Threshold of a stochastic SIR epidemic model with Lévy jumps

•The dynamics of a stochastic SIR epidemic model with Lévy jumps is investigated.•We find R˜0 as the threshold of this stochastic SIR model, which determines the extinction and prevalence of the disease.•R˜0 is smaller than the basic reproduction number R0R0 of the corresponding deterministic model.•The results show that Lévy jumps have significant effects on the dynamics behaviors of the model.•We simulate the theoretical results numerically.

This paper mainly investigates the effect of the Lévy jumps on the dynamics of a stochastic SIR epidemic model. Taking the accumulated jump size into account, a threshold of the considered model has been found out, denoted by R˜0, which can determine the extinction and persistence in mean of the epidemic. More specifically, if R˜0<1, the disease ultimately vanishes from the population; whereas if R˜0>1, the disease persists in the population. Numerical simulations have been carried out to illustrate the theoretical results.