The asymptotic behavior of a stochastic vaccination model with backward bifurcation

In this paper, we aim to explore the effect of environmental fluctuation on a deterministic vaccination model that exhibits backward bifurcation. First, we prove that the stochastic model admits a unique and global positive solution. Next, by investigating the asymptotic behavior of the stochastic model around the disease free equilibrium, we find that due to the influence of environmental fluctuation the disease dies out even though it may still persist for the deterministic model with backward bifurcation. We also estimate the probability distribution for the disease extinction time. Then, we derive sufficient conditions for the solution of the stochastic model to fluctuate around the endemic equilibrium, and thus we prove the ergodicity of the stochastic model. Our theoretical results are verified by computer simulations and numerical comparison results of the stochastic model and deterministic version is also given. Finally, numerical simulations of a stochastic reaction diffusion model with multiplicative noise are presented to illustrate the combined effects of spatial movement of individuals and environmental noise on the spread of disease.