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.