| We consider periodic optimal control problems in seasonal epidemics. For example, given the seasonality of the transmission rate, vaccine waning, and other factors, we seek for the best way to distribute a vaccine throughout the year. Possible options include uniform vaccination) maintain a fixed vaccination rate throughout the whole year, or pulse vaccination) vaccinate the entire population on a certain day sometime before the peak of the season. We show that both are suboptimal.
Mathematically, the project involves a non-standard periodic optimal control problem. Since pulses (delta functions) are possible solutions, the problem also involves optimizing over measures. In general, epidemic models, which include seasonal forcing, exhibit rich dynamical behavior, including nonlinear phenomena such as bifurcations, sub-harmonic and chaotic oscillations, and multistability. Accordingly, analysis of epidemic models with seasonal forcing is mathematically more demanding than analysis of non-seasonal models. From an application point of view, the motivation for this study originates from our intense work with the Ministry of Health during COVID-19 pandemic, and so our initial focus is on epidemic models. Yet, periodic optimal control problems arise in many systems, and we are also considering applications in chemical systems. Concretely, we are currently wrapping up initial results to attain the simplest problem and seeking a student to promote the next stages of the project. The project has analytical, numerical, and applicative elements, with varying emphasis between sub-projects. Non-math students are certainly welcome to apply. |
