Lyapunov exponent and almost sure asymptotic stability of a stochastic SIRS model

Epidemiological models with bilinear incidence rate usually have an asymptotically stable trivial equilibrium corresponding to the disease-free state, or an asymptotically stable nontrivial equilibrium (i.e.interior equilibrium) corresponding to the endemic state.In this paper, we consider an epidemiological model, which is a SIRS (susceptible-infected-removed-susceptible) model influenced by random perturbations.We prove that the solutions of the system are positive for all positive initial conditions and that the solutions are global, that is, there is no finite explosion time.We present necessary and sufficient condition for the almost sure asymptotic stability of the steady state of the stochastic system.