The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences

In this paper, we consider a class of DI SIR epidemic models with saturated incidences and parameter perturbation. We investigate the asymptotic behavior according to the perturbation and the reproductive number R0R0. When the perturbation is large, the infective in every group decays exponentially to zero while the susceptible converges weakly to stationary distribution regardless of the magnitude of R0R0. When the perturbation is small, we get the same exponential stability and weak convergence if R0≤1R0≤1, and we use a new class of stochastic Lyapunov functions to obtain the ergodicity and positive recurrence if R0>1R0>1.