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?

传染病的传播模型可以多种方式表达，包括作为普通微分方程（ODE）或功能微分方程（FDE）。 FDE的示例包括延迟微分方程（DDE）和部分微分方程（PDE）。 FDE模型通常是对简单ODE模型的直接概括，包括可能与系统历史记录相关的其他功能。 在过去的50年中，对FDE的分析进行了开发，但仍然比ODE分析要复杂得多。 在用FDE概括特定的ODE模型时，必须确定系统动力学的变化很重要。 其他结构是否会导致与ODE不同的行为？ 我的工作将集中在这些FDE的全球行为上。 这项工作的关键工具是Lyapunov的直接方法。 这可以被认为是将系统投射到高维碗上，并表明解决方案向下移动到底部，表明该系统在全球渐近稳定。 自2010年以来，使用Lyapunov功能来确定流行病学中DDE和PDE模型的全局动力学取得了长足进展。 可以将使用的Lyapunov功能视为是由适用于相关ODE模型的Lyapunov函数构建的，并添加了其他术语来处理系统的历史记录。 到目前为止，该方法已经以临时的方式使用，并且已经在许多单独的模型上使用。 我将研究该方法本身，确定原始ODE系统和Lyapunov函数的一般条件，在该功能下，FDE概括允许Lyapunov功能。 另一方面，通常是通过hopf分叉的延迟来消除稳定的，该分叉需要一对复杂的特征值的实际部分才能更改符号。 一个有趣的例子是SIRS模型，该模型在起火项中的延迟对动态没有重大影响，而免疫丧失项的延迟可以通过HOPF分叉导致不稳定。 我将研究该系统和类似的系统，这些系统显示了这种二分法，以深入了解以下问题。 为什么有些延迟会破坏系统的稳定，而其他延迟则没有？ ODE的全球稳定性何时意味着相关FDE的全球稳定性？ 我们是否可以从最近的成功中推断出可用的方法，从而使现有的抽象理论可以在实践中应用于研究界感兴趣的系统？