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Corrigendum: Dynamics of a susceptible—infected—susceptible epidemic reaction—diffusion model

Part of: Partial differential equations Parabolic equations and systems Mathematical sociology Applications - Dynamical systems and ergodic theory

Published online by Cambridge University Press:  30 March 2022

Keng Deng  and
Yixiang Wu
Keng Deng
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA
Yixiang Wu
Affiliation:
Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, Tennessee 37132, USA
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Abstract

This corrigendum corrects a result in [1].

Keywords

SIS epidemic reaction-diffusion modelspatial heterogeneitydisease-free equilibriumendemic equilibriumglobal attractivity

MSC classification

Primary: 92D30: Epidemiology
Secondary: 91D25: Spatial models 35K57: Reaction-diffusion equations 37N25: Dynamical systems in biology 35B35: Stability
Type
Corrigendum
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 2 , April 2023 , pp. 718 - 720
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

1. Uniqueness of EE

In Theorem 3.10 of [Reference Deng and Wu1], we claimed that the model has a unique EE if $\mathcal {R}_0>1$. However, there is an error in the proof of the case $d_S< d_I$. The correct statement and proof of Theorem 3.10 should be as follows. This correction has no impact on the results of the subsequent paper [Reference Wu and Zou2], which considered the same model.

Theorem If $\mathcal {R}_0>1$ and $d_S\ge d_I$, the model has a unique EE; if $\mathcal {R}_0>1$ and $d_S< d_I$, the model has an EE.

In the proof of Theorem 3.10, the following changes should be made:

  • • Page 938, Line 2 from the bottom. ‘it suffices to show that problem (3.15), (3.16) has a unique positive solution’ should be ‘it suffices to study the positive solution of problem (3.15), (3.16)’.

  • • Page 939, Line 9. ‘there exists a unique $\tau _0>0$’ should be ‘there exists $\tau _0>0$’.

We are able to prove the uniqueness of EE if $d_S< d_I$ with an additional assumption $N/|\Omega |\ge \gamma /\beta$. In the proof of the following result, for any $u, v\in C(\overline \Omega )$, we write $u\le v$ if $u(x)\le v(x)$ for all $x\in \overline \Omega$, $u<< v$ if $u(x)< v(x)$ for all $x\in \overline \Omega$ and $u< v$ if $u\le v$ but $u\neq v$.

Theorem Suppose $d_S< d_I$ and $N/|\Omega |\ge \gamma /\beta$. Then the EE of the model is unique.

Proof. By Proposition 3.2 and Remark 3.3, the assumption $N/|\Omega |\ge \gamma /\beta$ implies $\mathcal {R}_0>1$ and hence the model has at least one EE. To prove the uniqueness of the EE, by lemma 3.5, we only need to show that the positive solution of the following problem is unique:

(1.1)\begin{align} & d_I\Delta I + I\left( \frac{N}{|\Omega|}\beta-\gamma +d\frac{ \beta}{|\Omega|}\int_\Omega Idx-\frac{d_I \beta }{d_S} I \right)=0,\quad x\in\Omega,\end{align}
(1.2)\begin{align} & \frac{\partial I}{\partial n}=0, \quad x\in\partial\Omega, \end{align}

where $d=d_I/d_S-1$.

We first claim that for any positive solution $I$ of (1.1)–(1.2), it satisfies that

(1.3)\begin{equation} \frac{d_I}{d_S}I > \frac{d}{|\Omega|}\int_\Omega I dx. \end{equation}

To see this, we rewrite (1.1) as

(1.4)\begin{equation} d_I\Delta I + \beta I\left( \frac{N}{|\Omega|}-\frac{\gamma}{\beta} +\frac{ d}{|\Omega|}\int_\Omega Idx-\frac{d_I}{d_S} I \right)=0. \end{equation}

Let $I(x_0)=\min \{I(x): x\in \overline \Omega \}$ for some $x_0\in \overline \Omega$. Then similar to the proof of lemma 3.6, by using the maximum principle we can show

\[ \frac{d_I}{d_S}I(x_0) \ge \frac{N}{|\Omega|}-\frac{\gamma(x_0)}{\beta(x_0)}+\frac{d}{|\Omega|}\int_\Omega I dx. \]

Since $N/|\Omega |\ge \gamma /\beta$, we have

(1.5)\begin{equation} \frac{d_I}{d_S}I \ge \frac{d}{|\Omega|}\int_\Omega I dx. \end{equation}

If equality holds in (1.5), then $I$ is constant. However, since $d_I/d_S>d$, this is impossible. Thus the claim is valid.

Now suppose $I_1$ and $I_2$ are two distinct positive solutions of (1.1)–(1.2). Define

\[ k= \max\{\tilde k\ge 0: \tilde k I_1\le I_2 \}. \]

Interchanging $I_1$ and $I_2$ if necessary, we have $k\in (0, 1)$. Moreover, $kI_1 \le I_2$ and $kI_1(x_1)=I_2(x_1)$ for some $x_1\in \overline \Omega$.

Define a function $h:E\subset C_+(\overline \Omega ) \rightarrow C_+(\overline \Omega )$ by

\[ h(I)=(a-d_I\Delta)^{{-}1} I\left( a+ \frac{N}{|\Omega|}\beta-\gamma +d\frac{ \beta}{|\Omega|}\int_\Omega Idx-\frac{d_I \beta }{d_S} I\right) \]

for $I\in E$, where

\[ E=\left\{I\in C_+(\overline\Omega): \frac{d_I}{d_S}I \ge \frac{d}{|\Omega|}\int_\Omega I dx \text{ and } \|I\|\le C\right\}, \]

$a$ is a positive constant, and $C=\max \{\|I_i\|, i=1, 2\}$. We may choose $a$ large enough so that both $I_1$ and $I_2$ are fixed points of $h$. Moreover, since $d>0$, if $a$ is large, then $h$ is strictly increasing on $E$, i.e., $h(u) > > h(v)$ for $u> v$ and $u, v\in E$. For any $I\in E$, using (1.5), we can show $\tilde kh(I)\le h(\tilde kI)$ for any $\tilde k\in (0, 1)$. In addition, if (1.3) holds, then $\tilde kh(I)<< h(\tilde kI)$ for any $\tilde k\in (0, 1)$.

Hence, we have

\[ kI_1=kh(I_1)<{<} h(kI_1) \le h(I_2)=I_2, \]

which contradicts $kI_1(x_1)=I_2(x_1)$. This proves the uniqueness of the positive solution of (1.1)–(1.2).

References

Deng, K. and Wu, Y.. Dynamics of a susceptible—infected—susceptible epidemic reaction—diffusion model. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 146 (2016), 929–946.CrossRefGoogle Scholar
Wu, Y. and Zou, X.. Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. Journal of Differential Equations 261 (2016), 4424–4447.CrossRefGoogle Scholar