Dynamical behavior of a stochastic SVIR epidemic model with vaccination

In this paper, we investigate the dynamical behavior of SVIR models in random environments. Firstly, we show that if R0s<1, the disease of stochastic autonomous SVIR model will die out exponentially; if R̃0s>1, the disease will be prevail. Moreover, this system admits a unique stationary distribution and it is ergodic when R̃0s>1. Results show that environmental white noise is helpful for disease control. Secondly, we give sufficient conditions for the existence of nontrivial periodic solutions to stochastic SVIR model with periodic parameters. Finally, numerical simulations validate the analytical results.