Stability in mathematical biology

流行病学系统经常表现出在很长一段时间内稳定的模式。由于进行与疫情爆发有关的实验是不可行的，数学建模已成为在人口层面上调查支配疾病动态的机制的重要工具。通常情况下，接受调查的人群会根据流行病学状况分为几个亚组：例如，易感人群、暴露人群、传染病人群和接种疫苗人群。这些人群的规模随时间的变化通常可以用一个常微分方程式来模拟。我的研究计划的主要目的是发展研究流行病模型稳定性的建设性方法。目前，证明稳定性的理论已经很发达，但理论和应用之间存在着很大的差距。理论结果往往是这样的：“如果满足一定条件的函数存在，则系统是稳定的。”但这并不能洞察如何找到或构造必要的函数。这将提供对诸如设计疫苗接种战略等问题的洞察，这些战略导致在多个城市或国家同步传播一种疾病。

Epidemiological systems frequently display patterns that are stable over long time periods.  Since it is not feasible to perform experiments relating to epidemic outbreaks, mathematical modelling has become an important tool for the investigation of the mechanisms that govern disease dynamics at the population level. Typically, a population under investigation is divided into a small number of subgroups based on epidemiological status: susceptible, exposed, infectious and vaccinated, for example.  The change in the sizes of these groups over time can often be modeled with an ordinary differential equation. The main objective of my research program is to develop constructive approaches for investigating the stability properties of epidemic models.  At present, the theory of demonstrating stability is well-developed, however, there is a significant gap between the theory and its application. Theoretical results tend to be of the form: "If a function satisfying certain conditions exists, then the system is stable."  This provides no insight into how the necessary function can be found or constructed. This will provide insight into such issues as designing vaccination strategies which lead to a synchronized eradiaction of a disease over multiple cities or countries.