The threshold and density function analysis of a stochastic SIVS model with saturation incidence

Mathematical model is the main tool to study the dynamics of infectious diseases, which has played an important role in controlling the spread of infectious diseases. We consider a stochastic SIVS model with saturation incidence in this paper. First of all, we establish the threshold [Formula: see text] for extinction and persistence for the stochastic epidemic model. Additionally, we give the specific expression of the probability density function of the stochastic model near the unique endemic quasi-equilibrium by solving the Fokker–Planck equation. In the end, the supporting theoretical results are verified by numerical simulation.