We applied a generalized SEIR epidemiological model to the recent SARS-CoV-2 outbreak in the world, with a focus on Italy and its Lombardy, Piedmont, and Veneto regions. We focused on the application of a stochastic approach in fitting the model parameters using a Particle Swarm Optimization (PSO) solver, to improve the reliability of predictions in the medium term (30 days). We analyzed the official data and the predicted evolution of the epidemic in the Italian regions, and we compared the results with the data and predictions of Spain and South Korea. We linked the model equations to the changes in people’s mobility, with reference to Google’s COVID-19 Community Mobility Reports. We discussed the effectiveness of policies taken by different regions and countries and how they have an impact on past and future infection scenarios. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.