What if vaccinated individuals could be infected?

The COVID-19 pandemic has turned the spotlight on sanitary systems, on the weakness of surveillance network and on the need of models able to describe the different issues of the pandemic and to propose, by analysing possible scenarios, containment measures suitably tailored for the situation. In this paper, starting from the specific current emergency, an improvement of the well known epidemic SEIR model is proposed, adding the categories of subjects that have received the vaccination, P, and V, corresponding to those that have received one or both doses respectively. Based on the current knowledge, the vaccination avoids the severe effects of the virus but not the contagion; moreover it is estimated that after about 8 months the immune memory can decrease, both for vaccinated subjects and healed ones. These two aspects are considered in this work along with the possible scheduling of the containment measures.