## 2 Materials and Methods
### Estimation of the Risk of Airborne Infection  
The risk of airborne infection for a certain disease in a room can be assessed by using Wells-Riley models. These are all based on the concept of quanta, a term which [PERSON] [10] defined as a certain amount of pathogen which could cause an infection. [PERSON] used a fully-mixed box model and proposed that the probability of infection of the individuals within the room could be modelled with a Poisson distribution of parameter \(\lambda\), where \(\lambda\) is the total amount of quanta inhaled by an individual during a certain period. AssumingWells' hypothesis, it can be shown that the probability \(P\) of having at least one infected individual in the room can be expressed as  
\[P=1-e^{-\lambda} \tag{6}\]  
Therefore, a quantum (\(\lambda=1\)) can be interpreted as the infectious dose with a 63% chance of infection within a certain population. All of the Wells-Riley models stem from Equation (6) and mainly differ on the way in which \(\lambda\) is calculated. The Gammaitoni-Nucci approach [26; 27], which extends the original steady state model proposed by [PERSON] and his collaborators [11], is based on a mass balance of quanta, which can be expressed as  
\[\frac{dN(t)}{dt}=\sigma(t)qI-\frac{Q}{V}N(t) \tag{7}\]  
where \(N\) is the number of quanta in the room, \(t\) is the time and \(Q\) is the volumetric airflow entering and leaving the room, which is supposed to be pathogen-free. \(qI\) expresses the generation of quanta within the room as the product of the number of infected individuals \(I\) and the quanta generation rate per person \(q\), which is supposed to be the same for each infected individual. Additionally, \(\sigma\) is a Heaviside operator (unitary step function), which allows to consider time periods where the room is unoccupied. \[\sigma(t)= \begin{cases}1&\quad\text{if the room is occupied}\\ 0&\quad\text{if the room is unoccupied}\end{cases}  \tag{8}\]  
It should be noted that the original Gammaitoni-Nucci model did not include a Heaviside operator. We have introduced this modification by emulating what [12; 15] did for the Rudnick-Milton method. Assuming that only \(N\) is a function of time, Equation (7) can easily be solved with a certain initial condition for quanta (\(N_{\text{o}}\)). Furthermore, assuming that the quanta is evenly distributed in the room's volume, the quanta inhaled by an individual over a certain time period \(T\) can be expressed as  
\[\lambda=\frac{p}{V}\int_{0}^{T}\sigma N(t)\ dt \tag{9}\]  
where \(V\) is the room's volume and \(p\) is the breathing rate per person, approximately 8 L/s [25]. The Heaviside operator is introduced in order not to account for breathed quanta while the room is unoccupied. The Gammaitoni-Nucci model is not the best choice for naturally ventilated rooms, as it requires to know the ventilation rate. Nevertheless, it is useful for simulating different scenarios and associating ventilation rates with risk levels. We present a parametric study by selecting different ventilation rates and calculating \(N\) for each time step by solving Equation (7). After that, we numerically compute \(\lambda\) with Equation (9). Another approach for calculating \(\lambda\) is presented by [PERSON] and [PERSON] [25]. It uses CO\({}_{2}\) as a tracer for estimating the fraction of the air in the room that has already been breathed by someone (\(f\)), which is the air that can cause infection. By performing a mass balance to CO\({}_{2}\), it can be shown that said fraction can be calculated at any instant as  
\[f=\frac{C-C_{\text{atm}}}{C_{a}} \tag{10}\]  
where \(C\), \(C_{\text{atm}}\) and \(C_{a}\) are the CO\({}_{2}\) concentrations in the room, outside and in people's exhaled breath, respectively. We assume 380 ppm for \(C_{\text{atm}}\) and \(37,500\) ppm for \(C_{a}\)[12; 34]. A slight modification to the original Rudnick-Milton model is introduced in [12] and in [15] to account for intermittent occupation.