A state dependent switching optimal control as a formalization of political interventions

The worldwide emergency of the COVID-19 pandemic has shown how the social and economic behaviours, with impact on everyday life of citizens, play a crucial role on the speed and the intensity of the epidemic spread as well as the sanitary effort required for containing the number of hospitalised patients and the mortality rate. Political decisions assumed by the Governments to regulate, moderate and limit the individual contacts were the only effective approach until vaccination started to be massively available. Contacts limitations are always considered, in any Country, as an individual freedom limitation and then it has been seen as an unpopular approach. In this paper, starting from an ad-hoc designed simplified model, an optimal control based approach is used to show how the vaccination rate can effectively contribute to relax the individual limitations. A state based switching technique is adopted to replicate the Governments' decisions based on the epidemic evolution. Some numerical simulations are reported to support the results.