  Qatar has undergone distinct waves of COVID-19 infections, compounded by the
emergence of variants, posing additional complexities. This research uniquely
delves into the varied efficacy of existing vaccines and the pivotal role of
vaccination timing in the context of COVID-19. Departing from conventional
modeling, we introduce two models that account for the impact of vaccines on
infections, reinfections, and deaths. Recognizing the intricacy of these
models, we use the Bayesian framework and specifically utilize the
Metropolis-Hastings Sampler for estimation of model parameters. The study
conducts scenario analyses on two models, quantifying the duration during which
the healthcare system in Qatar could have potentially been overwhelmed by an
influx of new COVID-19 cases surpassing the available hospital beds.
Additionally, the research explores similarities in predictive probability
distributions of cumulative infections, reinfections, and deaths, employing the
Hellinger distance metric. Comparative analysis, employing the Bayes factor,
underscores the plausibility of a model assuming a different susceptibility
rate to reinfection, as opposed to assuming the same susceptibility rate for
both infections and reinfections. Results highlight the adverse outcomes
associated with delayed vaccination, emphasizing the efficacy of early
vaccination in reducing infections, reinfections, and deaths. Our research
advocates prioritizing early vaccination as a key strategy in effectively
combating future pandemics. This study contributes vital insights for
evidence-based public health interventions, providing clarity on vaccination
strategies and reinforcing preparedness for challenges posed by infectious
diseases. The data set and implementation code for this project is made
available at \url{https://github.com/elizabethamona/VaccinationTiming}.
