Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of stochastic differential equations with a degenerate diffusion, this paper provides sufficient conditions that are very close to the necessary ones for the permanence. In addition, this paper develops ergodicity of the underlying system. It is proved that the transition probabilities converge in total variation norm to the invariant measure. Our result gives a precise characterization of the support of the invariant measure. Rates of convergence are also ascertained. It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree.
['Nguyen Thanh Dieu', 'Dang Hai Nguyen', 'Nguyen Huu Du', 'G. Yin']
Classification of Asymptotic Behavior in A Stochastic SIR Model