具有年龄结构的多菌株传染病建模与最优控制研究

项目名称： 具有年龄结构的多菌株传染病建模与最优控制研究

项目编号： No.61203228

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 自动化学科

项目作者： 杨俊元

作者单位： 运城学院

项目金额： 24万元

中文摘要： 现有的肺结核、艾滋病和流感等多菌株传染病模型大多假定个体无差异，使得模型在信息量和合理性方面存在明显缺陷，具有年龄结构的模型可以克服这种不足。本项目针对一些具体的多菌株传染病不同菌株及菌株之间的重复感染、共同感染、变异、交叉免疫等疾病的传播机理，建立具有年龄结构的多菌株传染病动力学模型；用半群理论、构造Lyapunov泛函的方法、单调系统理论和分支理论研究所建立模型的动力学特征：基本再生数和侵入再生数、各平衡态的稳定性、一致持续生存、后向分支和Hopf分支等；再根据参数敏感性分析确立控制参数，利用Pontryagin最大值原理建立最优控制问题，计算最优控制时间和最优花费；最后采用最小二乘法估计参数，选择适当的差分格式并设计合理的计算方法，模拟达到对这种病的预测和控制的目的。该研究将为年龄结构多菌株传染病全局动力学研究提供理论方法,同时也可为其预防和控制提供理论依据。

中文关键词： 基本再生数；侵入再生数；博弈论；共同感染；积分半群

英文摘要： In most existing multi-strain epidemic models like tuberculosis, HIV, and influenza, there is no difference among the individuals. This leads to the deficiency of the models in the amount of information and rationality. Age-structured models can overcome this drawback. In this project, for some concrete diseases, we build some age-structured multi-strain models to reflect dynamics of various strains and the transmission mechanisms including reinfection, coinfection, mutation and cross-immunity. Their dynamics will be studied by applying semigroup theory, Lyapunov function method, theory of monotone systems, and bifurcation theory. Results on basic reproduction number and invasion number, stability of equilibria, persistence, backward bifurcation, and Hopf bifurcation will be obtained. We further use the sensitivity analysis of parameters to determine the control variables and establish optimal control problems by employing the Pontryagin maximum principle. The optimal control times and optimal cost will be computed. Finally, we use numerical simulations to achieve the goals of predicting and controlling disease. For this purpose, we estimate the parameters by the least squares method, and choose suitable ways to discretize systems and reasonable computation methods. This project provides with not only theoret

英文关键词： basic reproduction number；invasion reproduction number；game theory；coinfectiion；integral semigroup