<p>In this paper, we investigate a diffusive susceptible-infected-susceptible (SIS) epidemic model with a mass action infection mechanism on an evolving domain. The model incorporates logistic population growth for susceptible individuals and the evolution of the time-varying domain is governed by a scaling function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We establish the global existence and uniform boundedness of solutions to the system. Furthermore, we define the basic reproduction number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {R}_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, which relies on the evolution rate of the domain, the diffusion coefficient of the infected populations, etc. We show that the disease-free equilibrium (DFE) is globally stable if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {R}_{0}< 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, whereas an endemic equilibrium (EE) exists and is globally stable for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {R}_{0}> 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in homogeneous environments. We also analyze the asymptotic profiles of the EE for large and small diffusion rates of the susceptible and infected populations. Our numerical results for the specific scaling function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\rho (t) = 1+ (\rho _\infty - 1)(1 - e^{-\alpha ^* t})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>∞</mi> </msub> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msup> <mi>α</mi> <mo>∗</mo> </msup> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> indicate that domain expansion may hinder disease elimination compared to fixed domains.</p>

The diffusive SIS epidemic model with mass action infection mechanism on an evolving domain

期刊论文
发表日期：

2026年7月14日

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Delong Dong,
Danhua Jiang

摘要

In this paper, we investigate a diffusive susceptible-infected-susceptible (SIS) epidemic model with a mass action infection mechanism on an evolving domain. The model incorporates logistic population growth for susceptible individuals and the evolution of the time-varying domain is governed by a scaling function $\rho (t)$ $ρ (t)$ . We establish the global existence and uniform boundedness of solutions to the system. Furthermore, we define the basic reproduction number $\mathcal {R}_{0}$ $R_{0}$ , which relies on the evolution rate of the domain, the diffusion coefficient of the infected populations, etc. We show that the disease-free equilibrium (DFE) is globally stable if $\mathcal {R}_{0}< 1$ $R_{0} < 1$ , whereas an endemic equilibrium (EE) exists and is globally stable for $\mathcal {R}_{0}> 1$ $R_{0} > 1$ in homogeneous environments. We also analyze the asymptotic profiles of the EE for large and small diffusion rates of the susceptible and infected populations. Our numerical results for the specific scaling function $\rho (t) = 1+ (\rho _\infty - 1)(1 - e^{-\alpha ^* t})$ $ρ (t) = 1 + (ρ_{\infty} - 1) (1 - e^{- α^{*} t})$ indicate that domain expansion may hinder disease elimination compared to fixed domains.