非线性脉冲微分方程模型在传染病动力学中的应用研究

Research on the Application of Nonlinear Impulsive Differential Equation Models in Infectious Disease Dynamics

摘要: 本文基于经典的SIR模型，构建了包含医疗资源有限以及固定时刻脉冲策略的传染病模型，旨在刻画间歇性疫苗接种对传染病控制的影响。通过运用频闪映射、不动点定理以及Lambert W函数的性质，分析了疾病根除周期解的存在性。接着，结合Floquet乘子理论、比较定理和不等式放缩方法，研究了系统的全局渐近稳定性。随后，通过数值模拟分析了医疗资源约束对疾病传播的影响，并结合敏感性分析识别出影响传染病传播的关键参数。研究结果表明，在固定时间节点实施疫苗接种策略能够有效抑制疾病传播，且接种间隔越短，防控效果越显著。

Abstract: In the context of limited medical resources, this paper builds an infectious disease model based on the classical SIR model, incorporating a fixed-time impulse vaccination strategy to examine the impact of intermittent vaccination on disease control. By utilizing stroboscopic mapping, the fixed-point theorem, and the properties of the Lambert W function, the existence of periodic solutions for disease eradication is analyzed. Subsequently, the global asymptotic stability of the system is studied using Floquet multiplier theory, comparison theorems, and inequality scaling methods. Subsequently, numerical simulations were conducted to analyze the impact of healthcare resource constraints on disease transmission, and sensitivity analysis was employed to identify the key parameters influencing the spread of infectious diseases. The results indicate that implementing vaccination strategies at fixed time points can effectively suppress disease transmission, and shorter vaccination intervals lead to more significant control effects.