Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate

Stability of SIR models has been studied extensively within the framework of disease epidemiology. We formulate a nonlinear mathematical model to study the role of nonlinear incidence rates and the effect of time delay in a nonlinear logistically growing time delayed SIR   model with variable population size. The existence and stability of the possible equilibria are examined in terms of a certain threshold condition ℜ0ℜ0, the basic reproduction number. For mathematical tractability, the stability of the endemic equilibrium is shown using Lyapunov functional approach with linear incidence rate. Numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease, to support the analytical conclusion and illustrate possible behavioral scenarios of the model.