## 2 Materials and Methods  
In December, the MIUR published an official dataset relating to infections in schools (joining different data collection) for the period 14 September 2020-30 October 2020 [22]. The report spoke of approximately 65,000 Italian positive cases identified in the time window 14 September-30 October (we are only talking about primary and lower secondary school, because most of the high schools were in any case remote). The available data count 65,000 cases for the whole Italy, but they are underestimated because not all Italian schools have participated in this tracking activity, and not all schools have released their data to the ministry. It should also be considered that 75% of those under 19 years old are asymptomatic [www.iss.it] (accessed on 27 April 2021), and this large slice of young people is lost in the tracing activity. In total, 65,000 cases out of 360,000 total cases [23; 24] detected in the same period is a considerable percentage of 18% of the total. Furthermore, the major impact is to be considered in the contagion that then arose in the second instance within the family walls, leading after about two incubation cycles of the virus, to an uncontrolled growth of the curves (the one that we observed from 28 September 2020 onwards in our predictive model and in Figure 1). In order to understand the relation between schools and global infection, we considered the data officially released by the MIUR and we carried out a correlation analysis on the Lombardy Region (RL) and Campania Region (RC), two regions that have adopted two different policies of opening and closing schools. RL is characterized by 12 provinces for a total of 10 M inhabitants, while RC has 5 provinces with 5.8 M inhabitants. Moreover, we extended this analysis to a third Italian region: Emilia Romagna Region (REm) that is characterized by 9 provinces and a total population of 4.5 M inhabitants. At the end, the study covered 24 provinces out of 107 Italian provinces, and 20.3 M inhabitants out of 60.3 M total Italian inhabitants. The three regions applied the following opening/closure strategies:  
* RL reopened all primary and secondary schools in presence at 14 September 2020 (high level secondary school with 50% attendance and 50% online) [25];
* RC reopened all primary and secondary schools in presence at 24 September 2020 (high level secondary school with 50% attendance and 50% online) and then all levels were closed in advance starting from the October 16 and until 13 November [26];
* REm reopened all primary and secondary schools in presence at 14 September 2020 (high level secondary school with 50% attendance and 50% online). Specifically, a twofold correlation study was conducted:  
1. between school contagion index (both total and separate for primary and secondary school, respectively) and an index of global contagion at the provincial level (both for RL and RC). The correlation study was done with a global contagion index on the reference period from 14 September 2020 to 30 October 2020 and also considering the first two weeks after the reopening of schools (from 14 September to 28 September, where the contagion theoretically should not be detectable, given the latency time between positivity and the onset of symptoms and related diagnostic screening) then in the following two weeks (from 28 September to 12 October, when it is likely that contagion was triggered in schools and then it potentially spreads in the intrafamily context), and after four weeks of spreading;
2. between contagion index and mobility indexes derived from the COVID-19 Google Community Mobility Report [27], where mobility data at regional and national level in different sectors (e.g., mobility near parks and public gardens, pharmacies, at work level, train stations, residential, etc.) were analyzed.