key: cord-0926488-jb70w1ck
authors: Rowe, Bertrand R.; Canosa, André; Meslem, Amina; Rowe, Frantz
title: Increased airborne transmission of COVID-19 with new variants, implications for health policies
date: 2022-05-12
journal: Build Environ
DOI: 10.1016/j.buildenv.2022.109132
sha: b25cb3655d834ece9ed7c277ab70567b3f114a93
doc_id: 926488
cord_uid: jb70w1ck

New COVID-19 variants, either of higher viral load such as delta or higher contagiousness like omicron, can lead to higher airborne transmission than historical strains. This paper highlights their implications for health policies, based on a clear analytical understanding and modeling of the airborne contamination paths, of the dose following exposure, and the importance of the counting unit for pathogens, itself linked to the dose-response law. Using the counting unit of Wells, i.e. the quantum of contagium, we develop the conservation equation of quanta which allows deriving the value of the quantum concentration at steady state for a well-mixed room. The link with the monitoring concentration of carbon dioxide is made and used for a risk analysis of a variety of situations for which we collected CO(2) time-series observations. The main conclusions of these observations are that 1) the present norms of ventilation, are both insufficient and not respected, especially in a variety of public premises, leading to high risk of contamination and that 2) air can often be considered well-mixed. Finally, we insist that public health policy in the field of airborne transmission should be based on a multi parameter analysis such as the time of exposure, the quantum production rate, mask wearing and the infector proportion in the population in order to evaluate the risk, considering the whole complexity of dose evaluation. Recognizing airborne transmission requires thinking in terms of time of exposure rather than in terms of proximal distance.

Particles emitted by a human refer either to spherical microdroplets or to more or less 128

hydrated "dry nuclei", resulting from water evaporation of the respiratory fluids, which, beside 129 water, contains minor components like mucus, proteins and viruses [23] . VL is a key parameter of 130 particle infective power and depends on the mean number of viruses per unit volume of respiratory 131 fluids, which lead to a mean number per particle. This latter is statistical, i.e. it implies a large 132 distribution of particles with various viral contents. A mean VL per particle lower than unity implies 133 that some microparticles will contain a virus and others will not. Moreover, evaporation of exhaled 134 microdroplets can result in particles of lower size without virus loss. Since the smallest particles are 135 very abundant, they can be very efficient in airborne transmission. 136

These particles can be characterized by their size and composition, including VL which depends 137 on the viral strain. Their size depends mainly on their origin from the respiratory tract and of their 138 evolution in the ambient air, including evaporation. The largest droplets, behaving in a ballistic way, 139 are most often emitted by talking, sneezing, or coughing. The smallest ones come from various parts 140 of the respiratory tract, including the lungs. They have a large distribution of sizes, and many are 141 below 10 m, especially after evaporation of some of the largest ones. In a kind of reversible way, 142 the smallest ones (< 5 m) can penetrate deep in the lung when re-breathed and are known as 143 respirable aerosols [10, 24] . 144

One of the most sophisticated apparatuses used for the size characterization of these aerosols 145 is the specific wind tunnel developed by L. Morawska found four main modes in the distribution of particle size, centered around 0.8, 1.8, 3.5, and 5.5 m 148 respectively. definitions are much more difficult for biological pathogen agents that are not easy to measure and 154

have the possibility to replicate in the target host [28, 29] . Concerning aerosols and as stated by Haas 155 et al. [28] "precise information on the concentration of pathogens in aerosols has a lot of uncertainty 156 associated with it". Moreover, and for any kind of disease (i.e. respiratory, digestive etc.), the effect 157 of the dose could depend on the way of transmission: inoculation, ingestion, airborne etc. Having 158 defined a dose, the work of epidemiology is to assess quantitatively the risk for a given dose: by 159 nature, such an assessment is statistical; it results most often in a law linking the probability of 160 infection to the dose. 161

For airborne transmission of respiratory diseases, the definition of a dose is far from being 162 straightforward since measuring pathogen concentrations in the air is extremely difficult [28] . 163 Therefore, Wells [4] defined the quantum of contagium as a hypothetical quantity that has been 164 inhaled per susceptible individuals (men or animals) when 63.2% (correspondingly to a Poisson dose-165 response law, see sub-section 2.4) of these individuals display symptoms of infection. Quantum is 166 used throughout the present paper and contrary to what has been sometimes claimed [30] , it has no 167 dimension but is a counting unit (like moles compared to molecules). It considers a variety of 168 mechanisms: inhalation of airborne particles, pathogen inhibition by host defenses (see 169 supplementary materials1, hereafter SM1-7) or other losses, before any replication will start in an 170 infected cell. Therefore, it corresponds statistically to a number of pathogens higher than one. 171

However, these statistical concepts do not mean that very few pathogens are never enough to 172 start infection, as assumed sometimes. Indeed, the so-called "single hit" models make statistical risk 173 assessment considering a very small probability, although non-zero, of infection by a single pathogen 174 [28, [31] [32] [33] [34] . Further, and as stated by Haas et al. [28] , the term of Minimum Infective Dose is very 175 misleading since "Minimum" suggests some threshold effect for the infection. They emphasize that it 176 corresponds in fact to the average dose administered and most frequently relates to the value 177 required to cause half of the subjects to experience a response; they suggest that "median infectious 178

dose" should be more appropriate, and they show that it is not possible to infer the probability of 179 infection by a single pathogen from the magnitude of the median infectious dose. 180 2.3 Link between the quantum production rate and infectious aerosols 181

Evaluation of quantum concentration in air requires knowing the production rate of quanta by 182 an infector, defined per unit time (unit: h -1 for example). It can be deduced from epidemiological 183 observations [35] but also linked to the distributions of microdroplets emitted by humans, together 184

with the knowledge of VL in respiratory fluids and of the efficiency of the viral strain. 185

Following Buonanno et al. [36] the production rate of quanta can be written as: 186

where refers to unit volume viral load of respiratory fluid, is a proportionality factor between 188 the exhaled viral content (copies/unit time) and quanta, is the pulmonary exhaled volume rate 189 (volume/unit time), ( ) the size distribution of droplet concentration (diameter ) of volume . 190 The factor depends on the microbiological characteristics of the variant and can explain a higher 191 value of (and hence a higher contagiousness) even with a lower . 192

Equation (1) implies that the production rate of quanta can be considered as proportional to 193 VL in the respiratory fluids and to a factor (c) which depends on the virus microbiological 194 characteristics. Equation (1) assumes a single mean value of VL. This is a reasonable assumption 195 since the quantum production rate is a statistical mean quantity that does not consider the diversity 196 of particle emission processes, although VL depends probably on the particle origin from the 197 respiratory tract. Note also that the integral in (1) 

Note that if there is some air treatment (filtration or sterilization or both) for the volume , it 234

can be considered as an increase in the flow rate of fresh air and therefore results in an increase of 235 2 value. Indeed, it is also possible to introduce the virus lifetime as an increase in the ventilation 236 flow rate through equations (7) and (8). The virus lifetime depends on a variety of phenomena 237

including UV irradiation. 238

In a situation where the stationary state has already been reached in a homogeneous volume 239 at the beginning of exposure then, following equation (9 and 2), the inhaled dose is: 240

which yields for the probability of transmission: 242

Together with the quantum definition, these equations are the basis of the Wells-Riley model 244

[ Let be the ratio of the doses of exposure between and  in case of identical situations, 282 from section 2 (Eq. 1 and 2), R can be reduced to the ratio of VLs and of the proportionality factors c: 283

It is then easy to demonstrate that relative probabilities of being infected between 285 respectively  and IS variants follow the next equation: 286

It results that, from the recognized fact that ≫ , the airborne contamination by the  289 variant is much more efficient than with initial strains for comparable situations, as shown in Figure 1 Assuming an infector proportion , we can express the number of infectors as: 306

Strictly speaking it is the prevalence of infectors, including asymptomatic, that should be used 308 for r. It is anyway probable that the number of infectors is proportional to . As discussed in SM4 it 309 is extremely difficult to have the exact value of r from the values of positivity rate or incidence rate 310 reported by health agencies. Below we use a "reasonable" value for r consistent with the pandemic conforms to the norm and for a given value of , the difference between a school, a restaurant and a 317 commercial center comes essentially from the time of exposure . Note that this time is a total time 318 which does not need to be continuous but can be a summation of hourly and daily exposition in the 319 various spaces that the individual went through, due to the fact that the risk is essentially 320

probabilistic. Clearly the difference in quantum production rate between  variant and previous 321 strain, plays an enormous role in the dose, and hence in the probability of infection. However, it is 322 clear from equations (11) and (17) The CO2 time evolution followed the standard law: 395

where [CO2]0 is the CO2 concentration, expressed in ppm, at the beginning of the analytical fit (t = 0), 397

[CO2] is the stationary CO2 concentration (t = ), Q the ventilation flow rate (m 3 /h), V the room 398 volume and t the time at which the measurement was carried out. From this equation, it is 399 straightforward to determine the ventilation flow rate Q from an exponential fit of the measurement 400 when the volume V is known, at least when [CO2] is not ill-defined, a situation that occurs when the 401 number of people constantly changes with time like in the restaurant (see Table 1 ). 402

The CO2 time evolutions are illustrated in Figure 4 -(a-d) where the reference of the CO2 403 concentration has been taken as an outdoor [CO2]ext usual value of 400 ppm instead of considering 404

[CO2]0 as the reference. This makes it easier for the readers to return to the absolute value since the 405 initial [CO2]0 is never the same from one test to another. Table 1 ). These lecture rooms 413 are at a University building over 50 years old, which has not yet undergone any energy retrofit. The 414

ULR5 is equipped with air intake vents installed in window frames. As the building envelope is not 415 airtight and since the toilets facilities, equipped with mechanical air exhaust, are far away from ULR5, 416 little fresh air enters by the windows intake vents. In addition, exhaust flow rates at the level of the 417 building are too low compared to the regulatory ventilation needs. This explains the observed very 418

poor IAQ with maximum concentrations of CO2 exceeding 5000 ppm. This trend has been confirmed 419 in a similar lecture room (ULR4, not shown for brevity) where CO2 concentration measurements 420 during five consecutive scholar days lead to an air stuffiness index ICONE of 4, i.e. very high 421 confinement (see SM6). 422

The ULR20 is a lecture room, among three rooms of the same previous building, which were 423 fitted more than ten years ago with a common dynamic two-way ventilation system, using the level 424 of CO2 in the exhaust circuit to control the ventilation flow rate. CO2 rise from which the ventilation rate could be estimated. Finally, we carried out a CO2 monitoring during a week (Figure 4 -d) in a modern restaurant 467 situated in a coastal location of the Department of "Côtes d'Armor" in France. We used two Aranet 468 sensors each one set in one of the two lunchrooms of the restaurant which communicate to each 469 other through a large aperture. The two sensors were approximatively at a distance of 10 m to each 470 other and demonstrate a similar CO2 concentration along the week. This is a strong demonstration 471 that for this case, the well-mixed assumption holds. Interestingly, the restaurant is exposed to the 472 wind, which can cause large variations in air renewal flow rates. Observations correlate strongly with 473

an enhancement of ventilation with the strength of the wind (and inversely for CO2 concentration) 474 which is shown on each peak of the figure in Beaufort scale (Bt = 1-2 on Monday; 4-5 on Tuesday; 5-7 475 on Wednesday; 5-6 on Thursday and 5-7 on Friday). Not indicated is the direction of the wind which 476 has been changing continuously along the week. The high variability in peak CO2 from day to day can 477 be clearly seen in Figure 4- annual costs ranging from a few dollars to ten dollars per person constitutes less than 0.1% of typical 501 public spending on elementary and secondary education in the US. Such spending is judged a small 502 price to pay given the evidence of health and performance benefits. This observation is more than 503 ever true in this pandemic period and could be extended to other countries and other sectors than 504 education. In the same spirit, it is desirable to generalize the use of CO2 sensors, a very affordable 505 tool, in buildings to assist people in applying the suitable mitigation behaviours such as windows 506

opening for instance to accelerate indoor air renewal. 507 4.2 Risk assessment 508 For the various situations described above it is important to derive a risk probability for an 509 exposed person (susceptible) as a function of the observed CO2 concentration. healthy subject is exposed to successive doses corresponding to different periods of exposure Δ , 539 then the total dose is just the sum of the successive doses (cumulative risk): 540

Following these formulas, we can deduce some risk probabilities corresponding respectively to 542 the situations described in section 4.1. They are summarized in Table 2 : 543 • For the school, the situation would be catastrophic without the risk reduction due to the 551 mask. However, the precise quantitative impact of mask wearing is difficult to evaluate 552 as discussed in the SM5. Also the social acceptability of mask wearing by children merits 553 to be discussed. 554

• For the restaurant/bar, we have considered that customers are mainly workers who 555 spend about 80 minutes at lunch. The risk is negligible for a single meal. In Table 2 P is   556 bracketed since conditions varied depending on the day. If the restaurant is visited on a 557 daily basis (5 meals) risk could be raised to a few percent following equation 23. 558

However, the calculation does not consider that the mask is partly worn in the 559 restaurant. In any case our observation and calculation show that the risk here is not 560 especially high, which questions public policy in this field. 561

• For the other premises, which are located at the university, observations show a 562 considerable dispersion. The risk can be very high for a lecture room very poorly 563 ventilated (case URL5 of table 2) as well as reasonable in well ventilated area (case 564 UAW). It must also be considered that for the university premises we have not 565 considered either mask wearing or the cumulative aspect of the dose. As discussed in 566 SM5, using masks induces a risk reduction of a factor of about 9. This is however easily 567 counterbalanced within one week if students attend 9 lectures in the same room which is 568 quite possible. It remains that in poorly ventilated areas the risk is high. 569 570 5 Implications of increased airborne contamination for health policy 571

The previous sections highlight the multiparameter character of the risk, through the time of 572 exposure and the concentration of airborne infectious particles, itself linked to the proportion of 573 infectors and to the indoor ventilation flow rate. With new variants such as  or  (omicron) the 574 quantum emission rate can be estimated orders of magnitude higher than with the original strain 575 due to VL or microbiological characteristics. Then, the spread of the virus should be mainly airborne 576 even for close contact, and much more efficient. necessary, and we insist that authorities have to change their mind in matter of priority. 587

Amongst the various interventions of public policy discussed below we focus on the non-588 pharmaceutical ones. We first consider interventions directly targeting IAQ, i.e., mask, air filters and 589 sterilizers, and ventilation. In this context we will also discuss the influence of the way of life, which 590 depends on the country and the climate, and could lead to take immediate measures with strong 591 positive consequences. We will then turn to interventions that are not directly targeting IAQ but 592 nevertheless have implications on IAQ (e.g., living conditions during lockdown) or whose 593 effectiveness is dependent of our understanding of contamination routes (e.g., contact tracing When IAQ is deficient, especially in indoor situation, wearing a mask is certainly highly useful 608

[10,57,58], but their efficiency (especially for surgical ones) is not such that it could be the solution 609 alone. It is possible to calculate that the risk probability P could be decreased by a factor of ten when 610 J o u r n a l P r e -p r o o f both infectors and susceptibles wear it (see SM5). However, with new variants the quantum 611 production rate increase could counterbalance this advantage. Moreover, in most countries, after a 612 deny of mask interest, the choice of surgical ones in the general population has been made, although 613 they are much less efficient than N95 respirators [59,60]. In some situations, the public should be 614 informed of the better choice, depending on the IAQ (see sub-section 4.2). People must be told that 615

wearing mask under the nostrils is inefficient. 616 617

Therefore we conclude that wearing a mask alone, although useful, is insufficient to 618 counterbalance the very high VL due to delta variant or the microbiological characteristics of 619 omicron. Also social acceptability of masks on the long term is most doubtful. Therefore, we must 620 take further corrective measures to improve IAQ. 621 IAQ has been recognized as a concern for public health and is addressed by building norms. The problem is closely linked to building ventilation, which has been for centuries a natural 628 ventilation, i.e., fresh air intake by voluntary or involuntary leaks on the building envelope allowing 629 entrance and circulation of fresh air without real control. Since the first oil shock and the subsequent 630 implementation of increasingly restrictive energy regulations, including today new constraints linked 631 with environmental impact, things changed with buildings becoming more and more airtight and 632

with HVAC technologies allowing ventilation control. The admission of fresh air is therefore 633 minimized at the lowest value (hygienic flow rates) compatible with physicochemical IAQ, in order to 634 save energy but frequently this leads to non-compliance with regulatory hygienic flow rates. applicable since January 2022 for residential buildings, takes a step forward by setting up an 643 obligation to measure ventilation flow rates. However, one can object that this point is not subject to 644 a building acceptance certificate. Another criticism is that verification of the airflows is not entrusted 645

to an independent control office since ventilation system installers can make the flow rates 646 measurements themselves. distancing between individuals is higher than 1.5 m, correspondingly to the communication of 717 government and health agencies. However, a misunderstanding of the real mode of transmission in 718 this case has led to irrational measures such as organizing files in supermarket with obligation to use 719 entrances different from exit. Although it has not been yet studied in the literature, staying in the 720 wake of an infector in a file for several minutes is certainly riskier than crossing the infector. We 721 recommend that, although social distancing must be encouraged, such measures directed against 722 fast crossing should be removed since they are misleading for the public and could in fact induce 723 higher airborne transmission. 724 5.2 Implications for interventions that are not directly targeting IAQ 725

The most radical intervention to mitigate the pandemic has certainly been the various forms of 726 lockdowns that, notably in western societies, constitute a major limitation to liberties and was 727 unprecedented in non-war conditions. While first lockdowns might have been necessary, given the 728 lack of governmental readiness to fight such pandemics in western societies, we now realize that, 729

beyond the obvious socio-economic implications, it has a significant downside related to 730 psychological isolation and mental health. and alert facility managers and users in a way that could be similar and complement that outbreak 745 risks related to fomite-based transmission [88] . But mitigation measures such as contact tracing apps 746

will have little effect against long range transmission by aerosols. These apps have not been designed 747

to fight this transmission path of the pandemic and aerosol transmission was ignored at their 748 inception. When aerosols are emitted from delta variant, it is the exhaled microdroplets 749 concentrations in a given space that creates the major risk. Focusing on close crossing (less than one 750 meter for more than 15 minutes as we did in France with stopCOVID) in a public space can be 751 dangerous because people can feel safe (at least feel being well informed with their app), when in 752 fact what they should be warned (possibly by their smartphone, but even better by public screens or 753 specific systems) is about the situation over IAQ. Therefore, given the airborne danger of delta 754 variant, we consider that contact tracing apps are inappropriate for at least three reasons: First, to be 755 effective they require that a very large share of the population uses them for contact tracing which 756

has been considered unrealistic [89] and is still the case. In fact, whereas contact tracing apps have 757 been redesigned to be less intrusive (e.g Norway case) and their governmental communication to 758

influence their adoption adapted in to be less coercive (e.g. France case), a common nudging tactic to 759

influence their adoption has consisted in adding a number of features influencing individual benefits 760

such as giving information about risky regions or allowing to show conformity to vaccination plans to 761 access public places thus transforming a risk detection app into an information public health and a 762 sanitary pass app. As a result, after vaccination campaigns, these apps have been hugely 763

downloaded. However, the effective activation of the apps for personal risk detection is still very low. 764

Second, as we emphasize in the present paper, relevant parameters, notably time of exposure to 765 risk, and space, but not necessarily distance, to a likely infector, were not well understood by proactive. As the situation may vary a lot from place to place or evolve rapidly, the population at risk 779 could also use a warning emergency system conveying alerts through a dedicated device [94] or 780 some augmented reality app [88] . Given our experimentation measures, such public displays would 781 be highly trustworthy thanks to their high representational fidelity (notably current, nearly exact and 782 relevant (on these notions see [93])) of the CO2 measures and thus limit the use of such warning 783 emergency systems to those at risk and not coerce all the population to acquiesce to a rampant form 784 of data surveillance. The cost of such air monitoring equipment and public website will not be very 785 high and they are a more ethical and scientifically valid choice, given the prevalence of the aerosol 786 transmission path, than current digital policy based on smartphone close-contact tracing. 787

The present health policies in many countries suffer from an original sin which was the deny of 789 airborne transmission. The advent of strains such as  or  leads to much higher quantum production 790 rates, implying that spreading of the epidemic is now mainly airborne. level, but also to the probability that an infector is or has been in the room and to the time of 813 exposure. 814

Finally, digital means should be directed at informing people (e.g. with appropriate screens or 815 web applications possibly using augmented reality for particularly vulnerable persons, rather than 816 digitally tracing their (social) behavior and surveilling them). With the introduction of smartphone-817 based contact-tracing apps further embedded in sanitary passes, the pandemic has considerably 818 accelerated the pace of the transformation of western societies towards digital surveillance. While 819 some initial intentions were hoped to be good, such trend is dangerous and shows that ethical use of 820 the digital is still in its infancy. 821

We insist that thinking only in terms of social distancing or social interactions has become a 822 paradigm that needs to be changed. Scientific literature demonstrates that we can be infected by 823 close contact, but other situations can be dangerous due to the very nature of airborne transmission. 824

As viruses can stay infectious in the air, we should not only consider the possibility of contamination 825 in co-presence, typically when people face each other, but also when people follow each other in a 826 file or even when infected people have left a poorly ventilated room. These scenarios need to be 827 highlighted in public information. 828

And last but not least, when the present pandemic will be over, what will stay in the mid and 829 long term is the necessity to change our mind and norms in matter of IAQ, in order to include this 830 problem of airborne pathogen transmission, an enormous challenge for building technology. 

As discussed in the main paper the quantum of contagium, as defined by Wells [1] , considers a 1168 variety of mechanisms including pathogen inhibition by host defenses. These defenses include, 1169 beside microbiological phenomena (immune response and others), some physical processes 1170 described below that are important for contamination by the aerosol route. 1171

In a series of remarkable experiments with rabbits and mice, Wells demonstrated that, 1172

concerning aerosols, very fine particles (which include dry nuclei) have a much higher infectious 1173 power than coarse particle, at least for disease such as tuberculosis and influenza. Wells' explanation 1174 was that the human body has a very efficient system to prevent coarse particle larger than a few 1175 micrometers to penetrate deep in the respiratory system. Beside defenses against very coarse 1176 particles, specific to the upper respiratory tract (nostrils, nasal cavity, mouth, throat, pharynx), and 1177

voice box (larynx)), mucociliary clearance is a primary innate defense mechanism of the lung (see the 1178 reviews by Bustamante-Marin and Ostrowski [2] and Kuek [3] ) that helps to remove smaller particles 1179 and pathogens from the lower respiratory tract, using the epithelium formed by ciliated and 1180 secretory cells. These later provide a mucus which is expelled by cilia toward the digestive system 1181 after swallowing. It is known that most respirable pathogens do not provoke illness when ingested, 1182

and there is currently no evidence that COVID-19 could be transmitted by ingestion [4] . Note that the 1183 mechanism of very fine particles deposition into the lungs has been the subject of numerous studies 1184

for mineral toxic dusts, such as asbestos [5] . 1185

Nowadays, the formidable progress of microbiology allows studying the influence of cellular 1186 characteristics on the vulnerability of cells to coronaviruses, which start with binding of the viral 1187 spike (S) proteins to cellular receptors [6]. Following some data, it has been anticipated that 1188 infectivity was higher in the upper respiratory tract and that the nose was a primary target [7] . 1189

However the severity of the COVID-19 is linked to the occurrence of pneumonia, followed by acute 1190 diffuse alveolar damage, which can be due to direct lung infection by airborne microparticles [8, 9] or 1191 by indirect infection from the oropharynx to the lung by aspiration of the viral inoculum when 1192 breathing [7] . Also the study of nonhuman primate model reveals, after autopsy, the importance of 1193 lung lesions in macaques [10]. It seems reasonable to assume that, when the virus reaches the lungs 1194 directly, before some immunity able to inhibit viral reproduction has been acquired, it could result in 1195 devastating pneumonia, as sometimes reported in young, healthy subjects. 1196

It has to be noticed that as well the remarkable experimental results of Wells for particle size 1197 than the most recent findings of microbiology cannot be directly used to develop a quantitative 1198 model of transmission risk. Therefore, some concepts and approaches must be developed prior to 1199 the establishment of any risk model. components. We shall just present the equation used in the main paper for the case of a well-mixed 1206 room (homogeneous hypothesis) and the approach underlying much of the calculations used in 1207 inhomogeneous models in order to calculate the concentration field of infectious particles. 1208 In a homogeneous model, it is assumed that there is no spatial gradient of risk in a space 1209

where the infectors and the receivers either evolve or stay in place. In other words, it is assumed that 1210 the infectious microdroplets are evenly distributed. This is typical of two kinds of situations. It was assumed no sink term for the particles inside the volume. 1235

In the same way, an equation of conservation can be applied to the quanta of contagium as = 2 × × × 0

With and being respectively the diameter and volume mass of the particle, the air 1274 viscosity, 0 and respectively a typical order of magnitude of flow velocity and length. is a slip 1275 parameter which takes into account the value of the particle Knudsen number. For particles of the 1276 size considered in airborne transmission is very close to one. Note that 0 ⁄ is the hydrodynamic 1277 time and that for most problems dealing with the behavior of exhaled aerosol particles in indoor 1278 situation the Stokes number remains much smaller than one, except for large particles in the close 1279 contact case discussed in next section. 1280

Il is also worthwhile to point out that when inhomogeneous infection modeling is applied to a 1281 specific geometry of the environment, it can be applied as such for the design of a new building for 1282 example but is limited for applications in the real life of most existing buildings and therefore, on the 1283 short term, for driving public policy. What is more interesting is the modeling of airborne close 1284 contact discussed in the next section. 1285 1286 J o u r n a l P r e -p r o o f

It is now largely admitted that the transmission of COVID-19 disease by close contact is most 1288

often an airborne one, referred in the literature as "short-range airborne transmission" 1289 [20, 20, 21, 21] . Close to the emitter the turbulent expiratory plume (or puff for cough and sneeze) can 1290 have a much higher quantum (viral) load than in the ambient air of the indoor space considered. 1291

Several models of this phenomena have been proposed, some very simple [20] others more 1292

sophisticated. The recent one by Cortellessa et al. [21] employs CFD for the air flow and Lagrangian 1293 calculations for the particles to derive the dose and the risk as a function of the distance between 1294 infector and susceptible. Not only the distance but also the time of exposure is considered in order to 1295 assess the risk, although the time is limited to fifteen minutes. Large microdroplets which behave in a 1296 ballistic way are also considered and shown to prevail only at very short distance (< 60 cm), with a 1297 contribution to the dose being completely negligible at higher distances, demonstrating the airborne 1298 character of most airborne contamination in close contact, excepted intimate. 1299

In their paper, Cortellessa et al. also made a comparison with the homogeneous risk. However, 1300 the comparison is restricted to the same time of exposure of fifteen minutes, with an initial 1301 concentration of quanta equal to zero. Therefore, it does not consider long times of exposure for the 1302 homogeneous case at steady state, as found for example in schools but such an extension can easily 1303 be done. Indeed, a good comparison should have to include the probability of close contacts 1304 together with contact durations, which is not done. Such a close contact risk assessment is anyway 1305 extremely useful for public policy. These calculations demonstrate a very good agreement between both ways of determining the 1389 probability of infection from either PWR or P. The agreement is even better when the restricted 1390 population is enhanced, essentially due to the statistical effect of using larger Np numbers. It can also 1391 be shown that the contribution of one infector (nmax=1) in the summation is the main one in many 1392 situations although, however, summation cannot be limited to the first term in SM4-8 for several 1393 conditions as indicated by the ncut value and the "nmax contribution" columns. The lower the 1394 proportion of infectors r, the larger the contribution of 1 ( ) × (1) which makes a lot of sense 1395 since for small r the probability of having more than one infector in the restricted population Np 1396 becomes very small. 1397

To conclude we stress that we have restricted the demonstration to a limited number of 1398 configurations but it is worth pointing out that several parameters act in a similar way 1399 mathematically speaking. Then, changing the time of exposure or/and the quantum rate of infectors 1400 would lead to essentially the same kind of conclusions. 1401 1402 J o u r n a l P r e -p r o o f

To build a probabilistic model of infection it is necessary to know the production rate of quanta 1404 (as defined by Wells) by an infector. It is defined per unit time and per infector (unit: h -1 for example) 1405

and can be deduced from epidemiological observations [22] but also linked to the distributions of 1406 microdroplets emitted by humans, together with the knowledge of viral load in respiratory fluids and 1407 of the mean number of viruses required to infect 63% of susceptibles. 1408

As stated in the main paper and following Buonanno et al. [23] , the quantum production rate 1409

can be written as: where the subscripts i and j refer to the size mode and the expiratory activity respectively. 1424

From equation SM5-2 and Table SM5-1 it is clear that the production rate of quanta can vary 1425 widely depending on the expiratory activity but also on the virus strain through and . Note also 1426 that the level of activity (which implies a given metabolism) plays an important role on this rate [23]. 1427

Therefore, it can change with time for a given infector. 1428

For a given respiratory activity, equation (SM5-2) can be written as: 1429

where the subscript j has been omitted. 1431

In the absence of masks for the emitter (infector) and the receiver (susceptible) the dose 1432

inhaled by the receiver can be written: 1433 (SM5-4) = ∫ ∞ × × 0 1434 J o u r n a l P r e -p r o o f even for unstationary conditions as long as the virus lifetime ≫

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Identifying airborne transmission as 899 the dominant route for the spread of COVID-19

CoV-2 Variant Classifications and Definitions

WHO, Tracking SARS-CoV-2 variants

Viral infection and 908 transmission in a large well-traced outbreak caused by the Delta SARS-CoV-2 variant

The Delta SARS-CoV-2 variant has a higher viral load than the Beta and the historical 914 variants in nasopharyngeal samples from newly diagnosed COVID-19 patients

Infections caused by the Delta variant (B.1.617.2) of SARS-CoV-2 are associated 918 with increased viral loads compared to infections with the Alpha variant (B.1.1.7) or non-919 Variants of Concern

Modelling aerosol transport and virus exposure with numerical 926 simulations in relation to SARS-CoV-2 transmission by inhalation indoors

Physico-chemical characteristics of evaporating respiratory 929 fluid droplets

Thoracic and respirable particle definitions 931 for human health risk assessment

Modality of human expired aerosol size 934 distributions

Size distribution and sites of origin of droplets expelled from 937 the human respiratory tract during expiratory activities

Dose-response 939 relationships for environmentally mediated infectious disease transmission models

Quantitative Microbial Risk Assessment

Collection, particle sizing and detection of airborne viruses

Review and comparison between the Wells-Riley and dose-946 response approaches to risk assessment of infectious respiratory diseases

Estimation of Risk Due to Low-Doses of Microorganisms -A Comparison of 949 Alternative Methodologies

Virus Transmission and Epidemiology

The Beta Poisson dose-response model is not a single-hit 953 model

An experimental test of the independent action hypothesis 956 in virus-insect pathosystems

Using a mathematical model to evaluate the efficacy of TB 958 control measures

Estimation of airborne viral emission: Quanta 960 emission rate of SARS-CoV-2 for infection risk assessment

962 Close proximity risk assessment for SARS-CoV-2 infection

The airborne contagiousness of 965 respiratory viruses: A comparative analysis and implications for mitigation, Geoscience 966 Frontiers

An air distribution optimization of hospital wards for 968 minimizing cross-infection

The COVID-19 pandemic 970 is a global indoor air crisis that should lead to change: A message commemorating 30 years 971 of Indoor Air

What we know and should know about ventilation

Circulaire du 9 août 1978 modifiée relative à la révision du règlement sanitaire 974 départemental (RSDT)

Part 1: Indoor Environmental Input Parameters for Design and 977 Assessment of Energy Performance of Buildings Addressing Indoor Air Quality

Covid-19 : aération, ventilation et mesure du CO2 dans les 982 ERP

Décret n° 2012-14 du 5 janvier 2012 relatif à l'évaluation des moyens d'aération et à la 984 mesure des polluants effectuées au titre de la surveillance de la qualité de l'air intérieur de 985 certains établissements recevant du public

Critical review of standards for indoor thermal environment and air quality

Direction Départementale des Affaires Sanitaires et Sociales, Règlement Sanitaire 991 Départemental de l'Ille et Vilaine

La qualité de l'air intérieur des écoles françaises est-elle bonne ?

Assessment of ventilation and indoor air pollutants in nursery and 998 elementary schools in France

Air 1000 stuffiness index: from schools to dwellings

The ventilation problem in schools: literature review

Bâtir pour la santé des enfants, Medieco, Sciences & Techniques

Modélisation COVID-19: Dynamique du variant Delta en France 1006 métropolitaine

Vaccine Breakthrough Infections with SARS-CoV-2 Variants

Predominance of antibody-resistant SARS-CoV-2 variants 1016 in vaccine breakthrough cases from the

A paradigm shift to combat indoor respiratory infection Building 1024 ventilation systems must get much better

Respiratory virus shedding in 1027 exhaled breath and efficacy of face masks

Influenza Virus 1029 Aerosols in Human Exhaled Breath: Particle Size, Culturability, and Effect of Surgical Masks

A Quantitative Assessment of the Total Inward 1032 Leakage of NaCl Aerosol Representing Submicron-Size Bioaerosol Through N95 Filtering

Facepiece Respirators and Surgical Masks

Do 1035 N95 respirators provide 95% protection level against airborne viruses, and how adequate are 1036 surgical masks?

Air-Treatment Systems for Controlling Hospital-Acquired Infections, Heating, 1038 Piping and Air Conditioning Engineering

Boverket -the Swedish National Board of Housing Building and Planning, The Swedish 1047 Obligatory Ventilation Control

Inspection of ventilation systems

VENTILATION: WHY does no one take it seriously?

Control of airborne infectious diseases in ventilated spaces

A review of the 1056 performance of different ventilation and airflow distribution systems in buildings

Infection probability under different air distribution patterns

Réglementation Thermique aux dépens de la santé des enfants

Integrity testing of HEPA filters: A practical approach, Cleanroom Technology: The 1065

Predicted inactivation of viruses of relevance to biodefense by 1069 solar radiation

Calculating photolysis rates and estimating photolysis lifetimes

Transmission aéroportée du Covid-1073 19 : « Il est temps d'agir avant le retour du froid !

The Conundrum of Low

Mortality Burden in sub-Saharan Africa: Myth or Reality? COMMENT, Glob. Health Sc. Pract., 1079

What Could Explain the Lower COVID-19 Burden in Africa despite Considerable 1082 Circulation of the SARS-CoV-2 Virus?

Multiple Early Introductions of SARS-CoV-2 to Cape Town, South 1085 Africa

Ventilation during the coronavirus (COVID-19) pandemic

Infection Spread and High-Resolution 1090 Detection of Close Contact Behaviors

A ticking time bomb of future harm: Lockdown, 1092 child abuse and future violence

Do psychiatric patients experience more 1095 psychiatric symptoms during COVID-19 pandemic and lockdown? A case-control study with 1096 service and research implications for immunopsychiatry

Abuse, self-harm and suicidal ideation in the UK during 1099 the COVID-19 pandemic

Effect estimates of COVID-19 non-1101 pharmaceutical interventions are non-robust and highly model-dependent

Assessing mandatory stay-at-home 1104 and business closure effects on the spread of COVID-19

COVID-19 in primary schools: no significant transmission among children or 1107 from students to teachers

Factors Associated With 1110 Household Transmission of SARS-CoV-2 An Updated Systematic Review and Meta-analysis

Integrated environment-occupant-pathogen information 1113 modeling to assess and communicate room-level outbreak risks of infectious diseases

Contact tracing apps and values dilemmas: A privacy paradox in a neoliberal world

The First GAEN-Based COVID-19 Contact Tracing App 1119 in Norway Identifies 80% of Close Contacts in

Contact-tracing apps and alienation in the age of 1122 COVID-19

One for all, all for one: Social 1124 considerations in user acceptance of contact tracing apps using longitudinal evidence from 1125 Germany and Switzerland

Timeliness, Trustworthiness, and Situational Awareness: 1127 Three Design Goals for Warning with Emergency Apps, ICIS 2021 Proceedings

An Internet of Things Approach to Contact Tracing-The BubbleBox System, Inform

Exposure of staff to aerosols and bioaerosols in a dental office

Experimental and numerical study on the transport of droplet aerosols 1135 generated by occupants in a fever clinic

UFP) concentrations during the reopening of schools in the 1138 COVID-19 pandemic: The case of a metropolitan area in Central-Southern Spain, Environ

Indoor CO2 sensors for COVID-19 risk mitigation: Current guidance and 1141 limitations

Increased airborne transmission of COVID-19 with new variants

22 chemin des moines, 22750 Saint Jacut de la Mer (France)

LGCGM, 3 Rue du Clos Courtel

Airborne Contagion and Air Hygiene. An Ecological Study of Droplet Infections

Cold Spring 1539 Harb

First contact: the role of respiratory cilia in host-pathogen 1541 interactions in the airways

Can the new type of coronavirus be 1543 transmitted via food and objects

Methodologic issues in epidemiologic risk 1546 assessment

1549 SARS-CoV-2 Cell Entry Depends on ACE2 and TMPRSS2 and Is Blocked by a Clinically Proven 1550

SARS-CoV

Genetics Reveals a Variable Infection Gradient in the Respiratory Tract

Airborne Transmission of SARS-CoV-2: The World Should Face the 1560 Reality

Association of COVID-19 Disease Severity with Transmission Routes and 1562 Suggested Changes to Community Guidelines

Comparative pathogenesis of 1568 COVID-19, MERS, and SARS in a nonhuman primate model

Video 5-2: Beyond the Well-Mixed Room-Natural Convection

Infection probability under different air distribution patterns

Simple quantitative assessment of 1577 the outdoor versus indoor airborne transmission of viruses and covid-19

Simulation-based study of COVID-19 outbreak associated 1580 with air-conditioning in a restaurant

Droplet Transmission of SARS-CoV-2 by Direct Air Flow in a Restaurant in Korea

COVID-19 Outbreak 1585 Associated with Air Conditioning in Restaurant

Transmission of SARS-CoV-2 during 1589 a 2-h domestic flight to Okinawa

Modelling aerosol transport and virus exposure with numerical 1596 simulations in relation to SARS-CoV-2 transmission by inhalation indoors

Fundamentals of Air Pollution Engineering

Short-range airborne route dominates 1601 exposure of respiratory infection during close contact

Close proximity risk assessment for SARS-CoV-2 infection

Using a mathematical model to evaluate the efficacy of TB 1606 control measures

Estimation of airborne viral emission: Quanta 1608 emission rate of SARS-CoV-2 for infection risk assessment

Size distribution and sites of origin of droplets expelled from 1611 the human respiratory tract during expiratory activities

Aerosol Penetration Through Surgical Masks

Aerosol 1615 Penetration and Leakage Characteristics of Masks Used in the Health-Care Industry

A Quantitative Assessment of the Total Inward 1618 Leakage of NaCl Aerosol Representing Submicron-Size Bioaerosol Through N95 Filtering

Facepiece Respirators and Surgical Masks

Guide d'application pour la surveillance 1621 du confinement de l'air dans les établissements d'enseignement, d'accueil de la petite 1622 enfance et d'accueil de loisirs

Décret n° 2012-14 du 5 janvier 2012 relatif à l'évaluation des moyens d'aération et à la 1625 mesure des polluants effectuées au titre de la surveillance de la qualité de l'air intérieur de 1626 certains établissements recevant du public

Risk of indoor airborne infection transmission estimated from 1629 carbon dioxide concentration

A predictive model of the temperature-1631 dependent inactivation of coronaviruses

defined by Wells [1] . Let be the total number of quanta in the volume and the quantum 1237 concentration. Considering the quantum production rate per infector and introducing a quantum 1238 lifetime, which can be considered as the Note that if a device able to sterilize a flow rate 3 is used, the above equations hold just by 1252replacing 2 by = 2 + 3 . 1253These equations funded on the well mixed room hypothesis are the basis of the famous Wells-1254 Riley model and are convenient for a very large number of indoor situations. However, 1255inhomogeneous infection patterns are reported for a number of well-documented transmission 1256 events in closed spaces, especially in restaurants [14-16] but also in other places such as aircrafts 1257 [17] . Generally, in these specific well studied cases, inhomogeneity was created by the mechanical 1258 ventilation system of air conditioning (hereafter AC) with recirculation, inducing locally larger air 1259 velocity. One typical and largely mediatized event concerned a restaurant in Guangzhou, China. It has 1260 been the subject of numerical modeling [14] . Numerous published works in the field do not relate to 1261 a specific observed event but to hypothetical situations supposed to represent typical cases, such as 1262 a supermarket [18] . These models rely on CFD (Computational Fluid Dynamics) calculations of the air 1263 flow stream, using a variety of software, such as Open Foam for example. Then the microparticle 1264 behavior is estimated using a variety of methods (Lagrangian, Monte-Carlo). In the Lagrangian 1265 approach the movement of each particle is calculated using Newton's second law of motion, where, 1266 within forces acting on the particle, the drag one is determined from the calculated field of air 1267 velocity. Note that, for a Stokes number << 1, the particles are just assumed to follow the flow. The 1268Stokes number can be defined as the ratio of two times ℎ ⁄ , being the time of velocity 1269 accommodation of a particle to the flow velocity and ℎ the hydrodynamic time (equal to a typical 1270 length of the problem divided by the flow velocity). The Stokes number reads [19]: 1271

The problem of determining the exact proportion of infectors in a large population NTot 1311 ( = / ) is a difficult one. Two statistical results are most often available. The positivity rate is 1312 the number of populations tested positive related to the total number of people tested, and 1313 therefore is a proportion without dimension. The incidence rate is the number of new people tested 1314 positive in a population, which can then be reported to a target population (for example 10 5 1315 individuals) for a given period of time (for example one day or one week). It is therefore a temporal 1316 rate and, as such, has the dimension of (time) -1 . It is clear from these definitions that the results will 1317depend on which people are tested and also of the size of the target. Since many people are infected 1318but not tested and that people tested positive in the past remain infectious for some time, it can be 1319 anticipated that the real number of infectors could be much higher than what can be deduced from 1320 an analysis of the incidence rate: in principle, this rate can drop to zero with still infectors in the 1321 population. Further, since the population tested is often a symptomatic one, the positivity rate of 1322 testing could be much higher than the real proportion of infectors. Only a blind testing of a 1323representative population would lead a true value of . 1324Therefore the purpose of the present SM is just to show that it is possible to estimate the 1325 probability of infection of a susceptible target using a simplified expression (see SM4-3) which 1326 essentially considers the given proportion of infectors r in a population of NTot individuals, provided 1327 that the ventilation flow rate per person, qnorm, is known and the time of exposure t is fixed. Here NTot

will represent the inhabitants of a country, a region, a metropole or a city or it can also denote a 1329 fixed reference population like 100000, for instance. Then, NTot is large. The number of infected 1330 people in that population will be quoted I further in the text (see SM4-6 and beyond) with I = r ×NTot.

In the main paper we have derived an equation for the dose inhaled by a susceptible person: 1332which assumes that the total ventilation rate 2 is given by 

Another way to calculate this probability, which seems to be more realistic, is to make a weighted 1345 summation of probabilities to be infected in conditions where one, two, three etc. This expression is numerically evaluated below for a few examples and compared to equation SM4-3. 1368 We consider here situations for which the restricted population is smaller than the total number of 1369 infectors in the reference population NTot: When a mask is worn the proportion of particles going through the mask could be strongly 1436 dependent of the particle size. Therefore, it could be considered that the quantum production rate is 1437 reduced accordingly and that it is possible to define a quantum production rate depending on the 1438 mode: 1439 confinement level is then expressed by a score scaled in six levels from 0 to 5. The score 0 1467 corresponds to zero confinement (CO2 level always below 1000 ppm), this is the most favourable 1468 situation. Notes 2 and 3 correspond to low and regular confinement, whereas notes 4 and 5 1469 correspond to very high and extreme confinement, level 5 is the most unfavourable situation (CO2 1470 concentration always above 1700 ppm during occupancy). In this case, the decree [29] stipulates that 1471 additional investigations must be carried out and the local authority (the departmental Prefect) must 1472 be informed. Very high 5 extreme 1474The icone index can be calculated precisely using the following expression: 1475 ICONE = 8.3 log10(1 + f1 + 3 f2) 1476where f1 and f2 represent the proportions of CO2 concentration measurements comprised in between 1477 1000 and 1700 ppm or higher than 1700 ppm respectively. Hence, the ICONE index is zero when all 1478 measurements have been found below 1000 ppm (f1 = f2 = 0) as said earlier whereas it is 5 when all 1479 measurements are higher than 1700 ppm (f1 = 0 and f2 = 1). 1480 1481

The exhaled breathing of human beings contains a much higher concentration of carbon 1483 dioxide than the normal outdoor air. As a matter of consequence when persons are gathered in a 1484 room this leads to a noticeable increase of its concentration as it was recognized by previous authors 1485 [30] . Considering the situation depicted in figure SM2-1, a conservation equation for CO2 can be 1486 written in the same way than for particles or quanta: 1487 (SM7-1) × 2 = × × 2, ℎ − 2 × { 2 − 2, } 1488 with the same notation meaning than in SM2 for , , and 2 . 2 is the current concentration of 1489 CO2 which can be expressed in ppm (part per million) since air density is assumed constant. are respectively CO2 concentration in the air exhaled by a human (close to 40 000 ppm) 1491and outdoor fresh air (around 420 ppm). 1492The last term of the equation comes from the fact that the fresh outdoor air contains CO2. 1493It follows that the carbon dioxide concentration in the room, equal to 2 (0) at = 0, will evolve 1494 following the equation: 1495Note that most often a "clean" room with a null virus concentration