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. (C) 2012 Elsevier Ltd. All rights reserved.