A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment

Danhua Jiang; Zhi-Cheng Wang; Liang Zhang

doi:10.3934/dcdsb.2018176

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2018, Volume 23, Issue 10: 4557-4578. Doi: 10.3934/dcdsb.2018176

This issue Previous Article Identification of generic stable dynamical systems taking a nonlinear differential approach Next Article Prevalence of stable periodic solutions in the forced relativistic pendulum equation

A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment

School of Mathematics and Statistics, Lanzhou University, and Key Laboratory of Applied Mathematics and Complex Systems of Gansu province, Lanzhou, Gansu 730000, China

* Corresponding author: Zhi-Cheng Wang
* Corresponding author: Zhi-Cheng Wang

Received: September 2017

Revised: February 2018

Early access: May 2018

Published: September 2018

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Abstract

In this paper, we study the effects of diffusion and advection for an SIS epidemic reaction-diffusion-advection model in a spatially and temporally heterogeneous environment. We introduce the basic reproduction number $\mathcal{R}_{0}$ and establish the threshold-type results on the global dynamics in terms of $\mathcal{R}_{0}$ . Some general qualitative properties of $\mathcal{R}_{0}$ are presented, then the paper is devoted to studying how the advection and diffusion of the infected individuals affect the reproduction number $\mathcal{R}_{0}$ for the special case that $γ(x,t)-β(x,t) = V(x,t)$ is monotone with respect to spatial variable $x$ . Our results suggest that if $V_{x}(x,t)≥0,\not\equiv0$ and $V(x, t)$ changes sign about $x$ , the advection is beneficial to eliminate the disease, whereas if $V_{x}(x,t)≤0,\not\equiv0$ and $V(x, t)$ changes sign about $x$ , the advection is bad for the elimination of disease.

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Mathematics Subject Classification: 35K57, 37N25, 92B05.

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