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<p>This paper conducts an in-depth investigation of a new spatio-temporal model for the cocaine–heroin epidemiological model with vital dynamics incorporating the Laplacian operator. The study rigorously establishes the existence, uniqueness, nonnegativity, and boundedness of solutions of the proposed model. In addition, the local stability of both drug-free and drug-addiction equilibria is analyzed by studying the corresponding characteristic equations. The research provides conclusive evidence that when the basic reproductive number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3924_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{R}_{0}$</EquationSource> </InlineEquation> exceeds 1, then the drug-addiction equilibrium is globally asymptotically stable. Conversely, using comparative arguments, it is shown that if <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_3924_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{R}_{0}$</EquationSource> </InlineEquation> is less than 1, then the drug-free equilibrium is globally asymptotically stable. Furthermore, the paper includes a series of numerical simulations to visually convey and support the analytical results.</p>

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Dynamical analysis of a cocaine–heroin epidemiological model with spatial distributions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月7日

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  • Achraf Zinihi,
  • Moulay Rchid Sidi Ammi,
  • Matthias Ehrhardt,
  • Ahmed Bachir

摘要

This paper conducts an in-depth investigation of a new spatio-temporal model for the cocaine–heroin epidemiological model with vital dynamics incorporating the Laplacian operator. The study rigorously establishes the existence, uniqueness, nonnegativity, and boundedness of solutions of the proposed model. In addition, the local stability of both drug-free and drug-addiction equilibria is analyzed by studying the corresponding characteristic equations. The research provides conclusive evidence that when the basic reproductive number R 0 $\mathcal{R}_{0}$ exceeds 1, then the drug-addiction equilibrium is globally asymptotically stable. Conversely, using comparative arguments, it is shown that if R 0 $\mathcal{R}_{0}$ is less than 1, then the drug-free equilibrium is globally asymptotically stable. Furthermore, the paper includes a series of numerical simulations to visually convey and support the analytical results.