Vaccination, Booster Doses and Social Constraints: A Steady State and an Optimal Transient Approaches to Epidemics Containment

The problem of the definition of control actions to contain epidemic diseases is crucial in case of high infectivity, dangerous or fatal consequences, large inhabited areas involved. Unfortunately, during the last three years, the COVID-19 pandemic has represented a critical situation all over the world. On the basis of the experiences for known diseases and the literature on the epidemic modeling, various strategies have been proposed and applied in different countries, someone using all the possible efforts, some others just maintaining the global health compact within acceptable levels. The effectiveness of the approaches has been always measured on the basis of the reproduction number Rt, which is intrinsically a steady-state evaluation, since it does not takes into account the control variations. In the present paper, with reference to a mathematical model which takes into account the different level of vaccination in the population, an optimal condition based approach is adopted to define the actions of intervention, bringing to a switching optimal control scheme based on the time by time evolution of the disease. The two approaches are developed and compared, showing that the use of partial information can bring to counter-intuitive situations and supporting the necessity of a feedback action to better adapt the containment measure to the situation. Numerical simulations are performed to better show the claimed results