Stability in mathematical biology

流行病学系统经常显示出长期稳定的模式。由于无法进行与流行病爆发有关的实验，数学建模已成为在人口水平上研究控制疾病动态的机制的重要工具。通常情况下，根据流行病学状况将接受调查的人口分为几个亚组：例如，易感、暴露、感染和接种疫苗。这些群体的规模随时间的变化通常可以用常微分方程来建模。我的研究计划的主要目标是开发建设性的方法来调查流行病模型的稳定性。目前，稳定性论证理论已经较为成熟，但理论与实际应用之间还存在着很大的差距。理论结果趋向于这样的形式：“如果一个满足一定条件的函数存在，那么系统是稳定的。”这没有提供如何查找或构造必要函数的信息。这将有助于深入了解诸如设计疫苗接种战略等问题，从而在多个城市或国家同步消灭一种疾病。

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.