具有类年龄结构和空间异质性的传染病动力学的建模与研究

The emergence of new infectious diseases，the re-raging of old infectious diseases in the world, and the emergence and prevalence of infectious diseases caused by biological invasions pose a severe challenge to humanity. The prevention and control of the dissemination of infectious diseases have become the highest priority of global public security problems, and the age of population, the course of disease and spatial heterogeneity are important factors that affect the spread of infectious diseases. Based on the characteristics of the spread of some specific infectious diseases and the actual background of their control, the aim of this project is to explore and establish epidemic models with class-age structure and spatial heterogeneity in order to more accurately characterize the dynamics of the spread of infectious diseases, and study the dynamics of the models by using modern theory of differential equations and dynamical systems. Some specific infectious disease models with susceptible physiological age, incubation age, age since infection, age since vaccination and spatial heterogeneity, as well as hepatitis B virus infection models with infected age of cells, host immune response and spatial diffusion of free virus will be established, respectively. Furthermore, we will calculate the basic reproduction numbers of diseases, analyze the local and global stability of feasible equilibria, Hopf bifurcation, backward bifurcation, the existence of traveling wave solutions and the asymptotic wave velocity, reveal the impact of class-age structure and spatial heterogeneity on the spread and control of infectious diseases, assess the efficacy and potential risks of disease control measures, and provide important theoretical basis for the scientific decision-making of prevention and control of infectious diseases.

全球新传染病的不断出现、旧传染病的重新肆虐以及生物入侵人为造成的传染病发生和流行使人类面临严峻挑战，预防与控制传染病的传播已成为全球公共安全最为优先考虑的问题，而种群年龄、疾病病程和空间的异质性则是影响传染病传播的重要因素。本项目根据某些具体传染病的传播特性和控制的实际背景，探索建立具有类年龄结构和空间异质性的传染病模型以更精确地刻画传染病的传播规律，利用现代微分方程与动力系统理论研究模型的动力学性态。分别建立具有易感者年龄、疾病潜伏年龄、感染年龄、接种年龄和空间异质性的某些具体传染病模型以及具有细胞感染年龄、宿主免疫反应和病毒空间扩散的乙肝病毒感染模型，计算疾病基本再生数，分析模型可行平衡点的局部和全局稳定性、Hopf分支、后向分支、行波解及渐近波速等问题，揭示类年龄结构和空间异质性对传染病传播和控制的影响，评估传染病控制措施的有效性和潜在风险，为传染病防制的科学决策提供重要的理论依据。

本项目根据某些具体传染病的传播特性和控制的实际背景，主要研究了类年龄结构和空间异质性对某些具体传染病的时空传播动力学的影响，探索建立了能更精确地刻画传染病传播动力学特性的类年龄结构传染病模型。所谓类年龄，是指个体从进入某一类仓室（如染病者类、潜伏者类、疫苗接种者）开始在该类中所度过的时间。在建模过程中假定染病者在染病期内的传染力、因病死亡率和康复率等均随染病年龄（个体在染病者类中停留的时间）而变化。应用非线性分析、微分方程与动力系统等现代数学理论和方法对模型的全局动力学性态展开相关理论研究和数值模拟。本项目重点研究了疾病潜伏年龄、染病年龄、接种年龄、疾病复发、疫苗接种策略(接种不完全有效和疫苗接种获得的特异性免疫衰减)、疾病发生率和种群的空间异质性等多重因素对传染病的时空传播的影响以及细胞感染年龄、细胞间直接感染、宿主免疫反应和游离病毒的空间异质性等因素对HIV抗病毒治疗的影响；评估了传染病控制措施的有效性和潜在风险，为传染病预防与控制的科学决策提供重要的理论依据。通过计算得到了疾病基本再生数（病毒感染基本再生率）；分析了由系统产生的连续解半流的渐近光滑性；应用无穷维动力系统持续生存理论研究了模型的一致持续生存性；通过提出一种分析特征方程根的分布的新方法完整解决了模型的可行稳态解的局部渐近稳定性问题；通过构造适当的Volterra型Lyapunov泛函和应用LaSalle不变集原理得到了模型的各类可行稳态解的全局渐近稳定性阈值。对于某些具有logistic增长、细胞间直接感染和非线性感染率的类年龄结构病毒感染动力学模型，应用积分半群理论和非稠定半线性方程的Hopf分支理论给出了系统存在Hopf分支的充分性条件。