<p>This paper explores traveling wave phenomena in an SIS epidemic model incorporating nonlinear incidence and age structure. After establishing the model’s mathematical well-posedness (solution existence and uniqueness), we examine conditions for infection spread. Based on the basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {R}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, we demonstrate that if <InlineEquation ID="IEq2"> <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> and the wave speed satisfies <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$c > c^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mmultiscripts> <mi>c</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation>, traveling waves exist, describing the spatial propagation of infection. For the same <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> but wave speeds <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c \in (0,c^{})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mmultiscripts> <mi>c</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the infection cannot successfully propagate through space. Thus, both a sufficiently large reproductive number and a sufficiently fast wave speed are required for the disease to induce a significant and persistent spatial spread in the population.</p>

Analysis of traveling wave solutions for an SIS model with age structure

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发表日期：

2026年2月4日

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Soufiane Bentout

摘要

This paper explores traveling wave phenomena in an SIS epidemic model incorporating nonlinear incidence and age structure. After establishing the model’s mathematical well-posedness (solution existence and uniqueness), we examine conditions for infection spread. Based on the basic reproduction number $\mathcal {R}_0$ $R_{0}$ , we demonstrate that if $\mathcal {R}_0 > 1$ $R_{0} > 1$ and the wave speed satisfies $c > c^{*}$ $c > c^{*}$ , traveling waves exist, describing the spatial propagation of infection. For the same $\mathcal {R}_0 > 1$ $R_{0} > 1$ but wave speeds $c \in (0,c^{*})$ $c \in (0, c^{*})$ , the infection cannot successfully propagate through space. Thus, both a sufficiently large reproductive number and a sufficiently fast wave speed are required for the disease to induce a significant and persistent spatial spread in the population.