key: cord-0559589-j2a8emf9
authors: Poydenot, Florian; Abdourahamane, Ismael; Caplain, Elsa; Der, Samuel; Haiech, Jacques; Jallon, Antoine; Khoutami, Ines; Loucif, Amir; Marinov, Emil; Andreotti, Bruno
title: Risk assessment for long and short range airborne transmission of SARS-CoV-2, indoors and outdoors, using carbon dioxide measurements
date: 2021-04-29
journal: nan
DOI: nan
sha: fbda9f7e39ac551b23150fc3d7a6c6d0f7d4a235
doc_id: 559589
cord_uid: j2a8emf9

The quantitative analysis of viral transmission risk in public places such as schools, offices, university lecture halls, hospitals, museums, theaters or shopping malls makes it possible to identify the effective levers for a proactive policy of health security and to evaluate the reduction in transmission thus obtained. The contribution to the epidemic propagation of SARS-CoV-2 in such public spaces can be reduced in the short term to a level compatible with an epidemic decline, i.e. with an overall epidemic reproduction rate below one. Here, we revisit the quantitative assessment of indoor and outdoor transmission risk. We show that the long range aerosol transmission is controlled by the flow rate of fresh air and by the mask filtering quality, and is quantitatively related to the CO2 concentration, regardless the room volume and the number of people. The short range airborne transmission is investigated experimentally using dedicated dispersion experiments performed in two French shopping malls. Exhaled aerosols are dispersed by turbulent draughts in a cone, leading to a concentration inversely proportional to the squared distance and to the flow velocity. We show that the average infection dose, called the viral quantum, can be consistently determined from epidemiological and biological experimental data. Practical implications. The results provide a rational design of sanitary policies to prevent the dominant routes of viral transmission by reinforced ventilation, air purification, mechanical dispersion by fans and incentives for correct wearing of quality masks (surgical mask, possibly covered by a fabric mask, or non-medical FFP2 masks). Combined, such measures significantly reduce the airborne transmission risk of SARS-CoV-2, with a quantitative assessment.

Respiratory pathogens are transmitted via droplets emitted by coughing or sneezing. However, oral fluid droplets harbouring pathogenic particles are also generated during expiratory human activities (including breathing, speaking or laughing), which may cause asymptomatic and pre-symptomatic transmission. The atomization process producing aerosols occurs in the respiratory tract when an air flow of sufficient velocity leads to the fragmentation of a mucus film.

Pathogens responsible for illnesses such as influenza, tuberculosis, measles or SARS can be carried by these small droplets, which can remain airborne for long periods of time. [1] [2] [3] There is ample evidence that SARS-CoV-2, the virus caus- Viral particles have been directly evidenced in the air exhaled by patients, which can survive for several hours in a mucus droplet and remain airborne. SARS-CoV-2 has even been found in hospital COVID ward ventilation exhaust filters. 11 Animal model experiments have shown that SARS-CoV-2 can spread through the air in conditions where ballistic drops are excluded. 12 Transmission in the most detailed case studies 13, 14 can only be adequately explained through airborne spread. Long distance transmission in quarantine hotels has been documented, 15 where the absence of close contacts was established via video surveillance footage review and the contamination chain was supported by genomic evidence. Asymptomatic, infected individuals do not cough or sneeze, yet they account for at least 50 % of all transmissions 16 ; this suggests that they do not spread the disease via large ballistic droplets. Indoor transmission is 19 times more prevalent than outdoors. 17 Large ballistic droplets are not affected by the indoor or outdoor environment; aerosols however are. The lack of long range airborne infection outdoors provides a direct explanation for the risk difference indoors and outdoors. Moreover, good ventilation was shown to decrease transmission. 5 Healthcare workers wearing personal protective equipment designed to protect against ballistic droplets, but not aerosols, have been infected. 18 Finally, superspreading events, when a single patient infects a large number of people 19 can only be explained by a long range transmission. All these arguments provide evidence for the airborne transmission of SARS-CoV-2.

Here, we revisit the problem of measuring the viral transmission risk 20 in public places such as schools, offices, university lecture halls, museums, theaters or shopping centers, but also outdoors. Our aim is to characterize the dominant transmission routes in social activities and to identify efficient ways of reducing the risk of epidemic contamination in pub- F I G U R E 1 Graphical abstract. Outdoors, airborne viral transmission only takes place in the wake of an infected person: the exhaled breath is very concentrated in viral particles and is gradually dispersed by turbulent air fluctuations. The transmission risk typically decays as the inverse squared distance to the infected person. There is no long range transmission oudoors. Indoors, the same turbulent dispersion induces a short range risk but the finite volume leads to a finite dilution of viral particles, hence a long range contamination risk. lic spaces. We first define the risk of transmission in a public space and document its dependence on the number of people present, the average time they spend, the available volume in which aerosols are stored and the level of ventilation.

We then discuss what an acceptable residual risk would be and against which ethical standard it should be assessed. The risk assessment for long range airborne transmission, which is specific to indoor conditions and null outdoors, is determined and quantitatively related to the CO 2 concentration. We then show that short range airborne transmission, localised in the wake of people infected by COVID-19, obeys to the same physical laws indoor, and outdoor. We report experimental measurements of turbulent dispersion of a passive tracer (CO 2 ) performed in two French shopping centers: Forum des Halles in Paris and Carré-Sénart. In most such public spaces, the turbulent diffusion is due to a small permanent air flow leading to a rapid spatial decay of the tracer concentration.

The supplementary risk when staying in the wake from other people is determined quantitatively as a function of the distance downwind. From this risk assessment, we define quantitative standards (gauge, CO 2 level, ventilation, masks) which should be implemented in public spaces to reach the acceptable residual risk. In conclusion, we elaborate on various techniques available to reduce the viral transmission risk in public places, as a complement to vaccination.

We provide here an overview of the biology of SARS-CoV-2 for non-specialists. Although it provides the basis for risk assessment calculation, this section is mostly independent of the rest of the article and can be read afterwards as well.

SARS-CoV-2 is a virus enveloped by a lipid bilayer in which the E, M and S proteins are inserted. The lipid membrane comes from the cell in which the virus replicated before being released, -but not from its plasma membrane. Interferons are produced by infected cells but also by sentinel immune cells. After their secretion, they diffuse and bind their receptors on surrounding cells without discriminating whether they are infected or not, shutting down cell functions. As concentration is higher around the site of infection, diffusion leads to an efficient stochastic tracking of infected cells. If the interferon response is launched early, in a localized and circumscribed way, the viral spreading across the cell tissue can be stopped. 32, 33 However, an overly strong interferon response is not only antiviral but also destructive.

Moreover, the viral-host crosstalk is further complexified due to the fact that some of the viral proteins counteract the host interferon responses and in a reciprocal way, the host interferon responses may amplify the virus infectivity. 34

The nasal cavity is a battlefield where the replication of the virus and its inhibition are opposed in time and space. The cellular mechanisms allowing the replication and secretion of the virus as well as those leading to the interferon response are specific to each individual. Thus, the kinetics of the mechanisms allowing the replication and secretion of the virus characterizes the capacity of an individual to be more or less contaminator. Similarly, the kinetics of the interferon response correlates with an individual's susceptibility to infection. 35 An important element often forgotten modulates these kinetics in the nasal cavity and participates in the immune response:

the mucus. Mucus is composed mainly of water (95%), lipids and proteins (mucins). 36, 37 It is secreted by specialized cells 

Contamination risk assessment requires the estimate of the probability of contamination under a given intake viral dose.

The intake dose is the amount of viral particles inhaled by a person, cumulated over time. The simplest hypothesis is to assume an independent action of all inhaled viral particles, which means that a single virus can initiate the contamination.

The probability that at least one virus particle manages to enter a cell and replicate is independent of the presence or not of others viral particles. However, more than one is statistically needed, as the probability that a single virus overwhelms the host immunity defences successfully is small, typically be- The dose d is standardly expressed directly in "quanta", used as a convenient unit for a quantity of viral particles.

Cooperativity can be simply taken into account by assuming that K virions successfully overcoming the barriers are needed to replicate in cells and overwhelm the host antiviral defences. The probability of contamination then reads:

which resumes to the Poisson process for K = 1. As no evidence of cooperativity in viral infection has ever been provided, we will keep the simpler Poisson process hypothesis for the rest of the paper. It must not be confused with the Wells-Riley model, that will be discussed later on.

The viral load of an infected person depends on time. For simplicity, the concentration of viral particles in the exhaled air, noted D, can be assumed to present the same temporal profile amongst patients:

where D m is a characteristic concentration and t c the contam- 

The characteristic viral concentration D m is highly variable amongst infected people. In order to consider an average over all statistical realizations, we introduce the probability density function g ( D m ). As analyzed in Goyal et al., 44 F I G U R E 2 (a) Model dimensionless viral charge h (t ) as a function of time t , in days, after contamination. h (t ) is assumed to increase exponentially up to the maximal viral load and then to decay exponentially. 42 The best fit to the data provides the four fit parameters: the maximal viral charge is reached after 5.4 days; the exponential growth rate before this maximum is 2.8 days −1 ; the exponential decay rate after this maximum is −1.5 days −1 . (b) Histogram of the duration between contamination and symptoms 43 . The solid line is the best fit by a Weibull distribution. (c) Average viral charge from Jang et al. 42 as a function of the duration after symptoms. The best fit provides the average viral charge D m . 

From a scientific perspective, the risk induced by a certain public space should characterize its contribution to the epidemic reproduction rate R , defined as the average number of people infected by a virus carrier. This risk should not de-pend on the epidemic incidence rate but only on the statistical creation of chains of contamination inside this public space,

given the sanitary procedures. It is, indeed, the epidemic reproduction rate that determines whether the epidemic amplifies (R > 1) or decays (R < 1). Here, we therefore define the risk of a public space as the average number of infections r that a COVID-19 infected person would cause on the average by staying in it. In order to be useful, this risk r must be directly comparable to 1. The individual risk, defined for an altruist rational agent, would then be the average of the risk r over a time window corresponding to the infectious time, weighted by the duration of each social activity.

Considering the contamination as a Poisson process, the mean number Z of people contaminated in a certain public place hosting N people during a given period of time, amongst which M infected people reads:

where d i is the intake dose of the individual labelled i while 1/a i is their contamination dose. In the case where all N people are statistically subjected to the same intake dose d i = d , the risk r reads:

The quantity ∫ f (a) 1 − e −ad da is the probability to get infected when an intake dose d is inhaled. It is called the dose response function. We consider now the low limit where the intake dose, expressed in quantaād has a very low probability of being larger than 1. This excludes super-spreading events, which occur when an infected person with a large exhaled concentration D m attends an under-ventilated place, leading to multiple simultaneous infections. Then, performing the linearization 1 − exp(−ad ) ad , the equation simplifies to:

The average number of secondary infections is proportional to the intake dose expressed in quanta,ād , and to the number

We now introduce the dimensionless dilution factor between the viral concentration in the inhaled air and the viral concentration in exhaled air D. In the next section, we will discuss how can be related to CO 2 concentration and to mask efficiency. To determine the risk, as defined above, the M infected people are statistically picked up amongst the N people present. The inhaled dose of one particular person amongst N can, averaged over configurations, be expressed as:

where q is the inhalation rate, i.e. the product of the breathing rate by the tidal volume. On the average, for light exercise,

To evaluate the risk, the factor M (N − M ) must itself be averaged over M . For simplicity's sake, the fact that people from a same group are more likely to get infected together is ignored. The probability that M people are infected amongst N is governed by the binomial law, given the epidemic prevalence P . The mean number of infected people isM = N P .

The mean number of infected peopleZ involves the multi-

. At small epidemic prevalence, the risk is independent of P as expected:

it characterizes the creation of contamination chains.

The risk is proportional to the product of the mean inhalation rateq by the integrated viral concentrationD m T divided by the mean quantum of infectionā −1 . These four parameters can be combined into a single one, which is mean integrated quantum emission N defined by:

N may depend on the particular activity taking place in the public space. The risk, as defined above, finally reads:

Consider first the case of a public space where nobody wears a respiratory mask. In first approximation, both carbon dioxide and aerosol droplets are emitted by an infected person at a rate proportional to the respiratory rate, which depends on their activity. When gravity is negligible in front of turbulenceinduced drag forces, aerosol droplets are also dispersed in the air according to the same effective laws as CO 2 . We can safely 

where C a 37500 ppm is the average CO 2 concentration in exhaled air.

Finally, we introduce a filtering factor λ due to the effect of respiratory masks, which will be quantified below. The use of air purifiers can be similarly included in the risk formula:

just like masks, it does not change the CO 2 level, but reduces the contamination risk.

We obtain the final formula for the average risk in a public space, in the low risk limit:

where 

The long range airborne transmission risk can be modeled using the well-mixed hypothesis: the viral particles are assumed to be dispersed over the whole enclosed volume V . Denoting by Q the flow rate of fresh air, the conservation of CO 2 reads:

where C is the average CO 2 concentration. Figure 3 shows a sample of the number of people N (t ) in the Euralille shopping mall and the associated CO 2 concentration signal C (t ).

The CO 2 concentration is delayed with respect to number of people N so that ventilation is better controlled using N than C .

Expressing C as a function of with equation (13), we get:

The sedimentation flux of heavy particles can be included inside Q as well. It is a linear relaxation equation towards the steady state solution:

The exponential relaxation time is simply the ratio of the volume V to the fresh air flow rate Q .

Consider a micro-society which shares the same air, under permanent social conditions. The epidemics will grow exponentially with a rate noted σ. Then the average risk is directly the epidemic reproduction rate R :

Contrarily to the previous formula, the transmission time is comparable to the epidemic timescale σ −1 . The infection rate I , defined as the mean number of infected people per unit time obeys a Fredholm integral equation of second kind:

At small epidemic prevalence P , these equations admit an exact exponential solution. The growth rate σ is related to the epidemic reproduction rate R by the Lotka-Euler equation:

This relation is plotted in figure 4 for the rescaled viral load shown in figure 2. 

Relation between epidemic reproduction number R and the observed epidemic growth rate σ, as given by equation (20).

The air carrying viral particles is gradually diluted by turbulent dispersion after exhalation by an infected person. In its vicinity, the concentration of viral particles is therefore higher than far away. In an indoor space, the dilution is limited by the finite volume, which leads to an accumulation of viral particles. Indoor and outdoor spaces differ by the long range risk of transmission but share the short range one. Two transport mechanisms exist at short ranges: ballistic droplets exit the nose and mouth with an initial momentum and rapidly fall to the ground under their own weight, while aerosol droplets are carried by an airflow, whichever is greater between the exit velocity out of the body and the ambient airflow. We investigate here the short-range aerosol risk by measuring and modeling the turbulent dispersion of CO 2 in a generic situation. We make use of controlled experiments performed in different corridors of two French shopping malls, under various ventilation conditions.

The experimental setup is shown in figure 5 . A controlled 

Concentration profiles are averaged over the fraction of the time during which the "wind" is reasonably aligned with the sensor axis. They show a fast decay whose best fit by a power law typically gives x −2 . Denoting by R (x ) the typical radius of the contaminant dispersion cone (Fig. 6) , the mass conservation equation provides the scaling law:

where ρ CO2 = 1.83 kg/m 3 is the density of sublimated CO 2 .

The decay of C − C e is therefore consistent with the overall conical shape of the dispersion zone: the dispersion radius tive aerodynamic source size. We can therefore parametrize the dilution by:

where α is the dispersion cone slope, determined by the turbulent fluctuation rate u * /ū. 

The scaling law R (x ) ∼ a + x is a very striking approximation.

Indeed, it is known since an article by Taylor 50 Neglecting the longitudinal diffusion, the average concentration C obeys the convection-diffusion equation

Assuming that the turbulent diffusion coefficient D does not depend on r , the equation admits an exact solution:

where R obeys the equation:

It integrates into R 2 = 4(D /ū)x in the regime described by

Taylor, or equivalently:

The concentration therefore decays as 1/x and not 1/x 2 as observed.

The deviation to Taylor diffusion theory comes from the fact that the source diameter is in the inertial subrange of turbulence. [47] [48] [49] Following Kolmogorov scaling law, the contaminant is dispersed by eddies of typical size R , with a typical velocity difference increasing as R 1/3 . In this inertial regime, one therefore expects a scaling of the form:

F I G U R E 7 Time-averaged concentration excess profiles as a function of distance x to the source. The concentration is rescaled by m/(ρ CO 2ū ), where m is the mass injection rate andū the air velocity. The solid curve is the best fit by equation (22) . Crosses and dashed line: human breathing measurements ( m = 10 mg/s) and its best fit curve.

which integrates into:

As the separation of scales between the source size and the integral length of turbulence L is small, the dispersion takes place in a cross-over regime between Taylor and Kolmogorov scaling laws. As equations (26) and (28) (26) and (28) . The approximation by a tangent at the inflection point, which corresponds to a diffusion coefficient D ∼ u * R and α ∝ u * /ū provides a sufficient approximation for the problem tackled here.

In the previous paragraphs we have addressed the generic case in large public spaces where the dispersion is created by horizontal draughts. For completeness, we provide here a simple framework to determine the short range risk in smaller rooms. We consider the opposite limit where there is no mean flow at all, but convective plumes creating turbulent mixing. For simplicity, we can assume that dispersion is isotropic and write the diffusion equation in spherical coordinates:

Again, considering a constant source of mass rate m, a steady state solution gradually appears, which obey:

At large scale, D = u * L can be considered as constant so that C decreases as r −1 . In the intermediate range of scales, using the Villermaux approximation D = u * r , C decreases as r −2 .

The scaling laws derived before therefore still holds, but with a geometrically determined constant α.

The risk assessment presents an interest if the quanta generation rate is known with an accuracy comparable to that on the epidemic reproduction rate R , say 10 %. The quanta generation rate is an average over individual properties. We must therefore examine the possibility to measure this key quantity, statistical by nature, starting from epidemiologic and from virologic measurements.

The fast contamination in the Diamond Princess boat 20 has provided the first proof of airborne contamination by SARS-CoV-2. It has been deduced from the cold weather conditions (−5 • C) that the ventilation was mostly recycling the air, at (20) , the epidemic reproduction rate is therefore R = 2.6.

From equation (18) , this corresponds to a total viral emis- F I G U R E 9 Curve relating the proportion P of infected people onboard the French aircraft carrier Charles de Gaulle, as a function of time t . 53 The solid line is the best fit by an exponential growth, which gives a growth rate σ = 0.16 day −1 that corresponds to R = 2.6. The horizontal dashed line is the theoretical herd immunity limit P → 1 − 1/R . The dotted line is the numerical integration of equation 19. sion N = 460 quanta or, equivalently, to a typical viral emission rateqāD m = 3.2 quanta/hour. Considering the viral load curve h (t ), this value corresponds to an emission rate on the order of 18 quanta/hour at maximum.

Schools constitute the best documented social subsystem. 20 Contrarily to the previous case of boats, schools are not isolated from the society. United Kingdom provides the best data sets for both systematic PCR tests and ventilation conditions. 

The epidemic outbreak in a poorly ventilated restaurant emission rate is therefore V q t 2 d 45 quanta/hour. In both cases, the viral emission rate for these superspreading events is 2 to 3 times larger than average, which sounds reasonably consistent.

The viral dose can be expressed in RNA molecules (GU, for vi- The ratio between ID50 and TCID50 is larger than 1 and relates the viral dose able to infect an organism to that able to kill a culture cell. To infect an organism, the coronavirus must infect the cells of the epithelium, replicate and not be eliminated by the immune system of the upper respiratory tract, the mucus. For SARS-CoV-2, this ratio ranges from 10 to 1000 depending on individuals. 59 We are interested here in the average over the population, which presents large error bars around the best estimate determined on SARS-CoV-1 that may be used, 60 350. In conclusion, the infectious quantum, defined over a population is aroundā −1 = 1 quantum 5 10 5 GU within, say, a factor of 2.

In order to express the exhaled viral dose per unit time during breathing in quanta/h, the quantity of SARS-CoV-2 RNA per unit air volume exhaled by patients must be measured. In the study by Ma et al., 61 patients were asked to exhale toward a cooled hydrophobic film via a long straw to collect a sample of exhaled breath condensate: the SARS-CoV-2 concentration was in the range 10 5 − 2 10 7 GU/m 3 .

As a consequence, the viral emission rate of SARS-CoV-2 in the breath ranges from 0.1 to 20 quanta/h. This value re- lies heavily on the ratio between ID50 and TCID50 that we have used. In the same article, 61 the authors notice that the emission rate was correlated with the viral load in the nose and the throat but not in the lung. The viral exhalation rate varies in time and reaches its maximum during the asymptomatic/presymptomatic phases.

Coughing, sneezing, singing, speaking, laughing or breathing produce droplets of mucosalivary fluid in two range of sizes. Droplets above 100 µm are produced by fragmentation of a liquid sheet formed at the upper end of the respiratory tract (i.e. for sneeze) or from filaments between the lips (i.e. plosives consonants when speaking 62,63 ), with an average around 500 µm. [64] [65] [66] In the first case, the initial sheet is stretched and get pierced. The liquid accumulates by capillarity retraction in a rim, which destabilizes into ligaments. 67, 68 The latter form droplets by a capillary instability referred to as the beads-on-a-string. 69 In the second case, a film forms between the lips, which destabilizes into filaments, themselves exhibiting the beads-on-a-string instability.

Droplets below 20 µm form either by bubble bursting events in the lungs alveoli or by turbulent destabilization of liquid films covering the lower and upper airways. The average droplet size, around 4 µm results from an interplay between the fluid film thickness, the turbulent stress and the surface tension. Further fragmentation of these droplets can occur in the tract constrictions where air flows at large velocity. Pulmonary surfactant helps reducing the droplet size, and contributes to prevent accumulation of fluid in airways.

The main entry zone is through the nasal epithelium and more specifically a subset of cells of the nasal epithelium expressing both the ACE2 receptor and the TMPRSS2 protease.

Other entry zones exist as well as different receptors and proteases. 28 In first approximation, they can be considered as minor routes in the dissemination of the epidemic and in the risk of contamination. The emission of mucus droplets containing viral particles in the nasal cavity has not been much investigated so far.

Between 20 µm and 100 µm, almost no droplets form, indicating two well separated mechanisms. For each class of droplets, the distribution around the average has been fitted either by a log-normal distribution, following the idea of a break-up cascade, or by a Gamma distribution, based on the idea that the blobs that make up a ligament exhibit an aggregation process before breaking up. 70 

The evaporation of liquid droplets in the air is controlled by the ambient relative humidity RH. Mass transport of water molecules from the droplets to the surrounding air is diffusive; as the drops evaporate, they release latent evaporation heat, which is also conducted away. This cools down the droplets, which in turns lowers the saturation pressure in the immediate surrounding of the drop, slowing down evaporation. Due to this coupled transport, a drop of initial radius a 0 shrinks to a radius a (t ) as 71,72

where D eff = 1. The classical picture 73 considers evaporating droplets as independent. This is true for aerosol droplets dispersed inside a room, but not of droplets inside a cough or sneeze spray. In that case, RH is roughly uniform and close to 1 inside the aerosol jet, meaning that no evaporation takes places except at the spray boundaries. 74, 75 This makes these drops extremely long-lived, up to a hundred times the isolated drop lifetime. 72, 76, 77 However, virus-laden respiratory droplets do not vanish as they contain viral particles and are not composed of pure water. The mucosalivary fluid is a dilute solution of surfactants, proteins and electrolytes, initially composed of ∼ 99 % water in volume. The solutes stabilise drops at a finite radius a eq , at which they still contain water 78 :

The sum is done over all solutes i . c i is the mass concentration of solute i , M i its molar mass, ν i its degree of dissociation (2 for NaCl). We model respiratory fluid by a mixture of NaCl and the total average protein content 79, 80 : c electrolytes = 9 g/L (physiological NaCl concentration), c proteins = 70 g/L, M proteins = 70 kg/mol. This gives a eq ≈ 10 −1 a 0 (1 − RH) −1/3 : typically, at medium relative humidity, aerosol drops formed at 5 µm remain at 500 nm, which is significantly larger than the virus itself.

In aerosol droplets at equilibrium, virions are gradually inactivated by the damage done by dessication or by antiviral proteins in saliva. 81 The inactivation rate of envelopped, airborne viruses increases [82] [83] [84] with RH: this suggests that virions can associate with proteins which protects them both from dessication and antivirals. 80 By stabilising the droplet at a finite radius, solute reduce the evaporation time to

The solute effect on the evaporation time is small at low RH since the drop shrinks by a lot; at 99 % RH, a 4 µm drop has its evaporation time increased by 80 %.

In the literature 20, 54, [85] [86] [87] [88] [89] We have performed a quantitative study of mask wearing in different public spaces around Paris. Figure 14 shows Covid patients during spring 2020, 14 occupational infections occurred using surgical masks (amongst 233) and 0 with FFP2/FFP3 masks (amongst 180). This suggests a higher efficiency ratio between FFP2 and surgical mask than expected, due to leakages.

In this article, we provide an effective definition of the risk r associated with a public space, defined as the average secondary infections per infected person. Under the commonly accepted hypothesis of no cooperation between virions, the risk is computed in the low risk limit. It is related to the integrated quantum emissionN, to the mask filtration factor λ and to the CO 2 concentration, which measures quantitatively the dilution factor between exhaled and inhaled air. The first central result of the article, given by the equation (14), is the absorption of the ventilation flow rate, of the room volume and of the number of people present into a single measurable quantity. The disappearance of the number of people N from equation (14) results is non-trivial and comes from two factors balancing each others: on the one hand, CO 2 is exhaled by all individuals present, and not only people infected by the virus; on the other hand, the transmission risk increases linearly with the number of people susceptible to be infected.

Indoor and outdoor spaces both present a risk of airborne transmission at short range, in the dilution cone of the exhaled breath. The concentration in viral particles or in CO 2 decays as the inverse squared distance to the emitter, and as the inverse wind speed. Figure 15 summarizes the findings: the risk, higher in the wake of other people, at short distance, can be determined by adding a short range excess CO 2 concentration to the well-mixed case.

The viral load curve results into an important source of variability. We have defined here the integrated, maximal and average viral emission rate, which are significantly smaller than previous estimates, 3 quanta/hour for the raw strain, which corresponds to 16 quanta/hour at maximum andN = 400 quanta for the integrated quantum emission. These values must be multiplied by ∼ 1.5 for the B.1.1.7 strain and by ∼ 2.0 for the P.1 and B.1.617 strains. 46 The mean viral emission rate is consistent both with epidemic and molecular measurements. Importantly, the infection dose is around 5 10 5 GU and not between 10 and 100 as mentioned in different recent articles, 20, 26, 54, [85] [86] [87] [88] [89] once the ratio between plaque-forming units (PFU) and genome units (GU) taken into account.

The transmission of SARS-CoV-2 in public space is predominantly airborne. The infection risk can be reduced by a combination of five actions in public spaces:

• Correct wearing of good quality masks to increase the mean filtration factor λ.

• Ventilation with a sufficient fresh air flow rate per person to reduce the long range risk.

• Air purification to complement ventilation where needed.

• Turbulent dispersion, distancing and reduction of static crowds to reduce the short range risk, both indoors and outdoors.

• Monitor CO 2 to measure the risk and adjust policies.

The risk of fomite transmission after droplet deposition on surfaces is probably negligible but it is still interesting to mitigate it to prevent contaminations by other illnesses. It can be eliminated by regular hand and surface washing.

A direct application of this article is to compare the risk in different public spaces, to focus public policies to decrease the risk in such places. The uncertainty is mostly on the mask filtration factor λ and on the integrated quantum emission N = 400 quanta, but the relative risk can be precisely determined. Still, it is interesting to compute the maximal CO 2 concentration consistent with the receding regime (R < 1), following the Zero-Covid strategy. This should be considered as a first estimate, which must be updated when better data will be available. With a small safety factor, we can takē N = 1000 quanta for the strains that are about to become dominant in the second half of 2021 and a risk r = 0.5. In the absence of facial masks, the excess CO 2 concentration must be limited to 18 ppm and therefore to C = 430 ppm. This is almost impossible indoors without opening all windows. Having this value in mind, figure 15 shows that the transmission risk outdoor, without masks, is real at short distances and for long periods of time. Static crowds without masks must therefore be avoided. It is important for people to learn how to take into account the wind strength and direction for static outdoors activities, in particular if they sing, eat, or drink for a long duration. As the risk outdoors is entirely at short range, large fans may be used to reduce the risk of a bar terrace or static queue in front of a shop ( figure 16 panel a) . The more these fans induce turbulent fluctuations, rather than an average flow, the better they are. They must be oriented upwards to change the wake direction. For outdoors dancing floors, injection of air at high flow rate, say, 1−10 m 3 /hour/person may be sufficient to reduce the risk (Fig. 16 panel c) .

With the current average wearing of mask in public space ( Fig. 14) , which leads to a small filtration factor λ 2 = 0.2, the maximum excess CO 2 concentration should be 90 ppm and therefore to C = 500 ppm, which is hardly realistic. The reduction of risk in public spaces must therefore combine improvement of mask fitting and ventilation. For instance, with an objective of 10 % FFP2, 70 % well fitted surgical masks, allowing for 20 % community masks or badly fitted masks, one can easily decrease the filtration parameter λ by a factor of 2. Explanatory signs, scientific video clips on aerosol contamination, on the correct wearing of masks and on the filtration levels of the different types of masks would be efficient incen-tives to reach this goal. For an average filtration of λ 2 = 0.05, a maximal excess CO 2 concentration of 390 ppm becomes sufficient, which corresponds to C = 800 ppm. This constitutes a reasonable compromise between masks and ventilation.

Consider now public spaces like theaters or concert halls where FFP2 masks can be included in the ticket price with different sizes available. The factor λ 2 would be decreased by 40. The risk becomes nonetheless acceptable, but negligible for standard ventilation conditions. Surgical masks and FFP2 masks are expensive but can be decontaminated about four times and reused as long as the filtration layers do not show tears. Public spaces could offer a mask decontamination service using a combination of UV-C, heat and hydrogen peroxide vapor. 94 Ventilation with fresh air is energy consuming, both in winter and during heat waves. The filtering of viral particles in the recycled air is a lever difficult to implement because the comfort ventilation installations have not been dimensioned to receive Hepa filters: they are not able to overcome the induced pressure drop . Two innovative methods can be used and even combined to decontaminate the air: UV-C neon lights, which are already used reliably and regularly, [95] [96] [97] [98] [99] [100] [101] [102] [103] and ultrasounds between 25 and 100 MHz, 104 which is at the proof of concept stage. These technics are cost-effective in the long term for destroying nucleic acids, DNA or RNA from bacteria, viruses or other micro-organisms present in the air.

The absence of masks, when eating and drinking pose a specific problem of aerosol risk reduction. In particular, collective catering facilities are amongst the most important places of high transmission risk. It is possible to use Hepa filtered air purifiers arranged to provide air free of viral particles and suck out stale air (Fig. 16 b-d).

In conclusion, it is important to rise the limits of this study. The funding company had no such involvement in study design, in the collection, analysis, and interpretation of data, nor in the writing of the article. The authors had the full responsability in the decision to submit it for publication.

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Defining the sizes of airborne particles that mediate influenza transmission in ferrets

Particle sizes of infectious aerosols: implications for infection control. The Lancet Respiratory Medicine

It Is Time to Address Airborne Transmission of Coronavirus Disease

Identifying airborne transmission as the dominant route for the spread of COVID-19. Proceedings of the National Academy of Sciences

Is the coronavirus airborne? Experts can't agree

Exaggerated risk of transmission of COVID-19 by fomites. The Lancet Infectious Diseases

COVID-19 rarely spreads through surfaces. So why are we still deep cleaning?

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

Long-distance airborne dispersal of SARS-CoV-2 in COVID-19 wards

SARS-CoV and SARS-CoV-2 are transmitted through the air between ferrets over more than one meter distance

Probable airborne transmission of SARS-CoV-2 in a poorly ventilated restaurant. Building and Environment

Transmission of SARS-CoV-2 by inhalation of respiratory aerosol in the Skagit Valley Chorale superspreading event

Transmission of Severe Acute Respiratory Syndrome Coronavirus 2 during Border Quarantine and Air Travel

SARS-CoV-2 Transmission From People Without COVID-19 Symptoms

Outdoor Transmission of SARS-CoV-2 and Other Respiratory Viruses: A Systematic Review. The Journal of Infectious Diseases

A SARS-CoV-2 Cluster in an

Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China

A guideline to limit indoor airborne transmission of COVID-19. Proceedings of the National Academy of Sciences

SARS-CoV-2 (COVID-19) by the numbers

Molecular Architecture of the SARS-CoV-2 Virus

A unifying structural and functional model of the coronavirus replication organelle: Tracking down RNA synthesis

SARS-CoV-2 viral load in nasopharyngeal swabs in the emergency department does not predict COVID-19 severity and mortality

TMPRSS2 distribution and extrapulmonary organ injury in patients with COVID-19

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

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

A Single-Cell RNA Expression Map of Human Coronavirus Entry Factors

SARS-CoV-2 Entry Genes Are Most Highly Expressed in Nasal Goblet and Ciliated Cells within Human Airways

SARS-CoV-2 entry factors are highly expressed in nasal epithelial cells together with innate immune genes

The central role of the nasal microenvironment in the transmission, modulation, and clinical progression of SARS-CoV-2 infection

Viruses and interferon: a fight for supremacy

The host type I interferon response to viral and bacterial infections

SARS-CoV-2 Receptor ACE2 Is an Interferon-Stimulated Gene in Human Airway Epithelial Cells and Is Detected in Specific Cell Subsets across Tissues

Impaired type I interferon activity and inflammatory responses in severe COVID-19 patients

The biology of mucus: Composition, synthesis and organization. Advanced

Airway Mucus Function and Dysfunction. The New England journal of medicine

Structural organization and classification of the human mucin genes

Early humoral defence: Contributing to confining COVID-19 to conducting airways? Scandinavian

Exhaled aerosol increases with COVID-19 infection, age, and obesity. Proceedings of the National Academy of Sciences of the United States of America

Viral kinetics of SARS-CoV-2 over the preclinical, clinical, and postclinical period. International Journal of Infectious Diseases

Estimation of the incubation period of COVID-19 using viral load data

Wrong person, place and time: viral load and contact network structure predict SARS-CoV-2 transmission and super-spreading events

Virological assessment of hospitalized patients with COVID-2019

Genomics and epidemiology of the P.1 SARS-CoV-2 lineage in Manaus

Histogramme des fluctuations scalaires dans le mélange turbulent transitoire

On the geometry of turbulent mixing

S0022112099005674/type/journal_article

Short circuits in the Corrsin-Obukhov cascade

Diffusion by Continuous Movements

Transmission potential of the novel coronavirus (COVID-19) onboard the diamond Princess Cruises Ship, 2020. Infectious Disease Modelling

Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship

Centre d'épidémiologie et de santé publique des armées. Investigation de l'épidémie de COVID-19 au sein du Groupe Aéronaval. Paris: Service de santé des armées

Seasonal variation in airborne infection risk in schools due to changes in ventilation inferred from monitored carbon dioxide

COVID-19 Outbreak Associated with Air Conditioning in Restaurant

Development and evaluation of an efficient 3'-noncoding region based SARS coronavirus (SARS-CoV) RT-PCR assay for detection of SARS-CoV infections

DNA vaccine protection against SARS-CoV-2 in rhesus macaques

The pathogenicity of SARS-CoV-2 in hACE2 transgenic mice

Spike mutation D614G alters SARS-CoV-2 fitness

Development of a Dose-Response Model for SARS Coronavirus: Dose-Response Model for SARS-CoV

Coronavirus Disease 2019 Patients in Earlier Stages Exhaled Millions of Severe Acute Respiratory Syndrome Coronavirus 2 Per Hour

Speech can produce jet-like transport relevant to asymptomatic spreading of virus

Stretching and breakup of saliva filaments during speech: A route for pathogen aerosolization and its potential mitigation

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

Modality of human expired aerosol size distributions

The Fluid Dynamics of Disease Transmission

What Determines the Drop Size in Sprays? Physical Review

Violent expiratory events: on coughing and sneezing

Visualization of sneeze ejecta: steps of fluid fragmentation leading to respiratory droplets

Atmospheric and Oceanographic Sciences Library

Aerosol persistence in relation to possible transmission of SARS-CoV-2

On air-borne infection. Study II. Droplets and droplet nuclei

Fine structure of the vapor field in evaporating dense sprays

Growth of respiratory droplets in cold and humid air

Extended Lifetime of Respiratory Droplets in a Turbulent Vapor Puff and Its Implications on Airborne Disease Transmission

Evolution of spray and aerosol from respiratory releases: theoretical estimates for insight on viral transmission

Interaction of aerosol particles composed of protein and salts with water vapor: hygroscopic growth and microstructural rearrangement

Toward Understanding the Risk of Secondary Airborne Infection: Emission of Respirable Pathogens

Physico-chemical characteristics of evaporating respiratory fluid droplets

Antiviral Activities in Human Saliva

Dynamics of Airborne Influenza A Viruses Indoors and Dependence on Humidity

The effect of environmental parameters on the survival of airborne infectious agents

The effects of temperature and relative humidity on the viability of the SARS coronavirus

Exhaled CO2 as a COVID-19 Infection Risk Proxy for Different Indoor Environments and Activities

Predicted Infection Risk for Aerosol Transmission of SARS-CoV-2. medRxiv

Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments

Respiratory aerosols and droplets in the transmission of infectious diseases

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

The size and concentration of droplets generated by coughing in human subjects

Low-cost measurement of face mask efficacy for filtering expelled droplets during speech. Science Advances

Efficacy of masks and face coverings in controlling outward aerosol particle emission from expiratory activities

Healthcare workers' high COVID-19 infection rate: the source of infections and potential for respirators and surgical masks to reduce occupational infections

US Centers for Disease Control and Prevention. Decontamination and reuse of filtering facepiece respirators

IES committee report: Germicidal Ultraviolet (GUV) -frequently asked questions

Ultraviolet irradiation doses for coronavirus inactivationreview and analysis of coronavirus photoinactivation studies

Evaluation of the effectiveness of the SurePure Turbulator ultraviolet-C irradiation equipment on inactivation of different enveloped and non-enveloped viruses inoculated in commercially collected liquid animal plasma

Ten Facts about UV Radiation and COVID-19

UV-C irradiation is highly effective in inactivating SARS-CoV-2 replication. medRxiv

Effectiveness of N95 respirator decontamination and reuse against SARS-CoV-2 virus

A Critical Review on Ultraviolet Disinfection Systems against COVID-19 Outbreak: Applicability, Validation, and Safety Considerations

Deactivation of SARS-CoV-2 with pulsed-xenon ultraviolet light: Implications for environmental COVID-19 control

Ultraviolet germicidal irradiation handbook: UVGI for air and surface disinfection

Effect of receptors on the resonant and transient harmonic vibrations of Coronavirus

Detecting SARS-CoV-2 in the Breath of COVID-19 Patients. Frontiers in Medicine

Stability of SARS coronavirus in human specimens and environment and its sensitivity to heating and UV irradiation. Biomedical and environmental sciences

Aerosol and Surface Stability of SARS-CoV-2 as Compared with SARS-CoV-1

Viable SARS-CoV-2 in the air of a hospital room with COVID-19 patients

Persistence of severe acute respiratory syndrome coronavirus 2 in aerosol suspensions