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

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 ⩽ 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.