# 理论分析方法

In the previous section, epidemic models are studied in a well-mixed population. These models can also be analyzed by other mathematical approaches in a network-structured demographic. In this section, we illustrate these approaches in detail in the case of SIR model. It should be emphasized that these approaches can also be generalized to other epidemic models.

## SIR模型

### 同质平均场理论

Denoting s(t), i(t) and r(t) as the fraction of susceptible, infected and recovered individuals in SIR model, thus we have s(t) + i(t) + r(t) = 1. In the homogeneous mean field approach, we assume that each individual is connected to ⟨k⟩ neighbors in average . Thus the changes of s(t), i(t) and r(t) are expressed by

### 异质平均场理论

The homogeneous mean field approach considers each individual can infect a constant number of neighbors, which is not suitable for the heterogeneous networks. A heterogeneous mean field approach is proposed to overcome this drawback . In this approach, individuals are divided into different categories based on node degrees, i.e., $s_k(t)$，$i_k(t)$ and $r_k(t)$ represent the fraction of susceptible, infected and recovered individuals with degree $k$ respectively. Consequently, the heterogeneous mean field approach can be governed by

### 基于点对的平均场理论

In order to study the epidemic models on networks, approaches consider the evolution of the edges are proposed, known as the pair-based approaches . In this model, $[X]$ represents the expected number of individuals in different types. For example, $[S]$ expresses the expected number of susceptible individuals, $[SI]$ denotes the expected number of links connecting a susceptible individual to an infected individual, $[SIS]$ represents a triple with form of S−I−S. The changes of the variables can be described by the following differential equations:

### 基于个体的平均场理论

The approaches introduced above have considered the network structure to some extent. However, these approaches are aggregate representations of the network, as they cannot distinguish among individuals with the same node degrees and ignores the central properties of the individuals. Thus an individual-based approach is proposed in terms of continuous time Markov chain SIR model .

We first give the definition of continuous time Markov chain SIR model. Suppose there are $N$ individuals in the population, as each individual can be in one of the three states (i.e., S, I, R), thus there are $3^N$ possible network states. Denote $X=\{x_1,x_2,...,x_N\}$ as the network state, where $x\in\{S,I,R\}$。$W_X=p(X=\{x_1,x_2,...,x_N\})$ is the probability that the network is in state $X$. The state transition matrix is given by $Q=(q_{X,Y})_{3^N×3^N}$, in which $q_{X,Y}$ represents the transition rate from network state X to Y. At any time $t$, $W_X(t)$ represents the probability of the network is in state $X(\sum_{X\in\{X_1,...,X_{3^N}\}}W_X(t)=1)$), thus we can obtain the change of this probability with time as follows:

## SIS模型

### 同质平均场理论

\begin{equation} \left\{ \begin{aligned} &\frac{ds(t)}{dt}=-\left<k\right>\beta s(t)i(t)+\gamma i(t),\\ &\frac{di(t)}{dt}=\left<k\right>\beta s(t)i(t)-\gamma i(t).\\ \end{aligned} \right. \end{equation}

### 异质平均场理论

\begin{equation} \left\{ \begin{aligned} &\frac{ds_k(t)}{dt}=-k\beta s_k(t)\theta_k(t)+\gamma i_k(t),\\ &\frac{di_k(t)}{dt}=k\beta s_k(t)\theta_k(t)-\gamma i_k(t).\\ \end{aligned} \right. \end{equation}

### 基于个体的平均场理论

\begin{equation} \label{eq:sis_i} \left\{ \begin{aligned} &\frac{dS_v}{dt}=-S_v(t)\beta\sum_{z\in{V}}a_{v,z}I_z(t)+\gamma I_v(t),\\ &\frac{dI_v}{dt}=S_v(t)\beta\sum_{z\in{V}}a_{v,z}I_z(t)-\gamma I_v(t).\\ \end{aligned} \right. \end{equation} 可以得到，传播阈值为邻接矩阵的最大特征值的倒数。

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