Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates

We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R0 1. Second we show that the model is permanent if and only if R0 > 1. Moreover, using a threshold parameter R0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1 < R0 R0 and it loses stability as the length of the delay increases past a critical value for 1 < R0 < R0. Our result is an extension of the stability results in (J-J. Wang, J-Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonl. Anal. RWA. 11 (2009) 2390-2402).