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

Continuous-time stochastic processes for the spread of COVID-19 disease simulated via a Monte Carlo approach and comparison with deterministic models