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第5行： − + − + − + − + − + − + − + − + − + − + − 最近，基于代理的模型（ABMs）已用于交换较简单的仓室模型，例如[7]。 例如，ABMs已被用于提供公共卫生方面的（非药物）干预措施，以防止SARS-CoV-2的传播[8]。 虽然ABMs复杂且需要很高的计算能力，但是该模型仍然由于其简化和不切实际的假设而受到批评[9] [10]。 不过，在准确校准ABMs的情况下，它们对于提供有关缓解和抑制措施方面的决策还是有用的[11]。+ 第57行：
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第73行： − + − \end{aligned} \right.+ − \end{equation}+ + + + + + + + − + − \end{aligned} \right.+ − \end{equation} 给定初始值i（0），我们可以得出i（t）的表达式： \begin{equation} i(t)=\frac{i(0)(\beta-\gamma)e^{(\beta-\gamma)t}}{\beta-\gamma+\beta i(0)e^{(\beta-\gamma)t}}.+ − \end{equation} 在这里，我们将$\lambda=\beta/\gamma$定义为该模型的有效感染率。 从等式（6）可知，当$\lambda>1$时，I的稳态值为$i=(\beta-\gamma)/\beta=1-1/\lambda$，表明终态时人群中存在一定比例的受感染者，也就是说当$\lambda>1$时，整体人群处于地方性疾病状态。 但是，当$\lambda<1$时，我们有$i\to0(t\to\infty)$，这意味着最终人群中将没有I状态个体，整体人群被称为健康状态。 综上，$\lambda=1$是SIS模型的传播阈值，也称为基本再生数[217]。+ + + + + + + + + + + + + + + + + + 第91行：
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无编辑摘要
== 历史 History ==
== 历史 History ==
The modeling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic.[1]
The modeling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic.<ref name=":13">{{cite book | vauthors = Daley DJ, Gani J | date = 2005 | title = Epidemic Modeling: An Introduction. | location = New York | publisher = Cambridge University Press }}</ref>
传染病建模是一种工具，用于研究疾病传播的机制，预测爆发的未来过程以及评估控制流行病的策略。
传染病建模是一种工具，用于研究疾病传播的机制，预测爆发的未来过程以及评估控制流行病的策略。
The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt's analysis of causes of death is considered the beginning of the "theory of competing risks" which according to Daley and Gani [1] is "a theory that is now well established among modern epidemiologists".
The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt's analysis of causes of death is considered the beginning of the "theory of competing risks" which according to Daley and Gani <ref name=":13"/> is "a theory that is now well established among modern epidemiologists".
约翰·格伦特（John Graunt）在1662年根据《死亡率法案》撰写的《自然和政治观察》一书中，他是第一个系统地尝试量化死亡原因的科学家。他研究的法案是每周出版的死亡人数和原因清单。 格兰特对死亡原因的分析被认为是“竞争风险理论”的开端。根据戴利和加尼[1]的说法，“竞争风险理论”是“现代流行病学家中已经建立的一种理论”。
约翰·格伦特（John Graunt）在1662年根据《死亡率法案》撰写的《自然和政治观察》一书中，他是第一个系统地尝试量化死亡原因的科学家。他研究的法案是每周出版的死亡人数和原因清单。 格兰特对死亡原因的分析被认为是“竞争风险理论”的开端。根据戴利和加尼的说法，“竞争风险理论”是“现代流行病学家中已经建立的一种理论”。
The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox.[2] The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months.[3] Daniel Bernoulli's work preceded the modern understanding of germ theory.
The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox.<ref>{{cite journal | vauthors = Hethcote HW | date = 2000 | title = The mathematics of infectious diseases. | journal = Society for Industrial and Applied Mathematics | volume = 42 | pages = 599–653 }}</ref> The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months.<ref name="pmid15334536">{{cite journal | vauthors = Blower S, Bernoulli D | title = An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it. 1766 | journal = Reviews in Medical Virology | volume = 14 | issue = 5 | pages = 275–88 | date = 2004 | pmid = 15334536 | doi = 10.1002/rmv.443 }}</ref> Daniel Bernoulli's work preceded the modern understanding of germ theory.
最早的疾病传播数学模型是1760年由丹尼尔·伯努利（Daniel Bernoulli）进行的。 受过医师培训的伯努利创建了一个数学模型来捍卫预防天花的接种方法。[2] 该模型的计算结果表明，普遍接种预防天花的疫苗可以将预期寿命从26岁7个月增加到29岁9个月。[3] 丹尼尔·伯努利（Daniel Bernoulli）的工作先于对细菌理论的现代理解。
最早的疾病传播数学模型是1760年由丹尼尔·伯努利（Daniel Bernoulli）进行的。 受过医师培训的伯努利创建了一个数学模型来捍卫预防天花的接种方法。该模型的计算结果表明，普遍接种预防天花的疫苗可以将预期寿命从26岁7个月增加到29岁9个月。丹尼尔·伯努利（Daniel Bernoulli）的工作先于对细菌理论的现代理解。
In the early 20th century, William Hamer[4] and Ronald Ross[5] applied the law of mass action to explain epidemic behaviour.
In the early 20th century, William Hamer<ref>{{cite book | vauthors = Hamer W | date = 1928 | title = Epidemiology Old and New | location = London | publisher = Kegan Paul }}</ref> and Ronald Ross<ref>{{cite book|last1=Ross|first1=Ronald | name-list-style = vanc |title=The Prevention of Malaria|date=1910|url=http://catalog.hathitrust.org/Record/001587831}}</ref> applied the law of mass action to explain epidemic behaviour.
在20世纪初期，威廉·哈默（William Hamer）[4]和罗纳德·罗斯（Ronald Ross）[5]运用群众行动定律来解释流行病行为。
在20世纪初期，威廉·哈默（William Hamer）和罗纳德·罗斯（Ronald Ross）运用群众行动定律来解释流行病行为。
The 1920s saw the emergence of compartmental models. The Kermack–McKendrick epidemic model (1927) and the Reed–Frost epidemic model (1928) both describe the relationship between susceptible, infected and immune individuals in a population. The Kermack–McKendrick epidemic model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics.[6]
The 1920s saw the emergence of compartmental models. The Kermack–McKendrick epidemic model (1927) and the Reed–Frost epidemic model (1928) both describe the relationship between susceptible, infected and immune individuals in a population. The Kermack–McKendrick epidemic model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics.<ref name=":2">{{cite book | vauthors = Brauer F, Castillo-Chávez C | date = 2001 | title = Mathematical Models in Population Biology and Epidemiology. | location = New York | publisher = Springer }}</ref>
1920年代出现了仓室模型。 Kermack-McKendrick流行病模型（1927年）和Reed-Frost流行病模型（1928年）均描述了人群中易感，感染和免疫个体之间的关系。 Kermack–McKendrick流行病模型成功地预测了爆发行为，与许多已记录的流行病中观察到的行为非常相似。[6]
1920年代出现了仓室模型。 Kermack-McKendrick流行病模型（1927年）和Reed-Frost流行病模型（1928年）均描述了人群中易感，感染和免疫个体之间的关系。 Kermack–McKendrick流行病模型成功地预测了爆发行为，与许多已记录的流行病中观察到的行为非常相似。
Recently, agent-based models (ABMs) have been used in exchange for simpler compartmental models, e.g.,[7]. For example, epidemiological ABMs have been used to inform public health (nonpharmaceutical) interventions against the spread of SARS-CoV-2[8]. Epidemiological ABMs, in spite of their complexity and requiring high computational power, have been criticized for simplifying and unrealistic assumptions[9][10]. Still, they can be useful in informing decisions regarding mitigation and suppression measures in cases when ABMs are accurately calibrated[11].
Recently, agent-based models (ABMs) have been used in exchange for simpler compartmental models, e.g.,<ref>{{cite journal | vauthors = Eisinger D, Thulke HH | title = Spatial pattern formation facilitates eradication of infectious diseases | journal = The Journal of Applied Ecology | volume = 45 | issue = 2 | pages = 415–423 | date = April 2008 | pmid = 18784795 | pmc = 2326892 | doi = 10.1111/j.1365-2664.2007.01439.x }}</ref>. For example, epidemiological ABMs have been used to inform public health (nonpharmaceutical) interventions against the spread of SARS-CoV-2<ref>{{cite journal | vauthors = Adam D | title = Special report: The simulations driving the world's response to COVID-19 | journal = Nature | volume = 580 | issue = 7803 | pages = 316–318 | date = April 2020 | pmid = 32242115 | doi = 10.1038/d41586-020-01003-6 | url = https://www.nature.com/articles/d41586-020-01003-6 }}</ref>. Epidemiological ABMs, in spite of their complexity and requiring high computational power, have been criticized for simplifying and unrealistic assumptions<ref>{{Cite journal| vauthors = Squazzoni F, Polhill JG, Edmonds B, Ahrweiler P, Antosz P, Scholz G, Chappin É, Borit M, Verhagen H, Giardini F, Gilbert N | display-authors = 6 |date=2020|title=Computational Models That Matter During a Global Pandemic Outbreak: A Call to Action|url=http://jasss.soc.surrey.ac.uk/23/2/10.html|journal=Journal of Artificial Societies and Social Simulation|volume=23|issue=2|pages=10|issn=1460-7425}}</ref><ref>{{cite journal | vauthors = Sridhar D, Majumder MS | title = Modelling the pandemic | journal = Bmj | volume = 369 | pages = m1567 | date = April 2020 | pmid = 32317328 | doi = 10.1136/bmj.m1567 | url = https://www.bmj.com/content/369/bmj.m1567 }}</ref>. Still, they can be useful in informing decisions regarding mitigation and suppression measures in cases when ABMs are accurately calibrated<ref>{{cite journal | vauthors = Maziarz M, Zach M | title = Agent-based modelling for SARS-CoV-2 epidemic prediction and intervention assessment: A methodological appraisal | journal = Journal of Evaluation in Clinical Practice | volume = 26 | issue = 5 | pages = 1352–1360 | date = October 2020 | pmid = 32820573 | pmc = 7461315 | doi = 10.1111/jep.13459 | url = https://onlinelibrary.wiley.com/doi/abs/10.1111/jep.13459 }}</ref>.
最近，基于代理的模型（ABMs）已用于交换较简单的仓室模型，例如。 例如，ABMs已被用于提供公共卫生方面的（非药物）干预措施，以防止SARS-CoV-2的传播[8]。 虽然ABMs复杂且需要很高的计算能力，但是该模型仍然由于其简化和不切实际的假设而受到批评。 不过，在准确校准ABMs的情况下，它们对于提供有关缓解和抑制措施方面的决策还是有用的。
== 假设条件 Assumptions ==
== 假设条件 Assumptions ==
在处理大量人群时，例如在结核病中，经常使用确定性或仓室模型。 在确定性模型中，人群中的个体被分配到不同的亚团体或不同仓室中，每个亚团体或仓室代表流行病的一个特定阶段。
在处理大量人群时，例如在结核病中，经常使用确定性或仓室模型。 在确定性模型中，人群中的个体被分配到不同的亚团体或不同仓室中，每个亚团体或仓室代表流行病的一个特定阶段。
The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model.[6]
The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model.<ref name=":2" />
从一类状态转变为另一类状态的转变速率在数学上可以用导数表示，因此可以使用微分方程来建立模型。在建立这样的模型时，必须假设仓室的人口规模在时间上是可区分的，并且流行过程是确定性的。 换句话说，只能使用用于建立模型的历史记录来计算仓室的人口变化。[6]
从一类状态转变为另一类状态的转变速率在数学上可以用导数表示，因此可以使用微分方程来建立模型。在建立这样的模型时，必须假设仓室的人口规模在时间上是可区分的，并且流行过程是确定性的。 换句话说，只能使用用于建立模型的历史记录来计算仓室的人口变化。
== 流行病模型 Epidemic models ==
== 流行病模型 Epidemic models ==
在经典的流行病传播模型中，人们通常被划分为几种的状态。 信息传播过程中最常见的状态是：（i）S（Susceptible）：易感状态，表示处于该状态的人们不了解信息，将来会被感染； （ii）I（Infected）：受感染状态;（iii） R (Recoverd)：已恢复状态。这些状态的不同组合可以导致不同的模型，例如SI，SIS和SIR模型。
Generally, people are divided into different compartments in the epidemic models, individuals in the same compartment are assigned with the same state. The most common states used are: (i) S: the susceptible state; (ii) I: the infected state; (iii) R: the recovered state. Different combinations of these states can result in different models, such as SI, SIS and SIR model.
在经典的流行病传播模型中，人们通常被划分为几种的状态。 最常见的状态是：（i）S（Susceptible）：易感状态，表示处于该状态的人们处于疾病易感的状态，可以被感染； （ii）I（Infected）：受感染状态;（iii） R (Recoverd)：已恢复状态。这些状态的不同组合可以导致不同的模型，例如SI，SIS和SIR模型。
Herein we first consider these epidemic models in a well-mixed population.
在这里，我们首先考虑在均匀混合的人群中的这些流行病模型。
在这里，我们首先考虑在均匀混合的人群中的这些流行病模型。
=== SI模型 ===
=== SI模型 ===
最简单的情况是SI模型，其中考虑了两个状态，即S和I。将s（t）和i（t）表示为在时刻t处易感染个体和受感染个体的比例，因此我们具有$s(t)+i(t)=1$。假定受感染个体I感染易感染个体S的速率为$\beta$。 在SI模型中，当一位易感态个体一旦被感染，则其将永远处于感染状态。因此，SI模型可以用以下常微分方程表示： \begin{equation} \label{eq:si} \left\{ \begin{aligned} &\frac{ds(t)}{dt}=-\beta s(t)i(t),\\ &\frac{di(t)}{dt}=\beta s(t)i(t).
The simplest case is the SI model, in which two states are considered, S and I. Denote s(t) and i(t) as the proportion of susceptible and infected individuals at time t, thus we have $s(t)+i(t)=1$. Suppose the probability of infected individuals infecting the susceptible individuals is $\beta$. Therefore, the SI model can be illustrated by ordinary differential equations as follows:
最简单的情况是SI模型，其中考虑了两个状态，即S和I。将s（t）和i（t）表示为在时刻t处易感染个体和受感染个体的比例，因此我们具有$s(t)+i(t)=1$。假定受感染个体I感染易感染个体S的速率为$\beta$。 在SI模型中，当一位易感态个体一旦被感染，则其将永远处于感染状态。因此，SI模型可以用以下常微分方程表示：
\begin{equation} \label{eq:si}
\left\{
\begin{aligned}
&\frac{ds(t)}{dt}=-\beta s(t)i(t),\\
&\frac{di(t)}{dt}=\beta s(t)i(t).
\end{aligned} \right.
\end{equation}
=== SIS模型 ===
=== SIS模型 ===
在SIS模型里，人群被划分为两类：易感人群（S）和受感染人群（I）。受感染人群为传染的源头，它通过一定速率$\beta$将疾病传播给易感人群。受感染人群本身则以速率$\gamma$被治愈恢复为易感态；易感人群一旦被感染，就又成为了新的传染源。若将此模型用于信息传播过程中，则表示尚未知晓信息的个体（S）在对信息的了解后（变为I状态），经过一段后，将忽略该信息并再次对该信息变得敏感（处于S状态）。我们将s（t）和i（t）分别表示为人群中易感和受感染个体的比例。因此，SIS模型也可以用常微分方程表示： \begin{equation} \label{eq:sis} \left\{ \begin{aligned} &\frac{ds(t)}{dt}=-\beta s(t)i(t)+\gamma i(t),\\ &\frac{di(t)}{dt}=\beta s(t)i(t)-\gamma i(t).
The SIS model is used to characterize epidemics that have transient immunity, such as influenza. When it turns to the information process, it means that the individuals know about the information (in I state) will ignore the information and become susceptible again (in S state). We denote s(t) and i(t) as the fraction of susceptible and infected individuals in the population respectively. The transmission probability between I state individuals and S state individuals is $\beta$, and the recovered probability of the I state individuals is $\gamma$. Accordingly, the SIS model can also be expressed by the ordinary differential equations:
在SIS模型里，人群被划分为两类：易感人群（S）和受感染人群（I）。受感染人群为传染的源头，它通过一定速率$\beta$将疾病传播给易感人群。受感染人群本身则以速率$\gamma$被治愈恢复为易感态；易感人群一旦被感染，就又成为了新的传染源。若将此模型用于信息传播过程中，则表示尚未知晓信息的个体（S）在对信息的了解后（变为I状态），经过一段后，将忽略该信息并再次对该信息变得敏感（处于S状态）。我们将s（t）和i（t）分别表示为人群中易感和受感染个体的比例。因此，SIS模型也可以用常微分方程表示：
\begin{equation} \label{eq:sis} \left\{
\begin{aligned}
&\frac{ds(t)}{dt}=-\beta s(t)i(t)+\gamma i(t),\\
&\frac{di(t)}{dt}=\beta s(t)i(t)-\gamma i(t).
\end{aligned} \right.
\end{equation}
Given an initial value i(0), we can derive the expression of i(t):
给定初始值i（0），我们可以得出i（t）的表达式：
\begin{equation} \label{eq:it}
i(t)=\frac{i(0)(\beta-\gamma)e^{(\beta-\gamma)t}}{\beta-\gamma+\beta i(0)e^{(\beta-\gamma)t}}.
\end{equation}
We define $\lambda=\beta/\gamma$ as the effective infection rate of this model. From the above equation, we can find that when $\lambda>1$, the steady-state value of I is $i=(\beta-\gamma)/\beta=1-1/\lambda$, an endemic disease state is obtained in this case (indicates that there will be a number of infected individuals in the population at the final state). However, when $\lambda<1$, we have$i\to0(t\to\infty)$, which means that there will be no I state individuals in the population finally and is called a healthy state. Therefore, $\lambda=1$ is a threshold value of the SIS model, which is also known as the basic reproduction number <ref>Heffernan J M, Smith R J, Wahl L M. Perspectives on the basic reproductive ratio[J]. Journal of the Royal Society Interface, 2005, 2(4): 281-293.{{url = https://www.ncbi.nlm.nih.gov/pmc/articles/pmc1578275/ }}</ref>
在这里，我们将$\lambda=\beta/\gamma$定义为该模型的有效感染率。 从上式可知，当$\lambda>1$时，I的稳态值为$i=(\beta-\gamma)/\beta=1-1/\lambda$，表明终态时人群中存在一定比例的受感染者，也就是说当$\lambda>1$时，整体人群处于地方性疾病状态。 但是，当$\lambda<1$时，我们有$i\to0(t\to\infty)$，这意味着最终人群中将没有I状态个体，整体人群被称为健康状态。 综上，$\lambda=1$是SIS模型的传播阈值，也称为基本再生数。
=== SIR模型 ===
=== SIR模型 ===
The SIR model is introduced to explain the epidemics with permanent immunity in the population. Different from the SI and SIS model, a recovered state (R) is given in this model. Individuals in S state would be infected by the I state individuals with probability $\beta$, whereas the I state individuals would recover to R state with a recovery probability $\gamma$. Thus, the ordinary differential equations of SIR model are
SIR模型是用来解释具有永久免疫力的人群中的流行病。 与SI和SIS模型不同，此模型中还存在恢复状态（R）。 处于S状态的个体被I状态个体以速率$\beta$感染，而处于I状态的个体以恢复速率$\gamma$恢复到R状态，节点一旦成为恢复的R状态则成为永久免疫人群，不可以再被感染。 因此，SIR模型的常微分方程为
SIR模型是用来解释具有永久免疫力的人群中的流行病。 与SI和SIS模型不同，此模型中还存在恢复状态（R）。 处于S状态的个体被I状态个体以速率$\beta$感染，而处于I状态的个体以恢复速率$\gamma$恢复到R状态，节点一旦成为恢复的R状态则成为永久免疫人群，不可以再被感染。 因此，SIR模型的常微分方程为
\end{aligned} \right.
\end{aligned} \right.
\end{equation}
\end{equation}
$\lambda=\beta/\gamma$ can also be defined as the basic reproduction number of the SIR model. We can get similar conclusion as the SIS model, i.e., when $\lambda<1$时, $r=0$, indicating that information cannot spread in the population. When $\lambda>1$, we have $r>0$, which means the information can spread out as the increase of $\lambda$. As a consequence, $\lambda=1$ is a threshold value of SIR model.
同样也可以定义$\lambda=\beta/\gamma$为该模型的有效感染率，可以得到与SIS模型类似的结论，即当$\lambda<1$时，$r=0$，表明疾病无法在人群中传播。当$\lambda>1$时，有$r>0$。$\lambda=1$是SIR模型的传播阈值。
同样也可以定义$\lambda=\beta/\gamma$为该模型的有效感染率，可以得到与SIS模型类似的结论，即当$\lambda<1$时，$r=0$，表明疾病无法在人群中传播。当$\lambda>1$时，有$r>0$。$\lambda=1$是SIR模型的传播阈值。
== 参考文献 References ==
== 参考文献 References ==
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