  In this study, we explore the dynamic interplay between the timing of
vaccination campaigns and the trajectory of disease spread in a population.
Through comprehensive data analysis and modeling, we have uncovered a
counter-intuitive phenomenon: initiating a vaccination process at an
inopportune moment can paradoxically result in a more pronounced second peak of
infections. This "rebound" phenomenon challenges the conventional understanding
of vaccination impacts on epidemic dynamics. We provide a detailed examination
of how improperly timed vaccination efforts can inadvertently reduce the
overall immunity level in a population, considering both natural and
vaccine-induced immunity. Our findings reveal that such a decrease in
population-wide immunity can lead to a delayed, yet more severe, resurgence of
cases. This study not only adds a critical dimension to our understanding of
vaccination strategies in controlling pandemics but also underscores the
necessity for strategically timed interventions to optimize public health
outcomes. Furthermore, we compute which vaccination strategies are optimal for
a COVID-19 tailored mathematical model, and find that there are two types of
optimal strategies. The first type prioritizes vaccinating early and rapidly to
reduce the number of deaths, while the second type acts later and more slowly
to reduce the number of cases; both of them target primarily the elderly
population. Our results hold significant implications for the formulation of
vaccination policies, particularly in the context of rapidly evolving
infectious diseases.
