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第37行： − The above approach can be generalized to reacting particles with different states by adding a reaction term to the above equations <ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>. We now describe a generalization to this setting of the standard SIS model in discrete time, with probability per unit time $\beta$ of infection and probability $\mu$ of recovery. We consider $\mathcal{N}$ individuals diffusing in a heterogeneous network with $N$ nodes and degree distribution $P(k)$. − Each node i of the network has a number $I(i)$ of infectious and $S(i)$ of susceptible individuals. The occupation numbers $I(i)$ and $S(i)$ can have any integer value, including $I(i)=S(i)=0$, that is, void nodes with no individuals. − This modeling scheme describes spatially structured interacting subpopulations, such as city locations, urban areas, or defined geographical regions <ref>Grenfell B, Harwood J. (Meta) population dynamics of infectious diseases[J]. Trends in ecology & evolution, 1997, 12(10): 395-399.</ref><ref>Ecology, genetics and evolution of metapopulations[M]. Academic Press, 2004.</ref> and is usually referred to as the metapopulation approach. Each node of the network represents a subpopulation and the compartment dynamics accounts for the possibility that individuals in the same location may get into contact and change their state according to the infection dynamics. The interaction among subpopulations is the result of the movement of individuals from one subpopulation to the other. We have thus to associate with each individual’s class a diffusion probability $p_I$ and $p_S$ that indicates the probability for any individual to leave its node and move to a neighboring node of the network. In general the diffusion probabilities are heterogeneous and can be node dependent; however, for simplicity we assume that individuals diffuse with probability $p_I=p_S=1$ along any of the links departing from the node in which they are. This implies that at each time step an individual sitting on a node with degree k will diffuse into one of its nearest neighbors with probability $1/k$. In order to write the dynamical equations of the system we define the following quantities: + − 通过在上述方程中加入反应项，可以将上述方法推广到具有不同状态的粒子发生反应的情形<ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>。现在，我们描述在该情形下SIS传播模型<font color="#ff8000">Susceptible Infected Susceptible Model</font> 的情况，其中单位时间的感染概率为<math>\beta</math>，恢复概率为<math>\mu</math>。我们考虑在具有<math>N</math>个节点和度分布为<math>P(k)</math>的异质网络中扩散的<math>\mathcal{N}</math>个个体。<math>网络的每个节点$i$分别具有$I(i)$个感染态个体和$S(i)$个易感态个体（取值为非负整数值）。这种建模体系描述了空间结构相互作用的亚种群，例如城市位置、城市区域或确定的地理区域（Gren fall和Harwood，1997； Hanski和Gaggiotti，2004），并且通常被称为亚种群方法。网络中的每个节点代表一个亚种群，仓室动力学表示了处于相同位置的个体可能会相互接触并根据系统对应的感染动力学改变自身的状态的可能性。亚群之间的相互作用是个体从一个亚种群迁移到另一个亚种群的结果。为简单起见，我们可以假设个体从他们所在的节点离开沿着任意一条连边扩散的概率为$p_I=p_S=1$。这意味着每个时间步，度为$k$的节点上的一个个体将以概率$1/k$扩散到其最近的邻居之一。为了得到系统的动力学方程，我们定义如下的量：</math> 第49行：
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==亚种群模型==
==亚种群模型==
通过在上述方程中加入反应项，可以将上述方法推广到具有不同状态的粒子发生反应的情形<ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>。现在，我们描述在该情形下SIS传播模型<font color="#ff8000">Susceptible Infected Susceptible Model</font> 的情况，其中单位时间的感染概率为<math>\beta</math>，恢复概率为<math>\mu</math>。我们考虑在具有<math>N</math>个节点和度分布为<math>P(k)</math>的异质网络中扩散的<math>\mathcal{N}</math>个个体。网络的每个节点<math>i</math>分别具有<math>I(i)</math>个感染态个体和<math>S(i)</math>个易感态个体（取值为非负整数值）。这种建模体系描述了空间结构相互作用的亚种群，例如城市位置、城市区域或确定的地理区域<ref>Grenfell B, Harwood J. (Meta) population dynamics of infectious diseases[J]. Trends in ecology & evolution, 1997, 12(10): 395-399.</ref><ref>Ecology, genetics and evolution of metapopulations[M]. Academic Press, 2004.</ref>，并且通常被称为亚种群方法<font color="#ff8000">Metapopulation approach</font>。网络中的每个节点代表一个亚种群，仓室动力学<font color="#ff8000">Compartment Dynamics</font>表示了处于相同位置的个体可能会相互接触并根据系统对应的感染动力学改变自身的状态的可能性。亚种群之间的相互作用是个体从一个亚种群迁移到另一个亚种群的结果。为简单起见，我们可以假设个体从他们所在的节点离开沿着任意一条连边扩散的概率为<math>p_I=p_S=1</math>。这意味着在每个时间步，度为<math>k</math>的节点上的一个个体将以概率<math>1/k</math>扩散到其最近的邻居之一。为了得到系统的动力学方程，我们定义如下的量：
\begin{equation}
\begin{equation}
I_k=\frac{1}{N_k}\sum_{i\in\mathcal{V}(k)}I(i)
I_k=\frac{1}{N_k}\sum_{i\in\mathcal{V}(k)}I(i)
S_k=\frac{1}{N_k}\sum_{i\in\mathcal{V}(k)}S(i)
S_k=\frac{1}{N_k}\sum_{i\in\mathcal{V}(k)}S(i)
\end{equation}
\end{equation}
where the sums $\sum_{i\in\mathcal{V}(k)}$ are performed over nodes of degree $k$. These two quantities express the average number of susceptible and infectious individuals in nodes with degree $k$. Clearly, $\mathcal{N}_k=I_k+S_k$ is the average number of individuals in nodes with degree $k$. These quantities allow one to write the discrete-time equation describing the time evolution of $I_k(t)$ for each class of degree k as
where the sums $\sum_{i\in\mathcal{V}(k)}$ are performed over nodes of degree $k$. These two quantities express the average number of susceptible and infectious individuals in nodes with degree $k$. Clearly, $\mathcal{N}_k=I_k+S_k$ is the average number of individuals in nodes with degree $k$. These quantities allow one to write the discrete-time equation describing the time evolution of $I_k(t)$ for each class of degree k as