Dynamics of Functional Differential Equations with Applications in Epidemiology

泛函微分方程动力学及其在流行病学中的应用

• 批准号: RGPIN-2016-06134
• 负责人: McCluskey, Christopher
• 金额: $ 1.31万
• 依托单位: Wilfrid Laurier University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710169
• 关键词: Dynamics; Functional; Differential; Equations; Applications

Models of the spread of infectious disease can be expressed in many ways, including as ordinary differential equations (ODEs) or as functional differential equations (FDEs). Examples of FDEs include delay differential equations (DDEs) and partial differential equations (PDEs). Often FDE models are direct generalizations of simpler ODE models, including additional features that may be related to the history of the system. The analysis of FDEs has been developed over the last 50 years, but is still much more complicated than the analysis of ODEs. In generalizing a particular ODE model with an FDE, it is important to determine how the dynamics of the system change. Does the additional structure result in different behaviour from the ODE or not? My work will focus on the global behaviour of these FDEs. A key tool in this work is Lyapunov's Direct Method. This can be thought of as projecting the system onto a high-dimensional bowl and showing that solutions move down the bowl to the bottom, demonstrating that the system is globally asymptotically stable. Since 2010, there has been great progress on using Lyapunov functionals to determine the global dynamics of DDE and PDE models in epidemiology. The Lyapunov functional that is used can be thought of as being built from the Lyapunov function that works for the associated ODE model, with additional terms added to deal with the system's history. So far, the approach has been used in an ad hoc manner, and has worked on many individual models. I will study the method itself, determining general conditions on the original ODE system and Lyapunov function, under which FDE generalizations admit a Lyapunov functional. On the other hand, some systems are de-stabilized by the addition of delay, usually through a Hopf bifurcation that requires the real part of a pair of complex eigenvalues to change sign. An interesting example is the SIRS model where a delay in the incidence term has no substantial effect on the dynamics, whereas a delay in the loss-of-immunity term can lead to instability through a Hopf bifurcation. I will study this system and similar systems that show this dichotomy, to gain insight into the questions below. Why do some delays destabilize a system, while other delays do not? When does the global stability of an ODE imply the global stability of an associated FDE? Can we extrapolate from recent successes to build usable methods that allow the existing abstract theory to be applied in practice to systems that are of interest to the research community?

传染病传播的模型可以用多种方式表示，包括常微分方程或函数微分方程。fde的例子包括延迟微分方程（DDEs）和偏微分方程（PDEs）。通常，FDE模型是更简单的ODE模型的直接概括，包括可能与系统历史相关的附加特性。fde的分析已经发展了近50年，但仍然比ode的分析复杂得多。