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

► The dynamics of the SIR model is completely determined by a threshold R0. ► If R0⩽1 then the disease always dies out. ► If R0>1 then the delayed SIR epidemic model is permanent. ► Trajectories converge to equilibria more quickly than the discrete time delay case. 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 R0. If R0⩽1, then the disease-free equilibrium is globally attractive and the disease always dies out. Otherwise, if R0>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, R0>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.