  This paper investigates asymptotic behavior of a stochastic SIR epidemic
model, which is a system with degenerate diffusion. It gives sufficient
conditions that are very close to the necessary conditions 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.
