We propose a multi-layer network model for the spread of COVID-19 that accounts for interactions within the family, between schoolmates, and casual contacts in the population . We utilize the proposed model-calibrated on epidemiological and demographic data-to investigate current questions concerning the implementation of non-pharmaceutical interventions (NPIs) during the vaccination campaign . Specifically, we consider scenarios in which the most fragile population has already received the vaccine, and we focus our analysis on the role of schools as drivers of the contagions and on the implementation of targeted intervention policies oriented to children and their families . We perform our analysis by means of a campaign of Monte Carlo simulations . Our findings suggest that, in a phase with NPIs enacted but in-person education, children play a key role in the spreading of COVID-19 . Interestingly, we show that children's testing might be an important tool to flatten the epidemic curve, in particular when combined with enacting temporary online education for classes in which infected students are detected . Finally, we test a vaccination strategy that prioritizes the members of large families and we demonstrate its good performance . We believe that our modeling framework and our findings could be of help for public health authorities for planning their current and future interventions, as well as to increase preparedness for future epidemic outbreaks.