理论分析方法

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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.

在传播研究中，已有许多理论方法被用于描述网络上的传播动力学，用于分析和预测等。下面以SIR模型和SIS模型为例介绍相关理论。值得一提的是，这些方法也可以推广到其他流行病模型。

SIR模型

在均匀混合的人群中，SIR模型的传播演化过程可以用如下的平均场理论表示： \begin{equation} \left\{ \begin{aligned} &\frac{ds(t)}{dt}=-\beta s(t)i(t),\\ &\frac{di(t)}{dt}=\beta s(t)i(t)-\gamma i(t),\\ &\frac{dr(t)}{dt}=\gamma i(t). \end{aligned} \right. \end{equation} We focus our study on a network $G = (V, E)$ with the average degree of $\left<k\right>$. The transmission probability between I state individuals and S state individuals is $\beta$, and the recovery rate of the I state individuals is $\gamma$.

这是没有考虑网络结构的简化的理论。其实，这些模型还可以通过考虑网络的结构来进行更为精确地理论分析。这里我们以平均度为$\left<k\right>$的网络$G=(V,E)$来介绍考虑了网络结构后的相关理论。其中，S状态个体被I状态个体感染的速率为$\beta$，I状态个体的恢复速率为$\gamma$。

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 [1]. Thus the changes of s(t), i(t) and r(t) are expressed by

以$s(t)$，$i(t)$和$r(t)$表示为易感，感染和恢复个体的比例，因此，我们有$s(t)+i(t)+r(t)=1$。同质平均场理论假设每个个体平均与$\left<k\right>$个邻居相连。 因此$s(t)$，$i(t)$和$r(t)$的演化过程可以表示为 \begin{equation} \left\{ \begin{aligned} &\frac{ds(t)}{dt}=-\left<k\right>\beta s(t)i(t),\\ &\frac{di(t)}{dt}=\left<k\right>\beta s(t)i(t)-\gamma i(t),\\ &\frac{dr(t)}{dt}=\gamma i(t). \end{aligned} \right. \end{equation} From the equations of homogeneous mean field approach, we know that this approach considers the degree of the network to some extent compared to the classical approach in a well-mixed population. The threshold value of this approach is also different, which is $R_0=\left<k\right>\beta/\gamma$ ($R_0$ is the basic reproductive number) [2].

在同质平均场理论下计算得到的阈值为$R_0=\left<k\right>\beta/\gamma$（$R_0$为基本生殖数）。

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 [3][4]. 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

同质平均场理论适用于网络的度分布比较均匀的情况，当网络中中每个节点的度值相差悬殊，即为异质网络时，这个理论就不再适用。异质平均场理论的提出就是为了解决同质平均场理论的这项缺点。其中，根据节点度可以将个体划分为不同的类别，即$s_k(t)$，$i_k(t)$和$r_k(t)$，分别代表度为$k$的易感，感染和恢复态个体的比例。因此，异质平均场理论可以由如下演化方程表示 \begin{equation} \label{eq:sir_HMF} \left\{ \begin{aligned} &\frac{ds_k(t)}{dt}=-k\beta s_k(t)\theta_k(t),\\ &\frac{di_k(t)}{dt}=k\beta s_k(t)\theta_k(t)-\gamma i_k(t),\\ &\frac{dr_k(t)}{dt}=\gamma i_k(t). \end{aligned} \right. \end{equation} where $\theta_k(t)$ is the probability that a random selected link is pointing to an infected individual. The specific form of $\theta_k(t)$ in terms of uncorrelated and correlated network is [5],

其中$\theta_k(t)$指的是S态节点随机选择的一条边指向感染个体的概率。无关联网络的$\theta_k(t)$的具体形式为， \begin{equation} \theta_k(t)=\sum_{k'}(k'-1)p(k')i_{k'}(t)/\left<k\right> \end{equation} 若为有关联网络，则为[6] \begin{equation} \theta_k(t)=\sum_{k'}i_{k'}(t)((k'-1)/k')p(k'|k) \end{equation} The threshold value of this model can also be calculated. When G is an uncorrelated network, the threshold is $\lambda^{ucr}=\frac{\left<k\right>}{\left<k^2\right>-\left<k\right>}$, where $\left<k^2\right>$ is the second moment of the node degree distribution. When G is a correlated network, the threshold is $\lambda^{cr}=\frac{1}{\overline{\Lambda}_m}$, where $\overline{\Lambda}_m$ is the maximum eigenvalue of the connectivity matrix $\overline{C}_{kk'}=\beta(k(k'-1)/k')p(k'/k)$.

通过异质平均场理论的计算可以得到无关联网络（uncorrelated network）的传播阈值为$\lambda^{ucr}=\frac{\left<k\right>}{\left<k^2\right>-\left<k\right>}$，其中$\left<k^2\right>$为节点度分布的二阶矩。当网络中不同节之间为度度关联（correlated network）时，阈值为$\lambda^{cr}=\frac{1}{\overline{\Lambda}_m}$，其中$\overline{\Lambda}_m$为连通性矩阵$\overline{C}_{kk'}=\beta(k(k'-1)/k')p(k'/k)$的最大特征值。

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 [7][8][9]. 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:

此理论考虑了疾病传播过程中，节点之间的状态的动力学相关性，其中$[X]$代表不同状态类型的个体的期望数量。例如$[S]$表示易感个体数量的期望值，$[SI]$表示连接了易感个体与感染个体的连边的数量期望值，$[SIS]$表示形式为S-I-S的三元组。变量可以通过以下微分方程来描述： \begin{equation} \label{eq:sir_pair} \left\{ \begin{aligned} &\frac{d[S]}{dt}=-\beta[SI],\\ &\frac{d[I]}{dt}=\beta[SI]-\gamma[I],\\ &\frac{d[R]}{dt}=\gamma[I],\\ &\frac{d[SS]}{dt}=-2\beta[SSI],\\ &\frac{d[SR]}{dt}=-\beta[RSI]+\gamma[SI],\\ &\frac{d[IR]}{dt}=\beta[RSI]+\gamma([II]-[IR]),\\ &\frac{d[II]}{dt}=2\beta([ISI]+[SI])-2\gamma[II],\\ &\frac{d[SI]}{dt}=\beta([SSI]-[ISI]-[SI])-\gamma[SI]. \end{aligned} \right. \end{equation} System (11) can be closed at the level of pairs by assuming different distributions of neighbors, for instance, Poisson, binomial or multinomial distribution [10][11]. As triples are considered in this system, clustering effect of the network can also be involved by different pair approximation approaches. On the other hand, system (11) is a homogeneous pair-based approach, which can be generalized to a heterogeneous case. Like the heterogeneous mean field approach, the node degree is considered in the heterogeneous pair-based approach [12]. In addition, based on different closure method, the threshold values can be derived respectively.

通过假设邻居的不同分布（例如，泊松分布，二项式分布或多项式分布），可以在点对级别闭合方程组。 由于在该方程组中考虑了三元组，因此不同的点对近似方法可能涉及网络的集群效应。另一方面，方程组(\ref{eq:sir_pair})是基于同质网络下的结果，是可以推广到异质网络情况的。 类似异质平均场理论，异质点对近似平均场理论也会考虑到网络中节点度的异质性，根据节点度将个体划分为不同的类别。另外，基于不同的闭合方法，即点对近似，可以分别推导出对应方法下的传播阈值形式。

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 [13].

同质和异质平均场理论在某种程度上考虑了网络结构，但是这些理论方法无法表征网络中更具体地一些细节差异，例如无法区分具有相同节点度的个体的差异，并且忽略了个体的中心性。因此，考虑了个体特征的连续时间马尔可夫链SIR理论被提出。

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:

在连续时间马尔可夫链SIR理论中，假设人群共有$N$个个体，每个个体都可以处于三种状态之一（即S，I，R），因此存在$3^N$种可能的网络状态。用$X=\{x_1,x_2,...,x_N\}$表示网络状态，其中$x\in\{S,I,R\}$。$W_X=p(X=\{x_1,x_2,...,x_N\})$是网络处于状态$X$的概率。状态转移矩阵由$Q=(q_{X,Y})_{3^N×3^N}$给出，其中$q_{X,Y}$表示从网络状态X到Y的转移速率。在任何时间$t$处，$W_X(t)$表示网络处于状态$X(\sum_{X\in\{X_1,...,X_{3^N}\}}W_X(t)=1)$的概率，可以得出该概率随时间的变化如下 \begin{equation} \frac{dW^T(t)}{dt}=W^T(t)Q \end{equation} where $W^T(t)$ is the transpose of $W(t)$. Given an initial value $W^T(0)$, the solution of Eq. (12) is obtained

其中$W^T(t)$是$W(t)$的转置。给定初始值$W^T(0)$，则上式方程的解可以表示为 \begin{equation} W^T(t)=W^T(0)e^{Qt} \end{equation} According to Eq. (13), the solution complexity of continuous time Markov chain SIR model is exponential $O(3^N)$. Therefore, an individual-based approach is given to decrease the complexity to $O(N)$. The $Q_{3^N×3^N}$ matrix is decomposed to $N$ infinitesimal matrices, in which each matrix is within three states [14][15]. By using the effective average infection rate instead of random infection rate, the state change of each individual $v$ is

根据上式可以得出连续时间马尔可夫链SIR理论的求解复杂度是指数的$O(3^N)$。因此，给出了一种基于个体的理论方法来降低复杂度到$O(N)$，矩阵$Q_{3^N×3^N}$被分解为$N$个无穷小矩阵，其中每个矩阵都处于三个状态之内。通过使用有效平均感染率（effective average infection rate）代替随机感染率（random infection rate），每个个体$v$的状态变化可以表示为 \begin{equation} \label{eq:sir_i} \left\{ \begin{aligned} &\frac{dS_v}{dt}=-S_v(t)\beta\sum_{z\in{V}}a_{v,z}I_z(t),\\ &\frac{dI_v}{dt}=S_v(t)\beta\sum_{z\in{V}}a_{v,z}I_z(t)-\gamma I_v(t),\\ &\frac{dR_v}{dt}=\gamma I_v(t).\\ \end{aligned} \right. \end{equation} where $A=(a_{v,z})_{N×N}$ is the adjacent matrix of the network. The threshold value for this model is $R_0=1/\lambda_{max,A}$, where $\lambda_{max,A}$ is the maximum eigenvalue of $A$.

其中$A=(a_{v,z})_{N×N}$是网络的邻接矩阵。此模型的传播阈值为$R_0=1/\lambda_{max,A}$，其中$\lambda_{max,A}$是矩阵$A$的最大本征值。

SIS模型

这里，我们同样以平均度为$\left<k\right>$的网络$G=(V,E)$来简要介绍考虑了网络结构后的其中一些理论。其中，S状态个体被I状态个体感染的速率为$\beta$，I状态个体恢复为S态的速率为$\gamma$。

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}

其中$\theta_k(t)=\sum_{k'}k'p(k')i_{k'}(t)/\left<k\right>$。可以计算得到传播阈值为 \begin{equation} \lambda_c=\frac{\left<k\right>}{\left<k^2\right>} \end{equation} 其中$\left<k\right>$和$\left<k^2\right>$分别表示节点度分布的一阶矩和二阶矩。在均匀网络（如ER网）上，阈值可以计算得到为$\lambda_c=1/(\left<k\right>+1)$。

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。

参考文献 References

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  2. ↑ Barrat A, Barthelemy M, Vespignani A. Dynamical processes on complex networks[M]. Cambridge university press, 2008.
  3. ↑ Moreno Y, Pastor-Satorras R, Vespignani A. Epidemic outbreaks in complex heterogeneous networks[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2002, 26(4): 521-529.
  4. ↑ Yang R, Wang B H, Ren J, et al. Epidemic spreading on heterogeneous networks with identical infectivity[J]. Physics Letters A, 2007, 364(3-4): 189-193.
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  7. ↑ Morris A J. Representing spatial interactions in simple ecological models[D]. University of Warwick, 1997.
  8. ↑ Keeling M J. The effects of local spatial structure on epidemiological invasions[J]. Proceedings of the Royal Society of London. Series B: Biological Sciences, 1999, 266(1421): 859-867.
  9. ↑ House T, Keeling M J. Insights from unifying modern approximations to infections on networks[J]. Journal of The Royal Society Interface, 2011, 8(54): 67-73.
  10. ↑ Morris A J. Representing spatial interactions in simple ecological models[D]. University of Warwick, 1997.
  11. ↑ Keeling M J. The effects of local spatial structure on epidemiological invasions[J]. Proceedings of the Royal Society of London. Series B: Biological Sciences, 1999, 266(1421): 859-867.
  12. ↑ House T, Keeling M J. Insights from unifying modern approximations to infections on networks[J]. Journal of The Royal Society Interface, 2011, 8(54): 67-73.
  13. ↑ Youssef M, Scoglio C. An individual-based approach to SIR epidemics in contact networks[J]. Journal of theoretical biology, 2011, 283(1): 136-144.
  14. ↑ Youssef M, Scoglio C. An individual-based approach to SIR epidemics in contact networks[J]. Journal of theoretical biology, 2011, 283(1): 136-144.
  15. ↑ Van Mieghem P, Omic J, Kooij R. Virus spread in networks[J]. IEEE/ACM Transactions On Networking, 2008, 17(1): 1-14.



此页参考来源： Zhang Z K, Liu C, Zhan X X, et al. Dynamics of information diffusion and its applications on complex networks[J]. Physics Reports, 2016, 651: 1-34.

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