  There is increasing evidence that one of the most difficult problems in
trying to control the ongoing COVID-19 epidemic is the presence of a large
cohort of asymptomatic infectives. We develop a SIR-type model taking into
account the presence of asymptomatic, or however undetected, infective, and the
substantially long time these spend being infective and not isolated. We
discuss how a SIR-based prediction of the epidemic course based on early data
but not taking into account the presence of a large set of asymptomatic
infectives would give wrong estimate of very relevant quantities such as the
need of hospital beds, the time to the epidemic peak, and the number of people
which are left untouched by the first wave and thus in danger in case of a
second epidemic wave. In the second part of the note, we apply our model to the
COVID-19 epidemics in Italy. We obtain a good agreement with epidemiological
data; according to the best fit of epidemiological data in terms of this model,
only 10\% of infectives in Italy is symptomatic.
