Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model
Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model
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DOI：
10.1016/j.nonrwa.2011.03.011
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发表时间：
2011-10-01
影响因子：
2
通讯作者：
Kuniya, Toshikazu
中科院分区：
文献类型：
--
作者：
Kuniya, Toshikazu
对于一类由偏微分方程组描述的具有年龄结构的SIR传染病模型，在基本再生数R-0大于1的情形下，地方病平衡点的全局渐近稳定性问题一直是一个悬而未决的问题.在本文中，我们构造了一个多群体传染病模型作为该模型的推广，并研究了其每个平衡点的全局渐近稳定性。通过对多群模型关于年龄变量的离散化，在一定的参数假设下，将PDE系统改写为ODE系统，然后应用经典的Lyapunov函数方法和最近发展起来的图论方法，结合最大值函数的思想，证明了离散化系统的每个平衡点的全局渐近稳定性完全由R-0决定，即无病平衡点是全局渐近稳定的，如果R-0 1。数值算例表明，随着离散步长的减小，离散常微分方程组的R-0数值解与原偏微分方程组的R-0数值解更加接近。(C)2011爱思唯尔有限公司保留所有权利。
For an age-structured SIR epidemic model, which is described by a system of partial differential equations, the global asymptotic stability of an endemic equilibrium in the situation where the basic reproduction number,R-0 is greater than unity has been an open problem for decades. In the present paper, we construct a multigroup epidemic model regarded as a generalization of the model, and study the global asymptotic stability of each of its equilibria. By discretizing the multigroup model with respect to the age variable under some parameter assumptions, we first rewrite the PDE system into an ODE system, and then, applying the classical method of Lyapunov functions and a recently developed graph-theoretic approach with an original idea of maximum value functions, we prove that the global asymptotic stability of each equilibrium of the discretized system is completely determined by R-0, namely, the disease-free equilibrium is globally asymptotically stable if R-0 1. A numerical example illustrates that a numerical solution of R-0 for the discretized ODE system becomes closer to that for the original PDE system as the step size of the discretization decreases. (C) 2011 Elsevier Ltd. All rights reserved.