<p>In this paper, we investigate traveling wave solutions for a delayed reaction-diffusion epidemic model incorporating both local and nonlocal diffusion mechanisms. The model describes the dynamics of susceptible, infected, and recovered populations, with infection spreading influenced by delayed interactions. The susceptible and recovered populations follow a nonlocal diffusion process, while the infected population undergoes local diffusion. We derive comprehensive results regarding the existence and nonexistence of traveling wave solutions. Specifically, we demonstrate that if the basic reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12064_2025_443_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12064_2025_443_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <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>, the system does not admit any traveling wave solutions. Conversely, when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12064_2025_443_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <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 identify a critical wave speed <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12064_2025_443_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho ^>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ρ</mi> <mo>∗</mo> </msup> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, such that for any <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12064_2025_443_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho \ge \rho ^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>≥</mo> <msup> <mi>ρ</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, the system admits a non-critical bounded traveling wave solution. For <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12064_2025_443_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho <\rho ^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo><</mo> <msup> <mi>ρ</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, however, the model does not admit bounded, non-negative traveling wave solutions. Numerical simulations are performed to validate these theoretical results, highlighting the influence of both diffusion and delay mechanisms on wave propagation in the SIR model.</p>

Traveling wave solutions in a delayed local and nonlocal diffusion model for a generalized SIR epidemic model

期刊论文
发表日期：

2025年7月17日

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Rassim Darazirar

摘要

In this paper, we investigate traveling wave solutions for a delayed reaction-diffusion epidemic model incorporating both local and nonlocal diffusion mechanisms. The model describes the dynamics of susceptible, infected, and recovered populations, with infection spreading influenced by delayed interactions. The susceptible and recovered populations follow a nonlocal diffusion process, while the infected population undergoes local diffusion. We derive comprehensive results regarding the existence and nonexistence of traveling wave solutions. Specifically, we demonstrate that if the basic reproduction number $R_0$ $R_{0}$ satisfies $R_0< 1$ $R_{0} < 1$ , the system does not admit any traveling wave solutions. Conversely, when $R_0 > 1$ $R_{0} > 1$ we identify a critical wave speed $\rho ^*>0$ $ρ^{*} > 0$ , such that for any $\rho \ge \rho ^*$ $ρ \geq ρ^{*}$ , the system admits a non-critical bounded traveling wave solution. For $\rho <\rho ^*$ $ρ < ρ^{*}$ , however, the model does not admit bounded, non-negative traveling wave solutions. Numerical simulations are performed to validate these theoretical results, highlighting the influence of both diffusion and delay mechanisms on wave propagation in the SIR model.