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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676868
• 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许多方式表示，包括常微分方程（ODE）或泛函微分方程（FDE）。 FDE的例子包括延迟微分方程（DDE）和偏微分方程（PDE）。 通常FDE模型是简单ODE模型的直接概括，包括可能与系统历史相关的附加特征。 在过去的50年里，FDE的分析已经发展起来，但仍然比ODE的分析复杂得多。在用FDE推广特定的ODE模型时，确定系统的动态如何变化是很重要的。 额外的结构是否会导致与常微分方程不同的行为？ 我的工作将集中在这些FDE的全球行为。 在这项工作中的一个关键工具是李雅普诺夫的直接方法。 这可以被认为是将系统投影到一个高维碗上，并显示解沿着碗向下移动到底部，证明系统是全局渐近稳定的。自2010年以来，利用李雅普诺夫泛函来确定流行病学中DDE和PDE模型的全局动力学特性取得了很大进展。 所使用的李雅普诺夫泛函可以被认为是从适用于相关ODE模型的李雅普诺夫函数构建的，添加了额外的项来处理系统的历史。 到目前为止，该方法一直以临时的方式使用，并且适用于许多单独的模型。 我将研究方法本身，确定原常微分方程系统和李雅普诺夫函数的一般条件，在此条件下，FDE推广允许李雅普诺夫泛函。另一方面，一些系统通过增加延迟而去稳定，通常通过要求一对复特征值的真实的部分改变符号的Hopf分支。 一个有趣的例子是SIRS模型，其中在发病率项的延迟没有实质性的影响的动态，而在免疫力损失项的延迟可能会导致不稳定通过一个霍普夫分岔。 我将研究这个系统和类似的系统，显示这种二分法，以深入了解下面的问题。为什么有些延迟会使系统不稳定，而其他延迟则不会？*什么时候一个常微分方程的全局稳定性意味着一个相关的频微分方程的全局稳定性？*我们能否从最近的成功中推断出建立可用的方法，使现有的抽象理论在实践中应用于研究界感兴趣的系统？