Threshold behavior of a stochastic SIS model with Le´vy jumps

In this paper, the dynamics of a stochastic SIS model with Lévy jumps are investigated. We first prove that this model has a unique global positive solution starting from the positive initial value. Then, taking the accumulated jump size into account, we find a threshold of the model, denoted by R˜0, which completely determines the extinction and prevalence of the disease: if R˜0<1, the disease dies out exponentially with probability one; if R˜0>1, the solution of the model tends to a point in time average which leads to the stochastical persistence of the disease. From the view of epidemiology, the existence of threshold is useful in determining treatment strategies and forecasting epidemic dynamics. Moreover, we find that Lévy noise can suppress disease outbreak. Finally, we introduce some numerical simulations to support the main results obtained.