Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1081-1106.

• Articles •

### DYNAMICS FOR AN SIR EPIDEMIC MODEL WITH NONLOCAL DIFFUSION AND FREE BOUNDARIES

Meng ZHAO1, Wantong LI2, Jiafeng CAO3

1. 1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;
2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;
3. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
• Received:2020-04-04 Revised:2020-07-01 Online:2021-08-25 Published:2021-09-01
• Contact: Wantong LI E-mail:wtli@lzu.edu.cn
• Supported by:
Zhao was supported by a scholarship from the China Scholarship Council, Li was partially supported by NSF of China (11731005), and Cao was partially supported by NSF of China (11901264).

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.

CLC Number:

• 35K57
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