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Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay. (English) Zbl 1251.92036
Summary: 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 ${R}_{0}⩽1$, then the disease-free equilibrium is globally attractive and the disease always dies out. Otherwise, if ${R}_{0}>1$, then the disease-free equilibrium is unstable, and meanwhile there exists uniquely an endemic equilibrium. We then prove that for any time delay $h>0$, the delayed SIR epidemic model is permanent if and only if there exists an endemic equilibrium. In other words, ${R}_{0}>1$ 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.
##### MSC:
 92D30 Epidemiology 34K20 Stability theory of functional-differential equations 34K60 Qualitative investigation and simulation of models
##### Keywords:
asymptotic stability; permanence