Study of multiple scales epidemiological models using Tikhonov-Vasilieva and geometric singular perturbation theory

The presented project aims at a systematic study of multiple scale models in epidemiology. Mathematically we plan to address and systematically study aspects of two drawbacks of the classical singular perturbation theory. The Tikhonov theorem only ensures the convergence on finite intervals of time t and thus the provided approximation is useless if one wants to investigate the long term dynamics of the original system. In other words, one cannot draw any valid conclusions about the long term dynamics of the micro problem from the macroscopic equation. The other problem with the standard theory is that it requires an isolated slow manifold on which we approximate the full system. In practice, however, there can be multiple slow manifolds that can intersect with each other. The solutions’ behaviour close to such points of intersection is quite complicated and far from being intuitive. The research will use a combination of the results from Tikhonov-Vasilievaand geometric singular perturbation theory to build on the older and recent developments in the field to develop a systematic theory of multiple-scale theory of epidemiological models taking into account the problems mentioned above.