The respiratory route is the main mode of transmission for the virus causing COVID- 19 28 (SARS-CoV-2) [1, 2, 3] . The virus is transported on particles that can enter the respiratory 29 tract. Whereas larger particles (droplets) are only able to stay in the air for a short time and 30 just in the near field (approx. 1.5 m), because they settle down quickly, smaller particles 31 (called aerosols) are also concentrated in the near field and in addition can follow the air flow 32 and cause infections in the far field. Epidemiologically, short-range transmission (through 33 aerosols or droplets) is distinguished from long-range transmission (aerosol) . 34 In order to perform an infection risk assessment for the airborne transmission in the far field 35 and to introduce appropriate preventive measures, it would be necessary to know the amount The so-called aerosols (liquid or solid particles in a dispersed phase with a fluid) as well as 50 droplets differ by size. The particles, which are transported in a fluid over a longer distance, 51 are called aerosols. Droplets are stronger influenced by gravitation and are depositing more 52 rapidly. Depending on the fluid velocity the size of particles, which can be transported in air 53 for a longer distance, is different. In internal spaces with typical air velocities of up to 0.2 m/s 54 particles smaller than 10 μm will be distributed by air very well, with a higher air velocity 55 larger particles may be transported in air as well. 56 SARS-CoV-2 was found to be transmitted via close contact as well as over distance in 57 internal spaces, whereby in distant transmission so-called super-spreading events are more 58 probable [1, 2, 4] . 59
In 1978, Riley et al. [5] evaluated a measles outbreak in a suburban elementary school. Based 60 on the number of susceptible persons (S), which have been infected (D) during each stage of 61 infection, the risk (P) for an infection in this stage has been calculated regarding equation (1) . 62 Therefore, the risk for an infection has been defined as the percentage of infected persons 63 from the number of pupils not already infected or vaccinated. 64
(1)
A Poisson-distribution of the risk of infection has been assumed as well as a stationary and 65 evenly distributed concentration of the pathogens in the room air. Equation (2) shows the 66 Poisson-distribution. 67
Therefore, Wells defined in 1955 [6] a size called quantum as the number of emitted 68 infectious units, where the probability to get infected is 1 − −1 = 63.2%. Hence, a quantum 69 can be seen as a combination of the amount of emitted aerosols with the virus transported on 70 them and a critical dose, which may result in an infection in 63.2 % of the exposed persons. 71
Using the quantum concept as well as equation (2) has been combined by Riley [5] to 72 equation (3) . 73
In equation (3), the number of infectious persons (I), the quanta emission rate depending on 74 the activity (q), the pulmonary ventilation rate of exposed susceptible persons (Qb), the 75 duration of stay (t) and the volume flow of pathogen free air (Q) was used. The quotient q/Q 76 represents the quanta concentration. 77
In poorly ventilated rooms, the assumption of a stationary concentration of quanta is not 78 justified, because of the long time, which is necessary until the stationary concentration is 79 reached. The normalized time-dependent concentration process can be calculated according to 80 equation (4) and is shown in Figure 1 . How rapidly the concentration of a human emitted 81 contamination in a room raises depends on the air exchange rate (ACH) and the time (t). This 82 relative concentration (crel) can be seen as an increase in the concentration compared to the 83 volume flow. 84 perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. . In all published studies identified ideal mixing ventilation was assumed, which means that 89 aerosols are evenly distributed in the room air. To avoid this assumption Noakes and Sleigh 90 [7] divided the room air into different zones, which are themselves considered to be well 91 mixed and have a uniform concentration. This should make it possible to calculate local 92 differences in concentration and thus locally differing infection risks. Furthermore, other 93 studies, which focus on the unsteady conditions mostly use the boundary condition of a 94 starting concentration of ( = 0) = 0 ³ . Gammaitoni and Nucci [8] implemented the 95 starting condition of ( = 0) = 0 as well as the number of exposed susceptible people, 96 which may also change over time depending on their immune status. the equation (5) . If R0 is known, q can be estimated as proposed by Dai and Zhao [9] . For 104 SARS-CoV-2 the average basic reproduction number has been estimated to be 3.28 [10] , 3.32 105 [11] and 3.77 [12] . The virus can be transported on particles in air and the emission of aerosols can be used as an 113
indicator for the emission of virus, but a correlation between q and the aerosol emission rate 114 (E) has not been investigated so far. In measurements at the Hermann-Rietschel-Institute 115 (HRI) of Technical University of Berlin [14, 15] the particle emission rates during breathing, 116 speaking, coughing as well as singing was measured. During breathing through the nose about 117 25 particles/s was emitted and during coughing about 13,700 particles/cough, whereas it can 118 be seen that depending on the activity a wide range of particle emission rates can be found. 119
The transmission of a pathogen via aerosols is also influenced by the stability of the virus in 120 the environment. In an experimental study van Doremalen et al [16] 
Besides the number of emitted pathogen-laden aerosols, the number of inhaled pathogens is 131 playing an important role as well with regard to the assessment of the risk of infection. The 132 pulmonary ventilation rate may differ with different activities. Gupta et al. [17] performed a 133 study with 25 healthy adults and found a sine wave for mere breathing, but a more constant 134 volume flow during talking. 135 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. . In measurements with athletes as well as sedentary persons a maximum volume flow for the 136 athletes of 200 l/min (12 m³/h) was found by Córdova and Latasa [18] . 137
To measure the airflow without movement restrictions, a helmet was used by Jiang et al. [19] For children, the lung volume is smaller. Therefore, the respiratory rate for children aged 14 149 can be assumed to be 0.45 m 3 /h for low activity (breathing while sitting, standing, talking) 150 [22] . 151 152
The Wells-Riley equation can be summarized as equation (7). To calculate the predicted 156 infection risk via aerosols (PIRA) in the far field of a room the concentration of quanta (c(t)) 157 and the respiratory rate (Qb) has to be known. The integration of c(t) can be understood as the 158 amount of particles inhaled per m 3 /h. With Qb the number of inhaled quanta can therefore be 159 calculated. 160 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. .
Equation (5) can be used for the definition of the quantum emission. This leads to a quanta 161 emission rate of q = 40 1/h at an assumed mean R0 = 3.35. The mathematical approximation 162 presented by Dai und Zhao [9] can be optimized by equation (8), see Figure 2 . For Figure 2  163 the quanta emission rate has been correlated with R0 of tuberculosis [23, 24] , Influenza [23, 164 25] , MERS [26, 27] and SARS-CoV [23, 28] . 165
Equation (8) 
171 All rights reserved. No reuse allowed without permission.
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The copyright holder for this this version posted October 12, 2020. . https://doi.org/10.1101/2020.10.08.20209106 doi: medRxiv preprint q is influenced by the activity of the person as was shown by Buonanno et al. [29] . Therefore, 172 the measured aerosol emission rates [14, 15] were correlated with the calculated quanta 173 emission rates influenced by the activity by equation (9) . For low-activity (breathing, 174 talking, sitting, standing) a basic volume flow Qb,o and normal activity = low activity 175 (breathing, talking, sitting, standing) with a basic emission rate of E0 was used. Furthermore, 176 the basic q (q0) was calculated with usage of R0 regarding equation (8) . With these 177 specifications qa can be calculated. 178
,0 = 0.54 3 ℎ ; 0 = 100 [15] 179
The effect of e.g. mouth-nose protection can be considered by using their filtration efficiency 180 (FMNS) like in equation (10) which however will not be further considered in the following. 181 , =
•
182
It is known that the infectivity of an infected person depends on the disease progression over 183 time [30] . This is shown qualitatively in Figure 3 . With a simplified mathematical approach, 184 this can be integrated into the quanta source rate. An equation could be implemented to take 185 this into account. 186 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. 
The concentration of quanta during the increase cI(t) can be calculated according to equation 192 (12) with the number of infectious persons (n). 193
An additional case is considered that if the time t is longer than the age of the air τn, most of 194 the virus-laden aerosols have left the room with the exhaust, before the inactivation can take 195 place. Therefore, this concentration during the steady state situation is called cτ(t). 196
with τn regarding equation (6) 197 198 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. 
203
The Predicted Infection Risk via Aerosols can be calculated by equation (17) and (18). 204
For the calculation of PIRA the following assumptions must be considered: 206  the aerosols are ideally mixed in the room 207  the near field (up to approx. 1.5 m distance from the emitting person) can contain a 208 much higher virus-laden aerosol concentration 209  the air, which is introduced into the room, is free of virus-laden aerosols (e.g. outside 210 air) 211  no deposition of small particles is considered, because the settling time is longer than 212 the stability of the virus and the deposition rate would therefore be substantially 213 smaller than the inactivation 214
The PIRA calculation model was validated by using parameters of several known outbreaks 217 during the SARS-CoV-2 pandemic. Twelve different scenarios either scientifically published 218 or registered by the local health authorities were selected (A-L). The boundary conditions for 219 the calculations of these situations can be found in Table 1 . 220 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. . https://doi.org/10.1101/2020.10.08.20209106 doi: medRxiv preprint *due to partly missing information, assumptions were made, especially for window ventilation. The assumptions are based on information from the persons involved on how the windows were opened and closed in combination with weather data at the time.
** It was assumed that the local regulations for fresh air supply were fulfilled. ***Geometry and Ventilation Rate due to [31] ****Attack Rate was simplified as percentage of persons infected. No separation regarding infection attack rate (measured serologically) and illness attack rate (persons with symptoms or laboratory-confirmed) was performed.
***** It was assumed that a school lesson lasts 45 minutes. 
The infection events used for the validation of the model are shown in Table 1 with the 223 necessary parameters for the calculation. In the following, the comparison between the 224 documented Attack Rate (AR) and the PIRA is drawn. 225
The q used here was calculated according to equation (9) with the assumption that the cases 226 emitted particles as measured in [14, 15] . Due to the high spread of the particle emission E 227 and the unknown proportions of breathing, speaking, singing and shouting as well as the 228 respiratory volume flows, simplified a-priori assumptions were made. To take into account 229 the effects of the uncertainties regarding q, Qb and especially with window ventilation on Q, 230 these values were further varied -q by +/-20%, Qb by +/-20% and Q by +/-50%, 231
individually and in combination, which then lead to a minimum PIRA and maximum PIRA. 232 Table 2 ). 240 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. . https://doi.org/10.1101/2020.10.08.20209106 doi: medRxiv preprint 242 From the PIRA model, it can be calculated how much volume flow per hour of exposure time 243 is required to not exceed a certain PIRA. The results are shown in Figure 5 . It can be seen that 244 for a PIRA of 10% a volume flow of clean air of 750 m³/h and hour of exposure has to be 245 supplied to the room (see Table 3 ), whereas for two hours 1500 m³/h will be necessary for the 246 same PIRA. 247
As a regression of the calculated results presented in Figure 5 equation (19) was derived. 248
Using equation (19) the required volume flow per hour of exposure time can be calculated. 249
This information refers to the steady state if the product of ACH and t is higher than 5.0, see 250 
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The copyright holder for this this version posted October 12, 2020. . https://doi.org/10.1101/2020.10.08.20209106 doi: medRxiv preprint Table 3 lists practical examples of the required volume flows depending on the exposure time 257 and PIRA. 258 Table 3 In Table 4 perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. . https://doi.org/10.1101/2020.10.08.20209106 doi: medRxiv preprint In outbreak E, there is only little documentation of the infection process, and further contact 288 between some of the persons has occurred in a restaurant afterwards. Furthermore, it has not 289 been determined whether the infection can be attributed to only one person. The high AR after 290 a short time of exposure allows the conclusion that either two index persons were present or 291 that the exposure time was prolonged by the meeting in a restaurant. 292
In outbreak F, the group was not together for the entire time and some of the subjects 293 continued rehearsing in another room. For this reason, the exposure time for the whole group 294 was lower and therefore this may account for the lower AR than calculated by PIRA is given. 295
In outbreak L an air exchange rate of 21 1/h was assumed. A relatively small change in the 296 assumed volume flow has a significant influence on the result of PIRA (where the total 297 exposure time was used). Furthermore, it cannot be excluded that droplet transmission may 298 also have happened. 299
Secondly, many assumptions were made, therefore it is not clear if the formula is already 300 optimal, perhaps further optimization during the course of the pandemic is possible if further 301 knowledge is available. 302
Third, the calculation model does not consider the sedimentation behavior of particles. It is 303 known that at higher air velocities and especially at high turbulence the sedimentation 304 behavior increases. In typical indoor air flows this decrease is about 10 % per hour. Compared 305 to the uncertainty of the overall emission rate, this effect is not significant. 306
Fourth, the calculation model assumes a homogeneous distribution of the particles in the room 307 air. Practically however the ventilation effectiveness is locally very different. The differences 308 can be slightly greater than 100%. 309
Finally, it must be noted that the aerosol concentration is significantly higher in the near field 310 of the emitting person and the results of PIRA are not valid within the generally accepted 311 1.5 m distance rules. 312 313
It was shown in this investigation that it was possible to calculate the risk of an infection via 315 aerosols for situations where the long-distance transmission is more important. By using the 316 model presented here, a good agreement to previous infection outbreaks in different settings 317 and different attack rates was achieved. Previous retrospectively determined quanta emission 318 rates usually assumed a stationary state. However, if the concentration process is important 319 for the total amount of inhaled virus-laden aerosols (usually at ACH x t < 5), then a stationary 320 observation leads to an incorrect boundary condition. The time-dependent viability of the 321 virus also plays a significant role. Here, the influence of the viability is higher at low air 322 change rates compared with high ones, because the virus stays in the room air for a longer 323 time period and the proportion of inactivated pathogens increase. However, the effect of time-324 dependent viability is not that important that a low air change rate has an overall positive 325 effect. 326
To reduce the risk of infection via aerosols the necessary volume flow of virus-free air 327 depending on the exposure time can be seen in Figure 5 . This figure may be helpful to 328 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted October 12, 2020. . https://doi.org/10.1101/2020.10.08.20209106 doi: medRxiv preprint implement measures, like increasing the virus-free air supply rate. Furthermore, the number of 329 exposed persons has to be kept in mind. An infection risk of 60% may result in one infected 330 person in a two-person office, but in 60 infected persons in a room with 100 persons. 331
Predicting the infection risk via aerosols and knowing the important parameters can help in the 332 selection of appropriate preventive actions. 333 334 Declarations 335