Stability in a Ross epidemic model with road diffusion

Xiaomei Bao; Canrong Tian; Xiaomei Bao; Canrong Tian

doi:10.3934/math.2022157

AIMS Mathematics

2022, Volume 7, Issue 2: 2840-2857. doi: 10.3934/math.2022157

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Stability in a Ross epidemic model with road diffusion

Xiaomei Bao ¹,
Canrong Tian ^{2
,
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1.
School of Foreign Languages, Yancheng Institute of Technology, Yancheng, Jiangsu 224003, China
2.
School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, Jiangsu 224003, China

Received: 05 October 2021 Accepted: 15 November 2021 Published: 19 November 2021
MSC : 35B35, 35K60

Reaction-diffusion equations have been used to describe the dynamical behavior of epidemic models, where the spreading of infectious disease has the same speed in every direction. A natural question is how to describe the dynamical system when the spreading of infectious disease is directed diffusion. We introduce the road diffusion into a Ross epidemic model which describes the spread of infected Mosquitoes and humans. With the comparison principle the system is proved to have a unique global solution. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 while the endemic equilibrium asymptotically stable if the basic reproduction number is greater than 1.
- epidemic model,
- basic reproduction number,
- asymptotic stability,
- road diffusion
Citation: Xiaomei Bao, Canrong Tian. Stability in a Ross epidemic model with road diffusion[J]. AIMS Mathematics, 2022, 7(2): 2840-2857. doi: 10.3934/math.2022157

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Abstract

Reaction-diffusion equations have been used to describe the dynamical behavior of epidemic models, where the spreading of infectious disease has the same speed in every direction. A natural question is how to describe the dynamical system when the spreading of infectious disease is directed diffusion. We introduce the road diffusion into a Ross epidemic model which describes the spread of infected Mosquitoes and humans. With the comparison principle the system is proved to have a unique global solution. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 while the endemic equilibrium asymptotically stable if the basic reproduction number is greater than 1.

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