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为了对网络中的反应扩散系统进行理论上的分析描述，我们允许节点之间具有异质的连接模式。对这些系统的第一种分析方法是考虑将基于节点度的平均场理论方法<font color="#ff8000">Degree-based mean-field approach</font> （简称为DBMF方法）扩展到具有任意度分布的反应扩散系统网络中。为简单起见，我们首先考虑在一个简化的系统上应用DBMF方法，这样具有简单网络结构的系统可以描述为：系统的粒子之间没有相互作用或反应，粒子只会在网络上扩散，并且网络可以具有任意的拓扑形式。以这样的描述构建这个系统，我们如果以度<math> k </math>作为量化指标定义分类的话，可以将系统简单表示为：
\begin{equation}
\begin{equation}
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:
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>
\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)
\begin{equation}
\begin{equation}
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
其中$\bar{S}(t)=\sum_k P(k)S_k$。
其中$\bar{S}(t)=\sum_k P(k)S_k$。
==参考文献 References ==
==参考文献 References==
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