Many countries are currently facing high mortality caused by the circulation of SARS-CoV-2 among the elderly not yet vaccinated. Vaccine shortage poses relevant challenges to health authorities, called to act in a timely manner, and with scarcity of vaccine, and data. We have developed a model for estimating of the impact of vaccination on the mortality of the elderly following a schedule of mRNA SARS-CoV-2 vaccine that prioritize first dose administration, as alternative to the standard schedule of two doses administered 3 to 4 weeks apart. We studied the Italian scenario, considering it representative of other Countries facing similar conditions in terms of virus circulation, mortality, and vaccine shortage, in the period from February 10 to April 14 2021. Under different conditions of quantity of vaccine administration, the schedule prioritizing first doses showed always significant increase of protected individuals, and a decrease of deaths, up to 19.8% less than the standard schedule. These findings support the vaccination option of prioritizing the first dose in the elderly until vaccine supplies are adequate.