In this paper, we introduce a general novel compartmental model accounting for the effects of vaccine efficacy, deployment rates and timing of initiation of deployment. It consists of compartments corresponding to susceptible, vaccinated susceptible, infectious, vaccinated infectious, active, and dead populations with various vaccine efficacies and vaccination deployment rates. Abstract We simulate different scenarios and initial conditions, and we find that the abundance and higher rate of deployment of low efficacy vaccines would lower the cumulative number of deaths in comparison to slower deployment of high efficacy vaccines. However, the latter can lower the number of active cases and achieve faster and higher herd immunity. We also forecast that, at the same daily deployment rate, the earlier introduction of vaccination schemes with lower efficacy would also lower the number of deaths with respect to a delayed introduction of high efficacy vaccines, which can however, still achieve lower numbers of infections and better herd immunity.