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<p>Multi-group structures within the population can be used for modeling the spatially heterogeneous mode of transmission. This work examines whether traveling waves exist or not in a delayed two-group SIR epidemic model with a generalized incidence function. We show whether traveling waves exist or not, depending on the basic reproduction number, <i>R</i><sub>0</sub>. More precisely, we show that: (i) there is a minimal wave speed denoted <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2474_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(c^* &gt; 0\)</EquationSource> </InlineEquation> when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2474_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">\(R_0 &gt; 1\)</EquationSource> </InlineEquation>; (ii) when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2474_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(R_0 \leq 1\)</EquationSource> </InlineEquation>, the system has no nontrivial traveling wave solution. The system admits a nontrivial traveling wave solution with wave speed <i>c</i> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2474_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(c &gt; c^*\)</EquationSource> </InlineEquation>, and there is no nontrivial traveling wave solution when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2474_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(c &lt; c^*\)</EquationSource> </InlineEquation>. Finally, we provide some numerical simulations showing the existence of a traveling wave solution. Our study can provide valuable insights for controlling disease outbreaks in spatially heterogeneous environments.</p>

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Traveling wave solution for a delayed reaction-diffusion two-group SIR epidemic model with a generalized nonlinear incidence function

  • 期刊论文

  • 发表日期：

    2025年4月30日

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  • Salih Djilali,
  • Rassim Darazirar,
  • Ibrahim Alraddadi

摘要

Multi-group structures within the population can be used for modeling the spatially heterogeneous mode of transmission. This work examines whether traveling waves exist or not in a delayed two-group SIR epidemic model with a generalized incidence function. We show whether traveling waves exist or not, depending on the basic reproduction number, R0. More precisely, we show that: (i) there is a minimal wave speed denoted \(c^* > 0\) when \(R_0 > 1\) ; (ii) when \(R_0 \leq 1\) , the system has no nontrivial traveling wave solution. The system admits a nontrivial traveling wave solution with wave speed c for \(c > c^*\) , and there is no nontrivial traveling wave solution when \(c < c^*\) . Finally, we provide some numerical simulations showing the existence of a traveling wave solution. Our study can provide valuable insights for controlling disease outbreaks in spatially heterogeneous environments.