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<p>In this paper, the complex dynamic behaviors (transcritical bifurcation at disease-free equilibrium and Hopf bifurcations at endemic equilibrium) of a SIS model with distributed time lag are exhaustively studied theoretically and numerically. The normal form coefficients of transcritical bifurcation and Hopf bifurcations are obtained by employing center manifold reduction and normal form theory, and projection method combined with Fredholm Alternative techniques, respectively. Curiously, a pair of supercritical Hopf bifurcations in opposite directions emerge near the endemic equilibrium, and a stable endemic bubble forms during the change of bifurcating parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11222_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>. The numerical results maintain a high degree of agreement with the theoretical conclusions.</p>

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Endemic bubble on a simple SIS model with distributed time lag

  • 期刊论文

  • 发表日期：

    2025年5月20日

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  • Xiangming Zhang,
  • Danni Wang,
  • Jiajia Tang,
  • Hai-Feng Huo

摘要

In this paper, the complex dynamic behaviors (transcritical bifurcation at disease-free equilibrium and Hopf bifurcations at endemic equilibrium) of a SIS model with distributed time lag are exhaustively studied theoretically and numerically. The normal form coefficients of transcritical bifurcation and Hopf bifurcations are obtained by employing center manifold reduction and normal form theory, and projection method combined with Fredholm Alternative techniques, respectively. Curiously, a pair of supercritical Hopf bifurcations in opposite directions emerge near the endemic equilibrium, and a stable endemic bubble forms during the change of bifurcating parameter \(\mathcal {R}_{0}\) R 0 . The numerical results maintain a high degree of agreement with the theoretical conclusions.