Lockdown and social distancing, as well as testing and contact tracing, are the main measures assumed by the governments to control and limit the spread of COVID-19 infection. In reason of that, special attention was recently paid by the scientific community to the mathematical modeling of infection spreading by including in classical models the effects of the distribution of contacts between individuals. Among other approaches, the coupling of the classical SIR model with a statistical study of the distribution of social contacts among the population, led some of the present authors to build a Social SIR model, able to accurately follow the effect of the decrease in contacts resulting from the lockdown measures adopted in various European countries in the first phase of the epidemic. The Social SIR has been recently tested and improved through a fruitful collaboration with the Health Protection Agency (ATS) of the province of Pavia (Italy), that made it possible to have at disposal all the relevant data relative to the spreading of COVID-19 infection in the province (half a million of people), starting from February 2020. The statistical analysis of the data was relevant to fit at best the parameters of the mathematical model, and to make short-term predictions of the spreading evolution in order to optimize the response of the local health system.