Dynamics of a Stochastic Epidemic Model with Vaccination and General Incidence Rate

In this paper, we consider the dynamics of the spread of infectious diseases for a stochastic SIVS (Susceptible - Infected - Vaccinated - Susceptible) epidemic model perturbed by white noise with general incidence rate. By constructing a threshold \(\lambda \) $\lambda$ via parameters of the model, we can give conditions to determine whenever the disease is extinct or permanent. Precisely, it is proved that \(\lambda < 0\) $\lambda <0$ implies the disease-free point \((S_*; 0; V_*)\) $(S_{\ast };0;V_{\ast })$ to be exponentially stable, i.e., the disease will eventually disappear with an exponential rate meanwhile \(\lambda >0\) $\lambda >0$ shows that the epidemic exists permanently in the population.