Against the COVID-19 pandemic, non-pharmaceutical interventions have been widely applied and vaccinations have taken off. The upcoming question is how the interplay between vaccinations and social measures will shape infections and hospitalizations. Hence, we extend the Susceptible-Exposed-Infectious-Removed (SEIR) model including these elements. We calibrate it to data of Luxembourg, Austria and Sweden until 15 December 2020. Sweden results having the highest fraction of undetected, Luxembourg of infected and all three being far from herd immunity in December. We quantify the level of social interaction, showing that a level around 1/3 of before the pandemic was still required in December to keep the effective reproduction number Refft below 1, for all three countries. Aiming to vaccinate the whole population within 1 year at constant rate would require on average 1,700 fully vaccinated people/day in Luxembourg, 24,000 in Austria and 28,000 in Sweden, and could lead to herd immunity only by mid summer. Herd immunity might not be reached in 2021 if too slow vaccines rollout speeds are employed. The model thus estimates which vaccination rates are too low to allow reaching herd immunity in 2021, depending on social interactions. Vaccination will considerably, but not immediately, help to curb the infection; thus limiting social interactions remains crucial for the months to come.