<p>Cholera is an infectious disease with an incubation period, which can be characterized by age structure and time delay in mathematical modeling. In this work, we develop a mathematical model that incorporates the incubation period through an age structure. We then reformulate this model as a nonlocal time delay reaction-diffusion system. This allows us to analyze the combined effects of drug resistance, the incubation period, and spatial heterogeneity on cholera transmission. We begin by proving the model’s well-posedness and defining the reproduction numbers of <i>V. cholerae</i>, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2584_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$R_B$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>B</mi> </msub> </math></EquationSource> </InlineEquation>, and the basic reproduction number, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2584_Article_IEq2.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>. Threshold results indicate that the non-infected steady state exhibits global asymptotic stability when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2584_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$R_B < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>B</mi> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2584_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0 \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. However, if <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2584_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$R_B \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>B</mi> </msub> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, (or <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2584_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$R_B < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>B</mi> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2584_Article_IEq7.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 disease persists, with one or more disease-endemic steady states. Due to the challenges of analyzing global stability in a heterogeneous scenario, we first establish that in the homogeneous case, a unique globally asymptotically stable disease-endemic equilibrium exists. We then establish the global attractivity of the disease-endemic steady state in a specific heterogeneous scenario. These results provide theoretical insights into the effects of drug resistance and spatial heterogeneity on cholera transmission dynamics.</p>

Complete dynamics of a heterogeneous space diffusion cholera model with nonlocal time delay and drug resistance

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发表日期：

2025年9月13日

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Shuai Zhang,
Shijie Fan,
Peng Wu

摘要

Cholera is an infectious disease with an incubation period, which can be characterized by age structure and time delay in mathematical modeling. In this work, we develop a mathematical model that incorporates the incubation period through an age structure. We then reformulate this model as a nonlocal time delay reaction-diffusion system. This allows us to analyze the combined effects of drug resistance, the incubation period, and spatial heterogeneity on cholera transmission. We begin by proving the model’s well-posedness and defining the reproduction numbers of V. cholerae, $R_B$ $R_{B}$ , and the basic reproduction number, $R_0$ $R_{0}$ . Threshold results indicate that the non-infected steady state exhibits global asymptotic stability when $R_B < 1$ $R_{B} < 1$ and $R_0 \le 1$ $R_{0} \leq 1$ . However, if $R_B \ge 1$ $R_{B} \geq 1$ , (or $R_B < 1$ $R_{B} < 1$ , $R_0 > 1$ $R_{0} > 1$ ), the disease persists, with one or more disease-endemic steady states. Due to the challenges of analyzing global stability in a heterogeneous scenario, we first establish that in the homogeneous case, a unique globally asymptotically stable disease-endemic equilibrium exists. We then establish the global attractivity of the disease-endemic steady state in a specific heterogeneous scenario. These results provide theoretical insights into the effects of drug resistance and spatial heterogeneity on cholera transmission dynamics.