<p>In this paper, we deal with the sharp threshold results of a mathematical epidemic model for vector-borne diseases. We construct a degenerate reaction–diffusion equation system with spatially heterogeneous parameters to consider the situation where vectors move around randomly in a spatially heterogeneous environment. We then define the basic reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2545_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℜ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> as a threshold parameter to predict whether the malaria will spread or not. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2545_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0 < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℜ</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, whereas the system is uniformly persistent if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2545_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℜ</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We also confirm that in the case of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2545_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0=1$</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 will go extinct. In addition, the global asymptotic stability of the unique constant positive equilibrium is investigated by using a Lyapunov function method in a special case where all parameters are spatially homogeneous.</p>

Analysis of a partially diffusive vector-borne disease model with human-to-human infection in a spatially heterogeneous environment

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2025年7月14日

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Jinliang Wang,
Han Lu,
Toshikazu Kuniya

摘要

In this paper, we deal with the sharp threshold results of a mathematical epidemic model for vector-borne diseases. We construct a degenerate reaction–diffusion equation system with spatially heterogeneous parameters to consider the situation where vectors move around randomly in a spatially heterogeneous environment. We then define the basic reproduction number $\Re _0$ $ℜ_{0}$ as a threshold parameter to predict whether the malaria will spread or not. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if $\Re _0 < 1$ $ℜ_{0} < 1$ , whereas the system is uniformly persistent if $\Re _0>1$ $ℜ_{0} > 1$ . We also confirm that in the case of $\Re _0=1$ $ℜ_{0} = 1$ , the disease will go extinct. In addition, the global asymptotic stability of the unique constant positive equilibrium is investigated by using a Lyapunov function method in a special case where all parameters are spatially homogeneous.