关键词: SIR model, nonlocal diffusion, free boundary, spreading and vanishing

Abstract: This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease. This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al. (An SIR epidemic model with free boundary. Nonlinear Anal RWA, 2013, 14:1992-2001). We first prove that this problem has a unique solution defined for all time, and then we give sufficient conditions for the disease vanishing and spreading. Our result shows that the disease will not spread if the basic reproduction number $R_0<1$, or the initial infected area $h_0$, expanding ability $\mu$, and the initial datum $S_0$ are all small enough when $1 < R_0 < 1+\frac{d}{\mu_2+\alpha}$. Furthermore, we show that if $1 < R_0 < 1+\frac{d}{\mu_2+\alpha}$, the disease will spread when $h_0$ is large enough or $h_0$ is small but $\mu$ is large enough. It is expected that the disease will always spread when $R_0\geq1+\frac{d}{\mu_2+\alpha}$, which is different from the local model.

Key words: SIR model, nonlocal diffusion, free boundary, spreading and vanishing