The asymptotic behavior of a stochastic SIS epidemic model with vaccination

Abstract

In this paper, we discuss a stochastic SIS epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number $R_{0}$ . When the perturbation is large, the number of infected decays exponentially to zero and the solution converges to the disease-free equilibrium regardless of the magnitude of $R_{0}$ . Moreover, we get the same exponential stability and the convergence if $R_{0}<1$ . When the perturbation and the disease-related death rate are small, we derive that the disease will persist, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model on average in time if $R_{0}>1$ . Furthermore, we prove that the system is persistent in the mean. Finally, the results are illustrated by computer simulations.