Stationary distribution and density function of a stochastic SVIR epidemic model.

We consider the long-term properties of a stochastic SVIR epidemic model with saturation incidence rates and logistic growth in this paper. We firstly derive the fitness of a unique global positive solution. Then we construct appropriate Lyapunov functions and obtain condition R 0 s > 1 for existence of stationary distribution, and conditions for persistence in the mean. Moreover, conditions including R 0 e < 1 for exponential extinction to the infected individuals are figured out. Finally, by employing Fokker-Planck equation and stochastic analysis, we derive the probability density function around the quasi-endemic equilibrium point when critical value R 0 p > 1 is valid. Consequently, some examples and illustrative simulations are carried out to verify the main theoretical results. [ABSTRACT FROM AUTHOR]