时滞传染病动力学中的Lyapunov函数构造及稳定性分析

Differential equation is a central area of mathmetics, and one of its recent and most important applications is in mathematical biology and medicine. When one introduces a time delay into a system of differential equations, it is often interest to determine whether or not bifucations occur for various lengths of the delay. In particular, many important mathematical models in epidemiology and virology described by delay differential equations. .Usually, scientists have utilized certain geometric methods for studying whether and when a bifucation occurs about a steady state. We know, Lyapunov function based methods paly a central role in stability analysis. Given the complexity of dynamical behavior possible even in low dimensions, these methods are powerful because they provide an analysis and design approach for global stability of an equilibrium solution. Global properties are important to dynamical behavior of epidemiological models since they can help us to clear virus development and disease spread. Constructing Lyapunov functions (functional) is direct and powerful to deal with stability analysis for general nonlinear systems. Our research focuses on using Lyapunov function to analyze global stability of all steady states. Based on our pervious works, we would develop new techniques to construct perfect Lyapunov functions and estabilsh stability criteria for some well-known epidemical models and virus infection models.

传染病数学模型及其动力学性质的研究在预防疾病传播和提出控制策略方面具有重要意义.通过研究符合实际背景和生物意义的微分方程(包括时滞微分方程，年龄结构的偏微分方程）来研究疾病的传播机制，并给出传染病发展趋势的估计是数学流行病学研究的一个重要课题.其研究方法包括经典的动力学理论，分支理论及Lyapunov稳定性理论等等.在传染病动力学中，时滞在疾病的传播和扩散过程中起到重要的作用，时滞可以由许多因素引起，一般用时滞来模拟传染病的潜伏期，患者对疾病的感染期以及康复者对疾病的免疫期等等.时滞的引入使得我们的模型更接近实际，同时也使得对其数学的分析更加困难.我们希望通过直接构造Lyapunov函数（泛函）的方法和LaSalle不变性原理，获得系统平衡点依赖于基本再生数阀值的动力学性质.系统的全局动力学性质和相应的数据模拟对控制疾病发展和扩散，药物的开发等等给予指导作用.

Lyapunov函数（泛函）在解决种群动力系统的一致持续性，全局稳定性等方面起着重要的作用。在本项目的研究中，主要通过构造成熟的Lyapunov函数(泛函)的方法来研究几类具有时滞的传染病动力学模型的全局稳定性。时滞的引起会考虑多种实际因素，包括疾病的潜伏期，感染期，康复期，季节变化，病毒进化等等。这使得研究的模型符合实际也更加复杂化．针对这几类传染病动力学模型，我们在已有研究工作的基础上，找出构造Lyapunov函数(泛函)的一般性方法，并推广应用于种群动力学中的其他方程.