  After the introduction of drastic containment measures aimed at stopping the
epidemic contagion from SARS-CoV2, many governments have adopted a strategy
based on a periodic relaxation of such measures in the face of a severe
economic crisis caused by lockdowns. Assessing the impact of such openings in
relation to the risk of a resumption of the spread of the disease is an
extremely difficult problem due to the many unknowns concerning the actual
number of people infected, the actual reproduction number and infection
fatality rate of the disease. In this work, starting from a compartmental model
with a social structure and stochastic inputs, we derive models with multiple
feedback controls depending on the social activities that allow to assess the
impact of a selective relaxation of the containment measures in the presence of
uncertain data. Specific contact patterns in the home, work, school and other
locations have been considered. Results from different scenarios concerning the
first wave of the epidemic in some major countries, including Germany, France,
Italy, Spain, the United Kingdom and the United States, are presented and
discussed.
