Background The roll-out of COVID-19 vaccines is a multi-faceted challenge whose performance depends on pace of vaccination, vaccine characteristics and heterogeneities in individual risks. Methods We developed a mathematical model accounting for the risk of severe disease by age and comorbidity and transmission dynamics. We compared vaccine prioritisation strategies in the early roll-out stage and quantified the extent to which measures could be relaxed as a function of the vaccine coverage achieved in France. Findings Prioritizing at-risk individuals reduces morbi-mortality the most if vaccines only reduce severity, but is of less importance if vaccines also substantially reduce infectivity or susceptibility. Age is the most important factor to consider for prioritization; additionally accounting for comorbidities increases the performance of the campaign in a context of scarce resources. Vaccinating 90% of ≥65 y.o. and 70% of 18-64 y.o. before the autumn 2021 with a vaccine that reduces severity by 90% and susceptibility by 80%, we find that control measures reducing transmission rates by 15-27% should be maintained to remain below 1,000 daily hospital admissions in France with a highly transmissible variant (basic reproduction number R0=4). Assuming 90% of ≥65 y.o. are vaccinated, full relaxation of control measures might be achieved with a vaccine coverage of 89-100% in 18-64 y.o or 60-69% of 0-64 y.o. Interpretation Even in optimistic scenarios, current vaccination intentions in the French populations might not allow a complete relaxation of control measures. Vaccination of children, if possible, could help achieve this objective.