Stability in mathematical biology

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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.