## 3 Results  
### Comparing Lombardy, Campania and Emilia Romagna Contagion Indexes  
As for the Lombardy Region (RL), the identified cases are 13,967 out of 88,412 total cases (15.8%), in the reference period September 14 th-October 30 th. In the Campania Region (RC), the cases identified are 4620 about 42,815 total cases (10.8%). It is important to recall that the Lombardy Region and the Campania Region have used in October different school policies, the first leaving primary and secondary schools open in attendance and high secondary schools at 50% in attendance [25], while RC intervening instead with targeted closures: schools opened at September 24 (secondary schools at 50% in attendance), and then all levels were closed in advance starting from the October 16 and until November 13 [26]. Let us now focus on the data of the Lombardy Region at the provincial level (see Figure 2 for numerical details for all the RL provinces: VA Varese, SO Sondrio, PV Pavia, MN Mantova, MI Milan, MB Monza and Brianza, LO Lodi, LC Lecco, CR Cremona, CO Como, BS Brescia, and BG Bergamo). If we calculate a \"school contagion\" index and a \"global contagion\" index (normalized on the ISTAT2020 population [[[http://demo.istat.it](http://demo.istat.it)]([http://demo.istat.it](http://demo.istat.it))] (accessed on 27 April 2021), for each province as: cases/1000 inhabitants), it is interesting to note that there is a strong correlation between a high rate of contagion in school and high contagion rate then at the provincial level. This correlation is not found instead by looking at, for example, the population density as another variable. As an example, let us take the case of Varese (VA): VA was one of the Lombardy provinces that was mostly impacted by the second COVID-19 wave. VA has the highest school contagion rate (together with MB) and a very high global contagion rate, as depicted in Figure 2.  
Let us now consider what happened in the first two weeks after the school reopening (14 September-28 September) where theoretically the effects of the school were just beginning to be visible: it is noted how there is no correlation between the school contagion index and the global contagion index at two weeks (CI = \(-\)0.10). If, on the other hand, we look at the correlation between school contagion and the index of contagion two weeks after the reopening of schools, we have a clear correlation with CI = 0.69, which rises further considering the effect after four weeks with CI = 0.80. For the entire reference period the CI rises to CI = 0.89) as clearly depicted in the scatter plot of Figure 3 (the linear regression trend line as the R\({}^{2}\) coefficient of determination with a very high value R\({}^{2}\) = 0.94). The correlation indices are identical if the school contagion data is separated between primary and secondary school. It is interesting to observe the correlation between the contagion index and the population density (CI = 0.60) but not between the school index and the population density (CI = 0.37). If, on the other hand, we observe the correlation between the contagion index and the mobility indexes, we note that the lower mobility registered with government restrictions and DPCM (Decreto del Presidente del consiglio dei ministri) does not have an interdependence relation with the contagion more or less accentuatedin the various provinces of RL, with the exception of transit station mobility that shows a correlation CI = \(-0.57\). Since the number of datapoints is limited, we tested the statistical significance with F-Test (\(\alpha\) = 0.05) and we obtained the following values: F = 21.20 with a P(F) = 7.94 \(\times\) 10\({}^{-6}\) and an F critical = 2.82 (gdl = 11), so the null hypothesis is rejected (since F calculated is greater than the F critical). As for Campania Region (Figure 4) and its five provinces (SA Salerno, NA Napoli, CE Caserta, BN Benevento, AV Avellino), schools reopened two weeks later than RL and  
Figure 3: Lombardy Region data set for school contagion index vs. overall contagion index.