Dynamics of Stochastically Perturbed SIS Epidemic Model with Vaccination

Abstract

We introduce stochasticity into an SIS epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. In the deterministic models, the basic reproduction number $R_0$ is a threshold which determines the persistence or extinction of the disease. When the perturbation and the disease-related death rate are small, we carry out a detailed analysis on the dynamical behavior of the stochastic model, also regarding of the value of $R_0$. If $R_0\le 1$, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, if $R_0>1$, there is a stationary distribution, which means that the disease will prevail. The results are illustrated by computer simulations.