Global asymptotic stability of endemic equilibria for a diffusive SIR epidemic model with saturated incidence and logistic growth

Yutaro Chiyo; Yuya Tanaka; Ayako Uchida; Tomomi Yokota

doi:10.3934/dcdsb.2022163

Article Contents

2023, Volume 28, Issue 3: 2184-2210. Doi: 10.3934/dcdsb.2022163

This issue Previous Article Besicovitch almost periodic stochastic processes and almost periodic solutions of Clifford-valued stochastic neural networks Next Article Regularity of the solution of a nonlinear Volterra integral equation of the second kind

Global asymptotic stability of endemic equilibria for a diffusive SIR epidemic model with saturated incidence and logistic growth

1.
Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received: January 2022

Revised: July 2022

Early access: August 2022

Published: March 2023

The fourth author is supported by JSPS KAKENHI Grant Number JP21K03278

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Abstract

This paper deals with the diffusive epidemic model with saturated incidence and logistic growth,

$ \begin{align*} \begin{cases} \dfrac{\partial S}{\partial t} = d_S \Delta S - \dfrac{\beta S I}{1+\alpha I} + rS\left(1- \dfrac{S}{K} \right), &x \in \Omega, \ t>0, \\ \dfrac{\partial I}{\partial t} = d_I \Delta I + \dfrac{\beta S I}{1+\alpha I} - \gamma I, &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, K, r, \alpha, \beta, \gamma >0 $ are constants. Setting $ \mathcal{R}_0: = \frac{K \beta}{\gamma} $, Avila-Vales et al. [1] succeeded in showing that if $ \mathcal{R}_0\leq1 $, then the disease-free equilibrium $ (K, 0) $ of the model with saturated treatment is globally asymptotically stable, whereas in the case $ \mathcal{R}_0>1 $ the model admits a constant endemic equilibrium $ (S^*, I^*) $ ($ S^*, I^*>0 $), and it is unknown whether $ (S^*, I^*) $ is globally asymptotically stable or not. The purpose of this paper is to establish that the constant endemic equilibrium of the above model is globally asymptotically stable by constructing a strict Lyapunov functional. The construction is carried out by optimizing a function of two real variables through straightforward calculations, division into some cases and arrangement of several conditions. Moreover, to show that the functional is strict, some auxiliary function is introduced.

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

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Figure 4.1. The solution of (1.1) with the parameters given in (4.1). The distributions of $ S, I $ become milder with time. Also, the asymptotic distributions of $ S, I $ are stable when $ t \to \infty $

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Figure 4.2. The solution of (1.1) with the parameters given in (4.2). Even though the parameters do not satisfy the condition (1.3), the asymptotic distributions of $ S, I $ converge to constant values

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Table 1. Known results

	Local asymptotic stability	Global asymptotic stability
The DFE ($ \mathcal{R}_0 \le 1 $)	Pérez et al. [14], Avila-Vales et al. [1]	Avila-Vales et al. [1]
The EE ($ \mathcal{R}_0> 1 $)	Pérez et al. [14]	Unknown

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