Electron. J. Differential Equations, Vol. 2018 (2018), No. 170, pp. 1-21.

Epidemic reaction-diffusion systems with two types of boundary conditions

Kehua Li, Jiemei Li, Wei Wang

Abstract:
We investigate an epidemic reaction-diffusion system with two different types of boundary conditions. For the problem with the Neumann boundary condition, the global dynamics is fully determined by the basic reproduction number $\mathcal{R}_0$. For the problem with the free boundary condition, the disease will vanish if the basic reproduction number $\mathcal{R}_0<1$ or the initial infected radius $g_0$ is sufficiently small. Furthermore, it is shown that the disease will spread to the whole domain if $\mathcal{R}_0>1$ and the initial infected radius $g_0$ is suitably large. Main results reveal that besides the basic reproduction number, the size of initial epidemic region and the diffusion rates of the disease also have an important influence to the disease transmission.

Submitted November 12, 2017. Published October 11, 2018.
Math Subject Classifications: 58F15, 58F17, 53C35.
Key Words: SIRS model; reaction-diffusion system; global dynamics; Neumann boundary condition; free boundary condition.

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Kehua Li
School of Applied Mathematics
Xiamen University of Technology
Xiamen, Fujian 361024, China
email: khli@xmut.edu.cn
Jiemei Li
School of Mathematics and Physics
Lanzhou Jiaotong University
Lanzhou, Gansu 730070, China
email: lijiemei81@126.com
Wei Wang
Department of Mathematics
College of Mathematics and Systems Science
Shandong University of Science and Technology
Qingdao 266590, China
email: weiw10437@gmail.com

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