基本再生数是传染病动力学模型中的一个重要参量, 其计算对分析疾病传播动态以及评估防控策略等具有指导性意义. 本文讨论了传染病仓室模型的三种基本再生数推导方法, 计算了三种不同发生率SIR模型的基本再生数. 在经典SEIR模型的基础上增加年龄因素, 研究了两类具有年龄结构的传染病模型. 建立了具有连续分布年龄结构的SEIR模型, 并通过稳定年龄分布将偏微分方程模型转化为常微分方程模型进行求解, 由正平衡点的存在性推导出模型的基本再生数. 考虑到如麻疹、手足口病等儿童传染病的年龄因素和疫苗接种的影响, 以学龄前儿童作为研究对象进行年龄分组, 建立了具有分组年龄结构和接种免疫的SEIR模型, 并通过三种方法求得模型的基本再生数. 对模型进行数值模拟可以发现, 提高疫苗接种率能有效地控制传染病的发展.

The basic reproduction number is an important parameter in the epidemic models, and its calculation has guiding significance for analyzing the laws of disease transmission and adopting prevention and control strategies. This article discusses three basic reproduction number derivation methods for epidemic models and derives the basic reproduction numbers for three SIR models with different incidence rates. On the basis of the classic SEIR model, age factors were added to study two different infectious disease models with age structures. A SEIR model with a continuous age distribution structure was established, and the partial differential equation model was transformed into an ordinary differential equation model for solution by stabilizing the age distribution. The basic reproduction number of the model was derived from the existence of the positive steady-state solution. Considering the age factors of infectious diseases in children such as measles and hand-foot-and-mouth disease(HMFD), as well as the impact of vaccination, preschool children were selected as the research subjects for age grouping. A SEIR model with grouping age structure and vaccination immunity was established, and the basic reproduction number of the model was obtained through three methods. Numerical simulation of the model reveals that increasing vaccination rates can effectively control the development of infectious diseases.