<p>This study examines whether a traveling wave solution for an addiction reaction-diffusion epidemic model with dispersed delays exists or not. The primary characteristic of the model is the potential weakness in the standard comparison principle, which prevents many known conclusions from being applied. If the basic reproduction number of the model, indicated by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_736_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>, is larger than one, there is a minimal wave speed, denoted by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_736_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>c</mi> <mo>∗</mo> </msup> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$c^{}>0$</EquationSource> </InlineEquation>, such that the system admits a nontrivial traveling wave solution with wave speed <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_736_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>c</mi> </math></EquationSource> <EquationSource Format="TEX">$c$</EquationSource> </InlineEquation> when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_736_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>c</mi> <mo>≥</mo> <msup> <mi>c</mi> <mo>∗</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$c\geq c^{}$</EquationSource> </InlineEquation>. However, for either <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_736_Article_IEq5.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> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_736_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>c</mi> <mo><</mo> <msup> <mi>c</mi> <mo>∗</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$c< c^{*}$</EquationSource> </InlineEquation>, or <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10440_2025_736_Article_IEq7.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>, there is no nontrivial traveling wave solution.</p>

Traveling Waves in a Reaction-Diffusion Addiction Epidemic Model with Distributed Delays

期刊论文
发表日期：

2025年6月23日

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Hamid Toumi,
Khaled Boudjema Djeffal,
Abdelheq Mezouaghi,
Rassim Darazirar,
Salih Djilali

摘要

This study examines whether a traveling wave solution for an addiction reaction-diffusion epidemic model with dispersed delays exists or not. The primary characteristic of the model is the potential weakness in the standard comparison principle, which prevents many known conclusions from being applied. If the basic reproduction number of the model, indicated by $R_{0}$ $R_{0}$ , is larger than one, there is a minimal wave speed, denoted by $c^{*} > 0$ $c^{*}>0$ , such that the system admits a nontrivial traveling wave solution with wave speed $c$ $c$ when $c \geq c^{*}$ $c\geq c^{*}$ . However, for either $R_{0} > 1$ $R_{0}>1$ and $c < c^{*}$ $c< c^{*}$ , or $R_{0} < 1$ $R_{0}<1$ , there is no nontrivial traveling wave solution.