## 3 Results
### Risk of Infection
#### 3.2.2 Gammaitoni-Nucci Model  
In order to explore the relationship between ventilation rates and risk of airborne transmission, the Gammaitoni-Nucci model was used. For this section a quanta emission rate of 5 quanta/h was assumed. First, artificial quanta curves were computed for five consecutive days of occupation based on the solution of Equation (7). Ventilation rates of 0.5 h\({}^{-1}\), 1 h\({}^{-1}\), 3.80 h\({}^{-1}\) and 6.43 h\({}^{-1}\) were chosen, the last of which correspond to the minimum requirement of the ASHRAE 62.1 standard and WHO's recommendation for the pandemic, respectively. Then, the Gammaitoni-Nucci model was used to evaluate the risk of airborne transmission. Figure 11 shows the results obtained. It is surprising that even WHO's recommended ventilation rate implies a critical transmission situation. Furthermore, it is remarkable that even though the estimated ventilation rates for the classroom were (as a weighted daily average) below 3 h\({}^{-1}\), the transmission risk estimated via the Rudnick-Milton model was significantly lower than what is predicted by the Gammaitoni-Nucci model. This observation raises the question of weather the intense but short ventilation periods observed during normal operation of the classroom might be responsible for significantly lowering the transmission risk. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Monthly Mean \(R_{\mathbf{o}}\)**} \\ \hline \multicolumn{4}{c}{**Morning shift**} \\ \hline \multirow{3}{*}{**Month**} & \multicolumn{4}{c}{_q_ **[quanta/h]**} \\ \cline{2-5}  & 1 & 5 & 10 & 20 \\ \hline August & 0.18 & 0.88 & 1.74 & 3.38 \\ September & 0.10 & 0.47 & 0.94 & 1.85 \\ October & 0.11 & 0.52 & 1.03 & 2.03 \\ November & 0.07 & 0.37 & 0.73 & 1.45 \\ December & 0.04 & 0.21 & 0.43 & 0.85 \\ \hline \hline \multicolumn{5}{c}{**Afternoon shift**} \\ \hline \multirow{3}{*}{**Month**} & \multicolumn{4}{c}{_q_ **[quanta/h]**} \\ \cline{2-5}  & 1 & 5 & 10 & 20 \\ \hline August & 0.12 & 0.57 & 1.13 & 2.22 \\ September & 0.10 & 0.48 & 0.96 & 1.88 \\ October & 0.07 & 0.35 & 0.70 & 1.37 \\ November & 0.07 & 0.33 & 0.65 & 1.29 \\ December & 0.05 & 0.27 & 0.53 & 1.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean monthly estimation of the basic reproduction number for different quanta emission rates. To evaluate the previous observation, new artificial quanta curves with dynamic ventilation rates were generated. A basal low ventilation rate was selected and after a certain period a high ventilation rate was imposed for a short time. Then the risk was once again evaluated via the Gammaitoni-Nucci model. Figure 12 shows an example. Several scenarios were tested and are summarised in Table 6, in which the percentage decrease in the basic reproductive number with respect to the corresponding static basal ventilation rate is presented, as well as the absolute basic reproduction number achieved. The results clearly show the tremendous impact that periodic ventilation has on the long distance airborne risk of infection. Despite the absolute values of the basic reproductive number might still deem the situation unsafe, their relative reduction is almost always  
Figure 11: Transmission risk for different ventilation rates, evaluated with artificial quanta concentration curves and the Gammaitoni-Nucci model. Full curves represent quanta concentrations and dashed curves represent the probability of infection. The grey shaded background identifies occupation periods. The coloured boxes detail the probability of infection and the basic reproduction number by the end of the week. Figure 12: Comparison between static and dynamic ventilation rates, evaluated with artificial quanta concentration curves and the Gammaitoni-Nucci model. Full curves represent quanta concentrations and dashed curves represent the probability of infection. The grey shaded background identifies occupation periods. The coloured boxes detail the probability of infection and the basic reproduction number by the end of the week. above 50%.