpile group for different finite convex grids <math>\Gamma\subset\mathbb{Z}^2</math> of the standard square lattice <math>\mathbb{Z}^2</math>, Lang and Shkolnikov introduced the ''extended sandpile model'' in 2019.<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref> The extended sandpile model is defined nearly exactly the same as the ''usual sandpile model'' (i.e. the original Bak–Tang–Wiesenfeld model <ref name="Bak1987" />), except that vertices at the boundary <math>\partial\Gamma</math> of the grid are now allowed to carry a non-negative real number of grains. In contrast, vertices in the interior of the grid are still only allowed to carry integer numbers of grains. The toppling rules remain unchanged, i.e. both interior and boundary vertices are assumed to become unstable and topple if the grain number reaches or exceeds four.

为了更好地理解不同有限凸网格<math>\Gamma\subset\mathbb{Z}^2</math>的沙堆群的结构，Lang和Shkolnikov在2019年提出了“扩展沙堆模型”。<ref name=Lang2019>{{Cite journal|last1=Lang|first1=Moritz|last2=Shkolnikov|first2=Mikhail|date=2019-02-19|title=Harmonic dynamics of the abelian sandpile|journal=Proceedings of the National Academy of Sciences|language=en|volume=116|issue=8|pages=2821–2830|doi=10.1073/pnas.1812015116|pmid=30728300|pmc=6386721|issn=0027-8424}}</ref>扩展沙堆模型的定义与“通常的沙堆模型”几乎完全相同（即原始的Bak–Tang–Wiesenfeld模型<ref name="Bak1987" />），除了网格边界<math>\partial\Gamma</math>的顶点现在允许放置非负实数的沙粒。相比之下，网格内部的顶点仍然只允许放置整数个粒子。崩塌规则保持不变，即假设当沙粒数达到或超过4时，内部顶点和边界顶点都变得不稳定并发生崩塌。

Also the recurrent configurations of the extended sandpile model form an abelian group, referred to as the ''extended sandpile group'', of which the usual sandpile group is a [[discrete subgroup]]. Different to the usual sandpile group, the extended sandpile group is however a continuous [[Lie group]]. Since it is generated by only adding grains of sand to the boundary <math>\partial\Gamma</math> of the grid, the extended sandpile group furthermore has the [[topological group|topology]] of a [[torus]] of dimension <math>|\partial\Gamma|</math> and a volume given by the order of the usual sandpile group.<ref name="Lang2019" />
扩展沙堆模型的递归构型也形成了一个阿贝尔群，称为“扩展沙堆群”，通常的扩展沙堆群是一个[[离散子群]]。与通常的沙堆群不同，扩展沙堆群是一个连续的[[李群]]。只因为它是由添加沙粒到网格的边界<math>\partial\Gamma</math>上形成的，扩展后的沙堆群还具有维度<math>|\partial\Gamma|</math>的环面拓扑结构，并且按通常沙堆组的顺序给出的体积。<ref name="Lang2019" />
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）a volume given by the order of the usual sandpile group.翻译存疑。

Of specific interest is the question how the recurrent configurations dynamically change along the continuous [[geodesic]]s of this torus passing through the identity. This question leads to the definition of the sandpile dynamics
特别感兴趣的问题是循环构型如何通过恒等式，沿着这个环面的连续[[测地线]]动态变化的问题。这个问题引出了沙堆动力学的定义
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）identity在整篇文章中的翻译需进行统一，如何翻译？？同一性，恒等式？？
:<math>D_H(t)=(I-t\Delta H)^\circ</math> （扩展沙堆模型）
respectively
:<math>\tilde{D}_H(t)=(I+\lfloor-t\Delta H\rfloor)^\circ</math> （普通沙堆模型）

induced by the integer-valued [[harmonic function]] <math>H</math> at time <math>t\in\mathbb{R}\setminus\mathbb{Z}</math>, with <math>I</math> the identity of the sandpile group and <math>\lfloor.\rfloor</math> the floor function.<ref name="Lang2019" />
For low-order polynomial harmonic functions, the sandpile dynamics are characterized by the
smooth transformation and apparent conservation of the patches constituting the sandpile identity. For example, the harmonic dynamics induced by <math>H=xy</math> resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation. The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.<ref name="Lang2019" /> This proposes a natural [[renormalization]] for the extended and usual sandpile groups, meaning a mapping of recurrent configurations on a given grid to recurrent configurations on a sub-grid. Informaly, this renormalization simply maps configurations appearing at a given time <math>t</math> in the sandpile dynamics induced by some harmonic function <math>H</math> on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of <math>H</math> to the respective sub-grid.<ref name="Lang2019" />
由整值调和函数<math>H</math>在时间<math>t\in\mathbb{R}\setminus\mathbb{Z}</math>，沙堆群的同一性<math>I</math>和底函数<math>\lfloor.\rfloor</math>导出的。<ref name="Lang2019" />对于低阶多项式调和函数，沙堆动力学的特征是组成沙堆恒等式的斑块的光滑变换和明显守恒。例如，由<math>H=xy</math> 诱导的谐波动力学类似于动画中可视化的主对角线上恒等式的“平滑拉伸”。进一步推测了由相同的谐函数在不同尺寸的正方形网格上引起的动力学构型的弱收敛，这意味着可能存在标度限制。<ref name="Lang2019" />这为扩展的和普通的沙堆组提出了一个自然的[[重归一化]]，这意味着在给定网格上的重复配置映射到子网格上的重复配置。非正式地，重归一化简单地映射了沙堆动力学中给定时间<math>t</math>时的构型，动力学由大型网格上的谐波函数<math>H</math>导出到相应的构型，这种构型在<math>H</math>限制到各自子网格的沙堆动力学中时同时出现。<ref name="Lang2019" />
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）上面的长句翻译需要重新审校Informaly, this renormalization simply maps
=== The divisible sandpile 可分割的沙堆===
A strongly related model is the so called '''divisible sandpile model''', introduced by Levine and Peres in 2008,<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref> in which, instead of a discrete number of particles in each site <math>x</math>, there is a real number <math>s(x)</math> representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time <math>t</math>, it will be full for all later times.

Levine和Peres在2008年提出了一个与之密切相关的模型，即所谓的“可分割的沙堆模型”。<ref>{{Cite journal|last1=Levine|first1=Lionel|last2=Peres|first2=Yuval|date=2008-10-29|title=Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile|journal=Potential Analysis|language=en|volume=30|issue=1|pages=1–27|doi=10.1007/s11118-008-9104-6|issn=0926-2601|arxiv=0704.0688|s2cid=2227479}}</ref>与每个位置<math>x</math>上的沙粒数量为离散数不同，有一个实数<math>s(x)</math>代表位置的总质量。如果这个质量是负的，我们就可以把它理解为一个空洞。当一个位置上的质量大于1时，就会发生崩塌; 它将多余的部分均匀地分发给它的邻居，这就导致了如果一个位置在<math>t</math>的时刻是满的，它在以后的所有时间都是满的。

== References引用 ==
{{Reflist|30em}}

== Further reading ==
* {{cite book
| author = Per Bak
| year = 1996
| title = How Nature Works: The Science of Self-Organized Criticality
| publisher = Copernicus
| location = New York
| isbn = 978-0-387-94791-4
}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1987
| title = Self-organized criticality: an explanation of 1/''&fnof;'' noise
| journal = [[Physical Review Letters]]
| volume = 59
| issue = 4
| pages = 381–384
| doi = 10.1103/PhysRevLett.59.381
| bibcode=1987PhRvL..59..381B | pmid=10035754}}
* {{cite journal
|author1=Per Bak |author2=Chao Tang |author3=Kurt Wiesenfeld | year = 1988
| title = Self-organized criticality
| journal = [[Physical Review A]]
| volume = 38
| issue = 1
| pages = 364–374
| doi = 10.1103/PhysRevA.38.364
|pmid=9900174 | bibcode=1988PhRvA..38..364B
}}
* {{cite journal | last1=Cori | first1=Robert | last2=Rossin | first2=Dominique | last3=Salvy | first3=Bruno | title=Polynomial ideals for sandpiles and their Gröbner bases | zbl=1002.68105 | journal=Theor. Comput. Sci. | volume=276 | issue=1–2 | year=2002 | doi=10.1016/S0304-3975(00)00397-2 | pages=1–15| url=https://hal.archives-ouvertes.fr/hal-00016378/file/sandpile.pdf }}
* {{cite book|first=Caroline|last=Klivans|authorlink=Caroline Klivans|title= The Mathematics of Chip-Firing|title-link= The Mathematics of Chip-Firing|publisher=CRC Press|year=2018}}
* {{cite book | author1-last=Perkinson | author1-first=David | author2-last=Perlman | author2-first=Jacob | author3-last=Wilmes | author3-first=John | chapter=Algebraic geometry of sandpiles | pages=211–256 | editor1-last=Amini | editor1-first=Omid | editor2-last=Baker | editor2-first=Matthew | editor3-last=Faber | editor3-first=Xander | title=Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 | zbl=1281.14002 | series=Contemporary Mathematics | volume=605 | doi=10.1090/conm/605/12117 
| location=Providence, RI | publisher=[[American Mathematical Society]] | isbn=978-1-4704-1021-6 | year=2013 | citeseerx=10.1.1.760.283 | s2cid=7851577 }}

==External links==
*{{cite web|last1=Garcia-Puente|first1=Luis David|title=Sandpiles|url=https://www.youtube.com/watch?v=1MtEUErz7Gg|website=YouTube|publisher=[[Brady Haran]]|accessdate=15 January 2017|format=YouTube video}}

[[Category:Self-organization]]
[[Category:Phase transitions]]
[[Category:Dynamical systems]]
[[Category:Critical phenomena]]
[[Category:Nonlinear systems]]
[[Category:Cellular automaton rules]]

仓室模型

2020-11-12T08:38:29Z

Zcy：

本词条由Agnes初步翻译，由Zcy初步审校

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'''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

'''仓室模型Compartmental models'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）词条名Compartmental models in epidemiology翻译为复杂传染病是否妥当

The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]].<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>

The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.

这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population.

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.

模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。

==The SIR model==

SIR模型

The '''SIR model'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

'''SIR 模型'''是最简单的仓室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。

:'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I：染病者。这类人群已经感染疾病，并且有能力感染易感者。

:'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant".

R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant".

R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。

This model is reasonably predictive<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]].

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella.

这个模型可以合理地预测传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。

[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]

These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/>

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.

这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变），我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。

As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。

[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：

[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]

===Transition rates===

传染率

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N^2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''质量作用定律the law of mass action'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>).

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).

在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''爱尔郎分布Erlang distribution''')。

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

对于没有染病者康复的特殊情况（<math>γ=0 </math>），SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。

===The SIR model without vital dynamics===

缺少生命动力学的SIR模型

The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>

The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:

流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''仓室模型comparenmental models'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]
& \frac{dR}{dt} = \gamma I,
\end{align}
</math>

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。

This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]].

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。

This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以推导出隐式的'''解析解analytic solution'''。首先:

:<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

it follows that:

it follows that:

接下来是:

:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

其次，我们注意到不同种类人群的动态取决于以下比例:

:<math> R_0 = \frac{\beta}{\gamma},</math>

the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math>

the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>

所谓的'''基本再生数basic reprodution numer'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=\gamma^{-1}</math> ，那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>

By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

通过将第一个'''微分方程differential equation'''除以第三个微分方程，分离变量并进行积分，我们得到

:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

易感人群的初始比例<math>S_0=S(0)/N</math>，以及

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有

:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

(note that the infectious compartment empties in this limit).

(note that the infectious compartment empties in this limit).

(请注意，在这个极限时，没有染病者)。

This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely

This transcendental equation has a solution in terms of the Lambert function, namely

这个超越方程有一个Lambert函数解，即

:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。

The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

'''基本再生数basic reprodution numer'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:

:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

it yields that if:

it yields that if:

如果:

:<math> R_{0} \cdot S(0) > N,</math>

then:

then:

那么：

:<math> \frac{dI}{dt}(0) >0 ,</math>

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果

:<math> R_{0} \cdot S(0) < N,</math>

then

then

那么

:<math> \frac{dI}{dt}(0) <0 ,</math>

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important.

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''基本再生数basic reprodution numer'''和最初的易感者人群数量都极其重要。

====The force of infection====

感染力

Note that in the above model the function:

Note that in the above model the function:

注意，在上面的模型中，函数:

:<math> F = \beta I,</math>

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

建立了从易感者到染病者的传染率模型，这被称为'''感染力the force of infection'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''感染力the force of infection'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :

:<math> F = \beta \frac{I}{N} .</math>

Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref> and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。

====Exact analytical solutions to the SIR model====

SIR 模型的精确解析解

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

2014年，Harko 和合作者推导出了 SIR 模型的精确'''解析解analytical solution'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化

:<math>\mathcal{S}(u)= S(0)u </math>

:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

for

为了

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

with initial conditions

有初始条件

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''超越方程transcendental equation''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。

An equivalent analytical solution found by Miller<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref> yields

An equivalent analytical solution found by Miller yields

等价的'''解析解analytical solution'''由米勒发现

:<math>
\begin{align}
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
I(t) & = N-S(t)-R(t) \\[8pt]
R(t) & = R(0) + \rho \xi(t) \\[8pt]
\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*
\end{align}
</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" />

Effectively the same result can be found in the original work by Kermack and McKendrick.

实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。

===The SIR model with vital dynamics and constant population===

具有生命动力学和稳定人口的SIR模型

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:

考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R
\end{align}
</math>

for which the disease-free equilibrium (DFE) is:

for which the disease-free equilibrium (DFE) is:

其'''无病平衡点the disease-free quilibrium'''为:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

In this case, we can derive a basic reproduction number:

在这种情况下，我们可以得出一个基本再生数:

:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。

==Variations on the basic SIR model==

基础SIR模型的变化

===The SIS model===

SIS模型

[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]

Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。

[[File:SIS compartment model.svg|400px|center|SIS compartmental model]]

SIS compartmental model

【图5：SIS compartmental modelSIS传染病模型】

We have the model:

We have the model:

我们有以下模型:

:<math>
\begin{align}
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\end{align}
</math>

Note that denoting with ''N'' the total population it holds that:

Note that denoting with N the total population it holds that:

注意，用N表示人群总数:

:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

It follows that:

It follows that:

由此可见:

:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>:

i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:

也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:

:<math>
\begin{align}
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
\end{align}
</math>

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> such that the basic reproduction rate is greater than unity. The solution is given as

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

这个模型可以找到一个'''解析解analytical solution'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''平均场方程the mean-field euations''') ，使基本再生率大于单位数。给出了解如下:

:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to:

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:

作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:

:<math>
\frac{dI}{dt} \propto I\cdot (N-I).
</math>

In the long run, in this model, all individuals will become infected.

In the long run, in this model, all individuals will become infected.

从长远来看，在这种模式下，所有的个体都会被感染。

===The SIRD model===

SIRD模型

[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]

Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>

【图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>】

[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]

Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).

【图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。】

The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:

The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:

易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
& \frac{dR}{dt} = \gamma I, \\[6pt]
& \frac{dD}{dt} = \mu I,
\end{align}
</math>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.

这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。

===The MSIR model===

MSIR模型

For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。

[[File:MSIR.PNG|800px|center|MSIR compartmental model]]To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations:

MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

【图8：MSIR compartmental model. To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 仓室模型为了从数学上表示这一点，增加了一个额外的分类，M(t)。这导致了下列微分方程:】

::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]
\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

===Carrier state===

病原携带状态

Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。

[[File:SIR with carrier model.png|A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.]]

A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.

【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改，使单词“携带者”清晰可见。】

===The SEIR model===

SEIR模型

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).

对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。

[[File:SEIR.PNG|800px|center|SEIR compartmental model]]

SEIR compartmental model

【图10：SEIR compartmental modelSEIR传染病模型】

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]
\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R.
\end{align}
</math>

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。

For this model, the basic reproduction number is:

For this model, the basic reproduction number is:

对于这种模式，'''基本再生数basic reproduction number'''是:

:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件

:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

it holds that:

it holds that:

它认为:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:

: <math>
\begin{align}
\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]
\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1
\end{align}
</math>

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。

=== The SEIS model ===

SEIS模型

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。

:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:

在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:

::<math>
\begin{align}
\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I
\end{align}
</math>

=== The MSEIR model ===

MSEIR模型

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。

::<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

=== The MSEIRS model ===

MSEIRS模型

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。

:::<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

===Variable contact rates===

可变接触率

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。

In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力

:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

with period T equal to one year.

with period T equal to one year.

周期T等于一年。

Thus, our model becomes

Thus, our model becomes

因此，我们的模型变成了

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I
\end{align}
</math>

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:

:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。

==Modelling vaccination==

疫苗接种模型

The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}

The SIR model can be modified to model vaccination

SIR 模型可以经过修改，对疫苗接种进行建模

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

通常，在SIR 模型中引入一个额外的类别， <math>V</math> 为免疫类型。下面是一些例子。

===Vaccinating newborns===

新生儿疫苗接种

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:

:<math>
\begin{align}
\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]
\frac{dV}{dt} & = \nu N P - \mu V
\end{align}
</math>

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:

:<math> \lim_{t \to +\infty} V(t)= N P,</math>

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:

:<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

In other words, if

In other words, if

也就是说，如果

:<math> P < P^{*}= 1-\frac{1}{R_0} </math>

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。

===Vaccination and information===

疫苗接种与信息

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:

:<math> P=P(I), \quad P'(I)>0.</math>

In such a case the eradication condition becomes:

In such a case the eradication condition becomes:

在这种情况下，传染病根除的条件是:

:<math> P(0) \ge P^{*},</math>

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。

===Vaccination of non-newborns===

非新生儿疫苗接种

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\end{align}
</math>

leading to the following eradication condition:

leading to the following eradication condition:

推导出以下的根除条件:

:<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

===Pulse vaccination strategy===

脉冲疫苗接种策略

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]
\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots
\end{align}
</math>

It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:

It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:

很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：

:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

and that the eradication condition is:

and that the eradication condition is:

而根除条件是:

:<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

==The influence of age: age-structured models==

年龄的影响：年龄结构模型

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:

:<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

:<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:

:<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

:<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

where:

where:

其中:

:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:

:<math>i(t,0)=r(t,0)=0</math>

but that are nonlocal for the density of susceptible newborns:

but that are nonlocal for the density of susceptible newborns:

但对于易感新生儿的密度来说，这是非局部的:

:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:

:<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:

:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

and the demographic equilibrium is

and the demographic equilibrium is

人口均衡是

:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

automatically ensuring the existence of the disease-free solution:

automatically ensuring the existence of the disease-free solution:

能够自动保证无疾病时解的存在:

:<math>DFS(a)= (n^*(a),0,0).</math>

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

基本再生数可以通过求解适当的函数算子的谱半径来计算。

== Other considerations within compartmental epidemic models ==

分区传染病模型中的其他考虑事项

=== Vertical transmission ===

垂直传染

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.

就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。

=== Vector transmission ===

水平传染

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.

人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。

=== Others ===

其他

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

在建立流行病模型时可能需要考虑的其他事件包括以下事项:

*Non-homogeneous mixing
不同类型的疾病混合

*Variable infectivity
动态变化的传染率

*Distributions that are spatially non-uniform
空间上的不均匀分布

*Diseases caused by macroparasites
由大型寄生虫引起的疾病

==Deterministic versus stochastic epidemic models==
确定性和随机流行病模型
It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.

必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。

To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。仓室模型的可靠性仅限于区分人群类别的应用。

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.

平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。

==See also==

参见

*[[Mathematical modelling in epidemiology]]

*[[Modifiable areal unit problem]]

*[[Next-generation matrix]]

*[[Risk assessment]]

*[[Attack rate]]

==References==

参考文献

{{reflist}}

==Further reading==

深入阅读

*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}

*{{cite book |editor1-last=Vynnycky |editor1-first=E. |editor2-last=White |editor2-first=R. G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |isbn=978-0-19-856576-5 }}

==External links==

外部链接

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]

*{{cite web |title=Simulating an epidemic |work=[[3Blue1Brown]] |date=March 27, 2020 |url=https://www.youtube.com/watch?v=gxAaO2rsdIs |via=[[YouTube]] }}

==编辑推荐==

[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]

[https://mp.weixin.qq.com/s/8JceYjtmFcqUbJl5af4Etg集智俱乐部推文：从恐惧到仇恨: 美国新冠病毒肺炎流行病如何引发种族仇恨 | 网络科学论文速度36篇]

[https://swarma.org/?p=18877集智俱乐部推文：Science最新评论：传染病模型面临真实世界的考验]

[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型用Links建模网络动力学]

{{Computer modeling|state=expanded}}

{{DEFAULTSORT:Compartmental Models In Epidemiology}}

[[Category:Epidemiology]]

Category:Epidemiology

类别: 流行病学

[[Category:Scientific modeling]]

Category:Scientific modeling

类别: 科学建模

<noinclude>

This page was moved from [[wikipedia:en:Compartmental models in epidemiology]]. Its edit history can be viewed at [[复杂传染病/edithistory]]</noinclude>

[[Category:待整理页面]]

仓室模型

2020-11-01T14:12:30Z

Zcy：

本词条由Agnes初步翻译，由Zcy初步审校

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

'''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

'''房室模型Compartmental models'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）词条名Compartmental models in epidemiology翻译为复杂传染病是否妥当

The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]].<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>

The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.

这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population.

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.

模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。

==The SIR model==

SIR模型

The '''SIR model'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

'''SIR 模型'''是最简单的房室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。

:'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I：染病者。这类人群已经感染疾病，并且有能力感染易感者。

:'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant".

R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant".

R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。

This model is reasonably predictive<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]].

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella.

这个模型可以合理地预测传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。

[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]

These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/>

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.

这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变），我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。

As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。

[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：

[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]

===Transition rates===

传染率

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N^2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''质量作用定律the law of mass action'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>).

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).

在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''爱尔郎分布Erlang distribution''')。

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

对于没有染病者康复的特殊情况（<math>γ=0 </math>），SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。

===The SIR model without vital dynamics===

缺少生命动力学的SIR模型

The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>

The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:

流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''房室模型comparenmental models'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]
& \frac{dR}{dt} = \gamma I,
\end{align}
</math>

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。

This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]].

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。

This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以推导出隐式的'''解析解analytic solution'''。首先:

:<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

it follows that:

it follows that:

接下来是:

:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

其次，我们注意到不同种类人群的动态取决于以下比例:

:<math> R_0 = \frac{\beta}{\gamma},</math>

the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math>

the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>

所谓的'''基本再生数basic reprodution numer'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=\gamma^{-1}</math> ，那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>

By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

通过将第一个'''微分方程differential equation'''除以第三个微分方程，分离变量并进行积分，我们得到

:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

易感人群的初始比例<math>S_0=S(0)/N</math>，以及

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有

:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

(note that the infectious compartment empties in this limit).

(note that the infectious compartment empties in this limit).

(请注意，在这个极限时，没有染病者)。

This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely

This transcendental equation has a solution in terms of the Lambert function, namely

这个超越方程有一个Lambert函数解，即

:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。

The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

'''基本再生数basic reprodution numer'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:

:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

it yields that if:

it yields that if:

如果:

:<math> R_{0} \cdot S(0) > N,</math>

then:

then:

那么：

:<math> \frac{dI}{dt}(0) >0 ,</math>

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果

:<math> R_{0} \cdot S(0) < N,</math>

then

then

那么

:<math> \frac{dI}{dt}(0) <0 ,</math>

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important.

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''基本再生数basic reprodution numer'''和最初的易感者人群数量都极其重要。

====The force of infection====

感染力

Note that in the above model the function:

Note that in the above model the function:

注意，在上面的模型中，函数:

:<math> F = \beta I,</math>

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

建立了从易感者到染病者的传染率模型，这被称为'''感染力the force of infection'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''感染力the force of infection'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :

:<math> F = \beta \frac{I}{N} .</math>

Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref> and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。

====Exact analytical solutions to the SIR model====

SIR 模型的精确解析解

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

2014年，Harko 和合作者推导出了 SIR 模型的精确'''解析解analytical solution'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化

:<math>\mathcal{S}(u)= S(0)u </math>

:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

for

为了

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

with initial conditions

有初始条件

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''超越方程transcendental equation''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。

An equivalent analytical solution found by Miller<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref> yields

An equivalent analytical solution found by Miller yields

等价的'''解析解analytical solution'''由米勒发现

:<math>
\begin{align}
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
I(t) & = N-S(t)-R(t) \\[8pt]
R(t) & = R(0) + \rho \xi(t) \\[8pt]
\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*
\end{align}
</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" />

Effectively the same result can be found in the original work by Kermack and McKendrick.

实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。

===The SIR model with vital dynamics and constant population===

具有生命动力学和稳定人口的SIR模型

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:

考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R
\end{align}
</math>

for which the disease-free equilibrium (DFE) is:

for which the disease-free equilibrium (DFE) is:

其'''无病平衡点the disease-free quilibrium'''为:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

In this case, we can derive a basic reproduction number:

在这种情况下，我们可以得出一个基本再生数:

:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。

==Variations on the basic SIR model==

基础SIR模型的变化

===The SIS model===

SIS模型

[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]

Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。

[[File:SIS compartment model.svg|400px|center|SIS compartmental model]]

SIS compartmental model

【图5：SIS compartmental modelSIS传染病模型】

We have the model:

We have the model:

我们有以下模型:

:<math>
\begin{align}
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\end{align}
</math>

Note that denoting with ''N'' the total population it holds that:

Note that denoting with N the total population it holds that:

注意，用N表示人群总数:

:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

It follows that:

It follows that:

由此可见:

:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>:

i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:

也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:

:<math>
\begin{align}
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
\end{align}
</math>

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> such that the basic reproduction rate is greater than unity. The solution is given as

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

这个模型可以找到一个'''解析解analytical solution'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''平均场方程the mean-field euations''') ，使基本再生率大于单位数。给出了解如下:

:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to:

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:

作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:

:<math>
\frac{dI}{dt} \propto I\cdot (N-I).
</math>

In the long run, in this model, all individuals will become infected.

In the long run, in this model, all individuals will become infected.

从长远来看，在这种模式下，所有的个体都会被感染。

===The SIRD model===

SIRD模型

[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]

Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>

【图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>】

[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]

Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).

【图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。】

The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:

The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:

易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
& \frac{dR}{dt} = \gamma I, \\[6pt]
& \frac{dD}{dt} = \mu I,
\end{align}
</math>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.

这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。

===The MSIR model===

MSIR模型

For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。

[[File:MSIR.PNG|800px|center|MSIR compartmental model]]To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations:

MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

【图8：MSIR compartmental model. To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 房室模型为了从数学上表示这一点，增加了一个额外的分类，M(t)。这导致了下列微分方程:】

::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]
\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

===Carrier state===

病原携带状态

Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。

[[File:SIR with carrier model.png|A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.]]

A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.

【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改，使单词“携带者”清晰可见。】

===The SEIR model===

SEIR模型

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).

对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。

[[File:SEIR.PNG|800px|center|SEIR compartmental model]]

SEIR compartmental model

【图10：SEIR compartmental modelSEIR传染病模型】

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]
\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R.
\end{align}
</math>

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。

For this model, the basic reproduction number is:

For this model, the basic reproduction number is:

对于这种模式，'''基本再生数basic reproduction number'''是:

:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件

:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

it holds that:

it holds that:

它认为:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:

: <math>
\begin{align}
\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]
\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1
\end{align}
</math>

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。

=== The SEIS model ===

SEIS模型

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。

:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:

在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:

::<math>
\begin{align}
\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I
\end{align}
</math>

=== The MSEIR model ===

MSEIR模型

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。

::<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

=== The MSEIRS model ===

MSEIRS模型

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。

:::<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

===Variable contact rates===

可变接触率

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。

In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力

:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

with period T equal to one year.

with period T equal to one year.

周期T等于一年。

Thus, our model becomes

Thus, our model becomes

因此，我们的模型变成了

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I
\end{align}
</math>

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:

:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。

==Modelling vaccination==

疫苗接种模型

The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}

The SIR model can be modified to model vaccination

SIR 模型可以经过修改，对疫苗接种进行建模

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

通常，在SIR 模型中引入一个额外的类别， <math>V</math> 为免疫类型。下面是一些例子。

===Vaccinating newborns===

新生儿疫苗接种

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:

:<math>
\begin{align}
\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]
\frac{dV}{dt} & = \nu N P - \mu V
\end{align}
</math>

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:

:<math> \lim_{t \to +\infty} V(t)= N P,</math>

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:

:<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

In other words, if

In other words, if

也就是说，如果

:<math> P < P^{*}= 1-\frac{1}{R_0} </math>

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。

===Vaccination and information===

疫苗接种与信息

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:

:<math> P=P(I), \quad P'(I)>0.</math>

In such a case the eradication condition becomes:

In such a case the eradication condition becomes:

在这种情况下，传染病根除的条件是:

:<math> P(0) \ge P^{*},</math>

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。

===Vaccination of non-newborns===

非新生儿疫苗接种

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\end{align}
</math>

leading to the following eradication condition:

leading to the following eradication condition:

推导出以下的根除条件:

:<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

===Pulse vaccination strategy===

脉冲疫苗接种策略

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]
\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots
\end{align}
</math>

It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:

It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:

很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：

:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

and that the eradication condition is:

and that the eradication condition is:

而根除条件是:

:<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

==The influence of age: age-structured models==

年龄的影响：年龄结构模型

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:

:<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

:<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:

:<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

:<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

where:

where:

其中:

:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:

:<math>i(t,0)=r(t,0)=0</math>

but that are nonlocal for the density of susceptible newborns:

but that are nonlocal for the density of susceptible newborns:

但对于易感新生儿的密度来说，这是非局部的:

:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:

:<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:

:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

and the demographic equilibrium is

and the demographic equilibrium is

人口均衡是

:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

automatically ensuring the existence of the disease-free solution:

automatically ensuring the existence of the disease-free solution:

能够自动保证无疾病时解的存在:

:<math>DFS(a)= (n^*(a),0,0).</math>

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

基本再生数可以通过求解适当的函数算子的谱半径来计算。

== Other considerations within compartmental epidemic models ==

分区传染病模型中的其他考虑事项

=== Vertical transmission ===

垂直传染

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.

就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。

=== Vector transmission ===

水平传染

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.

人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。

=== Others ===

其他

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

在建立流行病模型时可能需要考虑的其他事件包括以下事项:

*Non-homogeneous mixing
不同类型的疾病混合

*Variable infectivity
动态变化的传染率

*Distributions that are spatially non-uniform
空间上的不均匀分布

*Diseases caused by macroparasites
由大型寄生虫引起的疾病

==Deterministic versus stochastic epidemic models==
确定性和随机流行病模型
It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.

必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。

To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。房室模型的可靠性仅限于区分人群类别的应用。

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.

平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。

==See also==

参见

*[[Mathematical modelling in epidemiology]]

*[[Modifiable areal unit problem]]

*[[Next-generation matrix]]

*[[Risk assessment]]

*[[Attack rate]]

==References==

参考文献

{{reflist}}

==Further reading==

深入阅读

*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}

*{{cite book |editor1-last=Vynnycky |editor1-first=E. |editor2-last=White |editor2-first=R. G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |isbn=978-0-19-856576-5 }}

==External links==

外部链接

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]

*{{cite web |title=Simulating an epidemic |work=[[3Blue1Brown]] |date=March 27, 2020 |url=https://www.youtube.com/watch?v=gxAaO2rsdIs |via=[[YouTube]] }}

==编辑推荐==

[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]

[https://mp.weixin.qq.com/s/8JceYjtmFcqUbJl5af4Etg集智俱乐部推文：从恐惧到仇恨: 美国新冠病毒肺炎流行病如何引发种族仇恨 | 网络科学论文速度36篇]

[https://swarma.org/?p=18877集智俱乐部推文：Science最新评论：传染病模型面临真实世界的考验]

[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型用Links建模网络动力学]

{{Computer modeling|state=expanded}}

{{DEFAULTSORT:Compartmental Models In Epidemiology}}

[[Category:Epidemiology]]

Category:Epidemiology

类别: 流行病学

[[Category:Scientific modeling]]

Category:Scientific modeling

类别: 科学建模

<noinclude>

This page was moved from [[wikipedia:en:Compartmental models in epidemiology]]. Its edit history can be viewed at [[复杂传染病/edithistory]]</noinclude>

[[Category:待整理页面]]

仓室模型

2020-11-01T14:09:45Z

Zcy：

本词条由Agnes初步翻译，由Zcy初步审校

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

'''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

'''房室模型Compartmental models'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑
==[[用户:Zcy|Zcy]]（[[用户讨论:Zcy|讨论]]）词条名Compartmental models in epidemiology翻译为复杂传染病是否妥当

The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]].<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>

The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.

这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population.

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.

模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。

==The SIR model==

SIR模型

The '''SIR model'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

'''SIR 模型'''是最简单的房室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。

:'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I：染病者。这类人群已经感染疾病，并且有能力感染易感者。

:'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant".

R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant".

R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。

This model is reasonably predictive<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]].

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella.

这个模型可以合理地预测传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。

[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]

These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/>

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.

这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变），我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。

As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。

[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：

[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]

===Transition rates===

传染率

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N^2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''质量作用定律the law of mass action'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>).

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).

在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''爱尔郎分布Erlang distribution''')。

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

对于没有染病者康复的特殊情况（<math>γ=0 </math>），SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。

===The SIR model without vital dynamics===

缺少生命动力学的SIR模型

The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>

The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:

流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''房室模型comparenmental models'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]
& \frac{dR}{dt} = \gamma I,
\end{align}
</math>

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。

This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]].

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。

This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以推导出隐式的'''解析解analytic solution'''。首先:

:<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

it follows that:

it follows that:

接下来是:

:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

其次，我们注意到不同种类人群的动态取决于以下比例:

:<math> R_0 = \frac{\beta}{\gamma},</math>

the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math>

the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>

所谓的'''基本再生数basic reprodution numer'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=\gamma^{-1}</math> ，那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>

By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

通过将第一个'''微分方程differential equation'''除以第三个微分方程，分离变量并进行积分，我们得到

:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

易感人群的初始比例<math>S_0=S(0)/N</math>，以及

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有

:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

(note that the infectious compartment empties in this limit).

(note that the infectious compartment empties in this limit).

(请注意，在这个极限时，没有染病者)。

This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely

This transcendental equation has a solution in terms of the Lambert function, namely

这个超越方程有一个Lambert函数解，即

:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。

The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

'''基本再生数basic reprodution numer'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:

:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

it yields that if:

it yields that if:

如果:

:<math> R_{0} \cdot S(0) > N,</math>

then:

then:

那么：

:<math> \frac{dI}{dt}(0) >0 ,</math>

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果

:<math> R_{0} \cdot S(0) < N,</math>

then

then

那么

:<math> \frac{dI}{dt}(0) <0 ,</math>

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important.

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''基本再生数basic reprodution numer'''和最初的易感者人群数量都极其重要。

====The force of infection====

感染力

Note that in the above model the function:

Note that in the above model the function:

注意，在上面的模型中，函数:

:<math> F = \beta I,</math>

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

建立了从易感者到染病者的传染率模型，这被称为'''感染力the force of infection'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''感染力the force of infection'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :

:<math> F = \beta \frac{I}{N} .</math>

Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref> and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。

====Exact analytical solutions to the SIR model====

SIR 模型的精确解析解

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

2014年，Harko 和合作者推导出了 SIR 模型的精确'''解析解analytical solution'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化

:<math>\mathcal{S}(u)= S(0)u </math>

:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

for

为了

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

with initial conditions

有初始条件

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''超越方程transcendental equation''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。

An equivalent analytical solution found by Miller<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref> yields

An equivalent analytical solution found by Miller yields

等价的'''解析解analytical solution'''由米勒发现

:<math>
\begin{align}
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
I(t) & = N-S(t)-R(t) \\[8pt]
R(t) & = R(0) + \rho \xi(t) \\[8pt]
\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*
\end{align}
</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" />

Effectively the same result can be found in the original work by Kermack and McKendrick.

实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。

===The SIR model with vital dynamics and constant population===

具有生命动力学和稳定人口的SIR模型

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:

考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R
\end{align}
</math>

for which the disease-free equilibrium (DFE) is:

for which the disease-free equilibrium (DFE) is:

其'''无病平衡点the disease-free quilibrium'''为:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

In this case, we can derive a basic reproduction number:

在这种情况下，我们可以得出一个基本再生数:

:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。

==Variations on the basic SIR model==

基础SIR模型的变化

===The SIS model===

SIS模型

[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]

Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。

[[File:SIS compartment model.svg|400px|center|SIS compartmental model]]

SIS compartmental model

【图5：SIS compartmental modelSIS传染病模型】

We have the model:

We have the model:

我们有以下模型:

:<math>
\begin{align}
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\end{align}
</math>

Note that denoting with ''N'' the total population it holds that:

Note that denoting with N the total population it holds that:

注意，用N表示人群总数:

:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

It follows that:

It follows that:

由此可见:

:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>:

i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:

也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:

:<math>
\begin{align}
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
\end{align}
</math>

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> such that the basic reproduction rate is greater than unity. The solution is given as

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

这个模型可以找到一个'''解析解analytical solution'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''平均场方程the mean-field euations''') ，使基本再生率大于单位数。给出了解如下:

:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to:

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:

作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:

:<math>
\frac{dI}{dt} \propto I\cdot (N-I).
</math>

In the long run, in this model, all individuals will become infected.

In the long run, in this model, all individuals will become infected.

从长远来看，在这种模式下，所有的个体都会被感染。

===The SIRD model===

SIRD模型

[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]

Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>

【图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>】

[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]

Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).

【图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。】

The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:

The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:

易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
& \frac{dR}{dt} = \gamma I, \\[6pt]
& \frac{dD}{dt} = \mu I,
\end{align}
</math>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.

这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。

===The MSIR model===

MSIR模型

For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。

[[File:MSIR.PNG|800px|center|MSIR compartmental model]]To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations:

MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

【图8：MSIR compartmental model. To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 房室模型为了从数学上表示这一点，增加了一个额外的分类，M(t)。这导致了下列微分方程:】

::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]
\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

===Carrier state===

病原携带状态

Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。

[[File:SIR with carrier model.png|A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.]]

A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.

【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改，使单词“携带者”清晰可见。】

===The SEIR model===

SEIR模型

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).

对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。

[[File:SEIR.PNG|800px|center|SEIR compartmental model]]

SEIR compartmental model

【图10：SEIR compartmental modelSEIR传染病模型】

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]
\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R.
\end{align}
</math>

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。

For this model, the basic reproduction number is:

For this model, the basic reproduction number is:

对于这种模式，'''基本再生数basic reproduction number'''是:

:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件

:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

it holds that:

it holds that:

它认为:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:

: <math>
\begin{align}
\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]
\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1
\end{align}
</math>

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。

=== The SEIS model ===

SEIS模型

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。

:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:

在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:

::<math>
\begin{align}
\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I
\end{align}
</math>

=== The MSEIR model ===

MSEIR模型

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。

::<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

=== The MSEIRS model ===

MSEIRS模型

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。

:::<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

===Variable contact rates===

可变接触率

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。

In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力

:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

with period T equal to one year.

with period T equal to one year.

周期T等于一年。

Thus, our model becomes

Thus, our model becomes

因此，我们的模型变成了

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I
\end{align}
</math>

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:

:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。

==Modelling vaccination==

疫苗接种模型

The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}

The SIR model can be modified to model vaccination

SIR 模型可以经过修改，对疫苗接种进行建模

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

通常，在SIR 模型中引入一个额外的类别， <math>V</math> 为免疫类型。下面是一些例子。

===Vaccinating newborns===

新生儿疫苗接种

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:

:<math>
\begin{align}
\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]
\frac{dV}{dt} & = \nu N P - \mu V
\end{align}
</math>

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:

:<math> \lim_{t \to +\infty} V(t)= N P,</math>

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:

:<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

In other words, if

In other words, if

也就是说，如果

:<math> P < P^{*}= 1-\frac{1}{R_0} </math>

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。

===Vaccination and information===

疫苗接种与信息

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:

:<math> P=P(I), \quad P'(I)>0.</math>

In such a case the eradication condition becomes:

In such a case the eradication condition becomes:

在这种情况下，传染病根除的条件是:

:<math> P(0) \ge P^{*},</math>

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。

===Vaccination of non-newborns===

非新生儿疫苗接种

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\end{align}
</math>

leading to the following eradication condition:

leading to the following eradication condition:

推导出以下的根除条件:

:<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

===Pulse vaccination strategy===

脉冲疫苗接种策略

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]
\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots
\end{align}
</math>

It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:

It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:

很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：

:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

and that the eradication condition is:

and that the eradication condition is:

而根除条件是:

:<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

==The influence of age: age-structured models==

年龄的影响：年龄结构模型

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:

:<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

:<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:

:<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

:<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

where:

where:

其中:

:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:

:<math>i(t,0)=r(t,0)=0</math>

but that are nonlocal for the density of susceptible newborns:

but that are nonlocal for the density of susceptible newborns:

但对于易感新生儿的密度来说，这是非局部的:

:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:

:<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:

:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

and the demographic equilibrium is

and the demographic equilibrium is

人口均衡是

:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

automatically ensuring the existence of the disease-free solution:

automatically ensuring the existence of the disease-free solution:

能够自动保证无疾病时解的存在:

:<math>DFS(a)= (n^*(a),0,0).</math>

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

基本再生数可以通过求解适当的函数算子的谱半径来计算。

== Other considerations within compartmental epidemic models ==

分区传染病模型中的其他考虑事项

=== Vertical transmission ===

垂直传染

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.

就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。

=== Vector transmission ===

水平传染

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.

人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。

=== Others ===

其他

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

在建立流行病模型时可能需要考虑的其他事件包括以下事项:

*Non-homogeneous mixing
不同类型的疾病混合

*Variable infectivity
动态变化的传染率

*Distributions that are spatially non-uniform
空间上的不均匀分布

*Diseases caused by macroparasites
由大型寄生虫引起的疾病

==Deterministic versus stochastic epidemic models==
确定性和随机流行病模型
It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.

必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。

To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。房室模型的可靠性仅限于区分人群类别的应用。

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.

平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。

==See also==

参见

*[[Mathematical modelling in epidemiology]]

*[[Modifiable areal unit problem]]

*[[Next-generation matrix]]

*[[Risk assessment]]

*[[Attack rate]]

==References==

参考文献

{{reflist}}

==Further reading==

深入阅读

*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}

*{{cite book |editor1-last=Vynnycky |editor1-first=E. |editor2-last=White |editor2-first=R. G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |isbn=978-0-19-856576-5 }}

==External links==

外部链接

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]

*{{cite web |title=Simulating an epidemic |work=[[3Blue1Brown]] |date=March 27, 2020 |url=https://www.youtube.com/watch?v=gxAaO2rsdIs |via=[[YouTube]] }}

==编辑推荐==

[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]

[https://mp.weixin.qq.com/s/8JceYjtmFcqUbJl5af4Etg集智俱乐部推文：从恐惧到仇恨: 美国新冠病毒肺炎流行病如何引发种族仇恨 | 网络科学论文速度36篇]

[https://swarma.org/?p=18877集智俱乐部推文：Science最新评论：传染病模型面临真实世界的考验]

[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型用Links建模网络动力学]

{{Computer modeling|state=expanded}}

{{DEFAULTSORT:Compartmental Models In Epidemiology}}

[[Category:Epidemiology]]

Category:Epidemiology

类别: 流行病学

[[Category:Scientific modeling]]

Category:Scientific modeling

类别: 科学建模

<noinclude>

This page was moved from [[wikipedia:en:Compartmental models in epidemiology]]. Its edit history can be viewed at [[复杂传染病/edithistory]]</noinclude>

[[Category:待整理页面]]

仓室模型

2020-11-01T13:51:37Z

Zcy：

本词条由Agnes初步翻译，由Zcy初步审校

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

'''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

'''房室模型Compartmental models'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]].<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>

The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.

这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population.

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.

模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。

==The SIR model==

SIR模型

The '''SIR model'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

'''SIR 模型'''是最简单的房室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。

:'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I：染病者。这类人群已经感染疾病，并且有能力感染易感者。

:'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant".

R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant".

R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。

This model is reasonably predictive<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]].

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella.

这个模型可以合理地预测传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。

[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]

These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/>

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.

这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变），我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。

As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。

[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：

[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]

===Transition rates===

传染率

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N^2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''质量作用定律the law of mass action'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>).

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).

在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''爱尔郎分布Erlang distribution''')。

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

对于没有染病者康复的特殊情况（<math>γ=0 </math>），SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。

===The SIR model without vital dynamics===

缺少生命动力学的SIR模型

The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>

The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:

流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''房室模型comparenmental models'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]
& \frac{dR}{dt} = \gamma I,
\end{align}
</math>

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。

This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]].

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。

This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以推导出隐式的'''解析解analytic solution'''。首先:

:<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

it follows that:

it follows that:

接下来是:

:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

其次，我们注意到不同种类人群的动态取决于以下比例:

:<math> R_0 = \frac{\beta}{\gamma},</math>

the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math>

the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>

所谓的'''基本再生数basic reprodution numer'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=\gamma^{-1}</math> ，那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>

By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

通过将第一个'''微分方程differential equation'''除以第三个微分方程，分离变量并进行积分，我们得到

:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

易感人群的初始比例<math>S_0=S(0)/N</math>，以及

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有

:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

(note that the infectious compartment empties in this limit).

(note that the infectious compartment empties in this limit).

(请注意，在这个极限时，没有染病者)。

This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely

This transcendental equation has a solution in terms of the Lambert function, namely

这个超越方程有一个Lambert函数解，即

:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。

The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

'''基本再生数basic reprodution numer'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:

:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

it yields that if:

it yields that if:

如果:

:<math> R_{0} \cdot S(0) > N,</math>

then:

then:

那么：

:<math> \frac{dI}{dt}(0) >0 ,</math>

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果

:<math> R_{0} \cdot S(0) < N,</math>

then

then

那么

:<math> \frac{dI}{dt}(0) <0 ,</math>

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important.

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''基本再生数basic reprodution numer'''和最初的易感者人群数量都极其重要。

====The force of infection====

感染力

Note that in the above model the function:

Note that in the above model the function:

注意，在上面的模型中，函数:

:<math> F = \beta I,</math>

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

建立了从易感者到染病者的传染率模型，这被称为'''感染力the force of infection'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''感染力the force of infection'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :

:<math> F = \beta \frac{I}{N} .</math>

Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref> and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。

====Exact analytical solutions to the SIR model====

SIR 模型的精确解析解

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

2014年，Harko 和合作者推导出了 SIR 模型的精确'''解析解analytical solution'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化

:<math>\mathcal{S}(u)= S(0)u </math>

:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

for

为了

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

with initial conditions

有初始条件

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''超越方程transcendental equation''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。

An equivalent analytical solution found by Miller<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref> yields

An equivalent analytical solution found by Miller yields

等价的'''解析解analytical solution'''由米勒发现

:<math>
\begin{align}
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
I(t) & = N-S(t)-R(t) \\[8pt]
R(t) & = R(0) + \rho \xi(t) \\[8pt]
\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*
\end{align}
</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" />

Effectively the same result can be found in the original work by Kermack and McKendrick.

实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。

===The SIR model with vital dynamics and constant population===

具有生命动力学和稳定人口的SIR模型

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:

考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R
\end{align}
</math>

for which the disease-free equilibrium (DFE) is:

for which the disease-free equilibrium (DFE) is:

其'''无病平衡点the disease-free quilibrium'''为:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

In this case, we can derive a basic reproduction number:

在这种情况下，我们可以得出一个基本再生数:

:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。

==Variations on the basic SIR model==

基础SIR模型的变化

===The SIS model===

SIS模型

[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]

Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。

[[File:SIS compartment model.svg|400px|center|SIS compartmental model]]

SIS compartmental model

【图5：SIS compartmental modelSIS传染病模型】

We have the model:

We have the model:

我们有以下模型:

:<math>
\begin{align}
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\end{align}
</math>

Note that denoting with ''N'' the total population it holds that:

Note that denoting with N the total population it holds that:

注意，用N表示人群总数:

:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

It follows that:

It follows that:

由此可见:

:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>:

i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:

也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:

:<math>
\begin{align}
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
\end{align}
</math>

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> such that the basic reproduction rate is greater than unity. The solution is given as

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

这个模型可以找到一个'''解析解analytical solution'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''平均场方程the mean-field euations''') ，使基本再生率大于单位数。给出了解如下:

:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to:

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:

作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:

:<math>
\frac{dI}{dt} \propto I\cdot (N-I).
</math>

In the long run, in this model, all individuals will become infected.

In the long run, in this model, all individuals will become infected.

从长远来看，在这种模式下，所有的个体都会被感染。

===The SIRD model===

SIRD模型

[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]

Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>

【图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>】

[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]

Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).

【图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。】

The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:

The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:

易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
& \frac{dR}{dt} = \gamma I, \\[6pt]
& \frac{dD}{dt} = \mu I,
\end{align}
</math>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.

这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。

===The MSIR model===

MSIR模型

For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。

[[File:MSIR.PNG|800px|center|MSIR compartmental model]]To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations:

MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

【图8：MSIR compartmental model. To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 房室模型为了从数学上表示这一点，增加了一个额外的分类，M(t)。这导致了下列微分方程:】

::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]
\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

===Carrier state===

病原携带状态

Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。

[[File:SIR with carrier model.png|A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.]]

A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.

【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改，使单词“携带者”清晰可见。】

===The SEIR model===

SEIR模型

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).

对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。

[[File:SEIR.PNG|800px|center|SEIR compartmental model]]

SEIR compartmental model

【图10：SEIR compartmental modelSEIR传染病模型】

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]
\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R.
\end{align}
</math>

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。

For this model, the basic reproduction number is:

For this model, the basic reproduction number is:

对于这种模式，'''基本再生数basic reproduction number'''是:

:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件

:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

it holds that:

it holds that:

它认为:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:

: <math>
\begin{align}
\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]
\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1
\end{align}
</math>

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。

=== The SEIS model ===

SEIS模型

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。

:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:

在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:

::<math>
\begin{align}
\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I
\end{align}
</math>

=== The MSEIR model ===

MSEIR模型

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。

::<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

=== The MSEIRS model ===

MSEIRS模型

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。

:::<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

===Variable contact rates===

可变接触率

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。

In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力

:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

with period T equal to one year.

with period T equal to one year.

周期T等于一年。

Thus, our model becomes

Thus, our model becomes

因此，我们的模型变成了

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I
\end{align}
</math>

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:

:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。

==Modelling vaccination==

疫苗接种模型

The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}

The SIR model can be modified to model vaccination

SIR 模型可以经过修改，对疫苗接种进行建模

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

通常，在SIR 模型中引入一个额外的类别， <math>V</math> 为免疫类型。下面是一些例子。

===Vaccinating newborns===

新生儿疫苗接种

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:

:<math>
\begin{align}
\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]
\frac{dV}{dt} & = \nu N P - \mu V
\end{align}
</math>

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:

:<math> \lim_{t \to +\infty} V(t)= N P,</math>

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:

:<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

In other words, if

In other words, if

也就是说，如果

:<math> P < P^{*}= 1-\frac{1}{R_0} </math>

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。

===Vaccination and information===

疫苗接种与信息

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:

:<math> P=P(I), \quad P'(I)>0.</math>

In such a case the eradication condition becomes:

In such a case the eradication condition becomes:

在这种情况下，传染病根除的条件是:

:<math> P(0) \ge P^{*},</math>

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。

===Vaccination of non-newborns===

非新生儿疫苗接种

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\end{align}
</math>

leading to the following eradication condition:

leading to the following eradication condition:

推导出以下的根除条件:

:<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

===Pulse vaccination strategy===

脉冲疫苗接种策略

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]
\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots
\end{align}
</math>

It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:

It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:

很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：

:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

and that the eradication condition is:

and that the eradication condition is:

而根除条件是:

:<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

==The influence of age: age-structured models==

年龄的影响：年龄结构模型

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:

:<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

:<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:

:<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

:<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

where:

where:

其中:

:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:

:<math>i(t,0)=r(t,0)=0</math>

but that are nonlocal for the density of susceptible newborns:

but that are nonlocal for the density of susceptible newborns:

但对于易感新生儿的密度来说，这是非局部的:

:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:

:<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:

:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

and the demographic equilibrium is

and the demographic equilibrium is

人口均衡是

:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

automatically ensuring the existence of the disease-free solution:

automatically ensuring the existence of the disease-free solution:

能够自动保证无疾病时解的存在:

:<math>DFS(a)= (n^*(a),0,0).</math>

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

基本再生数可以通过求解适当的函数算子的谱半径来计算。

== Other considerations within compartmental epidemic models ==

分区传染病模型中的其他考虑事项

=== Vertical transmission ===

垂直传染

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.

就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。

=== Vector transmission ===

水平传染

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.

人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。

=== Others ===

其他

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

在建立流行病模型时可能需要考虑的其他事件包括以下事项:

*Non-homogeneous mixing
不同类型的疾病混合

*Variable infectivity
动态变化的传染率

*Distributions that are spatially non-uniform
空间上的不均匀分布

*Diseases caused by macroparasites
由大型寄生虫引起的疾病

==Deterministic versus stochastic epidemic models==
确定性和随机流行病模型
It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.

必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。

To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。房室模型的可靠性仅限于区分人群类别的应用。

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.

平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。

==See also==

参见

*[[Mathematical modelling in epidemiology]]

*[[Modifiable areal unit problem]]

*[[Next-generation matrix]]

*[[Risk assessment]]

*[[Attack rate]]

==References==

参考文献

{{reflist}}

==Further reading==

深入阅读

*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}

*{{cite book |editor1-last=Vynnycky |editor1-first=E. |editor2-last=White |editor2-first=R. G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |isbn=978-0-19-856576-5 }}

==External links==

外部链接

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]

*{{cite web |title=Simulating an epidemic |work=[[3Blue1Brown]] |date=March 27, 2020 |url=https://www.youtube.com/watch?v=gxAaO2rsdIs |via=[[YouTube]] }}

==编辑推荐==

[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]

[https://mp.weixin.qq.com/s/8JceYjtmFcqUbJl5af4Etg集智俱乐部推文：从恐惧到仇恨: 美国新冠病毒肺炎流行病如何引发种族仇恨 | 网络科学论文速度36篇]

[https://swarma.org/?p=18877集智俱乐部推文：Science最新评论：传染病模型面临真实世界的考验]

[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型用Links建模网络动力学]

{{Computer modeling|state=expanded}}

{{DEFAULTSORT:Compartmental Models In Epidemiology}}

[[Category:Epidemiology]]

Category:Epidemiology

类别: 流行病学

[[Category:Scientific modeling]]

Category:Scientific modeling

类别: 科学建模

<noinclude>

This page was moved from [[wikipedia:en:Compartmental models in epidemiology]]. Its edit history can be viewed at [[复杂传染病/edithistory]]</noinclude>

[[Category:待整理页面]]

仓室模型

2020-11-01T13:46:12Z

Zcy：

本词条由Agnes初步翻译，由Zcy初步审校

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

'''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

'''房室模型Compartmental models'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]].<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>

The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.

这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population.

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.

模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。

==The SIR model==

SIR模型

The '''SIR model'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

'''SIR 模型'''是最简单的房室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。

:'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I：染病者。这类人群已经感染疾病，并且有能力感染易感者。

:'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant".

R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant".

R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。

This model is reasonably predictive<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]].

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella.

这个模型可以合理地预测传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。

[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]

These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/>

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.

这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变），我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。

As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。

[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：

[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]

===Transition rates===

传染率

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N^2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''质量作用定律the law of mass action'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>).

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).

在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''爱尔郎分布Erlang distribution''')。

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

对于没有染病者康复的特殊情况（<math>γ=0 </math>），SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。

===The SIR model without vital dynamics===

缺少生命动力学的SIR模型

The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>

The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:

流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''房室模型comparenmental models'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]
& \frac{dR}{dt} = \gamma I,
\end{align}
</math>

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。

This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]].

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。

This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以推导出隐式的'''解析解analytic solution'''。首先:

:<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

it follows that:

it follows that:

接下来是:

:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

其次，我们注意到不同种类人群的动态取决于以下比例:

:<math> R_0 = \frac{\beta}{\gamma},</math>

the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math>

the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>

所谓的'''基本再生数basic reprodution numer'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=|\gamma^{-1}</math> ，那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>

By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

通过将第一个'''微分方程differential equation'''除以第三个微分方程，分离变量并进行积分，我们得到

:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

易感人群的初始比例<math>S_0=S(0)/N</math>，以及

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有

:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

(note that the infectious compartment empties in this limit).

(note that the infectious compartment empties in this limit).

(请注意，在这个极限时，没有染病者)。

This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely

This transcendental equation has a solution in terms of the Lambert function, namely

这个超越方程有一个Lambert函数解，即

:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。

The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

'''基本再生数basic reprodution numer'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:

:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

it yields that if:

it yields that if:

如果:

:<math> R_{0} \cdot S(0) > N,</math>

then:

then:

那么：

:<math> \frac{dI}{dt}(0) >0 ,</math>

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果

:<math> R_{0} \cdot S(0) < N,</math>

then

then

那么

:<math> \frac{dI}{dt}(0) <0 ,</math>

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important.

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''基本再生数basic reprodution numer'''和最初的易感者人群数量都极其重要。

====The force of infection====

感染力

Note that in the above model the function:

Note that in the above model the function:

注意，在上面的模型中，函数:

:<math> F = \beta I,</math>

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

建立了从易感者到染病者的传染率模型，这被称为'''感染力the force of infection'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''感染力the force of infection'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :

:<math> F = \beta \frac{I}{N} .</math>

Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref> and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。

====Exact analytical solutions to the SIR model====

SIR 模型的精确解析解

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

2014年，Harko 和合作者推导出了 SIR 模型的精确'''解析解analytical solution'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化

:<math>\mathcal{S}(u)= S(0)u </math>

:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

for

为了

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

with initial conditions

有初始条件

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''超越方程transcendental equation''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。

An equivalent analytical solution found by Miller<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref> yields

An equivalent analytical solution found by Miller yields

等价的'''解析解analytical solution'''由米勒发现

:<math>
\begin{align}
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
I(t) & = N-S(t)-R(t) \\[8pt]
R(t) & = R(0) + \rho \xi(t) \\[8pt]
\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*
\end{align}
</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" />

Effectively the same result can be found in the original work by Kermack and McKendrick.

实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。

===The SIR model with vital dynamics and constant population===

具有生命动力学和稳定人口的SIR模型

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:

考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R
\end{align}
</math>

for which the disease-free equilibrium (DFE) is:

for which the disease-free equilibrium (DFE) is:

其'''无病平衡点the disease-free quilibrium'''为:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

In this case, we can derive a basic reproduction number:

在这种情况下，我们可以得出一个基本再生数:

:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。

==Variations on the basic SIR model==

基础SIR模型的变化

===The SIS model===

SIS模型

[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]

Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。

[[File:SIS compartment model.svg|400px|center|SIS compartmental model]]

SIS compartmental model

【图5：SIS compartmental modelSIS传染病模型】

We have the model:

We have the model:

我们有以下模型:

:<math>
\begin{align}
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\end{align}
</math>

Note that denoting with ''N'' the total population it holds that:

Note that denoting with N the total population it holds that:

注意，用N表示人群总数:

:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

It follows that:

It follows that:

由此可见:

:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>:

i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:

也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:

:<math>
\begin{align}
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
\end{align}
</math>

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> such that the basic reproduction rate is greater than unity. The solution is given as

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

这个模型可以找到一个'''解析解analytical solution'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''平均场方程the mean-field euations''') ，使基本再生率大于单位数。给出了解如下:

:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to:

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:

作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:

:<math>
\frac{dI}{dt} \propto I\cdot (N-I).
</math>

In the long run, in this model, all individuals will become infected.

In the long run, in this model, all individuals will become infected.

从长远来看，在这种模式下，所有的个体都会被感染。

===The SIRD model===

SIRD模型

[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]

Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>

【图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>】

[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]

Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).

【图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。】

The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:

The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:

易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
& \frac{dR}{dt} = \gamma I, \\[6pt]
& \frac{dD}{dt} = \mu I,
\end{align}
</math>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.

这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。

===The MSIR model===

MSIR模型

For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。

[[File:MSIR.PNG|800px|center|MSIR compartmental model]]To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations:

MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

【图8：MSIR compartmental model. To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 房室模型为了从数学上表示这一点，增加了一个额外的分类，M(t)。这导致了下列微分方程:】

::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]
\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

===Carrier state===

病原携带状态

Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。

[[File:SIR with carrier model.png|A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.]]

A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.

【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改，使单词“携带者”清晰可见。】

===The SEIR model===

SEIR模型

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).

对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。

[[File:SEIR.PNG|800px|center|SEIR compartmental model]]

SEIR compartmental model

【图10：SEIR compartmental modelSEIR传染病模型】

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]
\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R.
\end{align}
</math>

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。

For this model, the basic reproduction number is:

For this model, the basic reproduction number is:

对于这种模式，'''基本再生数basic reproduction number'''是:

:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件

:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

it holds that:

it holds that:

它认为:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:

: <math>
\begin{align}
\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]
\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1
\end{align}
</math>

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。

=== The SEIS model ===

SEIS模型

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。

:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:

在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:

::<math>
\begin{align}
\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I
\end{align}
</math>

=== The MSEIR model ===

MSEIR模型

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。

::<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

=== The MSEIRS model ===

MSEIRS模型

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。

:::<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

===Variable contact rates===

可变接触率

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。

In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力

:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

with period T equal to one year.

with period T equal to one year.

周期T等于一年。

Thus, our model becomes

Thus, our model becomes

因此，我们的模型变成了

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I
\end{align}
</math>

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:

:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。

==Modelling vaccination==

疫苗接种模型

The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}

The SIR model can be modified to model vaccination

SIR 模型可以经过修改，对疫苗接种进行建模

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

通常，在SIR 模型中引入一个额外的类别， <math>V</math> 为免疫类型。下面是一些例子。

===Vaccinating newborns===

新生儿疫苗接种

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:

:<math>
\begin{align}
\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]
\frac{dV}{dt} & = \nu N P - \mu V
\end{align}
</math>

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:

:<math> \lim_{t \to +\infty} V(t)= N P,</math>

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:

:<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

In other words, if

In other words, if

也就是说，如果

:<math> P < P^{*}= 1-\frac{1}{R_0} </math>

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。

===Vaccination and information===

疫苗接种与信息

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:

:<math> P=P(I), \quad P'(I)>0.</math>

In such a case the eradication condition becomes:

In such a case the eradication condition becomes:

在这种情况下，传染病根除的条件是:

:<math> P(0) \ge P^{*},</math>

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。

===Vaccination of non-newborns===

非新生儿疫苗接种

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\end{align}
</math>

leading to the following eradication condition:

leading to the following eradication condition:

推导出以下的根除条件:

:<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

===Pulse vaccination strategy===

脉冲疫苗接种策略

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]
\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots
\end{align}
</math>

It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:

It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:

很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：

:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

and that the eradication condition is:

and that the eradication condition is:

而根除条件是:

:<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

==The influence of age: age-structured models==

年龄的影响：年龄结构模型

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:

:<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

:<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:

:<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

:<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

where:

where:

其中:

:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:

:<math>i(t,0)=r(t,0)=0</math>

but that are nonlocal for the density of susceptible newborns:

but that are nonlocal for the density of susceptible newborns:

但对于易感新生儿的密度来说，这是非局部的:

:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:

:<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:

:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

and the demographic equilibrium is

and the demographic equilibrium is

人口均衡是

:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

automatically ensuring the existence of the disease-free solution:

automatically ensuring the existence of the disease-free solution:

能够自动保证无疾病时解的存在:

:<math>DFS(a)= (n^*(a),0,0).</math>

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

基本再生数可以通过求解适当的函数算子的谱半径来计算。

== Other considerations within compartmental epidemic models ==

分区传染病模型中的其他考虑事项

=== Vertical transmission ===

垂直传染

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.

就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。

=== Vector transmission ===

水平传染

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.

人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。

=== Others ===

其他

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

在建立流行病模型时可能需要考虑的其他事件包括以下事项:

*Non-homogeneous mixing
不同类型的疾病混合

*Variable infectivity
动态变化的传染率

*Distributions that are spatially non-uniform
空间上的不均匀分布

*Diseases caused by macroparasites
由大型寄生虫引起的疾病

==Deterministic versus stochastic epidemic models==
确定性和随机流行病模型
It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.

必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。

To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。房室模型的可靠性仅限于区分人群类别的应用。

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.

平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。

==See also==

参见

*[[Mathematical modelling in epidemiology]]

*[[Modifiable areal unit problem]]

*[[Next-generation matrix]]

*[[Risk assessment]]

*[[Attack rate]]

==References==

参考文献

{{reflist}}

==Further reading==

深入阅读

*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}

*{{cite book |editor1-last=Vynnycky |editor1-first=E. |editor2-last=White |editor2-first=R. G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |isbn=978-0-19-856576-5 }}

==External links==

外部链接

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]

*{{cite web |title=Simulating an epidemic |work=[[3Blue1Brown]] |date=March 27, 2020 |url=https://www.youtube.com/watch?v=gxAaO2rsdIs |via=[[YouTube]] }}

==编辑推荐==

[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]

[https://mp.weixin.qq.com/s/8JceYjtmFcqUbJl5af4Etg集智俱乐部推文：从恐惧到仇恨: 美国新冠病毒肺炎流行病如何引发种族仇恨 | 网络科学论文速度36篇]

[https://swarma.org/?p=18877集智俱乐部推文：Science最新评论：传染病模型面临真实世界的考验]

[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型用Links建模网络动力学]

{{Computer modeling|state=expanded}}

{{DEFAULTSORT:Compartmental Models In Epidemiology}}

[[Category:Epidemiology]]

Category:Epidemiology

类别: 流行病学

[[Category:Scientific modeling]]

Category:Scientific modeling

类别: 科学建模

<noinclude>

This page was moved from [[wikipedia:en:Compartmental models in epidemiology]]. Its edit history can be viewed at [[复杂传染病/edithistory]]</noinclude>

[[Category:待整理页面]]

仓室模型

2020-11-01T13:34:34Z

Zcy：

本词条由Agnes初步翻译，由Zcy初步审校

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

'''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

'''房室模型Compartmental models'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]].<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>

The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.

这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population.

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.

模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。

==The SIR model==

SIR模型

The '''SIR model'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

'''SIR 模型'''是最简单的房室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。

:'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I：染病者。这类人群已经感染疾病，并且有能力感染易感者。

:'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant".

R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant".

R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。

This model is reasonably predictive<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]].

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella.

这个模型可以合理地预测传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。

[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]

These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/>

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.

这些变量（S、I和R）表示特定时间每个类别人群的数量。为了表示易感者、染病者和康复者数量会随时间变化（总的人群规模保持不变），我们将这些类别人群的精确数量设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发和控制传染病的大规模爆发。

As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

从变量t可以看出，该模型是动态的，每种类别人群的数量可以随时间波动。这种动态的重要性在传染期较短的地方性疾病中表现得最为明显，如在1968年引进疫苗之前英国的麻疹。由于易感者数量<math> (S (t)) </math>随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，由于更多的人受到感染，转变为染病者和康复者的类别，易感者数量迅速下降。这种疾病只有易感者数量增加时才能再次爆发，例如：由于后代的出生，增加易感者人群数量。

[[File:Graph SIR model without vital dynamics.png|thumb|图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者]]

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

人群中的每个成员通常由易感者转变为感染者，然后转化为康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：

[[File: SIR Flow Diagram.png|thumb| 图3：States in an SIR epidemic model and the rates at which individuals transition between them SIR流行病模型中的状态，以及个人在不同类别之间转换的比率]]

===Transition rates===

传染率

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

为了完整说明这个模型，箭头上标明了类别之间的传染率。在S和I之间，传染率假定为<math> d\left( S/N \right) /dt=-\beta SI/N^2 </math>，其中 <math> N </math> 是总人口，<math> β </math>是平均每人每次接触的数量乘以易感者和感病者之间接触传播疾病的概率，<math> SI/N 2 </math> 是易感个体和染病个体之间接触的比例，这种接触导致易感个体转化为染病个体。(这在数学上与化学中的'''质量作用定律the law of mass action'''类似，在这个定律中，分子之间的随机碰撞导致化学反应，反应比例与两种反应物的浓度成正比)。

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>).

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).

在 I 和 R 之间，假设转化率与染病者的数目成正比，即<math>γI </math>。这相当于假设一个染病者在任何时间间隔<math>dt </math>内恢复的概率简化为<math>γdt </math>。如果平均每个人在时间段<math>D </math>内具有传染性，那么<math>γ= 1/D </math>。这也相当于假设一个人在感染状态下的持续时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和贴近现实的分布来修正I-R 传染率(例如'''爱尔郎分布Erlang distribution''')。

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

对于没有染病者康复的特殊情况（<math>γ=0 </math>），SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个个体最终都会被感染。

===The SIR model without vital dynamics===

缺少生命动力学的SIR模型

The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>

The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:

流行病导致的动态变化，例如流感，往往比出生和死亡的导致的动态变化更快，因此，出生和死亡往往被简单的'''房室模型comparenmental models'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]
& \frac{dR}{dt} = \gamma I,
\end{align}
</math>

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

其中，<math>S</math> 是易感人群的数量，<math>I</math> 是染病人群的数量，<math>R</math> 是康复人群的数量(死亡或康复) ，<math>N</math> 是这三者的总和。

This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]].

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

这个模型是William Ogilvy Kermack和Anderson Gray McKendrick首次提出的，我们这里的模型是Kermark - McKendrick理论的一个特例，它是在McKendrick与Ronald Ross合作的基础上提出的。

This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以推导出隐式的'''解析解analytic solution'''。首先:

:<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

it follows that:

it follows that:

接下来是:

:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

用数学形式来体现人口 <math> N </math>的稳定性。注意，上述关系意味着只需要在方程中研究三个变量中的两个。

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

其次，我们注意到不同种类人群的动态取决于以下比例:

:<math> R_0 = \frac{\beta}{\gamma},</math>

the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math>

the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>

所谓的'''基本再生数basic reprodution numer'''(亦称基本再生率)。该比率是在所有受试者都是易感类别的人群中，由一次感染引起的新感染的预期数量(这些新感染有时称为二次感染)得出的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=|\gamma^{-1}</math> ，那么这个想法可能更容易被理解。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}.</math>

By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

通过将第一个'''微分方程differential equation'''除以第三个微分方程，分离变量并进行积分，我们得到

:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

易感人群的初始比例<math>S_0=S(0)/N</math>，以及

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

<math>s_\infty = S(\infty) / N</math> and <math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively in the limit <math>t \to \infty,</math> one has

在<math>t \to \infty,</math>的极限条件下，易感个体和染病个体的比例分别为<math>s_\infty = S(\infty) / N</math> 和 <math>r_\infty = R(\infty) / N</math>。此时有

:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

(note that the infectious compartment empties in this limit).

(note that the infectious compartment empties in this limit).

(请注意，在这个极限时，没有染病者)。

This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely

This transcendental equation has a solution in terms of the Lambert function, namely

这个超越方程有一个Lambert函数解，即

:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

这表明，在流行病结束时，除非<math>s_0=0</math>，否则一定有个体没有转化为康复状态，一些人仍然是易感类别。这意味着，传染病的结束是由于染病者的减少，而不是完全由于没有易感者。

The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

'''基本再生数basic reprodution numer'''和最初的易感人群比例的作用都极其重要。事实上，将易感者人群数量变化的等式可以重写成如下形式:

:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

it yields that if:

it yields that if:

如果:

:<math> R_{0} \cdot S(0) > N,</math>

then:

then:

那么：

:<math> \frac{dI}{dt}(0) >0 ,</math>

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

也就是说，随着染病者数量的增加(达到人口相当大的一个比例) ，将会有一场流行病的爆发。相反，如果

:<math> R_{0} \cdot S(0) < N,</math>

then

then

那么

:<math> \frac{dI}{dt}(0) <0 ,</math>

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important.

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

也就是说，这种情况与易感者的初始规模无关，这时疾病将不会引起流行病的爆发。因此，很明显，'''基本再生数basic reprodution numer'''和最初的易感者人群数量都极其重要。

====The force of infection====

感染力

Note that in the above model the function:

Note that in the above model the function:

注意，在上面的模型中，函数:

:<math> F = \beta I,</math>

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

建立了从易感者到染病者的传染率模型，这被称为'''感染力the force of infection'''。然而，对于大部分传染病来说，更现实的做法是考虑一种'''感染力the force of infection'''，这种传染力并不取决于染病者人群的绝对数量，而是取决于染病者人群的比例(就总人口<math>N</math>而言) :

:<math> F = \beta \frac{I}{N} .</math>

Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref> and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso和后来的其他作者提出了非线性的感染力，以建模更现实的传染过程。

====Exact analytical solutions to the SIR model====

SIR 模型的精确解析解

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

2014年，Harko 和合作者推导出了 SIR 模型的精确'''解析解analytical solution'''。在没有考虑正常出生和死亡生命动力学的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下时间参数化

:<math>\mathcal{S}(u)= S(0)u </math>

:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

for

为了

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

with initial conditions

有初始条件

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''超越方程transcendental equation''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。

An equivalent analytical solution found by Miller<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref> yields

An equivalent analytical solution found by Miller yields

等价的'''解析解analytical solution'''由米勒发现

:<math>
\begin{align}
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
I(t) & = N-S(t)-R(t) \\[8pt]
R(t) & = R(0) + \rho \xi(t) \\[8pt]
\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*
\end{align}
</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化，一个人预期接触到的染病者数量。这两个解是通过<math>e^{-\xi(t)} = u</math>相关联的。

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" />

Effectively the same result can be found in the original work by Kermack and McKendrick.

实际上，同样的结果可以在Kermack 和 McKendrick的原著中找到。

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

注意到原微分方程右边的所有项都与<math>I</math>成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，通过时间重缩放，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和<math>\xi</math>)仅仅是线性关系。

===The SIR model with vital dynamics and constant population===

具有生命动力学和稳定人口的SIR模型

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:

考虑有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的人群，以及正在传播的传染病。具有质量作用传递的模型是:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R
\end{align}
</math>

for which the disease-free equilibrium (DFE) is:

for which the disease-free equilibrium (DFE) is:

其'''无病平衡点the disease-free quilibrium'''为:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

In this case, we can derive a basic reproduction number:

在这种情况下，我们可以得出一个基本再生数:

:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

这种基本再生数具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，表明<math>R_{0}</math>可以理解为在完全易感人群中，由一个染病者引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。

==Variations on the basic SIR model==

基础SIR模型的变化

===The SIS model===

SIS模型

[[File: SIR-Modell.png |thumb|图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染]]

Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

有些传染病，例如普通感冒和流感，并不能产生持久的免疫力。这种传染病在感染康复后不会产生免疫力，个体会再次变为易感染类型。

[[File:SIS compartment model.svg|400px|center|SIS compartmental model]]

SIS compartmental model

【图5：SIS compartmental modelSIS传染病模型】

We have the model:

We have the model:

我们有以下模型:

:<math>
\begin{align}
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\end{align}
</math>

Note that denoting with ''N'' the total population it holds that:

Note that denoting with N the total population it holds that:

注意，用N表示人群总数:

:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

It follows that:

It follows that:

由此可见:

:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>:

i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:

也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:

:<math>
\begin{align}
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
\end{align}
</math>

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> such that the basic reproduction rate is greater than unity. The solution is given as

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

这个模型可以找到一个'''解析解analytical solution'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''平均场方程the mean-field euations''') ，使基本再生率大于单位数。给出了解如下:

:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人群数量，<math>\chi = \beta-\gamma</math>， <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者人群数量是<math>S(t) = N - I(t)</math>。

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to:

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:

作为一种特殊情况，通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>，因此可以简化为:

:<math>
\frac{dI}{dt} \propto I\cdot (N-I).
</math>

In the long run, in this model, all individuals will become infected.

In the long run, in this model, all individuals will become infected.

从长远来看，在这种模式下，所有的个体都会被感染。

===The SIRD model===

SIRD模型

[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]

Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>

【图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>】

[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]

Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).

【图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。】

The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:

The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:

易感-染病-康复-死亡-模型区分了康复者(特别是指从疾病中存活下来并且现已免疫的个体)和死亡者。该模型使用了下列微分方程组:

:<math>
\begin{align}
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
& \frac{dR}{dt} = \gamma I, \\[6pt]
& \frac{dD}{dt} = \mu I,
\end{align}
</math>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.

这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。

===The MSIR model===

MSIR模型

For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开始加入一个M类型(用于母系免疫)来显示。

[[File:MSIR.PNG|800px|center|MSIR compartmental model]]To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations:

MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

【图8：MSIR compartmental model. To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 房室模型为了从数学上表示这一点，增加了一个额外的分类，M(t)。这导致了下列微分方程:】

::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]
\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

===Carrier state===

病原携带状态

Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。他们可能回归到染病者状态并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是Mary Mallon，她将伤寒传染给了22个人。携带者被标记为C。

[[File:SIR with carrier model.png|A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.]]

A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.

【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改，使单词“携带者”清晰可见。】

===The SEIR model===

SEIR模型

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).

对于许多传染病，有一段很长的疾病潜伏期，在此期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(暴露者类型)。

[[File:SEIR.PNG|800px|center|SEIR compartmental model]]

SEIR compartmental model

【图10：SEIR compartmental modelSEIR传染病模型】

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

假设疾病潜伏期是一个随机变量，这个随机变量服从一个带有参数<math>''a''</math>的指数分布。（疾病平均潜伏期是<math>a^{-1}</math>），并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:

:<math>
\begin{align}
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]
\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R.
\end{align}
</math>

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

我们有<math>S+E+I+R=N,</math>，但是这是常量，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。

For this model, the basic reproduction number is:

For this model, the basic reproduction number is:

对于这种模式，'''基本再生数basic reproduction number'''是:

:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，可以证明，这独立于生物学意义上的初始条件

:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

it holds that:

it holds that:

它认为:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

当接触率 < math > beta (t) </math >周期性变化时， DFE 全局吸引性的条件是，以下线性系统具有周期系数:

: <math>
\begin{align}
\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]
\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1
\end{align}
</math>

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。

=== The SEIS model ===

SEIS模型

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

SEIS 模型类似于之前的 SEIR 模型，只是最终不会转化为免疫属性。

:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:

在这个模型中，感染不会留下任何免疫性，因此染病个体恢复到易感状态，移回 s (t)状态。下列微分方程描述了这一模型:

::<math>
\begin{align}
\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I
\end{align}
</math>

=== The MSEIR model ===

MSEIR模型

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

考虑被动免疫和潜伏期的因素情况下的传染病，建立了 MSEIR 模型。

::<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
::<math>
\begin{align}
\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]
\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]
\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]
\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]
\frac{dR}{dT} & = \gamma I - \mu R
\end{align}
</math>

=== The MSEIRS model ===

MSEIRS模型

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

MSEIRS 模型与 MSEIR 模型相似，但 r 类型的免疫效果是暂时的，因此当临时免疫结束时，个体将恢复其易感性。

:::<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

===Variable contact rates===

可变接触率

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。

In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性（“季节性”）变化接触率的感染力

:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

with period T equal to one year.

with period T equal to one year.

周期T等于一年。

Thus, our model becomes

Thus, our model becomes

因此，我们的模型变成了

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I
\end{align}
</math>

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(康复的动力学很容易从<math>R=N-S-I</math>得出)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会出现非常有趣和复杂的非线性参数共振现象。显然，如果:

:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

然而，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，输出是一个周期函数，其周期是输入周期的倍数。

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

这就可以解释某些传染病的每年(常见为两年)流行病爆发，是由于接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。

==Modelling vaccination==

疫苗接种模型

The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}

The SIR model can be modified to model vaccination

SIR 模型可以经过修改，对疫苗接种进行建模

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

通常，在SIR 模型中引入一个额外的类别， <math>V</math> 为免疫类型。下面是一些例子。

===Vaccinating newborns===

新生儿疫苗接种

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

当出现传染病时，主要任务之一是通过预防措施消除传染病，或者如果可能的话，通过建立大规模疫苗接种计划消除传染病。假设一种新生儿疫苗(通过疫苗给予终身免疫力)接种率为 <math>P \in (0,1)</math>的疾病，:

:<math>
\begin{align}
\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]
\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]
\frac{dV}{dt} & = \nu N P - \mu V
\end{align}
</math>

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:

:<math> \lim_{t \to +\infty} V(t)= N P,</math>

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

因此，我们需要处理<math>S</math>和<math>I</math>的长期行为，它认为:

:<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

In other words, if

In other words, if

也就是说，如果

:<math> P < P^{*}= 1-\frac{1}{R_0} </math>

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

疫苗接种计划没有成功地根除这种疾病，相反，它将继续流行下去，尽管它的流行水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本再生数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。

===Vaccination and information===

疫苗接种与信息

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

现代社会正面临着“理性”豁免的挑战，即家庭由于对感染风险的感知和对疫苗损害风险的“理性”比较，从而决定不给儿童接种疫苗。为了评估这种行为是否真的是理性的，也就是说，它是否同样能够根除疾病，人们可以简单地假定疫苗接种率是染病人群数量的递增函数:

:<math> P=P(I), \quad P'(I)>0.</math>

In such a case the eradication condition becomes:

In such a case the eradication condition becomes:

在这种情况下，传染病根除的条件是:

:<math> P(0) \ge P^{*},</math>

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

也就是说疫苗接种率基线应高于“强制接种”的阈值，而在豁免情况下，该阈值不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而没有考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。

===Vaccination of non-newborns===

非新生儿疫苗接种

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

如果以接种率<math>ρ</math>为非新生儿接种疫苗，易感者和接种者的方程必须修改如下:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\end{align}
</math>

leading to the following eradication condition:

leading to the following eradication condition:

推导出以下的根除条件:

:<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

===Pulse vaccination strategy===

脉冲疫苗接种策略

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

这一策略在一段时间内对易感人群中特定年龄群(如儿童或老年人)反复接种疫苗。利用这一策略，可立即消除易感人群，从而有可能从整个人群中消除传染病(如麻疹)。每<math>T</math>个时间单位，在相对较短的时间内（相对于疾病的动态），对一定比例<math> p </math>的易感受试者进行接种。这就得出了以下易感者和疫苗接种者的脉冲微分方程:

:<math>
\begin{align}
\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]
\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots
\end{align}
</math>

It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:

It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:

很容易看出，通过设置{{math|1=''I'' = 0}}，可以通过以下方式获得易感个体的动态：

:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

and that the eradication condition is:

and that the eradication condition is:

而根除条件是:

:<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

==The influence of age: age-structured models==

年龄的影响：年龄结构模型

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑流行病不同类别人群的年龄<math>s(t,a),i(t,a),r(t,a)</math>(限制在易感者-染病者-康复者的模型中) ，这样:

:<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

:<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:

:<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

:<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

where:

where:

其中:

:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

是感染力，接触核心<math> k(a,a_1;t) </math>取决于年龄之间的相互作用。

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

新生婴儿的初始条件(即对于<math> a = 0 </math>) ，可以直接用于传染和康复:

:<math>i(t,0)=r(t,0)=0</math>

but that are nonlocal for the density of susceptible newborns:

but that are nonlocal for the density of susceptible newborns:

但对于易感新生儿的密度来说，这是非局部的:

:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

其中，<math>\varphi_j(a), j=s,i,r</math>是成人的生育能力。

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> ，得到:

:<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

在三个流行病类别中，最简单的相同生育情况下，为了实现人口均衡，我们需要以下充分必要条件，将生育率<math>\varphi(.)</math>与死亡率<math>\mu(a)</math>联系起来:

:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

and the demographic equilibrium is

and the demographic equilibrium is

人口均衡是

:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

automatically ensuring the existence of the disease-free solution:

automatically ensuring the existence of the disease-free solution:

能够自动保证无疾病时解的存在:

:<math>DFS(a)= (n^*(a),0,0).</math>

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

基本再生数可以通过求解适当的函数算子的谱半径来计算。

== Other considerations within compartmental epidemic models ==

分区传染病模型中的其他考虑事项

=== Vertical transmission ===

垂直传染

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.

就艾滋病和乙型肝炎等疾病而言，受感染父母的子女有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，通过将一部分新生成员包括在染病人群中，可以在模型中考虑其他成员涌入染病类别人群的情况。

=== Vector transmission ===

水平传染

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.

人与人之间间接传播的疾病，如通过蚊子传播的疟疾，通过中间媒介传播。在这些情况下，传染病会从人传播到昆虫，并且传播模型必须包括这两种物种，通常比直接传播模型需要更多的类别区分。

=== Others ===

其他

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

在建立流行病模型时可能需要考虑的其他事件包括以下事项:

*Non-homogeneous mixing
不同类型的疾病混合

*Variable infectivity
动态变化的传染率

*Distributions that are spatially non-uniform
空间上的不均匀分布

*Diseases caused by macroparasites
由大型寄生虫引起的疾病

==Deterministic versus stochastic epidemic models==
确定性和随机流行病模型
It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.

必须强调的是，这里提出的确定性模型只有在人群数量足够大的情况下才有效，因此应该谨慎使用。

To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

更准确地说，这些模型仅在热力学极限中有效，在该热力学极限中，总体数量是无限大的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会传播，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以采用实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。房室模型的可靠性仅限于区分人群类别的应用。

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.

平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。

==See also==

参见

*[[Mathematical modelling in epidemiology]]

*[[Modifiable areal unit problem]]

*[[Next-generation matrix]]

*[[Risk assessment]]

*[[Attack rate]]

==References==

参考文献

{{reflist}}

==Further reading==

深入阅读

*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}

*{{cite book |editor1-last=Vynnycky |editor1-first=E. |editor2-last=White |editor2-first=R. G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |isbn=978-0-19-856576-5 }}

==External links==

外部链接

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]

*{{cite web |title=Simulating an epidemic |work=[[3Blue1Brown]] |date=March 27, 2020 |url=https://www.youtube.com/watch?v=gxAaO2rsdIs |via=[[YouTube]] }}

==编辑推荐==

[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]

[https://mp.weixin.qq.com/s/8JceYjtmFcqUbJl5af4Etg集智俱乐部推文：从恐惧到仇恨: 美国新冠病毒肺炎流行病如何引发种族仇恨 | 网络科学论文速度36篇]

[https://swarma.org/?p=18877集智俱乐部推文：Science最新评论：传染病模型面临真实世界的考验]

[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型用Links建模网络动力学]

{{Computer modeling|state=expanded}}

{{DEFAULTSORT:Compartmental Models In Epidemiology}}

[[Category:Epidemiology]]

Category:Epidemiology

类别: 流行病学

[[Category:Scientific modeling]]

Category:Scientific modeling

类别: 科学建模

<noinclude>

This page was moved from [[wikipedia:en:Compartmental models in epidemiology]]. Its edit history can be viewed at [[复杂传染病/edithistory]]</noinclude>

[[Category:待整理页面]]

仓室模型

2020-11-01T02:37:13Z

Zcy：

本词条由Agnes初步翻译

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

'''Compartmental models''' simplify the [[mathematical modelling of infectious disease|mathematical modelling of infectious diseases]]. The population is assigned to compartments with labels - for example, '''S''', '''I''', or '''R''', ('''S'''usceptible, '''I'''nfectious, or '''R'''ecovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

'''房室模型Compartmental models'''简化了传染病传播的数学模型。人群被划分为带有标签的类别，例如，S，I，和 R，(易感者，染病者和康复者)。不同类别人群的标签会发生变化。标签的变化反映了不同类别人群之间的转化模式，例如SEIS模型代表易感者类型可以转变为暴露者类型、暴露者类型可以转变为染病者类型，染病者类型可以再次转变回易感者类型。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

The origin of such models is the early 20th century, with an important work being that of [[Kermack–McKendrick theory|Kermack and McKendrick in 1927]].<ref name="Kermack–McKendrick">{{cite journal |last1=Kermack |first1=W. O. |last2=McKendrick |first2=A. G. |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=1927 |doi=10.1098/rspa.1927.0118|bibcode=1927RSPSA.115..700K |doi-access=free }}</ref>

The origin of such models is the early 20th century, with an important work being that of Kermack and McKendrick in 1927.

这类模型起源于20世纪初，克马克和麦克德里克在1927年的一项重要工作。

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.

这些模型通常使用确定参数的常微分方程进行分析，但也可以使用具有随机参数的随机框架进行分析，这种随机框架更加贴近现实，但分析起来要复杂得多。

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the [[Basic reproduction number|reproductive number]]. Such models can show how different [[Public health intervention|public health interventions]] may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of [[Vaccine|vaccines]] in a given population.

Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.

模型试图预测疾病的传播方式、感染总人数、流行病持续时间等，并估计各种流行病学参数，如再生数。这些模型可以显示不同的公共卫生干预措施如何对疾病传播产生影响，例如，控制疾病传播最有效的方式是在指定人群中发放数量有限的疫苗。

==The SIR model==

SIR模型

The '''SIR model'''<ref name="Harko">{{Cite journal|last1=Harko|first1=Tiberiu|last2=Lobo|first2=Francisco S. N.|last3=Mak|first3=M. K. |title=Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates |journal=Applied Mathematics and Computation|language=en|volume=236|pages=184–194|year=2014 |doi=10.1016/j.amc.2014.03.030|bibcode=2014arXiv1403.2160H |arxiv=1403.2160 }}</ref><ref name="Beckley">{{cite journal |last1=Beckley |first1=Ross |last2= Weatherspoon|first2=Cametria |last3=Alexander |first3=Michael |last4=Chandler |first4= Marissa|last5=Johnson |first5= Anthony|last6=Batt |first6= Ghan S.|date=2013 |title=Modeling epidemics with differential equations |url=http://www.tnstate.edu/mathematics/mathreu/filesreu/GroupProjectSIR.pdf |journal=Tenessee State University internal report |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:

'''SIR 模型'''是最简单的房室模型之一，许多模型都是这种基本模型的衍生物。该模型由三种类型的人群组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

S：易感者。当一个易感者和一个感病者产生“传染性接触”时，易感者就会感染这种疾病，并被归入染病者类别。

:'''I''': The number of '''i'''nfectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.

I：染病者。这类人群已经感染疾病，并且有能力感染易感者。

:'''R''' for the number of '''r'''emoved (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "'''r'''ecovered" or "'''r'''esistant".

R: for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to the total population. This compartment may also be called "recovered" or "resistant".

R：康复者或病死者。这些人已经被感染过，并且已经从疾病中康复并被归入康复者类别，或者已经因感染疾病死亡。假如死亡数量与总人口数量相比可以忽略不计，这种类别人群也可称为“康复者”或“抵抗者”。

This model is reasonably predictive<ref name="Yang2020">{{cite journal |last1= Yang|first1= Wuyue|last2= Zhang|first2= Dongyan|last3= Peng|first3= Liangrong|last4= Zhuge|first4= Changjing|last5= Liu|first5= Liu|date=2020 |title= Rational evaluation of various epidemic models based on the COVID-19 data of China|url=https://arxiv.org/pdf/2003.05666.pdf |journal=arXiv:2003.05666v1 (q-bio.PE) |volume= |issue= |pages= |doi= |access-date=July 19, 2020}}</ref> for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as [[measles]], [[mumps]] and [[rubella]].

This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella.

这个模型可以合理地预测传染病在人与人之间传播的情况，以及康复后会产生持续性抗体的疾病，如麻疹、腮腺炎和风疹等疾病。

[[File:SIR model simulated using python.gif|thumb|图1：Spatial SIR model simulation. Each cell can infect its eight immediate neighbors.空间 SIR 模型仿真。每个单元都能感染它的八个相邻单元。]]

These variables ('''S''', '''I''', and '''R''') represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): '''S'''(''t''), '''I'''(''t'') and '''R'''(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.<ref name="Yang2020"/>

These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.

这些变量（S、I和R）表示特定时间每个类别内的人数。为了表示易感者、感病者和康复者数量可能随时间变化(总种群规模保持不变) ，我们将精确数字设为时间''t''的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发并控制它们。

As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.

从''t''的变量函数中可以看出，该模型是动态的，因为每个分类中的数量可能随时间而波动。这一动态方面的重要性尤其体现在传染时间较短的地方性疾病中，如1968年引进疫苗之前英国的麻疹。由于易感者(S (t))随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，易感者人数迅速下降，因为更多的人受到感染，从而进入感病者和康复者的类别。这种疾病只有易感者数量增加时才能再次爆发，例如：当后代出生在易感者区域中时。

[[File:Graph SIR model without vital dynamics.svg|thumb|Yellow=Susceptible, Maroon=Infectious, Teal=Recovered]]

Yellow=Susceptible, Maroon=Infectious, Teal=Recovered

【图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者】

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

人群中的每个成员通常由易感者变到感染者，再回归康复者。这可以显示为一个流程图，在这个流程图中，方框代表不同的类别，箭头代表类别之间的过渡，即：

[[File:SIR Flow Diagram.svg|600px|center|SIR compartment model]]

SIR compartment model

【图3：SIR compartment modelSIR传染病模型】

===Transition rates===

传染率

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N2'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N2'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).

为了完整说明这个模型，箭头应该标明类别之间的传染率。在S和I之间，传染率假定为d (S/N)/dt =-SI/N 2 ，其中 ''N'' 是总人口，β是平均每人每次接触的人数，乘以易感者和感病者之间接触传播疾病的概率，SI/N 2 是易感个体和感病个体之间接触之后导致易感个体感染的百分比。(这与化学中的'''质量作用定律the law of mass action'''在数学计算上类似，即分子之间的随机碰撞导致化学反应，反应速率与两种反应物的浓度成正比)。

Between ''I'' and ''R'', the transition rate is assumed to be proportional to the number of infectious individuals which is γ''I''. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval ''dt'' is simply γ''dt''. If an individual is infectious for an average time period ''D'', then γ = 1/''D''. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an [[exponential distribution]]. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the [[Erlang distribution]]<ref name="Krylova2013">{{cite journal |last1=Krylova |first1=O. |last2=Earn |first2=DJ |date= May 15, 2013|title= Effects of the infectious period distribution on predicted transitions in childhood disease dynamics |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673147/ |journal=J R Soc Interface|volume=10 |issue=84 |pages= |doi=10.1098/rsif.2013.0098 |access-date=June 13, 2020|doi-access=free }}</ref>).

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).

在 I 和 R 之间，假设传染率与感病者的数目成正比，即γI。这相当于假设一个感病者在任何时间间隔内恢复的概率仅为γdt。如果平均每个人在时间段D内具有传染性，那么γ= 1/D。这也相当于假设一个人在感染状态下的时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和现实的分布来修正I-R 传染率(例如'''爱尔郎分布Erlang distribution''')。

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a [[logistic distribution|logistic]] solution, in which every individual eventually becomes infected.

For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

对于感病者全都没有康复的特殊情况（γ=0），SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个人最终都会被感染。

===The SIR model without vital dynamics===

缺少生命动力学的SIR模型

The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>

The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:

流行病动力学，例如流感，往往比出生和死亡的动力学变化更快，因此，出生和死亡往往被简单的'''传染病模型comparenmental models'''所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:

:<math>

<math>

<数学>

\begin{align}

\begin{align}

开始{ align }

& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]

& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]

& frac { dS }{ dt } =-frac { beta i s }{ n } ，[6 pt ]

& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]

& \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt]

& frac { dI }{ dt } = frac { beta i s }{ n }-gamma i，[6 pt ]

& \frac{dR}{dt} = \gamma I,

& \frac{dR}{dt} = \gamma I,

& \frac{dR}{dt} = \gamma I,

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</数学>

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.

其中，S</math> 是易感人群的存量，<math>I</math> 是感染人群的存量，<math>R</math> 是康复人群的存量(死亡或康复) ，<math>N</math> 是这三者的总和。

This model was for the first time proposed by [[William Ogilvy Kermack]] and [[Anderson Gray McKendrick]] as a special case of what we now call [[Kermack–McKendrick theory]], and followed work McKendrick had done with [[Ronald Ross]].

This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.

这个模型是一个特例，是在继承了麦克德里克对罗纳德·罗斯所做研究的基础上，由威廉·奥格威·科尔马克和安德森·格雷·麦克德里克首次提出，现在称作科尔马克-麦克德里克理论。

This system is [[non-linear]], however it is possible to derive its analytic solution in implicit form.<ref name="Harko"/> Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以用隐式形式得到它的'''解析解analytic solution'''。首先，请注意:

:<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0,</math>

0，</math > frac { dS }

it follows that:

it follows that:

接下来是:

:<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

<math> S(t) + I(t) + R(t) = \text{constant} = N,</math>

S (t) + i (t) + r (t) = text { constant } = n，</math >

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.

用数学术语来表达人口 <math> N </math>的稳定性。注意，上述关系意味着人们只需要研究三个变量中的两个的方程。

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

Secondly, we note that the dynamics of the infectious class depends on the following ratio:

其次，我们注意到传染种类的动态取决于以下比例:

:<math> R_0 = \frac{\beta}{\gamma},</math>

<math> R_0 = \frac{\beta}{\gamma},</math>

< math > r _ 0 = frac { beta }{ gamma } ，</math >

the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible.<ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref><ref name=nunn2006>{{cite book |author1=Sonia Altizer |author2=Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others ''before'' the infectious has been removed is: <math>T_{r}/T_{c}.</math>

the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>

所谓的'''基本再生数basic reprodution numer'''(亦称基本再生率)。这个比率是根据一群易感者中单次感染后的预期的新感染人数(这些新感染有时称为二次感染)计算出来的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=|\gamma^{-1}</math> ，那么这个想法可能更容易被看出来。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}。</数学>

By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

通过将第一个'''微分方程differential equation'''除以第三个，分离变量并进行积分，我们得到了

:<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

<math> S(t) = S(0) e^{-R_0(R(t) - R(0))/N}, </math>

< math > s (t) = s (0) e ^ {-r _ 0(r (t)-r (0))/n } ，</math >

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.

其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and

为易感人群的初始比例编写<math>S_0=S(0)/N</math>，以及

<math>s_\infty = S(\infty) / N</math> and

<math>s_\infty = S(\infty) / N</math> and

[math] s infty = S(\infty)/N</math>并且

<math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively

<math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively

敏感个体和被感染个体的比例分别为R(\infty)/N</math>

in the limit <math>t \to \infty,</math> one has

in the limit <math>t \to \infty,</math> one has

在这个极限里，<math>t \to \infty,</math> 一个人又

:<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>

< math > s_\infty = 1-r _\infty = s_0 e ^{-R_0(r _\infty -r _0)}</math >

(note that the infectious compartment empties in this limit).

(note that the infectious compartment empties in this limit).

(请注意，在这个限度内，没有感病者)。

This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely

This transcendental equation has a solution in terms of the Lambert function, namely

这个超越方程有一个关于Lambert函数的解决方案，即

:<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

<math>s_\infty = 1-r_\infty = - R_0^{-1}\, W(-s_0 R_0 e^{-R_0(1-r_0)}).</math>

< math > s _ infty = 1-r _ infty =-r _ 0 ^ {-1} ，w (- s _ 0 r _ 0 e ^ {-r _ 0(1-r _ 0)}) </math >

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.

这表明，在流行病结束时，除非<math>s_0=0</math>，否则只要有人没有康复，一些人仍然易受感染。这意味着传染病的结束是由于感病者的减少，而不是由于完全没有易感者。

The role of both the [[basic reproduction number]] and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:

'''基本再生数basic reprodution numer'''的作用和最初的易感性都极其重要。事实上，将传染性个体的等式重写如下:

:<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>

<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} -1 \right) \gamma I,</math>

it yields that if:

it yields that if:

它产生的条件是:

:<math> R_{0} \cdot S(0) > N,</math>

<math> R_{0} \cdot S(0) > N,</math>

<math> R_{0} \cdot S(0) > N,</math>

then:

then:

然后:

:<math> \frac{dI}{dt}(0) >0 ,</math>

<math> \frac{dI}{dt}(0) >0 ,</math>

</math> \frac{dI}{dt}(0) > ,</math>

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if

也就是说，随着感病者数量的增加(达到人口相当大的一个比例) ，将会有一场适当的流行病爆发。相反，如果

:<math> R_{0} \cdot S(0) < N,</math>

<math> R_{0} \cdot S(0) < N,</math>

<math> R_{0} \cdot S(0) < N,</math>

then

then

那么

:<math> \frac{dI}{dt}(0) <0 ,</math>

<math> \frac{dI}{dt}(0) <0 ,</math>

<math> \frac{dI}{dt}(0) <0 ,</math>

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the [[basic reproduction number]] and the initial susceptibility are extremely important.

i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.

也就是说，如果与易感者的初始规模无关，这种疾病将永远不会引起适当的流行病爆发。因此，很明显，'''基本再生数basic reprodution numer'''和最初的易感性都极其重要。

====The force of infection====

感染力

Note that in the above model the function:

Note that in the above model the function:

注意，在上面的模型中，函数是:

:<math> F = \beta I,</math>

<math> F = \beta I,</math>

<math> F = \beta I,</math>

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):

建立了从易感者到感病者的传染率模型，因此称之为'''感染力the force of infection'''。然而，对于大类传染病来说，更现实的做法是考虑一种'''感染力the force of infection'''，这种传染力并不取决于感染对象的绝对数量，而是取决于感染对象的比例(就总人口而言<math>N</math>) :

:<math> F = \beta \frac{I}{N} .</math>

<math> F = \beta \frac{I}{N} .</math>

<math> F = \beta \frac{I}{N} .</math>

Capasso<ref name="Capasso">{{cite book |first=V. |last=Capasso |title=Mathematical Structure of Epidemic Systems |location=Berlin |publisher=Springer |year=1993 |isbn=3-540-56526-4 }}</ref> and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.

卡帕索和后来的其他作者提出了非线性的感染力以建立更现实的传染过程。

====Exact analytical solutions to the SIR model====

SIR 模型的精确解析解

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

2014年，Harko 和合作者推导出了 SIR 模型的精确'''解析解analytical solution'''。在没有重要动力学设置的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下参数化时间

:<math>\mathcal{S}(u)= S(0)u </math>

<math>\mathcal{S}(u)= S(0)u </math>

<math>\mathcal{S}(u)= S(0)u </math>

:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>

:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

for

为了

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

with initial conditions

和初始条件

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''超越方程transcendental equation''' <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么遵守<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。

An equivalent analytical solution found by Miller<ref>{{cite journal | author = Miller, J.C. | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | year=2012 | at= section 4.1| doi = 10.1007/s11538-012-9749-6 | pmid = 22829179 | pmc = 3506030 }}</ref><ref>{{cite journal | author = Miller, J.C. | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | year=2017 | at = section 2.1.3| doi = 10.1016/j.idm.2016.12.003 | pmid = 29928728 | pmc = 5963332 }}</ref> yields

An equivalent analytical solution found by Miller yields

等价的'''解析解analytical solution'''由米勒发现

:<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

S(t) & = S(0) e^{-\xi(t)} \\[8pt]

S(t) & = S(0) e^{-\xi(t)} \\[8pt]

S (t) & = s (0) e ^ {-xi (t)}[8 pt ]

I(t) & = N-S(t)-R(t) \\[8pt]

I(t) & = N-S(t)-R(t) \\[8pt]

I (t) & = N-S (t)-r (t)[8 pt ]

R(t) & = R(0) + \rho \xi(t) \\[8pt]

R(t) & = R(0) + \rho \xi(t) \\[8pt]

R (t) & = r (0) + rho xi (t)[8 pt ]

\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*

\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*

Xi (t) & = frac { beta }{ n } int _ 0 ^ t i (t ^ *) ，dt ^ *

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.

在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化一个人预期收到的传染数量。这两个解是通过<math>e^{-\xi(t)} = u</math>关联的。

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick" />

Effectively the same result can be found in the original work by Kermack and McKendrick.

实际上，同样的结果可以在科马克和麦克德里克的原著中找到。

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

注意到原微分方程右边的所有项都与“数学”成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，时间重新调整，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和上方的<math>\xi</math>)仅仅是线性关系。

===The SIR model with vital dynamics and constant population===

具有生命动力学和稳定人口的SIR模型

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:

Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:

考虑人口有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的特点，以及正在传播的传染病。具有质量作用传递的模型是:

:<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]

\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]

\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]

\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]

\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]

\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]

\frac{dR}{dt} & = \gamma I - \mu R

\frac{dR}{dt} & = \gamma I - \mu R

\frac{dR}{dt} & = \gamma I - \mu R

\end{align}

\end{align}

结束{ align }

</math>

</math>

</math>

for which the disease-free equilibrium (DFE) is:

for which the disease-free equilibrium (DFE) is:

其'''无病平衡点the disease-free quilibrium'''为:

:<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>

In this case, we can derive a [[basic reproduction number]]:

In this case, we can derive a basic reproduction number:

在这种情况下，我们可以得出一个基本再生数:

:<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:

这种基本再生数具有临界性质。事实上，我们可以独立于具有生物学意义的初始值证明:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.

EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，人们可以表明<math>R_{0}</math>可以理解为在一个完全易感人群中，由一个感染对象引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。

==Variations on the basic SIR model==

基础SIR模型的变化

===The SIS model===

SIS模型

[[File:SIS System Graph.svg|thumb|Yellow=Susceptible, Maroon=Infected]]

Yellow=Susceptible, Maroon=Infected

【图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染】

Some infections, for example, those from the [[common cold]] and [[influenza]], do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.

有些传染病，例如普通感冒和流感，并不能提供任何持久的免疫力。这种感染在感染康复后不会产生免疫力，个体易再次感染。

[[File:SIS compartment model.svg|400px|center|SIS compartmental model]]

SIS compartmental model

【图5：SIS compartmental modelSIS传染病模型】

We have the model:

We have the model:

我们有一个模型:

:<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]

\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]

\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]

\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I

\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I

\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</math>

Note that denoting with ''N'' the total population it holds that:

Note that denoting with N the total population it holds that:

注意，用N表示它所拥有的总人口:

:<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.

It follows that:

It follows that:

由此可见:

:<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,

i.e. the dynamics of infectious is ruled by a [[logistic function]], so that <math>\forall I(0) > 0</math>:

i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:

也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:

:<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]

& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]

& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]

& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.

& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.

& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</math>

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations),<ref name=Hethcote1989>{{cite book |last=Hethcote |first=Herbert W. |chapter=Three Basic Epidemiological Models |title=Applied Mathematical Ecology |editor1-last=Levin |editor1-first=Simon A. |editor2-last=Hallam |editor2-first=Thomas G. |editor3-last=Gross |editor3-first=Louis J. |series=Biomathematics |volume=18 |publisher=Springer |location=Berlin |year=1989 |pages=119–144 |isbn=3-540-19465-7 |doi=10.1007/978-3-642-61317-3_5 }}</ref> such that the basic reproduction rate is greater than unity. The solution is given as

It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as

这个模型可以找到一个'''解析解analytical solution'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''平均场方程the mean-field euations''') ，使基本再生率大于单位数。给出了解决方案如下:

:<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.

其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人口，<math>\chi = \beta-\gamma</math>，和 <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者是<math>S(t) = N - I(t)</math>。

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the ''SI model''.<ref>[http://math.colgate.edu/~wweckesser/math312Spring06/handouts/IMM_SI_Model.pdf (p. 19) The SI Model]</ref> The differential equation system using <math>S=N-I</math> thus reduces to:

As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:

作为一个特例，人们通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型具有<math>R=0</math>，即没有康复者。这就是SI模型。微分方程系统使用<math>S=N-I</math>因此可以简化为:

:<math>

<math>

<math>

\frac{dI}{dt} \propto I\cdot (N-I).

\frac{dI}{dt} \propto I\cdot (N-I).

\frac{dI}{dt} \propto I\cdot (N-I).

</math>

</math>

</math>

In the long run, in this model, all individuals will become infected.

In the long run, in this model, all individuals will become infected.

从长远来看，在这种模式下，所有的个体都会被感染。

===The SIRD model===

SIRD模型

[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]

Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>

【图6：Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>】

[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]

Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).

【图7：Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>)SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。】

The ''Susceptible-Infectious-Recovered-Deceased-Model'' differentiates between ''Recovered'' (meaning specifically individuals having survived the disease and now immune) and ''Deceased''.{{cn|date=June 2020}} This model uses the following system of differential equations:

The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:

易感-感病-康复-死亡-模型区分恢复(特别是指从疾病中存活下来并且现在免疫)和死亡。该模型使用了下列微分方程组:

:<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]

& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]

& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]

& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]

& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]

& frac { dI }{ dt } = frac { beta i s }{ n }-gamma i-mu i，[6 pt ]

& \frac{dR}{dt} = \gamma I, \\[6pt]

& \frac{dR}{dt} = \gamma I, \\[6pt]

& \frac{dR}{dt} = \gamma I, \\[6pt]

& \frac{dD}{dt} = \mu I,

& \frac{dD}{dt} = \mu I,

1 = mu i,

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</math>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>

where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.

这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。

===The MSIR model===

MSIR模型

For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.

对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，但由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开头加入一个M类(用于母系免疫)来显示。

[[File:MSIR.PNG|800px|center|MSIR compartmental model]]To indicate this mathematically, an additional compartment is added, ''M''(''t'')''.'' This results in the following differential equations:

MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:

【图8：MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 分隔模型为了从数学上表示这一点，增加了一个额外的分隔，M(t)。这导致了下列微分方程:】

::<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]

\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]

{ dM }{ dT } & = Lambda-delta m-mu m [8 pt ]

\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]

\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S\\[8pt]

Frac { dS }{ dT } & = delta m-frac { beta SI }{ n }-mu s [8 pt ]

\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]

\frac{dI}{dT} & = \frac{\beta SI}{N} - \gamma I - \mu I\\[8pt]

Frac { dI }{ dT } & = frac { beta SI }{ n }-gamma i-mu i [8 pt ]

\frac{dR}{dT} & = \gamma I - \mu R

\frac{dR}{dT} & = \gamma I - \mu R

\frac{dR}{dT} & = \gamma I - \mu R

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</math>

===Carrier state===

载体状态

Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.

Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.

一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。然后他们可能回归到感病者并出现症状（如肺结核），或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是玛丽·马伦，她将伤寒传染给了22个人。携带者被标记为C。

[[File:SIR with carrier model.png|A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.]]

A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.

【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改使单词“携带者”清晰可见。】

===The SEIR model===

SEIR模型

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).

For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).

对于许多重要的感染，有一个重要的疾病潜伏期，在这期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(为暴露)。

[[File:SEIR.PNG|800px|center|SEIR compartmental model]]

SEIR compartmental model

【图10：SEIR compartmental modelSEIR传染病模型】

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:

假设疾病潜伏期是一个带有参数<math>''a''</math>，那么它就是一个服从指数分布的随机变量。平均疾病潜伏期是<math>a^{-1}</math>)，并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:

:<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]

\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]

{ dS }{ dt } & = Lambda-mu s-frac { beta i s }{ n }[8 pt ]

\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]

\frac{dE}{dt} & = \frac{\beta I S}{N} - (\mu +a ) E \\[8pt]

Frac { dE }{ dt } & = frac { beta i s }{ n }-(mu + a) e [8 pt ]

\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]

\frac{dI}{dt} & = a E - (\gamma +\mu ) I \\[8pt]

Frac { dI }{ dt } & = a e-(gamma + mu) i [8 pt ]

\frac{dR}{dt} & = \gamma I - \mu R.

\frac{dR}{dt} & = \gamma I - \mu R.

\frac{dR}{dt} & = \gamma I - \mu R.

\end{align}

\end{align}

结束{ align }

</math>

</math>

</math>

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.

我们有<math>S+E+I+R=N,</math>但是这只是常数，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。

For this model, the basic reproduction number is:

For this model, the basic reproduction number is:

对于这种模式，'''基本再生数basic reproduction number'''是:

:<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (''N'',0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions

类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，我们可以证明，独立于生物学上有意义的初始条件

:<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>

<math> \左(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \右\} </math>

it holds that:

it holds that:

它认为:

:<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

<math> R_0 \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = (N,0,0,0),</math>

左(s (t) ，e (t) ，i (t) ，r (t)右) = DFE = (n，0,0,0,0) ，</math >

:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

当周期性变化的接触率 < math > beta (t) </math > DFE 全局吸引性的条件是下列具有周期系数的线性系统:

: <math>

<math>

<math>

\begin{align}

\begin{align}

开始{ align }

\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]

\frac{dE_1}{dt} & = \beta(t) I_1 - (\gamma +a ) E_1 \\[8pt]

Frac { dE _ 1}{ dt } & = beta (t) i _ 1-(gamma + a) e _ 1[8 pt ]

\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1

\frac{dI_1}{dt} & = a E_1 - (\gamma +\mu ) I_1

压裂{ dI _ 1}{ dt } & = a e _ 1-(gamma + mu) i _ 1

\end{align}

\end{align}

结束{ align }

</math>

</math>

</math>

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).

是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。

=== The SEIS model ===

SEIS模型

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.

SEIS 模型类似于 SEIR 模型(上图) ，只是最终没有获得免疫性。

:::<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the ''S''(''t'') compartment. The following differential equations describe this model:

In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:

在这个模型中，感染不会留下任何免疫性，因此个体已经恢复到易感状态，移回 s (t)隔间。下列微分方程描述了这一模型:

::<math>

<math>

<math>

\begin{align}

\begin{align}

开始{ align }

\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]

\frac{dS}{dT} & = \Lambda - \frac{\beta SI}{N} - \mu S + \gamma I \\[6pt]

Frac { dS }{ dT } & = Lambda-frac { beta SI }{ n }-mu s + gamma i [6 pt ]

\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]

\frac{dE}{dT} & = \frac{\beta SI}{N} - (\epsilon + \mu)E \\[6pt]

Frac { dE }{ dT } & = frac { beta SI }{ n }-(epsilon + mu) e [6 pt ]

\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I

\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I

(γ + mu) i

\end{align}

\end{align}

结束{ align }

</math>

</math>

</math>

=== The MSEIR model ===

MSEIR模型

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.

以疾病为例，考虑被动免疫因素和潜伏期，建立了 MSEIR 模型。

:::<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>

<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>

<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>

::<math>

<math>

<math>

\begin{align}

\begin{align}

开始{ align }

\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]

\frac{dM}{dT} & = \Lambda - \delta M - \mu M \\[6pt]

{ dM }{ dT } & = Lambda-delta m-mu m [6 pt ]

\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]

\frac{dS}{dT} & = \delta M - \frac{\beta SI}{N} - \mu S \\[6pt]

压裂{ dS }{ dT } & = delta m-frac { beta SI }{ n }-mu s [6 pt ]

\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]

\frac{dE}{dT} & = \frac{\beta SI}{N} - (\varepsilon + \mu)E \\[6pt]

Frac { dE }{ dT } & = frac { beta SI }{ n }-(varepsilon + mu) e [6 pt ]

\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]

\frac{dI}{dT} & = \varepsilon E - (\gamma + \mu)I \\[6pt]

(γ + mu) i [6 pt ]

\frac{dR}{dT} & = \gamma I - \mu R

\frac{dR}{dT} & = \gamma I - \mu R

\frac{dR}{dT} & = \gamma I - \mu R

\end{align}

\end{align}

\结束{ align }

</math>

</math>

=== The MSEIRS model ===

MSEIRS模型

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.

MSEIRS 模型与 MSEIR 模型相似，但 r 类的免疫是暂时的，因此当暂时免疫结束时，个体将恢复其易感性。

:::<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>

===Variable contact rates===

多样化接触率

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.

众所周知，患病的概率在时间上并不是一成不变的。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。

In addition, Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate

此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性/“季节性”)变化接触率的感染力

:<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>

with period T equal to one year.

with period T equal to one year.

周期T等于一年。

Thus, our model becomes

Thus, our model becomes

因此，我们的模型变成了

:<math>

<math>

<math>

\begin{align}

\begin{align}

\开始{ align }

\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]

\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]

Frac { dS }{ dt } & = mu n-mu s-beta (t) frac { i }{ n } s [8 pt ]

\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I

\frac{dI}{dt} & = \beta(t) \frac{I}{N} S - (\gamma +\mu ) I

{ i }{ n } s-(gamma + mu) i

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</math>

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:

(康的动力学很容易遵循<math>R=N-S-I</math>)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会经历非常有趣和复杂的非线性参数共振现象。显然，如果:

:<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

<math>\frac 1 T \int_0^T \frac{\beta(t)}{\mu+\gamma} \, dt < 1 \Rightarrow \lim_{t \to +\infty} (S(t),I(t)) = DFE = (N,0), </math>

< math > frac 1t int _ 0 ^ t frac { beta (t)}{ mu + gamma } ，dt < 1 right tarrow lim _ { t to + infty }(s (t) ，i (t)) = DFE = (n，0) ，</math >

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.

反之，如果积分大于1，疾病就不会消失，可能会有这样的共振。例如，将周期性变化的接触率作为系统的输入，则输出是一个周期是输入周期的倍数的周期函数。

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.

这就可以解释某些传染病的多年(常见为两年)流行病爆发，即接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。

==Modelling vaccination==

疫苗接种模型

The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}

The SIR model can be modified to model vaccination<ref>

SIR 模型可以修改为疫苗接种模型<ref>

</ref>. Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

</ref>. Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.

</ref >.典型地，这些引入了一个额外的类别，SIR 模型， <math>V</math> ，为免疫个体。下面是一些例子。

===Vaccinating newborns===

新生儿疫苗接种

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:

面对传染病，主要任务之一是通过预防措施消除传染病，如果可能的话，通过建立大规模疫苗接种计划。假设一种新生儿接种疫苗(通过疫苗给予终身免疫力)的疾病，其发病率为 <math>P \in (0,1)</math>:

:<math>

<math>

<math>

\begin{align}

\begin{align}

开始{ align }

\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]

\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]

\frac{dS}{dt} & = \nu N (1-P) - \mu S - \beta \frac{I}{N} S \\[8pt]

\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]

\frac{dI}{dt} & = \beta \frac{I}{N} S - (\mu+\gamma) I \\[8pt]

Frac { dI }{ dt } & = beta frac { i }{ n } s-(mu + gamma) i [8 pt ]

\frac{dV}{dt} & = \nu N P - \mu V

\frac{dV}{dt} & = \nu N P - \mu V

\frac{dV}{dt} & = \nu N P - \mu V

\end{align}

\end{align}

\结束{ align }

</math>

</math>

</math>

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:

其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:

:<math> \lim_{t \to +\infty} V(t)= N P,</math>

<math> \lim_{t \to +\infty} V(t)= N P,</math>

< math > lim _ { t to + infty } v (t) = np，</math >

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:

因此，我们将处理<math>S</math>和<math>I</math>的长期行为，它认为:

:<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>

:<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

<math> R_0 (1-P) > 1 , \quad I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>

< math > r0(1-P) > 1，quad i (0) > 0 right tarrow lim _ t to + infty }左(s (t) ，i (t)右) = EE = 左(frac { n }{ r0(1-P)} ，n 左(r0(1-P)-1右)。数学

In other words, if

In other words, if

也就是说，如果

:<math> P < P^{*}= 1-\frac{1}{R_0} </math>

<math> P < P^{*}= 1-\frac{1}{R_0} </math>

1-frac {1}{ r _ 0} </math >

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.

疫苗接种计划在根除这种疾病方面并不成功，相反，它将继续流行下去，尽管它的水平低于缺乏疫苗接种的情况。这意味着数学模型表明，对于基本传染数可能高达18的疾病，应该至少为94.4% 的新生儿接种疫苗，以根除这种疾病。

===Vaccination and information===

疫苗接种与信息

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:

现代社会正面临着“理性”豁免的挑战，这个家庭决定不给孩子接种疫苗，是因为对感染的可感知风险和接种疫苗造成的损害进行了“合理”的比较。为了评估这种行为是否真的是理性的，即：如果它同样可以导致根除该疾病，人们可以简单地假定疫苗接种率是传染病人数的增加函数:

:<math> P=P(I), \quad P'(I)>0.</math>

<math> P=P(I), \quad P'(I)>0.</math>

P = p (i) ，p’(i) > 0

In such a case the eradication condition becomes:

In such a case the eradication condition becomes:

在这种情况下，根除条件是:

:<math> P(0) \ge P^{*},</math>

<math> P(0) \ge P^{*},</math>

<math> P(0) \ge P^{*},</math>

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.

也就是说基线疫苗接种率应高于“强制接种”的门槛，而在豁免情况下，“强制接种”不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而不是考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。

===Vaccination of non-newborns===

非新生儿疫苗接种

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:

如果还有非新生儿的疫苗接种率，易感者和接种者的等式必须修改如下:

:<math>

<math>

<math>

\begin{align}

\begin{align}

开始{ align }

\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]

\frac{dS}{dt} & = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S \\[8pt]

Frc { dS }{ dt } & = mu n (1-P)-mu s-rho s-beta frc { i }{ n } s [8 pt ]

\frac{dV}{dt} & = \mu N P + \rho S - \mu V

\frac{dV}{dt} & = \mu N P + \rho S - \mu V

\frac{dV}{dt} & = \mu N P + \rho S - \mu V

\end{align}

\end{align}

结束{ align }

</math>

</math>

</math>

leading to the following eradication condition:

leading to the following eradication condition:

导致以下的根除条件:

:<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>

===Pulse vaccination strategy===

脉冲接种疫苗策略

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:

随着时间的推移，这种策略反复对特定年龄组群(如幼儿或老年人)的易感人群进行疫苗接种。使用这种策略，易感者类别立即康复，从而有可能从整个人群中消灭传染病病毒(如麻疹)。每T时间单位一个常数分数p的易感受试者接种疫苗在一个相对较短的时间内(相对于疾病的动态)。这就导致了以下易受感染和接种疫苗的受试者的脉冲微分方程:

:<math>

<math>

<math>

\begin{align}

\begin{align}

开始{ align }

\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]

\frac{dS}{dt} & = \mu N - \mu S - \beta \frac{I}{N} S, \quad S(n T^+) = (1-p) S(n T^-), & & n=0,1,2,\ldots \\[8pt]

Frc { dS }{ dt } & = mu n-mu s-beta frc { i }{ n } s，方 s (n t ^ +) = (1-p) s (n t ^ -) ，& n = 0,1,2，ldots [8 pt ]

\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots

\frac{dV}{dt} & = - \mu V, \quad V(n T^+) = V(n T^-) + p S(n T^-), & & n=0,1,2,\ldots

Frc { dV }{ dt } & =-mu v，quad v (n t ^ +) = v (n t ^ -) + p s (n t ^ -) ，& n = 0,1,2，ldots

\end{align}

\end{align}

结束{ align }

</math>

</math>

</math>

It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:

It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:

很容易看出，通过设置，人们得到的易感体动力学是由以下因素决定的:

:<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}} E^{-\mu MOD(t,T)} </math>

< math > s ^ * (t) = 1-frac { p }{1-(1-p) e ^ {-mu t } e ^ {-mu MOD (t，t)} </math >

and that the eradication condition is:

and that the eradication condition is:

而根除条件是:

:<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

<math> R_0 \int_0^T S^*(t) \, dt < 1 </math>

< math > r0 int _ 0 ^ t s ^ * (t) ，dt < 1 </math >

==The influence of age: age-structured models==

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:

年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑到流行病的年龄<math>s(t,a),i(t,a),r(t,a)</math>(将我们限制在易感者-感病者-康复的计划) ，这样:

:<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>

:<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>

:<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

<math>R(t)=\int_0^{a_M} r(t,a)\,da</math>

< math > r (t) = int _ 0 ^ { a _ m } r (t，a) ，da </math >

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:

(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:

(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:

:<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_0^{a_M} k(a,a_1;t)i(a_1,t)\,da_1 </math>

< math > partial _ ts (t，a) + partial _ a s (t，a) =-mu (a，t) s (a，t)-s (a，t) int _ 0 ^ { a _ m } k (a，a _ 1; t) i (a _ 1，t) ，da _ 1 </math >

:<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \gamma(a)i(a,t) </math>

< math > partial _ ti (t，a) + partial _ a i (t，a) = s (a，t) int _ {0} ^ { a _ m }{ k (a，a _ 1; t) i (a _ 1，t) da _ 1}-mu (a) i (a，t)-gamma (a) i (a，t) </math >

:<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \gamma(a)i(a,t) </math>

< math > partial _ tr (t，a) + partial _ a r (t，a) =-mu (a) r (a，t) + gamma (a) i (a，t) </math >

where:

where:

其中:

:<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

<math>F(a,t,i(\cdot,\cdot))=\int_0^{a_M} k(a,a_1;t)i(a_1,t) \, da_1 </math>

(a，t，i (cdot，cdot)) = int _ 0 ^ { a _ m } k (a，a _ 1; t) i (a _ 1，t) ，da _ 1 </math >

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.

当然，虽然联系核心<math> k(a,a_1;t) </math>在于年龄之间的相互作用，但这将取决于感染力。

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:

新生婴儿的初始条件(即对于 a = 0) ，可以直接用于传染和康复:

:<math>i(t,0)=r(t,0)=0</math>

<math>i(t,0)=r(t,0)=0</math>

<math>i(t,0)=r(t,0)=0</math>

but that are nonlocal for the density of susceptible newborns:

but that are nonlocal for the density of susceptible newborns:

但对于易感新生儿的密度来说，这是非本地的:

:<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

<math>s(t,0)= \int_0^{a_M} \left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) \, da </math>

< math > s (t，0) = int _ 0 ^ { a _ m } left (varphi _ s (a，t) + varphi _ i (a) i (a，t) + varphi _ r (a) r (a，t) right) ，da </math >

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.

其中，<math>\varphi_j(a), j=s,i,r</math>是成人的受精能力。

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:

此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> 1得到:

:<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>

< math > partial _ tc (t，a) + partial _ a n (t，a) =-mu (a) n (a，t) </math >

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:

在三个流行等级生育率相等的最简单情况下，我们得到了人口平衡与生育率相关的下列充要条件出生率<math>\varphi(.)和死亡率</math><math>\mu(a)</math>灵活性间的联系，必须坚持:

:<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

<math> 1 = \int_0^{a_M} \varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right) \, da </math>

1 = int _ 0 ^ { a _ m } varphi (a) exp left (- int _ 0 ^ a { mu (q) dq } right) ，da </math >

and the demographic equilibrium is

and the demographic equilibrium is

人口平衡是

:<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

<math>n^*(a)=C \exp\left(- \int_0^a \mu(q) \, dq \right),</math>

= c exp left (- int _ 0 ^ a mu (q) ，dq right) ，</math >

automatically ensuring the existence of the disease-free solution:

automatically ensuring the existence of the disease-free solution:

自动确保无疾病解决方案的存在:

:<math>DFS(a)= (n^*(a),0,0).</math>

<math>DFS(a)= (n^*(a),0,0).</math>

<math>DFS(a)= (n^*(a),0,0).</math>

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

基本再生数可以用一个适当的函数算符的谱半径来计算。

== Other considerations within compartmental epidemic models ==

其他传染病模型的考虑因素

=== Vertical transmission ===

垂直传染

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>

In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.

在艾滋病和乙型肝炎等一些疾病的情况下，受感染父母的后代有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，可以通过将一部分新生儿成员包括在受感染的隔间中来考虑额外成员流入受感染类别的情况。

=== Vector transmission ===

水平传染

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>

Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.

由人间接传播的疾病，即。疟疾通过蚊子传播，通过媒介传播。在这种情况下，从人到昆虫的感染转移和一个流行病模型必须包括这两个物种，通常需要更多的区域比模型直接传染。

=== Others ===

其他

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />

Other occurrences which may need to be considered when modeling an epidemic include things such as the following:

在建立流行病模型时可能需要考虑的其他事件包括以下事项:

*Non-homogeneous mixing

*Variable infectivity

*Distributions that are spatially non-uniform

*Diseases caused by macroparasites

==Deterministic versus stochastic epidemic models==

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.<ref>{{cite journal |author=Bartlett MS |s2cid=91114210 |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}</ref>

It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.

重要的是要强调，这里提出的确定性模型只有在足够大人口的情况下才有效，因此应谨慎使用。

To be more precise, these models are only valid in the [[thermodynamic limit]], where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.

更准确地说，这些模型只适用于中东热力学极限，那里的人口实际上是无限的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会繁殖，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以呈现实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。传染病模型的可靠性仅限应用于传染病。

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.<ref>{{cite journal |author=Croccolo F. and Roman H.E. |title=Spreading of infections on random graphs: A percolation-type model for COVID-19 |journal=Chaos, Solitons & Fractals |volume=139 |pages=110077 |year=2020 |doi=10.1016/j.chaos.2020.110077 }}</ref>

One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.

平均场模型的一个可能的扩展是基于渗流理论的概念来考虑流行病在网络上的传播。

==See also==

参见

*[[Mathematical modelling in epidemiology]]

*[[Modifiable areal unit problem]]

*[[Next-generation matrix]]

*[[Risk assessment]]

*[[Attack rate]]

==References==

参考文献

{{reflist}}

==Further reading==

深入阅读

*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}

*{{cite book |editor1-last=Vynnycky |editor1-first=E. |editor2-last=White |editor2-first=R. G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |isbn=978-0-19-856576-5 }}

==External links==

外部链接

* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]

*{{cite web |title=Simulating an epidemic |work=[[3Blue1Brown]] |date=March 27, 2020 |url=https://www.youtube.com/watch?v=gxAaO2rsdIs |via=[[YouTube]] }}

==编辑推荐==

[https://mp.weixin.qq.com/s__biz=MzIzMjQyNzQ5MA==&mid=2247501936&idx=1&sn=c59b0236ed6082358b98b70411197d53&chksm=e89792fddfe01beb4320fe3af406b56025f724346e16e6e4a60bf33a219e543a4b7db18b72f8&scene=21#wechat_redirect集智俱乐部推文：超越SIR模型：信息与疾病传播的复杂建模]

[https://mp.weixin.qq.com/s/8JceYjtmFcqUbJl5af4Etg集智俱乐部推文：从恐惧到仇恨: 美国新冠病毒肺炎流行病如何引发种族仇恨 | 网络科学论文速度36篇]

[https://swarma.org/?p=18877集智俱乐部推文：Science最新评论：传染病模型面临真实世界的考验]

[https://campus.swarma.org/course/1105集智学园课程：网络上病毒传播的SIR模型用Links建模网络动力学]

{{Computer modeling|state=expanded}}

{{DEFAULTSORT:Compartmental Models In Epidemiology}}

[[Category:Epidemiology]]

Category:Epidemiology

类别: 流行病学

[[Category:Scientific modeling]]

Category:Scientific modeling

类别: 科学建模

<noinclude>

This page was moved from [[wikipedia:en:Compartmental models in epidemiology]]. Its edit history can be viewed at [[复杂传染病/edithistory]]</noinclude>

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2020-11-01T02:37:00Z

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