Threshold behaviour of a stochastic SIRS Le?vy jump model with saturated incidence and vaccination

A stochastic SIRS system with Le ' vy process is formulated in this paper, and the model incorporates the saturated incidence and vaccination strategies. Due to the introduction of Le ' vy jump, the jump stochastic integral process is a discontinuous martingale. Then the Kunita's inequality is used to estimate the asymptotic pathwise of the solution for the proposed model, instead of BurkholderDavis-Gundy inequality which is suitable for continuous martingales. The basic reproduction number R0s of the system is also derived, and the sufficient conditions are provided for the persistence and extinction of SIRS disease. In addition, the numerical simulations are carried out to illustrate the theoretical results. Theoretical and numerical results both show that Le ' vy process can suppress the outbreak of the disease.