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 R¯0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1