Epidemics in networks

Master’s Dissertation supervised by Maria Soledad Aronna presented to the School of Applied Mathematics (FGV/EMAp) for the fulfillment of the requirements to obtain the Master’s Degree in Mathematical Modelling.

This study presents a mathematical model for optimal vaccination strategies in interconnected metropolitan areas, considering commuting patterns. The epidemiological model employed is a compartmental SEIR model, with a vaccination rate for each city acting as a control function, and the commuting patterns are incorporated through a weighted adjacency matrix and a parameter that weighs day and night periods. The optimal control (OC) problem is formulated to minimize a functional cost that balances the number of hospitalizations and vaccines, including restrictions of a weekly availability cap and an application limit of vaccines per unit of time, leading to mixed control-state and pure-state constraints. The key findings of this work are the bounds for the basic reproduction number, particularly in the case of a metropolitan area, and the OC problem analysis. Theoretical analysis and numerical simulations provide insights into disease dynamics and the effectiveness of control measures. The research highlights the importance of prioritizing vaccination in the capital to reduce disease spread faster, as we depicted in our numerical simulations, especially when it has higher contact rates. This model serves as a tool to improve resource allocation in epidemic control across metropolitan regions. Future research will refine this strategy and explore theoretical aspects of constrained control-affine problems. The study contributes significantly to our understanding of epidemic dynamics in metropolitan areas and the role of OC in mitigating disease outbreaks.