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<p>This paper proposes a nonlinear state-dependent hybrid model with a dynamic threshold strategy. We first introduce the concepts of impulse and phase sets, and define the Poincaré map to analyze its properties. Next, we investigate the existence and stability of boundary periodic solution and positive order-k periodic solutions. The existence of boundary periodic solutions suggests that comprehensive treatment can be administered at fixed intervals, thereby reducing the need for frequent monitoring of tumor size. Subsequently, we theoretically prove the existence and types of bifurcations, and derive the formula for calculating the maximum Lyapunov exponent to verify the presence of chaos. Finally, we perform numerical simulations to analyze the types of bifurcations using <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2024_10847_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(P_{max}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mrow> <mi mathvariant="italic">max</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, <i>a</i>, <i>K</i>, and <i>AT</i> as bifurcation parameters, along with the corresponding maximum Lyapunov exponent diagrams. The chaotic phenomena observed in the bifurcation diagrams indicate irregular variations in tumor size, which pose challenges for tumor monitoring. The study results demonstrate that relying solely on surgery, without immune memory, cannot fully control the tumor. However, the combination of surgery and immunotherapy not only effectively controls cancer but also helps maintain immune system functionality.</p>

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Global analysis and optimal therapy in immunogenic tumors: a nonlinear state-dependent hybrid model with a dynamic threshold policy

  • 期刊论文

  • 发表日期：

    2025年1月9日

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  • Wenjie Qin,
  • Xingxiao Wu

摘要

This paper proposes a nonlinear state-dependent hybrid model with a dynamic threshold strategy. We first introduce the concepts of impulse and phase sets, and define the Poincaré map to analyze its properties. Next, we investigate the existence and stability of boundary periodic solution and positive order-k periodic solutions. The existence of boundary periodic solutions suggests that comprehensive treatment can be administered at fixed intervals, thereby reducing the need for frequent monitoring of tumor size. Subsequently, we theoretically prove the existence and types of bifurcations, and derive the formula for calculating the maximum Lyapunov exponent to verify the presence of chaos. Finally, we perform numerical simulations to analyze the types of bifurcations using \(P_{max}\) P max , a, K, and AT as bifurcation parameters, along with the corresponding maximum Lyapunov exponent diagrams. The chaotic phenomena observed in the bifurcation diagrams indicate irregular variations in tumor size, which pose challenges for tumor monitoring. The study results demonstrate that relying solely on surgery, without immune memory, cannot fully control the tumor. However, the combination of surgery and immunotherapy not only effectively controls cancer but also helps maintain immune system functionality.