Dynamic Analysis and Optimal Control of a Stochastic SVEIRS Model With Saturation Incidence

ABSTRACT This paper proposes a stochastic version based on a deterministic SVEIRS (Susceptible‐Vaccinated‐Exposed‐Infectious‐Recovered‐Susceptible) model with saturation incidences and analyzes their dynamic properties and control strategies. For the deterministic model, we simply study the asymptotic properties of disease‐free and endemic equilibria and discuss the parametric branching. Focusing on the stochastic model, we prove the existence and uniqueness of the global positive solution. Further, the extinction and persistence are obtained under certain threshold conditions. By constructing suitable Lyapunov functions, we explore the existence of an ergodic distribution. Moreover, the control variables are introduced into the stochastic model to prevent the spread of the epidemic. Finally, numerical simulations illustrate the theoretical results, such as the dynamic curves of extinction, persistence, and ergodic distribution under given parameters. The results show the treatment and vaccine controls significantly inhibit the disease.