空间扩散与年龄结构的肺结核动力学建模与分析

As a chronic infectious disease caused by tubercle bacilli in the lungs, Tuberculosis becomes one of most serious infectious diseases which leads to the largest number of incidence and mortality in China. By means of epidemic dynamical method to study the spread of Tuberculosis is an attractive research topic in the field of mathematical biology whether in China or in the world. Although there are some excellent results, most of the models are limited by the research methods, the special factors and the age structure of Tuberculosis are always ignored. In view of such case, we will focus on the construction of the Tuberculosis dynamical models with spatial diffusion and age structure. Firstly, we will integrate the theory of dynamical system, partial differential equations and operator semi-group to investigate the stability and persistence in our models, define the basic reproductive number, and discuss the existence, uniqueness and stability of the steady states, the existence of the traveling waves, and analyze the biological mechanism of related threshold parameters. Secondly, based on the statistical data of Tuberculosis in China, we will employ statistical method to estimate and test the parameters in our models, explore the epidemic regularity of Tuberculosis in China, predict its development trend and find out the optimal control measures. We believe that this proposed research will not only systemically develop the theory and method of the reaction diffusion model with age structure in theory, but also reveal the epidemic regularity of Tuberculosis in China and its development trend and provide a theoretical basis for the prevention and control of Tuberculosis in practice. Thus this research is significant for both theoretical analysis and practical applications.

肺结核是由结核杆菌感染肺部而引起的慢性传染病，是我国发病及死亡人数较多的传染病之一。借助传染病动力学方法研究肺结核的传播已经成为国内外生物数学领域的研究热点，但已有的成果限于研究方法而忽略了肺结核传播的空间因素和年龄结构。有鉴于此，本项目将建立具有空间扩散与年龄结构的肺结核动力学模型。首先，利用动力系统、偏微分方程及算子半群理论，研究模型的定性、稳定性及持续性等问题，给出模型的基本再生数，稳态解的存在唯一性及局部或全局稳定性，以及行波解的存在性，分析相关阈值参数的生物学机制。其次，结合我国肺结核病的统计数据，利用统计方法对模型参数进行估计与假设检验，探索我国肺结核病的流行规律，预测其发展趋势，寻找最优控制措施。该项目的研究不仅在理论上系统地发展具有空间扩散和年龄结构模型的理论和方法，而且在实践上揭示我国肺结核病流行规律及发展趋势，为肺结核病的防控提供理论依据，因而具有丰富的理论和现实意义。

肺结核是我国发病及死亡人数较多的传染病之一，借助传染病动力学方法研究肺结核传播过程中空间扩散和年龄结构的影响，发现肺结核流行和传播规律，并为防控部门提供科学依据，具有丰富的理论和现实意义。从肺结核的发病机理与传播机制出发，结合肺结核传播过程中例如人群的迁移、肺结核接种疫苗、多族群及肺结核的信息传播等因素，建立几类肺结核传播的动力学模型。对所建立的模型运用非线性泛函分析、积分半群、算子谱理论、上下界、Lyapunov泛函等理论和方法进行分析，研究模型解的存在性和有界性，解半流的渐近光滑性和全局吸引子的存在性，给出基本再生数的数学表达式，研究平衡点的稳定性和行波解的存在性等动力学性态，分析参数对基本再生数的影响，进而提出合理的控制策略。通过该项目的研究，不仅在理论上系统地发展具有空间扩散和年龄结构模型的理论和方法，提供Lyapunov稳定性的框架性理论，而且为疾控中心等相关部门预防和控制肺结核的传播提供理论依据。本项目按照既定目标完成了一系列有关空间扩散和年龄结构的传染病建模及分析研究，已发表SCI学术论文11篇，参加国内外学术会议9次，做国内报告和国际报告各1次。