<p>In this paper, in order to study comprehensive effect of stage-structure, incomplete immunity and spatial diffusion on the transmission dynamics of sheep brucellosis, we formulate a reaction-diffusion brucellosis model with partial immunity and stage structure in heterogeneous environment. Firstly, the well-posedness of the system is investigated, including the existence of global solution and its ultimate boundedness, and then the basic reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3889_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$R_{0}$</EquationSource> </InlineEquation> is defined using the next generation operator. Further, the threshold criteria on the global dynamics of the model are established in terms of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3889_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$R_{0}$</EquationSource> </InlineEquation> in two special cases. That is, if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3889_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$R_{0} < 1$</EquationSource> </InlineEquation>, the disease-free steady state is globally asymptotically stable, while if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3889_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$R_{0} > 1$</EquationSource> </InlineEquation>, the model is uniformly persistent and there at least exists a endemic steady state. Furthermore, for the homogeneous space and heterogeneous diffusion model, by constructing suitable Lyapunov functions, we obtain the global asymptotic stability for the disease-free steady-state when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3889_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>≤</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$R_{0}\leq 1$</EquationSource> </InlineEquation> and the global asymptotic stability endemic steady states when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3889_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$R_{0}>1$</EquationSource> </InlineEquation>. Finally, two simulation examples are given to verify our theoretical results.</p>

Analysis of a diffusive brucellosis model with partial immunity and stage structure in heterogeneous environment

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2025年1月24日

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Tingting Zheng,
Yicheng Hao,
Yantao Luo,
Zhidong Teng

摘要

In this paper, in order to study comprehensive effect of stage-structure, incomplete immunity and spatial diffusion on the transmission dynamics of sheep brucellosis, we formulate a reaction-diffusion brucellosis model with partial immunity and stage structure in heterogeneous environment. Firstly, the well-posedness of the system is investigated, including the existence of global solution and its ultimate boundedness, and then the basic reproduction number $R_{0}$ $R_{0}$ is defined using the next generation operator. Further, the threshold criteria on the global dynamics of the model are established in terms of $R_{0}$ $R_{0}$ in two special cases. That is, if $R_{0} < 1$ $R_{0} < 1$ , the disease-free steady state is globally asymptotically stable, while if $R_{0} > 1$ $R_{0} > 1$ , the model is uniformly persistent and there at least exists a endemic steady state. Furthermore, for the homogeneous space and heterogeneous diffusion model, by constructing suitable Lyapunov functions, we obtain the global asymptotic stability for the disease-free steady-state when $R_{0} \leq 1$ $R_{0}\leq 1$ and the global asymptotic stability endemic steady states when $R_{0} > 1$ $R_{0}>1$ . Finally, two simulation examples are given to verify our theoretical results.