## 3 Results
### Countries That Participated in the Tournament
We further worked with the numbers comprised in Table 1 by rounding the mean changepoint value for all the 17 countries and then by calculating the difference, in terms of days, between that value and the date when they played their first match. This way, we obtained that the average date of the changepoint, for all the 17 countries of interest, falls 14.97 days (95% CI 12.29-17.47) after the beginning of their participation in the tournament (approximately two weeks). Finally, we made a step further and, taking the mean values for the coefficients \(b_{1}\) and \(b_{2}\), we estimated how the slopes for the two lines changed, on average, before and after the changepoint. We gathered that all the 17 countries had a decreasing number of daily cases until the changepoint and ended up with a reversed trend afterwards. Table 2 shows the halving time before, and the doubling time, after the changepoint, for each given country of this group. More precisely, the mean halving time before the changepoint is 18.07 days (95% CI 11.81-29.42), while the mean doubling time after the changepoint is 29.10 days (95% CI 14.12-49.78). The credible intervals are quite wide, but if we better investigate the values reported in Table 2, we recognize that most of the deviation depends on just three countries, namely: Spain, Ukraine, and Poland, with their exceptionally large values. To better highlight and summarize all the results we have discussed so far, we also present Figures 1 and 2, where the same results are portrayed from a clear graphical viewpoint. In particular, Figure 1 takes into account the inversion of the SARS-COV-2 case trend of the following countries: Austria, Belgium, Croatia, Czechia, Denmark, Finland, France, Germany, Hungary, and Italy. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Country**} & **Days Needed to Halve the Number of Cases (before \(\tau\))** & **Days Needed to Double the Number of Cases (after \(\tau\))** \\ \hline Austria & 12.69 & 8.18 \\ Belgium & 11.05 & 17.60 \\ Croatia & 11.77 & 28.32 \\ Czechia & 12.05 & 28.22 \\ Denmark & 11.49 & 12.71 \\ Finland & 21.81 & 15.84 \\ France & 11.86 & 6.32 \\ Germany & 10.14 & 13.17 \\ Hungary & 11.10 & 22.10 \\ Italy & 13.56 & 7.43 \\ Netherlands & 12.11 & 7.69 \\ N. Maced. & 14.88 & 13.20 \\ Poland & 9.67 & 92.80 \\ Slovakia & 14.91 & 41.59 \\ Spain & 103.50 & 10.62 \\ Switzerland & 10.03 & 8.56 \\ Ukraine & 14.43 & 159.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantifying the inversion from a decrease to an increase in the SARS-COV-2 case rate for the countries of Table 1.  
Figure 1: Inversion of the SARS-COV-2 case trend for Austria, Belgium, Croatia, Czechia, Denmark, Finland. France, Germany, Hungary, and Italy, occurring not later than 2–3 weeks after their first match. Yellow space: duration of the tournament. Red vertical line: first match. Purple vertical line: last match. Green vertical line: last hosted match. Blue vertical line: changepoint. Blue space: CI amplitude for the changepoint. Blue bell-shaped peaks: peaks of the probability density function for the changepoint. Green segment: case rate trend before the changepoint. Red segment: case rate trend after the changepoint. Grey segments: fitted lines drawn randomly from the posterior distribution, based on the corresponding CI. Black dots: number of daily CARS-COV-2 cases. Rightmost \(y\) axis: number of cases. Leftmost \(y\) axis: logarithm of the number of cases. Figure 2, instead, shows the inversion of the SARS-COV-2 case trend of the Netherlands, North Macedonia, Poland, Slovakia, Spain, Switzerland, and Ukraine. All the relevant information needed to interpret the two figures was inserted in the corresponding captions. We used two separate figures, just for the sake of manageability. At the end, also based on an analysis of these figures, we can maintain that these results are fully compatible with the tournament being a factor.