<p>We investigate the asymptotic behavior of a novel SIS reaction–diffusion epidemic model in a heterogeneous environment, incorporating nonlocal transmission instead of classical pointwise infection. Our approach examines how infected individuals transmit the disease over long distances through a nonlocal contagion mechanism. Specifically, we assume that susceptible individuals at location <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x$</EquationSource> <EquationSource Format="MATHML"><math> <mi>x</mi> </math></EquationSource> </InlineEquation> become infected through interactions with infected individuals located within a neighborhood <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega _1(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For this nonlocal transmission model, we successfully identify the basic reproduction number <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. We establish the global existence and uniform boundedness of classical solutions, and we prove their convergence to nontrivial equilibrium states as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t \rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. In particular, when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_0 > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we analyze the asymptotic profiles of endemic equilibria under various scenarios for the dispersal coefficients <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$d_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. These results demonstrate how nonlocal infection and environmental heterogeneity jointly shape the persistence and spatial distribution of the disease, providing a more realistic framework for understanding spatial epidemic dynamics.</p>

Dynamics of a Reaction-Diffusion SIS Epidemic Model with Nonlocal Transmission

期刊论文
发表日期：

2026年4月23日

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Salih Djilali

摘要

We investigate the asymptotic behavior of a novel SIS reaction–diffusion epidemic model in a heterogeneous environment, incorporating nonlocal transmission instead of classical pointwise infection. Our approach examines how infected individuals transmit the disease over long distances through a nonlocal contagion mechanism. Specifically, we assume that susceptible individuals at location $x$ $x$ become infected through interactions with infected individuals located within a neighborhood $\Omega _1(x)$ $Ω_{1} (x)$ . For this nonlocal transmission model, we successfully identify the basic reproduction number $R_0$ $R_{0}$ . We establish the global existence and uniform boundedness of classical solutions, and we prove their convergence to nontrivial equilibrium states as $t \rightarrow +\infty $ $t \to + \infty$ . In particular, when $R_0 > 1$ $R_{0} > 1$ , we analyze the asymptotic profiles of endemic equilibria under various scenarios for the dispersal coefficients $d_1$ $d_{1}$ and $d_2$ $d_{2}$ . These results demonstrate how nonlocal infection and environmental heterogeneity jointly shape the persistence and spatial distribution of the disease, providing a more realistic framework for understanding spatial epidemic dynamics.