一类具有饱和发生率的时滞SIR模型

A Delayed SIR Model with Saturated Incidence Rate

摘要: 本文研究了一类具有饱和发生率的时滞SIR模型，该模型通过引入饱和发生率，符合在实际生活中感染人数增长到一定程度后传播速度变缓的规律，使模型更加贴合实际情况。首先，以时滞为参数，利用二代再生矩阵的方法计算求得模型的基本再生数R₀，并通过线性化、特征理论等方法对无病平衡点与地方病平衡点处的特征方程进行分析，得出时滞对无病平衡点与地方病平衡点的稳定性的影响：当R₀ < 1时，无病平衡点在一定时滞范围内是渐近稳定的；当R₀ > 1时，地方病平衡点随时滞的变化而发生变化，并给出了地方病平衡点处Hopf分支的存在条件，最后利用MATLAB进行数值模拟，验证理论结果的正确性，得到时滞会使系统的稳定性发生变化。

Abstract: This paper studies a class of delayed SIR models with a saturated incidence rate. By introducing the saturated incidence rate, the model conforms to the real-world law that the transmission speed slows down when the number of infections grows to a certain level, thereby making the model more consistent with practical scenarios. Firstly, taking time delay as a parameter, the basic reproduction number R₀ of the model is calculated using the next-generation matrix method. Subsequently, the characteristic equations at the disease-free equilibrium point and the endemic equilibrium point are analyzed via linearization, characteristic theory, and other methods, leading to the derivation of the influence of time delay on the stability of these two equilibrium points: when R₀ < 1, the disease-free equilibrium point is asymptotically stable within a certain range of time delay; when R₀ > 1, the endemic equilibrium point varies with changes in time delay, and the existence condition for Hopf bifurcation at the endemic equilibrium point is provided. Finally, numerical simulations are performed using MATLAB to verify the correctness of the theoretical results, confirming that time delay can alter the stability of the system.