AI 中文总结
研究针对登革热传播，构建纳入医院容量和阈值触发喷雾策略的非光滑常微分方程模型，确立平衡点性质，通过数值延拓揭示分岔，研究周期解确定最优干预模式，为有限资源下登革热控制策略设计提供定量见解。
AI 中文摘要
登革热仍是热带地区主要的公共卫生挑战，反复爆发表明当前干预策略效果欠佳。现有数学模型常假定医院容量无限且喷雾持续进行，忽略实际约束。我们构建了一个非光滑常微分方程模型，纳入有限医院容量及报告感染超过可用容量规定比例时触发的喷雾策略。该模型呈现三种流行病学相关运行模式。我们确立了无病和地方病平衡点的存在性与局部稳定性。数值延拓证实了分析结果，揭示了切换阈值处的边界平衡分岔、医院容量超限时导致持续振荡爆发的霍普夫分岔以及流行阈值附近产生额外不稳定平衡点的折叠分岔。我们还研究了喷雾率和激活阈值的周期解，确定了局部最优干预模式。结果表明医院容量、反应性喷雾和干预阈值从根本上塑造了登革热动态，为在有限医疗资源下设计有效的状态依赖控制策略提供了定量见解。
英文摘要
Dengue remains a major public health challenge in tropical regions, and recurring outbreaks suggest that current intervention strategies are not yet fully effective. Existing mathematical models typically assume unlimited hospital capacity and continuously applied fogging, neglecting practical constraints that strongly influence disease control. We develop a non-smooth ordinary differential equation model of dengue transmission that incorporates finite hospital capacity and a threshold-triggered fogging strategy activated when reported infections exceed a prescribed fraction of the available capacity. The model exhibits three epidemiologically relevant operating regimes, reflecting changes in hospitalization and vector-control policies as the epidemic progresses. We establish the existence and local stability of the disease-free and endemic equilibria. Numerical continuation confirms the analytical results and reveals boundary-equilibrium bifurcations at the switching thresholds, a Hopf bifurcation after hospital capacity is exceeded leading to sustained oscillatory outbreaks, and a fold bifurcation near the epidemic threshold that generates additional unstable equilibria. We further investigate periodic solutions with respect to the fogging rate and activation threshold, identifying locally optimal intervention regimes that reduce epidemic peaks while avoiding unnecessarily intensive control efforts. The results demonstrate that hospital capacity, reactive fogging, and intervention thresholds fundamentally shape dengue dynamics and provide quantitative insights for designing effective state-dependent control strategies under limited healthcare resources.
