## 2 Materials and Methods
### Bayesian Changepoint Detection and Analysis  
Using a changepoint estimation technique, based on a Bayesian piecewise regression, we have looked for a changepoint in the trend of the infection curve, whether it was growing or falling. In particular, we fitted a Poisson generalized linear model where the dependent variable was the number of new daily confirmed SARS-COV-2 cases, and the independent variable was just the number of days since 28 May 2021 (until 25 July). The result was a model comprised of a changepoint and two segments, whose slopes represent, respectively, the increase/decrease in case rate before the changepoint and the increase/decrease in case rate after it. The fact that our interest was not in modeling the spread of the virus with the maximum precision but rather in finding the point in time when the infection rate inverted (or simply changed) its trend, with the added bonus of a Bayesian uncertainty estimation, is worth noting. The model takes the mathematical form below:  
\[\ln(E(Y|x))= a_{1}+xb_{1}\text{if }x<\tau\ln(E(Y|x))= a_{2}+xb_{2}\text{if }x>\tau \tag{1}\]  
It is worth noting that our dependent variable (the confirmed daily cases) was modelled as a Poisson distribution, whose mean depends on the regression coefficients \(a_{1}\) and \(b_{1}\), respectively, along with the intercept and angular coefficient before the changepoint \(\tau\) (while \(a_{2}\) and \(b_{2}\) play the same role after the changepoint). To be considered are the three following facts:  
First, since the two regression lines are joined at the changepoint \(\tau\); the second intercept term \(a_{2}\) is not estimated as it is bound to be \(a_{2}=\tau\left(b_{1}-b_{2}\right)+a_{1}\). Second, the formula above returns the exponential growth/decay trend, both before and after \(\tau\), as easily identifiable slopes. Third, to compute the number of days needed to halve/double the number of cases before/after a changepoint, the following two formulas can be used: specifically, Formula (2) can be used to compute the halving (\(H\)) and doubling (\(D\)) time, before a changepoint:  
\[H_{b}=\frac{\ln\frac{y_{\tau}}{2}-a_{1}}{b_{1}}-\tau\,;D_{\text{b}}=\frac{ \ln 2y_{\tau}-a_{1}}{b_{1}}-\tau\,;\text{ where }y_{\tau}=e^{a_{1}+vb_{1}} \tag{2}\]  
Formula (3), instead, can be used to compute the halving and doubling time, after a changepoint. \[H_{\text{a}}=\frac{\ln\frac{y_{\tau}}{2}-\ln y_{\tau}}{b_{2}}\,;\;D_{\text{a }}=\frac{\ln 2y_{\tau}-\ln y_{\tau}}{b_{2}}\,;\text{ where }y_{\tau}=e^{a_{1}+vb_{1}} \tag{3}\]To fit the model above, we used the R package mcp, using a Markov chain Monte Carlo method [18]. For starting the Bayesian estimation, the default priors for \(\tau,a_{1},a_{2}\), and \(b_{1}\) were chosen as suggested in [18], thus considering the prior of \(\tau\) as a uniform, and the parameters \(a_{1},b_{1}\), and \(b_{2}\) as normally distributed, as reported in the following formulas:  
\[\tau \sim \text{Uniform }(\min(x),\max(x)) \tag{4}\] \[a_{1},b_{1},b_{2} \sim \mathcal{N}(0,10) \tag{5}\]  
It is now worth noting that the mean value of the computed changepoint posterior distribution was used to calculate the distance in time between the date of a given changepoint and that of the first match played the corresponding team. Similarly, the mean values for the coefficients \(b_{1}\) and \(b_{2}\) were used to compute the steepness of the two slopes, respectively, before and after the changepoint. The values obtained from the Bayesian regression have 95% credible intervals, associated with them. The aggregated statistics we computed for the countries (average distance from changepoint, average doubling/halving time, etc.) have 95% confidence intervals, computed using bootstrap. This completes the description of our method from a statistical viewpoint. Nonetheless, it is appropriate to motivate the reason behind the use of this statistical methodology.