具有非线性发生率的SIRS流行病模型中产生5个极限环的余维5 Hopf分岔
A codimension-5 Hopf bifurcation yielding 5 limit cycles in a SIRS epidemic model with nonlinear incidence rate
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中文总结 AI 辅助
本文通过代数焦点化简和验证区间计算，严格证明SIRS模型在广义Hopf点可产生三、四、五个小振幅极限环，并给出余维6的代数障碍，支持五为最大余维的猜想。
中文摘要 AI 辅助
在《微分方程杂志》(384, 2024)最近发表的一篇论文中，Cui和Zhao研究了具有非线性发生率$\frac{kI^p}{1+\alpha I^q}$的SIRS流行病模型，其中$p>0$和$q\geq0$是任意实数。他们证明了该模型中Bogdanov-Takens分岔的余维至多为2，而Hopf分岔的余维则作为开放问题被留下。在本工作中，我们在整个分析过程中保持$p$和$q$自由，研究这一问题。一种策略性的无量纲化和参数化消除了正平衡点的指数依赖性，并将广义Hopf条件转化为代数焦点方程，这些方程与显式半代数容许区域耦合。利用代数焦点化简、严格半代数容许性和验证区间计算，我们严格证明了存在容许的非退化广义Hopf点，这些点产生三个、四个和五个小振幅极限环。特别地，我们严格验证了一个严格内部余维5弱焦点和满秩局部开折，从而证明了五个小振幅极限环可以从单个正平衡点分岔出来。我们进一步推导了余维6的精确代数障碍，并在广泛的连续和投影分量搜索中未发现容许候选。这些结果为“五为最大”的猜想提供了分析和数值证据，但未声称任何余维6不存在性定理。相同的参数化还提供了Bogdanov-Takens余维至多为2的简单证明。
英文摘要
In a recent paper published in the Journal of Differential Equations (384, 2024), Cui and Zhao investigated an SIRS epidemic model with the nonlinear incidence rate $\frac{kI^p}{1+αI^q}$, where $p>0$ and $q\geq0$ are arbitrary real numbers. They proved that the codimension of a Bogdanov-Takens bifurcation in this model is at most two, while the codimension of the Hopf bifurcation was left as an open problem. In this work, we investigate this problem while keeping $p$ and $q$ free throughout the analysis. A strategic nondimensionalization and parametrization remove the exponential dependence of the positive equilibrium and convert the generalized-Hopf conditions into algebraic focus equations coupled to explicit semialgebraic admissibility regions. Using algebraic focus reduction, strict semialgebraic admissibility, and validated interval computation, we rigorously prove the existence of admissible nondegenerate generalized Hopf points generating three, four, and five small-amplitude limit cycles. In particular, we rigorously certify a strict-interior codimension-five weak focus and a full-rank local unfolding, thereby proving that five small-amplitude limit cycles can bifurcate from a single positive equilibrium. We further derive exact algebraic obstructions to codimension six and find no admissible candidate in extensive continuation and projected-component searches. These results provide analytical and numerical evidence for the conjecture that five is maximal, but no codimension-six nonexistence theorem is claimed. The same parametrization also yields a simple proof that the Bogdanov-Takens codimension is at most two.
发表机构
- Western University（韦仕敦大学）
- Lanzhou University（兰州大学）
- Texas Tech University（德克萨斯理工大学）
机构由 AI 辅助整理，请以论文原文为准。
