Dynamics of an SIRS epidemic model with cross-diffusion

Yaru Hu; Jinfeng Wang

doi:10.3934/cpaa.2021179

Article Contents

2022, Volume 21, Issue 1: 315-336. Doi: 10.3934/cpaa.2021179

This issue Previous Article Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation Next Article Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

Dynamics of an SIRS epidemic model with cross-diffusion

Yaru Hu^1, and
Jinfeng Wang^2, ,

1.
School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China
2.
School of Mathematical Sciences and Y.Y. Tseng Functional Analysis Research Center, Harbin Normal University, Harbin, Heilongjiang, 150025, China

^* Corresponding author
^* Corresponding author

Received: June 2021

Revised: September 2021

Early access: October 2021

Published: January 2022

The authors are supported by National NSFC grant 11971135, NSFH grant LH2019A017, 2018-KYYWF-0999, XKYQ201403 and HSDSSCX2021-12

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Abstract

The dynamical behavior of an SIRS epidemic reaction-diffusion model with frequency-dependent mechanism in a spatially heterogeneous environment is studied, with a chemotaxis effect that susceptible individuals tend to move away from higher concentration of infected individuals. Regardless of the strength of the chemotactic coefficient and the spatial dimension $ n $, it is established the unique global classical solution which is uniformly-in-time bounded. The model still recognizes the threshold dynamics in terms of the basic reproduction number $ \mathcal{R}_{0} $ even in the case of chemotaxis effects: if $ \mathcal{R}_{0}<1 $, the unique disease free equilibrium is globally stable; if $ \mathcal{R}_{0}>1 $, the disease is uniformly persistent and there is at least one endemic equilibrium, which is globally stable in some special cases with weak chemotactic sensitivity. We also show the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible is small, which indicates that the disease always exists in the entire habitat in this case. Our results suggest that one cannot eradicate the SIRS disease model by only controlling the diffusion rate of susceptible individuals.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35K59.

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