  Two stochastic models are proposed to describe the evolution of the COVID-19
pandemic. In the first model the population is partitioned into four
compartments: susceptible $S$, infected $I$, removed $R$ and dead people $D$.
In order to have a cross validation, a deterministic version of such a model is
also devised which is represented by a system of ordinary differential
equations with delays. In the second stochastic model two further compartments
are added: the class $A$ of asymptomatic individuals and the class $L$ of
isolated infected people. Effects such as social distancing measures are easily
included and the consequences are analyzed. Numerical solutions are obtained
with Monte Carlo simulations. Quantitative predictions are provided which can
be useful for the evaluation of political measures, e.g. the obtained results
suggest that strategies based on herd immunity are too risky.
