<p>To explore the effects of latent periods in humans and vectors on the propagation of the Zika virus in a heterogeneous environment, a nonlocal and delayed reaction-diffusion model incorporating various transmission pathways and general incidences is developed. The well-posedness of the model is established, including the existence of global positive solutions and a global attractor. By applying spectral theory, we define the basic reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1352_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {R}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> and derive its variational expression. Specifically, if <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1352_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {R}_0\le 1$</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>, the disease-free steady state (DFSS) is globally asymptotically stable. In contrast, if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1352_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <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>, the disease is persistent, and the model exhibits at least one positive steady state (PSS). For the particular situation of spatial homogeneity, the global attractivity of the PSS is established through the construction of suitable Lyapunov functional. Finally, numerical simulations demonstrate the primary theoretical conclusions and analyze the effects of latent periods and diffusion coefficients on the spread of the virus, as well as the sensitivity of the key parameters of the model to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1352_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {R}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

Dynamic Analysis of a Nonlocal and Delayed Reaction-Diffusion Zika Virus Model with Spatial Heterogeneity

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2025年8月18日

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Jingyun Shen,
Shengfu Wang,
Hong Cao,
Linfei Nie

摘要

To explore the effects of latent periods in humans and vectors on the propagation of the Zika virus in a heterogeneous environment, a nonlocal and delayed reaction-diffusion model incorporating various transmission pathways and general incidences is developed. The well-posedness of the model is established, including the existence of global positive solutions and a global attractor. By applying spectral theory, we define the basic reproduction number $\mathcal {R}_0$ $R_{0}$ and derive its variational expression. Specifically, if $\mathcal {R}_0\le 1$ $R_{0} \leq 1$ , the disease-free steady state (DFSS) is globally asymptotically stable. In contrast, if $\mathcal {R}_0>1$ $R_{0} > 1$ , the disease is persistent, and the model exhibits at least one positive steady state (PSS). For the particular situation of spatial homogeneity, the global attractivity of the PSS is established through the construction of suitable Lyapunov functional. Finally, numerical simulations demonstrate the primary theoretical conclusions and analyze the effects of latent periods and diffusion coefficients on the spread of the virus, as well as the sensitivity of the key parameters of the model to $\mathcal {R}_0$ $R_{0}$ .