  This study presents a mathematical model for optimal vaccination strategies
in interconnected metropolitan areas, considering commuting patterns. It is a
compartmental model with a vaccination rate for each city, acting as a control
function. The commuting patterns are incorporated through a weighted adjacency
matrix and a parameter that selects day and night periods. The optimal control
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 capacity of vaccines per unit of time. The key findings
of this work are bounds for the basic reproduction number, particularly in the
case of a metropolitan area, and the study of the optimal control problem.
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 better control the
disease spread, as we depicted in our numerical simulations. This model serves
as a tool to improve resource allocation in epidemic control across
metropolitan regions.
