<p>This study examines traveling wave solutions of the SIS epidemic model with nonlocal dispersion and delay. The research shows that a key factor in determining whether traveling waves exist is the basic reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2055_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$R_{0}$</EquationSource> </InlineEquation>. In particular, the system permits nontrivial traveling wave solutions for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2055_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>σ</mi> <mo>≥</mo> <msup> <mi>σ</mi> <mo>∗</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$\sigma \geq \sigma ^{}$</EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2055_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$R_{0} > 1$</EquationSource> </InlineEquation>, whereas there are no such solutions for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2055_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>σ</mi> <mo><</mo> <msup> <mi>σ</mi> <mo>∗</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$\sigma < \sigma ^{}$</EquationSource> </InlineEquation>. This is because there is a minimal wave speed <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2055_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>σ</mi> <mo>∗</mo> </msup> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\sigma ^{*}> 0$</EquationSource> </InlineEquation>. On the other hand, there are no traveling wave solutions when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2055_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>≤</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$R_{0} \leq 1$</EquationSource> </InlineEquation>. In conclusion, we provide several numerical simulations that illustrate the existence of TWS.</p>

Minimal wave speed and traveling wave in nonlocal dispersion SIS epidemic model with delay

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2025年5月12日

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Rassim Darazirar,
Rasha M. Yaseen,
Ahmed A. Mohsen,
Aziz Khan,
Thabet Abdeljawad

摘要

This study examines traveling wave solutions of the SIS epidemic model with nonlocal dispersion and delay. The research shows that a key factor in determining whether traveling waves exist is the basic reproduction number $R_{0}$ $R_{0}$ . In particular, the system permits nontrivial traveling wave solutions for $σ \geq σ^{*}$ $\sigma \geq \sigma ^{*}$ for $R_{0} > 1$ $R_{0} > 1$ , whereas there are no such solutions for $σ < σ^{*}$ $\sigma < \sigma ^{*}$ . This is because there is a minimal wave speed $σ^{*} > 0$ $\sigma ^{*}> 0$ . On the other hand, there are no traveling wave solutions when $R_{0} \leq 1$ $R_{0} \leq 1$ . In conclusion, we provide several numerical simulations that illustrate the existence of TWS.