Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay

作者：Li Chun Hsien; Tsai Chiung Chiou; Yang Suh Yuh^*

来源：Communications in Nonlinear Science & Numerical Simulation, 2012, 17(9): 3696-3707.

DOI：10.1016/j.cnsns.2012.01.018

• SIR epidemic model
• Time delay
• Asymptotic stability
• Permanence

In this paper, we study the dynamics of an SIR epidemic model with a logistic process and a distributed time delay. We first show that the attractivity of the disease-free equilibrium is completely determined by a threshold R-0. If R-0 %26lt;= 1, then the disease-free equilibrium is globally attractive and the disease always dies out. Otherwise, if R-0 %26gt; 1, then the disease-free equilibrium is unstable, and meanwhile there exists uniquely an endemic equilibrium. We then prove that for any time delay h %26gt; 0, the delayed SIR epidemic model is permanent if and only if there exists an endemic equilibrium. In other words, R-0 %26gt; 1 is a necessary and sufficient condition for the permanence of the epidemic model. Numerical examples are given to illustrate the theoretical results. We also make a distinction between the dynamics of the distributed time delay system and the discrete time delay system.

• 出版日期2012-9
• 单位中央民族大学