更改

\frac{d\mathcal{N}_k}{dt}=-d_k\mathcal{N}_k(t)+k\sum_{k'}P(k'|k)d_{k'k}\mathcal{N}_{k'}(t)

\frac{d\mathcal{N}_k}{dt}=-d_k\mathcal{N}_k(t)+k\sum_{k'}P(k'|k)d_{k'k}\mathcal{N}_{k'}(t)

\end{equation}

\end{equation}

The first term of the equation considers that only a fraction $d_k$ of particles moves out of the node per unit time. The second term instead accounts for the particles diffusing from the neighbors into the node of degree k. This term is proportional to the number of links k times the average number of particles coming from each neighbor. This is equal to the average over all possible degrees $k'$ of the fraction of particles moving on that edge, $d_{k'k}$, according to the conditional probability $P(k'|k)$ that an edge belonging to a node of degree k is pointing to a node of degree $k'$. Here the term $d_{k'k}$ is the diffusion rate along the edges connecting nodes of degree k and $k'$. In the simplest case of homogeneous diffusion each particle diffuses with rate $r$ from the node in which it is and thus the diffusion per link $d_{k'k}=r/k'$. On uncorrelated networks $P(k'|k)=k'P(k')/\left<k\right>$ and hence one easily gets in the stationary state $d\mathcal{N}/dt=0$ the solution (Noh and Rieger, 2004; Colizza, Pastor-Satorras, and Vespignani, 2007)

The first term of the equation considers that only a fraction $d_k$ of particles moves out of the node per unit time. The second term instead accounts for the particles diffusing from the neighbors into the node of degree k. This term is proportional to the number of links k times the average number of particles coming from each neighbor. This is equal to the average over all possible degrees $k'$ of the fraction of particles moving on that edge, $d_{k'k}$, according to the conditional probability $P(k'|k)$ that an edge belonging to a node of degree k is pointing to a node of degree $k'$. Here the term $d_{k'k}$ is the diffusion rate along the edges connecting nodes of degree k and $k'$. In the simplest case of homogeneous diffusion each particle diffuses with rate $r$ from the node in which it is and thus the diffusion per link $d_{k'k}=r/k'$. On uncorrelated networks $P(k'|k)=k'P(k')/\left<k\right>$ and hence one easily gets in the stationary state $d\mathcal{N}/dt=0$ the solution <ref>Noh J D, Rieger H. Random walks on complex networks[J]. Physical review letters, 2004, 92(11): 118701.</ref><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>

方程中的第一项表示每单位时间内有$d_k$部分的粒子从度为$k$的节点中离开，第二项表示粒子从邻居节点扩散到度为$k$的节点中，该项与连边数$k$成正比。条件概率$P(k'|k)$表示的是一条边的一端连接度为$k$的节点时，其另一端指向度为$k'$的概率。$d_{k'k}$表示的是两端连接了度为$k$和$k'$节点的连边的扩散率。在均匀扩散的最简单情况下，每个粒子从其所在的节点以速率$r$扩散，因此沿着每条连边的扩散率为$d_{k'k}=r/k'$。在无关联的网络上，$P(k'|k)=k'P(k')/\left<k\right>$，因此在稳态$d\mathcal{N}/dt=0$时，很容易得到解（Colizza等，2007b； Noh和Rieger，2004）：

方程中的第一项表示每单位时间内有$d_k$部分的粒子从度为$k$的节点中离开，第二项表示粒子从邻居节点扩散到度为$k$的节点中，该项与连边数$k$成正比。条件概率$P(k'|k)$表示的是一条边的一端连接度为$k$的节点时，其另一端指向度为$k'$的概率。$d_{k'k}$表示的是两端连接了度为$k$和$k'$节点的连边的扩散率。在均匀扩散的最简单情况下，每个粒子从其所在的节点以速率$r$扩散，因此沿着每条连边的扩散率为$d_{k'k}=r/k'$。在无关联的网络上，$P(k'|k)=k'P(k')/\left<k\right>$，因此在稳态$d\mathcal{N}/dt=0$时，很容易得到解（Colizza等，2007b； Noh和Rieger，2004）：

== 亚种群模型 ==

== 亚种群模型 ==

The above approach can be generalized to reacting particles with different states by adding a reaction term to the above equations (Colizza, Pastor-Satorras, and Vespignani, 2007). 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 $P(k)$.

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.

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 (Grenfell and Harwood, 1997; Hanski and Gaggiotti, 2004) 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:

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:

通过将反应项添加到上述方程中，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（Colizza等，2007b）。现在，我们描述在该情形下的SIS模型情况，其中单位时间的感染概率为$\beta$，恢复概率为$\mu$。我们考虑在具有$N$个节点和度分布为$P(k)$的异质网络中扩散的$\mathcal{N}$个个体。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：

通过将反应项添加到上述方程中，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（Colizza等，2007b）。现在，我们描述在该情形下的SIS模型情况，其中单位时间的感染概率为$\beta$，恢复概率为$\mu$。我们考虑在具有$N$个节点和度分布为$P(k)$的异质网络中扩散的$\mathcal{N}$个个体。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：