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Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force
Dianli Zhao,Sanling Yuan
Keywords:Stochastic epidemic model, jump-diffusion infection force, the threshold, extinction.
Abstract:
      This paper formulates a stochastic SIR epidemic model by supposing that the infection force is perturbed by Brown motion and Levy jumps. The globally positive and bounded solution is proved firstly by constructing the suitable Lyapunov function. Then, a stochastic basic reproduction number $R_0^{L}$ is derived, which is less than that for the deterministic model and the stochastic model driven by Brown motion. Analytical results show that the disease will die out if $R_0^{L}<1$, and $R_0^{L}>1$ is the necessary and sufficient condition for persistence of the disease. Theoretical results and numerical simulations indicate that the effects of Levy jumps may lead to extinction of the disease while the deterministic model and the stochastic model driven by Brown motion both predict persistence. Additionally, the method developed in this paper can be used to investigate a class of related stochastic models driven by Levy noise.