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 R 0 . 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 R 0 . When the perturbation is small, we get the same exponential stability and weak convergence if R 0 ≤ 1 , and we use a new class of stochastic Lyapunov functions to obtain the ergodicity and positive recurrence if R 0 > 1 .