The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence

In this paper, a stochastic periodic SIVS epidemic model with nonlinear incidence and vaccination is investigated. The threshold conditions on the existence of stochastic positive periodic solutions and the extinction of disease with probability one are established by constructing the new stochastic Lyapunov functions and using the new technique to deal with the nonlinear incidence and vaccination for the stochastic epidemic model. The numerical simulations are given to illustrate the main theoretical results and present some new interesting conjectures.