Global behaviour of an SIR epidemic model with time delay

We study the stability of a delay susceptible-infective-recovered epidemic model with time delay. The model is formulated under the assumption that all individuals are susceptible, and we analyse the global stability via two methods - by Lyapunov functionals, and - in terms of the variance of the variables. The main theorem shows that the endemic equilibrium is stable. If the basic reproduction number R0 is less than unity, by LaSalle invariance principle, the disease-free equilibrium Es is globally stable and the disease always dies out. By applying the integral averaging theory, we also investigate the stability in variance of the model.