  We study a simple realistic model for describing the diffusion of an
infectious disease on a population of individuals. The dynamics is governed by
a single functional delay differential equation, which, in the case of a large
population, can be solved exactly, even in the presence of a time-dependent
infection rate. This delay model has a higher degree of accuracy than that of
the so-called SIR model, commonly used in epidemiology, which, instead, is
formulated in terms of ordinary differential equations. We apply this model to
describe the outbreak of the new infectious disease, Covid-19, in Italy, taking
into account the containment measures implemented by the government in order to
mitigate the spreading of the virus and the social costs for the population.
