A generalized stochastic SIR epidemic model with vaccination rules

In this paper, a generalized stochastic SIR epidemic model with vaccination rules is presented and the threshold behavior of the proposed epidemic model is investigated. Firstly, the stability of the equilibrium of the deterministic system is considered and the corresponding conditions are obtained. Secondly, the threshold of a stochastic SIR system for the extinction and the permanence in mean of epidemic disease are investigated. The results show that a larger stochastic disturbance can cause infections diseases to go to extinction. However, for a relatively small stochastic disturbance, the evolutionary dynamics of the epidemic diseases are overwhelmingly depend on the incidence function. This implies that the stochastic disturbance and the incidence function play an important role in diseases control. To test the theoretical results, a series of numerical simulations of these cases with respect to different noise disturbance coefficients are conducted.