Dynamics of Functional Differential Equations with Applications in Epidemiology

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。***在用FDE泛化特定的ODE模型时，确定系统动态如何变化是很重要的。额外的结构是否会导致与ODE不同的行为？我的工作将集中在这些fde的全球行为上。这项工作的一个关键工具是李亚普诺夫的直接方法。这可以被认为是将系统投射到一个高维的碗上，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，在利用Lyapunov泛函确定流行病学中DDE和PDE模型的全局动态方面取得了很大进展。所使用的Lyapunov函数可以被认为是从适用于相关ODE模型的Lyapunov函数构建而来的，并添加了额外的术语来处理系统的历史。到目前为止，该方法已经以一种特别的方式使用，并且已经在许多单独的模型上工作。我将研究方法本身，确定原始ODE系统和Lyapunov函数的一般条件，在这些条件下，FDE推广承认Lyapunov泛函。另一方面，一些系统通过延迟的增加而失稳，通常是通过Hopf分岔，该分岔要求一对复特征值的实部改变符号。一个有趣的例子是SIRS模型，其中发生率项的延迟对动力学没有实质性影响，而免疫丧失项的延迟可能通过Hopf分岔导致不稳定。我将研究这个系统和类似的显示这种二分法的系统，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ODE的全局稳定性意味着相关FDE的全局稳定性？***我们能否从最近的成功中推断出可用的方法，使现有的抽象理论能够在实践中应用于研究界感兴趣的系统？**

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?**