  The Severe Acute Respiratory Syndrome Corona Virus 2 (SARS-CoV-2), or
Covid-19, burst into a pandemic in the beginning of 2020. An unprecedented
worldwide effort involving academic institutions, regulatory agencies and
industry is facing the challenges imposed by the rapidly spreading disease.
Emergency use authorization for vaccines were granted in the beginning of
December 2020 in Europe and nine days later in the United States. The urge for
vaccination started a race, forcing governs and health care agencies to take
decisions on the fly regarding the vaccination strategy and logistics. So far,
the vaccination strategies and non-pharmaceutical interventions, such as social
distancing and the use of face masks, are the only efficient actions to stop
the pandemic. In this context, it is of fundamental importance to understand
the dynamical behavior of the Covid-19 spread along with possible vaccination
strategies. In this work a Susceptible - Infected - Removed - Sick with
vaccination (SIRSi-Vaccine) model is proposed. In addtion, the SIRSi-Vaccine
model also accounts for unreported, or asymptomatic, cases and the possibility
of temporary immunity, either after infection or vaccination. Disease free and
endemic equilibrium points existence conditions are determined in the (! ? ?)
vaccine-effort and social distancing parameter space. The model is adjusted to
the data from S\~ao Paulo, Santos and Campinas, three major cities in the State
of S\~ao Paulo, Brazil.
