数学物理学报, 2026, 46(5): 2003-2016

异质网络上随机 SIS 传染病模型的拟平稳分布与灭绝时间分析

靖晓洁,1, 刘桂荣,2,*, 靳祯,3

1 中北大学数学学院, 太原 030051

2 山西大学数学与统计学院, 太原 030006

3 山西大学复杂系统研究所, 太原 030006

A Stochastic SIS Epidemic Model in Heterogeneous Networks: Quasi-Stationarity and Extinction

Jing Xiaojie,1, Liu Guirong,2,*, Jin Zhen,3

1 School of Mathematics, North University of China, Taiyuan 030051

2 School of Mathematics and Statistics, Shanxi University, Taiyuan 030006

3 Complex Systems Research Center, Shanxi University, Taiyuan 030006

通讯作者: * 刘桂荣, E-mail:lgr5791@sxu.edu.cn

收稿日期: 2025-02-21   修回日期: 2026-04-27  

基金资助: 国家自然科学基金(12501698)
国家自然科学基金(12371494)
国家自然科学基金(12231012)
山西省基础研究计划(202403021222164)
山西省重点研发计划(202202020101010)

Received: 2025-02-21   Revised: 2026-04-27  

Fund supported: NSFC(12501698)
NSFC(12371494)
NSFC(12231012)
Fundamental Research Program of Shanxi Province of China(202403021222164)
Shanxi Provincial Key Research and Development Project(202202020101010)

作者简介 About authors

靖晓洁,E-mail:jingxiaojie@nuc.edu.cn

靳祯,E-mail:jinzhen@sxu.edu.cn

摘要

该文章将矩封闭方法应用于异质网络上的 SIS 传染病模型. 在正态分布和对数正态分布的假设下, 推导了传播过程的拟平稳分布以及传染病从拟平稳状态开始的平均灭绝时间. 最后采用随机模拟, 检验矩封闭方法的近似效果. 结果表明, 拟平稳分布的近似效果与模拟数据拟合良好, 其中对数正态分布的拟合效果尤为突出.

关键词： 传染病传播; 异质网络; 拟平稳分布; 灭绝时间; 矩封闭

Abstract

In this paper, the moment closure method is applied to the SIS epidemic model on heterogeneous networks. Assumptions of normal and log-normal distributions are made to capture the quasi-stationary behavior and the expected time to extinction of the disease starting from the quasi-stationary state. The approximate performance of the moment closure method is then tested by stochastic simulations. The results show that the approximation is in good agreement with the simulations for the quasi-stationary distribution, particularly for the log-normal distribution.

Keywords： Epidemic spreading; Heterogeneous network; Quasi-stationary distribution; Time to extinction; Moment closure

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本文引用格式

靖晓洁, 刘桂荣, 靳祯. 异质网络上随机 SIS 传染病模型的拟平稳分布与灭绝时间分析[J]. 数学物理学报, 2026, 46(5): 2003-2016

Jing Xiaojie, Liu Guirong, Jin Zhen. A Stochastic SIS Epidemic Model in Heterogeneous Networks: Quasi-Stationarity and Extinction[J]. Acta Mathematica Scientia, 2026, 46(5): 2003-2016

1 引言

近年来, 基于网络传染病模型的随机动力学研究受到关注 [1-4]. Dangerfield 等 [1]利用扩散近似, 研究了空间结构和随机性在 SIS 型传染病传播中的影响. Graham 和 House[2] 探究了异质网络上 SIR 型传染病在早期指数增长阶段的渐近方差. Ball 等 [3]给出了异质网络上 SIR 型传染病的最终规模. Achterberg 和 Van Mieghem[4] 推导了广义自适应网络上马尔可夫 SIS 模型的平均场近似. 然而, 关于一般异质网络上传播过程的拟平稳分布的研究仍为有限.

研究拟平稳分布常用的方法之一是矩封闭近似 [5–9], 即通过系统状态分布的矩来研究传染病传播中的变异性[10,11]. 一般而言, 矩方程是无限维的, 无法直接处理. 因此, 需要给定分布, 利用该分布矩之间的关系对方程组进行截断, 以获得低阶矩方程. 现有研究在均匀混合层面上评估了给定分布在矩封闭方法上的近似效果, 忽略了传播过程中个体异质性. 本文将该方法推广到异质网络上, 假设传播过程处于阈值之上 (即 $R_{0}>1$), 研究传染病的随机动力学行为.

针对异质网络上 SIS 过程, 本文在第 2 节推导了其 Kolmogorov 微分方程, 并给出了母函数的偏微分方程, 进而得到了过程状态各阶矩的微分方程. 在第 3 节中, 基于正态和对数正态假设, 推导了两组微分方程系统, 并给出了该过程的拟平稳分布和灭绝时间. 在第 4 节中, 借助数值模拟验证了近似的有效性. 最后, 第 5 节给出了结论.

2 SIS网络模型

假设网络为 $N$ 个节点的配置网络模型. 节点或者是易感状态(S), 或者是染病状态(I). $\beta$ 是传染率, $\gamma$ 是恢复率.

2.1 随机模型

设 $I_{k}(t)$ 和 $S_{k}(t)$ 分别为 $t$ 时刻度为 $k$ 的易感节点和染病节点数, 满足

$\begin{equation*} I_{k}(t)+S_{k}(t)=N_{k}, \end{equation*}$

$N_{k}$ 为度为 $k$ 的节点总数. 令 $I(t)=(I_{1}(t),I_{2}(t),\cdots,I_{K}(t))$, $K$ 为最大度. 设 $\left\{I(t)\right\}$ 为连续时间的 Markov 链, 状态空间为

$\begin{equation*} C=\left\{\left(i_{1},\cdots,i_{k},\cdots,i_{K}\right):\,i_{k}\in\{0,1,\cdots,N_{k}\},\,k=1,\cdots,K\right\}. \end{equation*}$

联合概率分布为

$\begin{equation*} p_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)=\mathrm{P}\{I_{1}(t)=i_{1},\cdots,I_{k}(t)=i_{k},\cdots,I_{K}(t)=i_{K}\}. \end{equation*}$

当 $\left(i_{1},\cdots,i_{k},\cdots,i_{K}\right)\notin C$ 时, $p_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)=0$. 对 $k=1,\cdots,K$, 表 1 列出了度为 $k$ 的节点的状态转移速率, 其中 $\mu_{D}$ 为网络平均度.

表1   转移速率

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根据文献[12], 前向 Kolmogorov 微分方程如下

$\begin{equation}\label{EQ2.1} \begin{aligned} \frac{ p_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)}{\text{d}t}=&\sum_{k=1}^{K}\biggl\{\beta k[N_{k}-(i_{k}-1)]\frac{\sum_{l\neq k}li_{l}+k(i_{k}-1)}{\mu_{D}N}p_{i_{1},\cdots,i_{k}-1,\cdots,i_{K}}(t)\\ &-\beta k(N_{k}-i_{k})\frac{\sum_{l}li_{l}}{\mu_{D}N}p_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)\\ &+\gamma(i_{k}+1)p_{i_{1},\cdots,i_{k}+1,\cdots,i_{K}}(t)-\gamma i_{k}p_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)\biggr\}. \end{aligned} \end{equation} $

下面分析传染病灭绝前的演化过程. 令 $I(t)=\sum_{k=1}^{K}I_{k}(t)$. 记条件概率为 $q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)$,

$\begin{equation}\label{EQ2.2} \begin{aligned} q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)={}&\left\{I_{1}(t)=i_{1},\cdots,I_{k}(t)=i_{k},\cdots,I_{K}(t)=i_{K}\mid I(t)>0\right\}\\ ={}&\frac{p_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)}{1-p_{0,\cdots,0,\cdots,0}(t)}, \end{aligned} \end{equation}$

$(i_{1},\cdots,i_{k},\cdots,i_{K})\in C\setminus \big\{(0,\cdots,0,\cdots,0)\big\}$. 基于 (2.1) 式,

$\begin{equation}\label{EQ2.3} \begin{aligned} \frac{p_{0,\cdots,0,\cdots,0}(t)}{\text{d}t}=\sum_{k=1}^{K}\gamma p_{e_{k}}(t), \end{aligned} \end{equation}$

其中 $p_{e_{k}}(t)=\mathrm{P}\{I_{1}(t)=0,\cdots,I_{k}(t)=1,\cdots,I_{K}(t)=0\}.$

对 (2.2) 式中的 $q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)$ 求导并使用 (2.3) 式, 得

$\begin{equation*} \begin{aligned} \frac{q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)}{t} =\frac{\dot{p}_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)}{1-p_{0,\cdots,0,\cdots,0}(t)} +\gamma\frac{p_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)}{1-p_{0,\cdots,0,\cdots,0}(t)}\sum_{k=1}^{K}q_{e_{k}}(t), \end{aligned} \end{equation*}$

其中 $q_{e_{k}}(t)=\mathrm{P}\left\{I_{1}(t)=0,\cdots,I_{k}(t)=1,\cdots,I_{K}(t)=0\mid I(t)>0\right\}.$ 使用 (2.1) 式, 得到下列系统

$\begin{equation}\label{EQ2.4} \begin{aligned} \frac{q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)}{\text{d}t}=&\sum_{k=1}^{K}\biggl\{\beta k[N_{k}-(i_{k}-1)] \frac{\sum_{l\neq k}li_{l}+k(i_{k}-1)}{\mu_{D}N}q_{i_{1},\cdots,i_{k}-1,\cdots,i_{K}}(t)\\ &-\beta k(N_{k}-i_{k})\frac{\sum_{l}li_{l}}{\mu_{D}N}q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t) +\gamma(i_{k}+1)q_{i_{1},\cdots,i_{k}+1,\cdots,i_{K}}(t)\\ &-\gamma i_{k}q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)\biggr\} +\gamma q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)\sum_{k=1}^{K}q_{e_{k}}(t). \end{aligned} \end{equation}$

2.2 矩方程

下面给出 $I_{k}\,(k=1,\cdots,K)$ 的均值和方差以及 $I_{k}\,(k=1,\cdots,K)$ 间协方差的变化率. 令 $M(u_{1},\cdots,u_{k},\cdots,u_{K},t)$ 为矩母函数, 定义为

$\begin{equation*} \begin{aligned} M(u_{1},\cdots,u_{k},\cdots,u_{K},t)=\sum_{i_{K}=0}^{N_{K}}\cdots\sum_{i_{k}=0}^{N_{k}}\cdots\sum_{i_{1}=0}^{N_{1}} q_{i_{1},\cdots,i_{k},\cdots,i_{K}}(t)e^{i_{1}u_{1}+\cdots+i_{k}u_{k}+\cdots+i_{K}u_{K}}, \end{aligned} \end{equation*}$

其中 $q_{0,\cdots,0,\cdots,0}(t)=0$. 使用 (2.4) 式, 得

$\begin{equation*} \begin{aligned} \frac{\partial M}{\partial t}=&\sum_{k=1}^{K}\frac{\beta kN_{k}}{\mu_{D}N}\left(e^{u_{k}}-1\right)\sum_{l=1}^{K}l\frac{\partial M}{\partial u_{l}} -\sum_{k=1}^{K}\frac{\beta k}{\mu_{D}N}\left(e^{u_{k}}-1\right)\sum_{l=1}^{K}l\frac{\partial^{2} M}{\partial u_{k}\partial u_{l}}\\[6pt] &+\sum_{k=1}^{K}\gamma\left(e^{-u_{k}}-1\right)\frac{\partial M}{\partial u_{k}}+\gamma(M-1)\sum_{k=1}^{K}q_{e_{k}}. \end{aligned} \end{equation*}$

累积母函数 $G(u_{1},\cdots,u_{k},\cdots,u_{K},t)=\ln M(u_{1},\cdots,u_{k},\cdots,u_{K},t)$, 故

$\begin{equation}\label{EQ2.8} \begin{aligned} \frac{\partial G}{\partial t}=&\sum_{k=1}^{K}\frac{\beta kN_{k}}{\mu_{D}N}(e^{u_{k}}-1)\sum_{l=1}^{K}l\frac{\partial G}{\partial u_{l}} -\sum_{k=1}^{K}\frac{\beta k}{\mu_{D}N}(e^{u_{k}}-1)\sum_{l=1}^{K}l\left(\frac{\partial^{2} G}{\partial u_{k}\partial u_{l}} +\frac{\partial G}{\partial u_{k}}\frac{\partial G}{\partial u_{l}}\right)\\[6pt] &+\sum_{k=1}^{K}\gamma(e^{-u_{k}}-1)\frac{\partial G}{\partial u_{k}} +\gamma(1-e^{-G})\sum_{k=1}^{K}q_{e_{k}}. \end{aligned} \end{equation}$

状态变量的各阶矩与累积母函数间的关系如下

$\begin{equation*} \begin{aligned} \qquad\qquad E(I_{m})={}&\frac{\partial G(u_{1},\cdots,u_{k},\cdots,u_{K},t)}{\partial u_{m}}\bigg|_{u_{1}=0,\cdots,u_{k}=0,\cdots,u_{K}=0}, \end{aligned} \end{equation*}$
$\begin{equation*} \begin{aligned} \text{var}(I_{m})={}&\frac{\partial^{2} G(u_{1},\cdots,u_{k},\cdots,u_{K},t)}{\partial u_{m}^{2}}\bigg|_{u_{1}=0,\cdots,u_{k}=0,\cdots,u_{K}=0},\\[8pt] \text{Cov}(I_{m},I_{n})={}&\frac{\partial^{2} G(u_{1},\cdots,u_{k},\cdots,u_{K},t)} {\partial u_{m}\partial u_{n}}\bigg|_{u_{1}=0,\cdots,u_{k}=0,\cdots,u_{K}=0},\\[8pt] T_{I_{l}I_{m}I_{n}}={}&\frac{\partial^{3} G(u_{1},\cdots,u_{k},\cdots,u_{K},t)} {\partial u_{l}\partial u_{m}\partial u_{n}}\bigg|_{u_{1}=0,\cdots,u_{k}=0,\cdots,u_{K}=0},\\[8pt] T_{I_{m}I_{n}^{2}}={}&\frac{\partial^{3} G(u_{1},\cdots,u_{k},\cdots,u_{K},t)} {\partial u_{m}\partial u_{n}^{2}}\bigg|_{u_{1}=0,\cdots,u_{k}=0,\cdots,u_{K}=0}, \end{aligned} \end{equation*}$

$l,m,n=1,\cdots,K$, 其中 $T_{I_{l}I_{m}I_{n}}$ 和 $T_{I_{m}I_{n}^{2}}$ 为三阶中心矩,

$\begin{equation*} \begin{aligned} T_{I_{l}I_{m}I_{n}}={}&E\left\{[I_{l}-E(I_{l})][I_{m}-E(I_{m})][I_{n}-E(I_{n})]\right\},\\[6pt] T_{I_{m}I_{n}^{2}}={}&E\left\{[I_{m}-E(I_{m})][I_{n}-E(I_{n})]^{2}\right\}. \end{aligned} \end{equation*}$

因此, $I_{m}\,(m=1,\cdots,K)$ 的均值和方差以及 $I_{m}\,(m=1,\cdots,K)$ 间协方差的微分方程如下

$\begin{equation}\label{EQ2.9} \begin{aligned} \frac{\text{d} E(I_{m})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}lE(I_{l}) -\frac{\beta m}{\mu_{D}N}\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})\\[4pt] &-\gamma E(I_{m})+\gamma E(I_{m})\sum_{l=1}^{K}q_{e_{l}},\\[4pt] \frac{\text{dVar}(I_{m})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}lE(I_{l}) +\frac{2\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})\\[4pt] &-\frac{\beta m}{\mu_{D}N}\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m}) -\frac{2\beta m}{\mu_{D}N}\text{Var}(I_{m})\sum_{l=1}^{K}lE(I_{l}) -\frac{2\beta m}{\mu_{D}N}\sum_{l=1}^{K}lT_{I_{l}I_{m}^{2}}\\[4pt] &+\gamma E(I_{m})-2\gamma\text{Var}(I_{m}) -\gamma E(I_{m})^{2}\sum_{l=1}^{K}q_{e_{l}} +\gamma\text{Var}(I_{m})\sum_{l=1}^{K}q_{e_{l}},\\[4pt] \frac{\text{dCov}(I_{m},I_{n})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{n}) +\frac{\beta n}{\mu_{D}N}\left[N_{n}-E(I_{n})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})\qquad\qquad\qquad\qquad\quad\\[4pt] &-\frac{\beta(m+n)}{\mu_{D}N}\text{Cov}(I_{m},I_{n})\sum_{l=1}^{K}lE(I_{l}) -\frac{\beta(m+n)}{\mu_{D}N}\sum_{l=1}^{K}lT_{I_{l}I_{m}I_{n}}\\[4pt] &-2\gamma \text{Cov}(I_{m},I_{n})-\gamma E(I_{m})E(I_{n})\sum_{l=1}^{K}q_{e_{l}} +\gamma \text{Cov}(I_{m},I_{n})\sum_{l=1}^{K}q_{e_{l}}, \end{aligned} \end{equation}$

$m,n=1,\cdots,K$.

根据文献[13,14], 可知 $R_{0}=\beta(\sigma^{2}_{D}+\mu^{2}_{D})/(\gamma\mu_{D})$, 其中 $\sigma^{2}_{D}$ 为度分布的方差. 再根据文献[5,15], 当 $R_{0}>1$ 时, 拟平稳分布 $q_{e_{l}}\,(l=1,\cdots,K)$ 较小. 因此, 可以通过忽略每个方程中的 $q_{e_{l}}(t)\,(l=1,\cdots,K)$ 项, 推导出拟平稳状态下各阶矩的近似表达式. 由此, 系统 (2.6) 化简为

$\begin{equation}\label{EQ2.10} \begin{aligned} \frac{\text{d}E(I_{m})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}lE(I_{l}) -\frac{\beta m}{\mu_{D}N}\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})-\gamma E(I_{m}),\\[4pt] \frac{\text{dVar}(I_{m})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}lE(I_{l}) +\frac{2\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})\\[4pt] &-\frac{\beta m}{\mu_{D}N}\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m}) -\frac{2\beta m}{\mu_{D}N}\text{Var}(I_{m})\sum_{l=1}^{K}lE(I_{l})\\[4pt] &-\frac{2\beta m}{\mu_{D}N}\sum_{l=1}^{K}lT_{I_{l}I_{m}^{2}} +\gamma E(I_{m})-2\gamma \text{Var}(I_{m}),\\[4pt] \frac{\text{dCov}(I_{m},I_{n})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{n}) +\frac{\beta n}{\mu_{D}N}\left[N_{n}-E(I_{n})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})\\[4pt] &-\frac{\beta(m+n)}{\mu_{D}N}\text{Cov}(I_{m},I_{n})\sum_{l=1}^{K}lE(I_{l}) -\frac{\beta(m+n)}{\mu_{D}N}\sum_{l=1}^{K}lT_{I_{l}I_{m}I_{n}}\\[4pt] &-2\gamma \text{Cov}(I_{m},I_{n}), \end{aligned} \end{equation}$

$m,n=1,\cdots,K$.

3 拟平稳分布和灭绝时间

由系统 (2.7) 可知, 低阶矩方程包含高阶矩. 为了封闭系统, 使用一阶矩和二阶矩近似三阶矩. 首先, 假设 SIS 传播过程服从多元正态分布, 即 $T_{I_{l}I_{m}I_{n}}=T_{I_{l}I_{m}^{2}}=0~(l,m,n=1,\cdots,K)$. 因此, 系统 (2.7) 为

$\begin{equation}\label{EQ3.1} \begin{aligned} \frac{\text{d}E(I_{m})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}lE(I_{l}) -\frac{\beta m}{\mu_{D}N}\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})-\gamma E(I_{m}),\\ \frac{\text{dVar}(I_{m})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}lE(I_{l}) +\frac{2\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})\\ &-\frac{\beta m}{\mu_{D}N}\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m}) -\frac{2\beta m}{\mu_{D}N}\text{Var}(I_{m})\sum_{l=1}^{K}lE(I_{l}) +\gamma E(I_{m})-2\gamma \text{Var}(I_{m}),\\ \frac{\text{dCov}(I_{m},I_{n})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}\left[N_{m}-E(I_{m})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{n}) +\frac{\beta n}{\mu_{D}N}\left[N_{n}-E(I_{n})\right]\sum_{l=1}^{K}l\text{Cov}(I_{l},I_{m})\\ &-\frac{\beta(m+n)}{\mu_{D}N}\text{Cov}(I_{m},I_{n})\sum_{l=1}^{K}lE(I_{l}) -2\gamma \text{Cov}(I_{m},I_{n}), \end{aligned} \end{equation}$

$m,n=1,\cdots,K$.

其次, 假设 SIS 传播过程服从多元对数正态分布. 在正态分布下, 各阶矩通过加法联系起来, 即加法矩. 而在对数正态假设下, 这些关系遵循乘法规则. 因此, 采用乘法矩而非加法矩的形式, 矩方程可以用 $\hat{V}_{I_{m}}\,(m=1,\cdots,K)$ 和 $\xi_{I_{m},I_{n}}\,(m,n=1,\cdots,K)$ 来表示, 其中

$\begin{equation*} \begin{aligned} E(I_{m}^{2})={}&\hat{V}_{I_{m}}E(I_{m})^{2},\\[4pt] E(I_{m}I_{n})={}&\xi_{I_{m},I_{n}}E(I_{m})E(I_{n}),\\[4pt] E(I_{m}I_{n}^{2})={}&\hat{V}_{I_{n}}\xi_{I_{m},I_{n}}^{2}E(I_{m})E(I_{n})^{2},\\[4pt] E(I_{l}I_{m}I_{n})={}&\xi_{I_{m},I_{n}}\xi_{I_{l},I_{m}}\xi_{I_{l},I_{n}}E(I_{l})E(I_{m})E(I_{n}), \end{aligned} \end{equation*}$

$l,m,n=1,\cdots,K$. 使用文献[16,17], $\hat{T}_{I_{l}I_{m}I_{n}}=\hat{T}_{I_{l}I_{m}^{2}}=1,\,l,m,n=1,\cdots,K$, 得

$\begin{equation*} \begin{aligned} T_{I_{l}I_{m}I_{n}}={}&\xi_{I_{m},I_{n}}\xi_{I_{l},I_{m}}\xi_{I_{l},I_{n}}E(I_{l})E(I_{m})E(I_{n}) -\xi_{I_{m},I_{n}}E(I_{l})E(I_{m})E(I_{n})-\xi_{I_{l},I_{m}}E(I_{l})E(I_{m})E(I_{n})\\[4pt] &-\xi_{I_{l},I_{n}}E(I_{l})E(I_{m})E(I_{n})+2E(I_{l})E(I_{m})E(I_{n}),\\[4pt] T_{I_{m}I_{n}^{2}}={}&\hat{V}_{I_{n}}\xi_{I_{m},I_{n}}^{2}E(I_{m})E(I_{n})^{2} -2\xi_{I_{m},I_{n}}E(I_{m})E(I_{n})^{2}-\hat{V}_{I_{n}}E(I_{m})E(I_{n})^{2}+2E(I_{m})E(I_{n})^{2}. \end{aligned} \end{equation*}$

因此, 系统 (2.7) 为

$\begin{equation}\label{EQ3.2} \begin{aligned} \frac{\text{d}E(I_{m})}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}N_{m}\sum_{l=1}^{K}lE(I_{l}) -\frac{\beta m}{\mu_{D}N}E(I_{m})\sum_{l=1}^{K}l\xi_{I_{l},I_{m}}E(I_{l})-\gamma E(I_{m}),\\ E(I_{m})^{2}\frac{\text{d}\hat{V}_{I_{m}}}{\text{d}t}={}&\frac{\beta m}{\mu_{D}N}N_{m}\big[1-2\hat{V}_{I_{m}}E(I_{m})\big]\sum_{l=1}^{K}lE(I_{l})\\ &+\frac{\beta m}{\mu_{D}N}E(I_{m})\big[2N_{m}+2\hat{V}_{I_{m}}E(I_{m})-1\big]\sum_{l=1}^{K}l\xi_{I_{l},I_{m}}E(I_{l})\\ &-\frac{2\beta m}{\mu_{D}N}\hat{V}_{I_{m}}E(I_{m})^{2}\sum_{l=1}^{K}l\xi_{I_{l},I_{m}}^{2}E(I_{l})+\gamma E(I_{m}),\\ E(I_{m})E(I_{n})\frac{\xi_{I_{m},I_{n}}}{\text{d}t}={}&\frac{\beta}{\mu_{D}N}\big[mN_{m}+n\xi_{I_{m},I_{n}}E(I_{m})\big]E(I_{n}) \sum_{l=1}^{K}l\xi_{I_{l},I_{n}}E(I_{l})\\ &+\frac{\beta}{\mu_{D}N}\big[nN_{n}+m\xi_{I_{m},I_{n}}E(I_{n})\big] E(I_{m})\sum_{l=1}^{K}l\xi_{I_{l},I_{m}}E(I_{l})\\ &-\frac{\beta}{\mu_{D}N}\big[mN_{m}E(I_{n})+nN_{n}E(I_{m})\big]\xi_{I_{m},I_{n}}\sum_{l=1}^{K}lE(I_{l})\\ &-\frac{\beta(m+n)}{\mu_{D}N}\xi_{I_{m},I_{n}}E(I_{m})E(I_{n})\sum_{l=1}^{K}l\xi_{I_{l},I_{m}}\xi_{I_{l},I_{n}}E(I_{l}), \end{aligned} \end{equation}$

$m,n=1,\cdots,K$.

令 $q_{i_{1},\cdots,i_{k},\cdots,i_{K}}$ 为该过程的拟平稳分布, 且 $i=(i_{1},\cdots,i_{k},\cdots,i_{K})^{\top}$, 其中 $\top$ 表示转置. 有如下结论成立.

定理 3.1 假设 SIS 传播过程服从多元正态分布, 则拟平稳分布可以近似地表示为

$\begin{equation}\label{EQ3.3} q_{i_{1},\cdots,i_{k},\cdots,i_{K}}^{\text{Nor}}\approx \frac{f^{\text{Nor}}(i)} {\text{p}\left\{I_{1}>0.5,\cdots,I_{k}>0.5,\cdots,I_{K}>0.5\mid I>0\right\}}, \end{equation}$

其中

$\begin{equation*} f^{\text{Nor}}(i)=f^{\text{Nor}}(i_{1},\cdots,i_{k},\cdots,i_{K}) =\frac{1}{\sqrt{(2\pi)^{K}|\Sigma|}}\exp\left\{-\frac{1}{2}(i-\mu)^{\top}\Sigma^{-1}(i-\mu)\right\}, \end{equation*}$

$\mu$ 和 $\Sigma$ 分别为均值向量和协方差矩阵, $|\Sigma|$ 和 $\Sigma^{-1}$ 分别表示 $\Sigma$ 的行列式值和逆, 并且

$\begin{equation*} \begin{aligned} \text{p}&\left\{I_{1}>0.5,\cdots,I_{k}>0.5,\cdots,I_{K}>0.5\mid I>0\right\}\\ &=1-\frac{1}{\sqrt{(2\pi)^{K}|\Sigma|}}\int_{-\infty}^{0.5}\cdots\int_{-\infty}^{0.5} \exp\left\{-\frac{1}{2}(i-\mu)^{\top}\Sigma^{-1}(i-\mu)\right\}\text{d}i_{1}\cdot \text{d}i_{M}. \end{aligned} \end{equation*}$

证 对于正态近似, $\left\{I(t)\right\}$ 的拟平稳分布为多元正态分布. 由系统 (3.1)可导出拟平稳状态下染病节点的均值、方差及协方差. 因此, 均值向量为

$\begin{equation*} \mu=(\mu_{1},\cdots,\mu_{k},\cdots,\mu_{K})^{\top}, \end{equation*}$

其中

$\begin{equation*} \mu_{k}=E(I_{k}),\,k=1,\cdots,K. \end{equation*}$

协方差矩阵为

$\begin{equation*} \Sigma=\left[ \begin{array}{ccccc} \sigma_{11}~&~\cdots~&~\sigma_{1k}~&~\cdots~&~\sigma_{1K}\\[4pt] \sigma_{21}~&~\cdots~&~\sigma_{2k}~&~\cdots~&~\sigma_{2K}\\[4pt] \vdots~&~~&~~&~~&~\vdots\\[4pt] \sigma_{K1}~&~\cdots~&~\sigma_{Kk}~&~\cdots~&~\sigma_{KK} \end{array}\right], \end{equation*}$

其中, 对于 $i,j=1,\cdots,K$,

$\begin{equation*} \sigma_{ij}= \begin{cases} \text{Var}(I_{i}), &i= j,\\[4pt] \text{Cov}(I_{i},I_{j}),&i\neq j. \end{cases} \end{equation*}$

因此, 概率密度函数为

$f^{\text{Nor}}(i)=f^{\text{Nor}}(i_{1},\cdots,i_{k},\cdots,i_{K}) =\frac{1}{\sqrt{(2\pi)^{K}|\Sigma|}}\exp\left\{-\frac{1}{2}(i-\mu)^{\top}\Sigma^{-1}(i-\mu)\right\}.$

为了保证 $I$ 是正的, 在 $I_{k}=0.5\,(k=1,\cdots,K)$ 处截断分布, 最终结果即为 (3.3) 式.

定理 3.2 假设 SIS 传播过程服从多元对数正态分布, 则拟平稳分布可以近似地表示为

$\begin{equation}\label{EQ3.4} q_{i_{1},\cdots,i_{k},\cdots,i_{K}}^{\text{Log}}\approx \frac{1}{\sqrt{(2\pi)^{K}|\hat{\Sigma}|}\prod_{k=1}^{K}i_{k}} \exp\left\{-\frac{1}{2}\big[\ln(i)-\hat{\mu}\big]^{\top}\hat{\Sigma}^{-1}\big[\ln(i)-\hat{\mu}\big]\right\}, \end{equation}$

其中 $\hat{\mu}$ 为均值向量, $\hat{\Sigma}=(\hat{\sigma}_{ij})$ 为协方差矩阵.

证

对于对数正态近似, $\left\{I(t)\right\}$ 的拟平稳分布为多元对数正态分布. 由系统 (3.2) 可导出拟平稳状态下染病节点的均值、方差及协方差. 因此, 均值向量为 $ \hat{\mu}=(\hat{\mu}_{1},\cdots,\hat{\mu}_{k},\cdots,\hat{\mu}_{K})^{\top}, $ 其中

$ \hat{\mu}_{k}=\ln\left(\frac{E(I_{k})}{\sqrt{\hat{V}_{I_{k}}}}\right),\,k=1,\cdots,K. $

协方差矩阵为 $\hat{\Sigma}=(\hat{\sigma}_{ij})$, 其中

$\begin{equation*} \hat{\sigma}_{ij}= \begin{cases} \ln(\hat{V}_{I_{i}}), &i=j,\\[4pt] \ln\big(\xi(I_{i},I_{j})\big),&i\neq j, \end{cases} \end{equation*}$

$i,j=1,\cdots,K$. 因此, $\left\{I(t)\right\}$ 的拟平稳分布由 (3.4) 式给出.

令 $\tau_{Q}$ 为传染病从拟平稳状态开始的灭绝时间. 关于 $\tau_{Q}$ 的结论如下.

定理 3.3 对于正态分布近似,

$\begin{equation}E(\tau^{\text{Nor}}_{Q})\approx \frac{1}{\gamma\sum_{k=1}^{K}q^{\text{Nor}}_{e_{k}}},\end{equation}$

其中

$\begin{equation}\label{EQ3.6} q^{\text{Nor}}_{e_{k}}\approx\frac{f^{\text{Nor}}(e_{k})}{\mathrm{P}\left\{I_{1}>0.5,\cdots,I_{k}>0.5,\cdots,I_{K}>0.5\mid I>0\right\}},\,k=1,\cdots,K \end{equation}$

和

$\begin{equation*} f^{\text{Nor}}(e_{k})=\frac{1}{\sqrt{(2\pi)^{K}|\Sigma|}}\exp\left\{-\frac{1}{2}(e_{k}-\mu)^{\top}\Sigma^{-1}(e_{k}-\mu)\right\}. \end{equation*}$

证 根据文献[18], $\tau_{Q}$ 服从指数分布, 其参数为 $\gamma\sum_{k=1}^{K}q_{e_{k}}$. 因此, $E(\tau_{Q})$ 为

$\begin{equation}\label{EQ3.7} E(\tau_{Q})=\frac{1}{\gamma\sum_{k=1}^{K}q_{e_{k}}}. \end{equation}$

根据 (3.3) 式, $q^{\text{Nor}}_{e_{k}}$ 的表达式如 (3.6) 式所示, 将其带入 (3.7) 式, 得到 $ E(\tau^{\text{Nor}}_{Q})\approx \frac{1}{\gamma\sum_{k=1}^{K}q^{\text{Nor}}_{e_{k}}}. $ 根据 (3.4) 式, $q^{\text{Log}}_{e_{k}}=0,\,k=1,\cdots,K$. 因此, 对数正态分布不能用来近似 $E(\tau_{Q})$.

4 数值模拟

本节使用蒙特卡洛模拟研究异质网络上的 SIS 传播过程. 当不同状态下不同度的个体的数量趋于稳定时, 可以估计出拟平稳分布. 为了说明理论结果, 采用一个简单网络, 以便快速得到传染病过程的拟平稳分布. 然后, 将平均灭绝时间的精确值与正态分布近似下的结果进行了比较. 最后, 对参数进行了敏感性分析.

4.1 拟平稳分布

给出网络度分布: $p_{5}=4/5,p_{40}=1/5,p_{k}=0~(k\neq5,40)$. 在不同规模的网络上进行蒙特卡洛模拟, 并设置初始染病者数量为10, 然后让传染病开始演化, 直至达到稳定状态. 概率直方图是基于10000次随机模拟生成. (3.3) 式给出第一种近似结果, (3.4) 式给出第二种近似结果, Ornstein-Uhlenbeck (O-U)过程近似给出第三种近似结果.

图 1 的最上层为基于蒙特卡洛模拟得到的拟平稳分布 $q_{i_{5},i_{40}}$, 中间和下层分别对应正态分布和对数正态分布假设下的结果. 由于三维图中难以直观分辨差异, 图 2 和图 3 展示拟平稳状态下 $I_{5}$ 和 $I_{40}$ 的边际分布, 其中还包含了基于 O-U 过程的近似结果.

图 1

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图 1   种群规模 $N=100$ 和 $N=1000$ 下拟平稳分布的理论近似与蒙特卡洛模拟结果. 参数取值为 $\beta=1.5,~\gamma=1$.


图 2

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图 2   小种群规模下 ($N=100,150,200$) 拟平稳分布的近似结果与蒙特卡洛模拟结果.


图 3

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图 3   大种群规模下 ($N=1000, 1500, 2000$) 拟平稳分布的近似结果与蒙特卡洛模拟结果.


图 2 显示, 在较小种群规模下, 对数正态近似、正态近似和 O-U 过程近似之间存在明显差异. 图 3 表明, 随着种群规模增大, 这三种近似结果趋于一致, 并逐渐接近蒙特卡洛模拟结果. 这一事实在表 2 和表 3 中的定量结果中得到了进一步验证.

表2   小种群规模下蒙特卡洛模拟与三种近似方法的比较($\beta=1.5,\ \gamma=1$).

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表3   大种群规模下蒙特卡洛模拟与三种近似方法的比较.

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表 2 和表 3 展示了不同种群规模下理论计算与蒙特卡洛模拟的结果, 发现当种群规模较小时, 拟平稳分布的对数正态分布近似远优于正态分布近似或 O-U 过程近似. 总体而言, 矩封闭方法能提供比 O-U 过程近似更精确的近似. 这两种方法的结果差异主要体现在均值的近似上. 例如, 从表 2 可以看出, 当 $N=100$ 时, 矩封闭方法的结果为 $E(I_{40})\approx19.64$ 和 $E(I_{5})\approx70.09$, O-U 过程近似给出的结果为 $E(I_{40})\approx19.65$ 和 $E(I_{5})\approx70.13$, 而模拟结果为 $E(I_{40})\approx19.64$ 和 $E(I_{5})\approx70.07$.

4.2 灭绝时间

状态空间 $C$ 包含一个吸收态 $\left\{(0,\cdots,0,\cdots,0)\right\}$ 和一个瞬时态集合 $T$. 记 $Q$ 为转移速率矩阵, $Q_{T}$ 为 $Q$ 在 $T$ 上的子矩阵. 由文献[19] 知, 传播过程的拟平稳分布 $\boldsymbol{q}$ 存在且唯一, 其元素满足 $ q_{\boldsymbol{i}}=\lim _{t \rightarrow \infty} \mathrm{P}\{\boldsymbol{I}(t)=\boldsymbol{i} \mid \boldsymbol{I}(t) \in T\},~~i\in T. $$\boldsymbol{q}$ 可通过下式确定 $ \boldsymbol{q} Q_{\text{d}t}=-\frac{1}{\lambda}\boldsymbol{q},~~\sum_{i\in T}q_{i}=1, $ 其中 $-(1/\lambda)$ 是 $Q_{T}$ 实部最大的特征值. 由此, $\tau_{Q}$ 服从参数为 $-(1/\lambda)$ 的指数分布. 因此,

$\begin{equation}\label{EQ4.1} E(\tau_{Q})=\lambda. \end{equation}$

图 4 展示了 $\ln\left(E(\tau_{Q})\right)$ 的精确值 (通过 (4.1) 式计算) 与近似值 (通过 (3.5) 式计算) 比较的结果. 可以看出, 尽管精确值与近似值存在偏差, 但其对灭绝时间的研究仍有参考意义.

图 4

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图 4   不同种群规模下 $\ln\left(E(\tau_{Q})\right)$ 的精确值与近似值的比较. 参数为 $\beta=0.5,\,\gamma=5$.


本小节中, 通过数值模拟研究了 $\beta$ 和 $\gamma$ 对拟平稳状态下 $I_{5}$ 和 $I_{40}$ 的均值与方差的影响. 从图 5 的正态分布近似中可以看出: 对于 $I_{5}$, 均值随 $\beta$ 增大而增大, 方差则先增大后减小; 均值和方差均随 $\gamma$ 增大而减小. 对于 $I_{40}$, 均值随 $\beta$ 增大而增大, 而方差随之减小; 均值随 $\gamma$ 增大而减小, 方差随 $\gamma$ 增大而增大. 从图 6 的对数正态分布近似中可以看出: $I_{5}$ 和 $I_{40}$ 的均值均随 $\beta$ 增大而增大, 但方差随之增大而减小; 两者均值随 $\gamma$ 增大而减小, 方差随之增大而增大.

图 5

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图 5   基于正态分布近似, $\beta$ 和 $\gamma$ 对拟平稳状态下 $I_{5}$ 和 $I_{40}$ 的均值与方差的影响. (a) 展示了 $\beta$ ($\beta\in[0.5,1.5]$) 对拟平稳状态下 $I_{5}$ 和 $I_{40}$ 的均值与方差的影响, 其中 $\gamma=5$. (b) 展示了 $\gamma$ ($\gamma\in$[5,6]) 对拟平稳状态下 $I_{5}$ 和 $I_{40}$ 的均值与方差的影响, 其中 $\beta=0.5$. 人口规模 $N=50$.


图 6

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图 6   基于对数正态分布近似, $\beta$ 和 $\gamma$ 对拟平稳状态下 $I_{5}$ 和 $I_{40}$ 的均值与方差的影响. (a) 和 (b) 的参数设置与图 5 相同.


图 7 展示了 $\beta$ 和 $\gamma$ 对 $\ln\left(E(\tau_{Q})\right)$ 的影响. 图 7(a) 显示了 $\ln\left(E(\tau_{Q})\right)$ 和 $\beta$ 的线性关系, 且 $\ln\left(E(\tau_{Q})\right)$ 随 $\beta$ 的增大而单调递增. 类似地, 图 7(b) 显示了 $\ln\left(E(\tau_{Q})\right)$ 随 $\gamma$ 的增大而线性减小. 此外可以看出, 影响最大的参数是 $\beta$, 其次是 $\gamma$.

图 7

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图 7   $\beta$ 和 $\gamma$ 对 $\ln\left(E(\tau_{Q})\right)$ 的影响. (a) 展示了 $\beta$ ($\beta\in[0.5,1.5]$) 对 $\ln\left(E(\tau_{Q})\right)$ 的影响. (b) 展示了 $\gamma$ ($\gamma\in$[5,6]) 对 $\ln\left(E(\tau_{Q})\right)$ 的影响. 人口规模 $N=50$.


5 结论

本文关注具有异质群体的随机 SIS 传染病模型的随机动力学, 给出了拟平稳分布的两种近似形式, 正态分布与对数正态分布. 模拟表明对数正态分布的近似效果在不同种群规模下均优于正态分布. 另一个结果是从拟平稳状态开始的平均灭绝时间. 理论分析上, 对数正态分布无法近似平均灭绝时间, 而正态分布在灭绝时间近似上存在显著误差. 对平均灭绝时间的参数敏感性分析表明, 在控制传染病时, 应首先考虑减少传染病暴露的干预措施.

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Theor Popul Biol, 2004, 65(1): 49-65

PMID:14642344     

The major role played by demographic stochasticity in determining the dynamics and persistence of childhood diseases, such as measles, chickenpox and pertussis, has long been realized. Techniques which can be used to estimate the magnitude of this stochastic effect are of clear importance. In this study, we assess and compare the use of two moment closure approximations to estimate the variability seen about the average behavior of stochastic models for the recurrent epidemics seen in childhood diseases. The performance of the approximations are assessed using analytic techniques available for the simplest epidemiological model and using numerical simulations in more complex settings. We also present epidemiologically important extensions of previous work, considering variability in the SEIR model and in situations for which there is seasonal variation in disease transmission. Important implications of stochastic effects for the dynamics of childhood diseases are highlighted, including serious deficiencies of deterministic descriptions of dynamical behavior.

Clancy D, Mendy S T.

The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection

J Math Biol, 2010, 61(4): 527-544

DOI:10.1007/s00285-009-0313-4      PMID:19941137     

In stochastic modelling of infectious spread, it is often assumed that infection confers permanent immunity, a susceptible-infective-removed (SIR) model. We show how results concerning long-term (endemic) behaviour may be extended to a susceptible-infective-removed-susceptible (SIRS) model, in which immunity is temporary. Since the full SIRS model with demography is rather intractable, we also consider two simpler models: the susceptible-infective-susceptible (SIS) model with demography, in which there is no immunity; and the SIRS model in a closed population. For each model, we first analyse a deterministic model, then approximate the quasi-stationary distribution (equilibrium distribution conditional upon non-extinction of infection) using a moment closure technique. We look in particular at the effect of the immune period upon infection prevalence and upon time to fade-out of infection. Our main findings are that a shorter average immune period leads to higher infection prevalence in quasi-stationarity, and to longer persistence of infection in the population.

Clancy D, Mendy S T.

Approximating the quasi-stationary distribution of the SIS model for endemic infection

Methodol Comput Appl, 2011, 13(3): 603-618

[本文引用: 1]

Isham V.

Assessing the variability of stochastic epidemics

Math Biosci, 1991, 107(2): 207-224

[本文引用: 1]

Whittle P.

On the use of the normal approximation in the treatment of stochastic processes

J R Stat Soc B, 1957, 19(2): 268-281

DOI:10.1111/j.2517-6161.1957.tb00263.x      URL     [本文引用: 1]

Allen, L J S. An Introduction to Stochastic Processes with Applications to Biology. Florida: CRC Press, 2010

[本文引用: 1]

Pastor-Satorras R, Vespignani A.

Epidemic spreading in scale-free networks

Phys Rev Lett, 2001, 86(14): 3200-3203

PMID:11290142      [本文引用: 1]

The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks, finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks.

Pastor-Satorras R, Vespignani A.

Epidemic dynamics in finite size scale-free networks

Phys Rev E, 2002, 65(3): Art 035108

DOI:10.1103/PhysRevE.65.035108      URL     [本文引用: 1]

Nåsell I.

An alternative to moment closure

Bull Math Biol, 2017, 79(9): 2088-2108

DOI:10.1007/s11538-017-0321-2      URL     [本文引用: 1]

Keeling M J.

Metapopulation moments: coupling, stochasticity and persistence

J Anim Ecol, 2000, 69(5): 725-736

DOI:10.1046/j.1365-2656.2000.00430.x      PMID:29314000      [本文引用: 1]

1. Spatial heterogeneity has long been viewed as a reliable means of increasing persistence. Here, an analytical model is developed to consider the variation and, hence, the persistence of stochastic metapopulations. This model relies on a novel moment closure technique, which is equivalent to assuming log-normal distributions for the population sizes. 2. Single-species models show the greatest persistence when the mixing between subpopulations is large, so spatial heterogeneity is of no benefit. This result is confirmed by stochastic simulation of the full metapopulation. 3. In contrast, natural-enemy models exhibit the greatest persistence for intermediate levels of coupling. When the coupling is too low, there are insufficient rescue effects between the subpopulations to sustain the dynamics, whereas when the coupling is too high all spatial heterogeneity is lost. 4. The difference in behaviour between the one- and two-species models can be attributed to the oscillatory nature of the natural-enemy system.

Keeling M J.

Multiplicative moments and measures of persistence in ecology

J Theor Biol, 2000, 205(2): 269-281

PMID:10873438      [本文引用: 1]

Ecologists and epidemiologists have begun focusing on demographic stochasticity and spatial heterogeneity as important biological factors. With high-powered computers simulation of such systems is a common modelling technique; however we lack a detailed understanding of the processes involved. Moment closure approximations provide a simple method which can be used to capture the main features of a wide variety of stochastic models and to gain a more intuitive understanding. In this paper we give an alternative variation based on multiplicative moments which is equivalent to taking a novel third-order cumulant approximation. The differential equations for these multiplicative moments are far more robust than their additive counterparts. We use this technique to consider the behaviour and persistence of finite metapopulations for two common ecological systems.Copyright 2000 Academic Press.

Britton T, Traoré A.

A stochastic vector-borne epidemic model: Quasi-stationarity and extinction

Math Biosci, 2017, 289: 89-95

DOI:S0025-5564(17)30268-7      PMID:28511957      [本文引用: 1]

We consider a stochastic model describing the spread of a vector borne disease in a community where individuals (hosts and vectors) die and new individuals (hosts and vectors) are born. The time to extinction of the disease, T, starting in quasi-stationary (conditional on non extinction) is studied. Properties of the limiting distribution are used to obtain an approximate expression for E(T), the mean-parameter in the exponential distribution of the time to extinction, for a finite population. It is then investigated numerically and by means of simulations how E(T) and its approximations depend on the different model parameters.Copyright © 2017 Elsevier Inc. All rights reserved.

Darroch J N, Seneta E.

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

J Appl Probab, 1967, 4(1): 192-196

DOI:10.2307/3212311      URL     [本文引用: 1]

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