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 18À64 y.o. before autumn 2021 with a vaccine that reduces severity by 90% and susceptibility by 80%, we find that control measures reducing transmission rates by 15À27% should be maintained to remain below 1000 daily hospital admissions in France with a highly transmissible variant (basic reproduction number R 0 = 4). Assuming 90% of 65 y.o. are vaccinated, full relaxation of control measures might be achieved with a vaccine coverage of 89À100% in 18À64 y.o or 60À69% of 0À64 y.o. Interpretation: Age and comorbidity-based vaccine prioritization strategies could reduce the burden of the disease. Very high vaccination coverage may be required to completely relax control measures. Vaccination of children, if possible, could lower coverage targets necessary to achieve this objective.