This paper addresses the problem of describing the spread of COVID-19 by a mathematical model introducing all the possible control actions as prevention (informative campaign, use of masks, social distancing, vaccination) and medication. The model adopted is similar to SEIQR, with the infected patients split into groups of asymptomatic subjects and isolated ones. This distinction is particularly important in the current pandemic, due to the fundamental the role of asymptomatic subjects in the virus diffusion. The influence of the control actions is considered in analysing the model, from the calculus of the equilibrium points to the determination of the reproduction number. This choice is motivated by the fact that the available organised data have been collected since from the end of February 2020, and almost simultaneously containment measures, increasing in typology and effectiveness, have been applied. The characteristics of COVID-19, not fully understood yet, suggest an asymmetric diffusion among countries and among categories of subjects. Referring to the Italian situation, the containment measures, as applied by the population, have been identified, showing their relation with the government's decisions; this allows the study of possible scenarios, comparing the impact of different possible choices.

Since the beginning of 2020 in few weeks all the world has been interested by the pandemic due to SARS-CoV 2, causing more than 3 millions of dead people and more than 146 millions of infected patients. The virus moves with people and the most effective containment measure appears to be the severe lockdown; on the other hand, for obvious social and economic reasons, it can not be applied for long periods. Moreover, the increasing knwoledge on the virus and on its trasmission modes suggested various strategies, such as the use of masks, social distancing, disinfection and the fast identification of infected patients, up to the recent vaccination campaign. In this paper, the COVID-19 spread is studied referring to the Italian situation; the control actions introduced during 2020-2021 are identified in terms of their actual effects, allowing to study possible intervention scenarios.

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