Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

Abstract

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any $\tau$, we show that the disease-free equilibrium is globally asymptotically stable; when , the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for any $\tau =0$; when $R_0>1$, the disease will persist. However, for any $\tau \ne \text{0}$, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.