Stochastic dynamical behavior of SIRS epidemic models with random perturbation

In this paper, we consider a stochastic SIRS model with parameter perturbation, which is a standard technique in modeling population dynamics. In our model, the disease transmission coefficient and the removal rates are all affected by noise. We show that the stochastic model has a unique positive solution as it is essential in any population model. Then we establish conditions for extinction
or persistence of the infectious disease. When the infective part is forced to expire, the susceptible part converges weakly to an inverse-gamma distribution with explicit shape and scale parameters. In case of persistence, by new stochastic Lyapunov functions, we show the ergodic property and positive
recurrence of the stochastic model. We also derive an estimate for the mean of the stationary distribution. The analytical results are all verified by computer simulations, including examples based on experiments in laboratory populations of mice.