Abstract
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. Finally, the models are calibrated on data referring to the second wave of infection in Italy.
Highlights
The pandemic of COVID-19 has scourged the world since the beginning of 2020.1 The responsible virus is the SARS-CoV-2, identified in China at the end of 2019 (Zhou et al 2020)
In Ansumali et al (2020), Calafiore et al (2020), Giordano et al (2020) some deterministic epidemic models for COVID-19 based on ordinary differential equations have been proposed
In this paper we would like to introduce some epidemic models based on stochastic processes, taking into account peculiarities of the COVID-19 disease
Summary
The pandemic of COVID-19 has scourged the world since the beginning of 2020.1 The responsible virus is the SARS-CoV-2, identified in China at the end of 2019 (Zhou et al 2020). The simplest epidemic model is called SIR model It looks at a population split out into three compartments: susceptible, infected and removed. In Ansumali et al (2020), Calafiore et al (2020), Giordano et al (2020) some deterministic epidemic models for COVID-19 based on ordinary differential equations have been proposed. In this paper we would like to introduce some epidemic models based on stochastic processes, taking into account peculiarities of the COVID-19 disease. In the second model we would like to include asymptomatic people, i.e. infectious individuals without severe or identifiable symptoms. They seem to play an important role in the diffusion of the virus because usually they don’t know to be infectious. D in 2; in Sect. 6 we show and comment the numerical simulations, in particular we test the proposed models by considering the second wave of COVID-19 in Italy
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