Asymptotic stability of disease-free equilibria in a diffusive logistic SIR epidemic model with saturated incidence and treatment rates

Ryota Nakiyama; Yuya Tanaka; Tomomi Yokota

doi:10.3934/dcdsb.2024105

Article Contents

2025, Volume 30, Issue 2: 642-667. Doi: 10.3934/dcdsb.2024105

This issue Previous Article Well-posedness and dynamics of wave equations with nonlinear damping and moving boundary Next Article

Asymptotic stability of disease-free equilibria in a diffusive logistic SIR epidemic model with saturated incidence and treatment rates

1.
Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
2.
Department of Mathematical Sciences, Kwansei Gakuin University, 1 Gakuen Uegahara, Sanda, Hyogo 669-1330, Japan

^*Corresponding author: Tomomi Yokota
^*Corresponding author: Tomomi Yokota

Received: January 2024

Revised: July 2024

Early access: August 2024

Published: February 2025

The second author is supported by JSPS KAKENHI Grant Number 22KJ2805. The third author is supported by JSPS KAKENHI Grant Number JP21K03278.

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Abstract

This paper deals with the Neumann initial-boundary value problem for the diffusive logistic epidemic model with saturated incidence and treatment rates,

$ \begin{align*} \begin{cases} \dfrac{\partial S}{\partial t} = d_S\Delta S-\dfrac{\beta SI}{1+\alpha I}+rS\left(1-\dfrac{S}{K}\right),\quad &x \in \Omega, \ t>0, \\ \dfrac{\partial I}{\partial t} = d_I\Delta I+\dfrac{\beta SI}{1+\alpha I}-\dfrac{\lambda I}{1+\varepsilon I},\quad &x \in \Omega, \ t>0, \end{cases} \end{align*} $

where $ \Omega \subset \mathbb{R}^N $ ($ N \in \mathbb{N} $) is a bounded domain with smooth boundary and $ d_S, d_I, r, K, \alpha, \beta, \varepsilon, \lambda >0 $ are constants. The main result of this paper asserts that the disease-free equilibrium $ (K,0) $ is (globally) asymptotically stable under some conditions for $ K, \alpha, \beta, \varepsilon $ and $ \lambda $, and initial data if $ \mathcal{R}_0: = \frac{K\beta}{\lambda}\le1 $.

Keywords:

Diffusive SIR epidemic model,
disease-free equilibrium,
logistic growth,
saturated incidence and treatment rates,
asymptotic stability.

Mathematics Subject Classification: Primary: 35K40; Secondary: 35B40, 92D30.

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Figure 1.1. When $ \varepsilon = 0 $ and also $ 0< \varepsilon\le \alpha $, there are two constant equilibria of (1) with parameters chosen in the blue coloured region

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Figure 1.2. When $ \varepsilon> \alpha $, there are two, three and four constant equilibria of (1) with parameters chosen in the blue, green and red coloured regions, respectively

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