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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 \(R_{0}\leqslant 1\), 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 involving functional-differential equations
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