标题： 多周期动力学和SIRS流行病模型中高余维分支的三次心理饱和接触率 显示英文标题

标题： Multicycle dynamics and high-codimension bifurcations in SIRS epidemic models with cubic psychological saturated incidence

摘要： 本研究探讨了具有三次饱和接触率的SIRS流行病模型中的分岔动力学，扩展了Lu、Huang、Ruan和Yu（《微分方程杂志》，267，2019年）建立的二次饱和框架。 我们严格证明了三阶Bogdanov-Takens分岔和退化Hopf分岔的存在性，首次在流行病建模中展示了三个极限环的共存。 通过创新地应用奇异性理论，我们通过奇点的局部展开和前缘非退化奇点的识别来表征分岔集的拓扑结构。 我们的结果表明，在单调和非单调饱和条件下，三次非线性比二次模型诱导出更丰富的动态结构。 数值模拟验证了单调参数区域中的三个极限环和非单调区域中的两个极限环。 本研究通过引入高阶相互作用和全面的奇异性分析，推进了现有的分岔研究，从而为解码关键的公共卫生策略设计中的复杂传播机制提供了数学基础。 显示英文摘要

摘要： This study investigates bifurcation dynamics in an SIRS epidemic model with cubic saturated incidence, extending the quadratic saturation framework established by Lu, Huang, Ruan, and Yu (Journal of Differential Equations, 267, 2019). We rigorously prove the existence of codimension-three Bogdanov-Takens bifurcations and degenerate Hopf bifurcations, demonstrating, for the first time in epidemiological modeling, the coexistence of three limit cycles. By innovatively applying singularity theory, we characterize the topology of the bifurcation set through the local unfolding of singularities and the identification of nondegenerate singularities for fronts. Our results reveal that cubic nonlinearities induce significantly richer dynamical structures than quadratic models under both monotonic and nonmonotonic saturation. Numerical simulations verify three limit cycles in monotonic parameter regimes and two limit cycles in nonmonotonic regimes. This work advances existing bifurcation research by incorporating higher-order interactions and comprehensive singularity analysis, thereby providing a mathematical foundation for decoding complex transmission mechanisms that are critical to the design of public health strategies.