具有年龄结构的周期传染病模型的研究及其应用

Seasonal fluctuation is a common phenomenon of infectious diseases. The transmission dynamics of many seasonal diseases depend on the physiological age and infection period. Based on the transmission of the hand-foot-mouth disease and tuberculosis, we will formulate the periodic epidemic models with physiological age and infection age structures. We will study the dynamical behavior of those models, including the definition and calculation of the basic reproduction number R0, the extinction and uniform persistence of the disease, the existence and stability of periodic solution, and complicated bifurcations, and discuss the effect of seasonality and age structures on the dynamics of the model. We further estimate the model parameters by using the recent statistical epidemic data, predict the transmission tendency of hand-foot-mouth disease and tuberculosis in our country, and give the assessment on the control measures and optimal strategies. The periodic models with more age structures will provide new idea to epidemiological modeling. The dynamical analysis of those models will explore new methods to solve these challenging problems in applied mathematics. The expected results of this project will promote the research of applied mathematics in epidemiology, and help a lot in the prevention of hand-foot-mouth disease and tuberculosis in our country.

季节性波动是疾病传播过程中的一个普遍现象，且许多季节性疾病的感染和发病都与年龄和感染时间相关。本项目将以手足口病和结核病的传播为背景，建立具有生理年龄和感染年龄结构的周期传染病模型，并研究其动力学性态。我们将给出这些模型基本再生数R0的定义和计算方法，研究疾病的消除、一致持续、周期解的存在性和稳定性及复杂的分支，分析季节性和年龄结构对模型动力学性态的影响。分别利用我国近年来手足口和结核病的统计数据确定模型的参数，对我国这两种病的流行趋势进行预测，并对目前实施的防控措施给出定量评估和优化建议。具有多种年龄结构的周期传染病模型将为传染病动力学开拓新的建模思路，其性态的分析将为解决这类十分困难的应用数学问题探索新的研究方法。项目的实施和成果将深化应用数学在传染病动力学中的研究，并为手足口和结核的建模和防治研究提供新思路。

许多季节性疾病的传播动力学都与时滞有关. 通过考虑疾病传播的染病年龄和季节性，本项目建立了几类在染病的方程中具有时滞的周期传染病模型. 此类模型由于在分析中的困难，已有的结果很少. 我们利用下一代再生算子的方法定义了模型的基本再生数，研究了模型关于基本再生数的动力学性态，如疾病的一致持续或消除、周期解的存在性. 进而通过数值模拟探讨季节性、时滞及其它关键参数对疾病传播动力学的影响. 项目的完成丰富了传染病动力学的研究理论，有着广泛的应用前景.