# Risk assessment for long and short range airborne transmission of SARS-
CoV-2, indoors and outdoors

Florian Poydenota Ismael Abdourahamanea Elsa Caplaina Samuel Dera Jacques
Haiechb Antoine Jallona Inés Khoutamia Amir Loucifa Emil Marinova Bruno
Andreottia<EMAIL_ADDRESS>aLaboratoire de Physique de l’École normale
supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris
Cité, F-75005 Paris, France.
bCogitamus Laboratory and CNRS UMR 7242 BSC, 300 Bd Sébastien Brant, CS 10413,
67412 Illkirch Cedex.

(August 27, 2024)

###### Abstract

Preventive measures to reduce infection are needed to combat the COVID-19
pandemic and prepare for a possible endemic phase. Current prophylactic
vaccines are highly effective to prevent disease but lose their ability to
reduce viral transmission as viral evolution leads to increasing immune
escape. Long term proactive public health policies must therefore complement
vaccination with available nonpharmaceutical interventions (NPI) aiming to
reduce the viral transmission risk in public spaces. Here, we revisit the
quantitative assessment of airborne transmission risk, considering asymptotic
limits that considerably simplify its expression. We show that the aerosol
transmission risk is the product of three factors: a biological factor that
depends on the viral strain, a hydrodynamical factor defined as the ratio of
concentration in viral particles between inhaled and exhaled air, and a face
mask filtering factor. The short range contribution to the risk, both present
indoors and outdoors, is related to the turbulent dispersion of exhaled
aerosols by air drafts and by convection (indoor), or by the wind (outdoors).
We show experimentally that airborne droplets and CO2 molecules present the
same dispersion. As a consequence, the dilution factor, and therefore the
risk, can be measured quantitatively using the CO2 concentration, regardless
of the room volume, the flow rate of fresh air and the occupancy. We show that
the dispersion cone leads to a concentration in viral particles, and therefore
a short range transmission risk, inversely proportional to the squared
distance to an infected person and to the flow velocity. The aerosolization
criterion derived as an intermediate result, which compares the Stokes
relaxation time to the Lagrangian time scale, may find application for a broad
class of aerosol-borne pathogens and pollutants.

Making rational public health policy decisions to prevent the dominant routes
of viral transmission requires a simple but quantitative assessment of the
airborne transmission risk in public places, both indoors and outdoors. Here
we show that CO2 and aerosol viral particles disperse following the same law,
which allows us to relate this aerosol transmission risk to the CO2
concentration. The results provide quantitative guidance useful to complement
vaccination, treatment for the vulnerable patient population and the test-
trace-isolate strategy by a national ventilation plan and gradual mandates of
FFP2 respirators in indoor places, when the epidemic circulates above a
threshold.

##  I Introduction

The SARS-CoV-2 pandemic enters its third year despite the design of highly
effective vaccines, which induce circulating antibodies and systemic T and B
cell responses that block viral spread and disease. Beside the lack of
vaccination at the global scale, they do not establish immunity at mucosal
surfaces against infection by variants like Omicron (B.1.1.529), which present
both increased intrinsic transmissibility and viral immune evasion. Widespread
transmission therefore contributes to a degree of unpredictability in the
evolution of the pandemic. As a consequence, vaccination must be complemented
in the long term by effective public health policies contributing to suppress
transmission at a low economic and social cost. To help making rational public
health policy decisions, we propose here a method to measure the risk for
long- and short-range airborne transmission of SARS-CoV-2, both indoors and
outdoors.

Respiratory viruses and bacteria can be transported by droplets emitted by
coughing or sneezing, which may cause symptomatic transmission, and during
expiratory human activities such as breathing, speaking or laughing, which may
cause asymptomatic and pre-symptomatic transmission. Pathogens responsible for
illnesses such as influenza, tuberculosis, measles or SARS, initially carried
by these droplets, can form an aerosol phase Zhou  _et al._ (2018); Fennelly
(2020) and cause airborne transmission by inhalation. The silent spread of
SARS-CoV-2 by asymptomatic infected individuals, who do not cough nor sneeze,
has been hypothesized as early as January 2020. Since June 2020, there is
ample evidence that this virus is primarily transmitted through aerosols
Morawska and Milton (2020); Zhang  _et al._ (2020); Greenhalgh  _et al._
(2021); Jimenez  _et al._ (2022). This is particularly obvious in public
places where face mask wearing has been mandatory Cheng  _et al._ (2021) as
the heavier, millimeter-sized droplets have a ballistic trajectory that is
relatively insensitive to the presence of air and are stopped by all types of
masks.

Many misconceptions regarding aerosols can be found in the medical and
scientific literature. Clinicians often use an incorrect definition, still
reported by the World Health Organization: aerosols would be particles smaller
than $5\;{\rm\mu m}$ that settle slowly enough to be transported over a few
meters Jimenez  _et al._ (2022); Randall  _et al._ (2021). As a consequence,
problematic criteria are introduced, such as the distance after which a
particle launched with an initial velocity in a fluid at rest stops Nazaroff
(2022), or the settling time of a single particle dropped from head height in
still air Merhi  _et al._ (2022). An aerosol is a locally homogeneous phase
constituted of solid or liquid particles suspended in a gas, either by thermal
fluctuations (for very small particles) or by turbulent fluctuations. The
aerosol phase tend to homogenize and to diffuse over the whole available space
by turbulent dispersion. The ”slow settling” misconception therefore omits the
fundamental constitutive mechanism of aerosols of any particle size:
turbulence.

The infection risk has been modeled in a series of papers, starting from the
seminal works of Wells and Riley Wells (1955); Riley  _et al._ (1978), which
parametrize the infection risk as a function of the global air dilution and
the disease infectiousness. The authors work upon the the well-mixed
hypothesis, in which air inside a room is instantly mixed by turbulence so
that all people inside breathe the same air. Beyond this hypothesis, fluid
dynamics models of indoor air circulation Ho (2021); Li  _et al._ (2021a);
Vuorinen  _et al._ (2020); Liu  _et al._ (2021); Wang  _et al._ (2021) and
heterogeneous airflows produced by respiratory activity have been introduced
Bourouiba (2021); Yang  _et al._ (2020); Giri  _et al._ (2022). The Wells-
Riley model has been extended to unsteady conditions by Rudnick and Milton
Rudnick and Milton (2003), who have pointed out that CO2 could be used as a
risk proxy Bivolarova  _et al._ (2017). Regarding SARS-CoV-2, most models
extend the Wells-Riley equation to account for unsteady conditions in
ventilation or occupancy Bazant  _et al._ (2021); Burridge  _et al._ (2021)
and to different viral emission rates depending on the respiratory activity
Bazant and Bush (2021), face coverings and particle removal and inactivation
Peng and Jimenez (2021); Peng  _et al._ (2022); Buonanno  _et al._ (2020). All
of these models use well-characterized “super-spreading” events to derive
estimates of the viral emission rate, or closed micro-societies like cruise
ships Azimi  _et al._ (2021); Parhizkar  _et al._ (2021).

These models have left three problems open up to now. (i) Can an analytical
closed formula be derived, simple enough to be used to produce regulatory
ventilation standards? (ii) Is the dispersion of CO2 and airborne viral
particles governed by the same law? (iii) What is the law governing the short
range contribution to the transmission, both indoor and outdoor? Here, we
propose a definition of the environmental risk of viral transmission in public
spaces such as schools, offices, university lecture halls, museums or shopping
centers, but also outdoor. We report experimental and theoretical results
showing that exhaled CO2 and airborne viral particles diffuse at the same
rate. We finally derive a quantitative analytical model which relates this
risk to the CO2 concentration.

Figure 1: Viral particles are exhaled by an infected person at a concentration
$C_{e}$. They are dispersed by drafts (velocity $\bar{v}$), which lead to a
decrease of their concentration. The dispersion cone must not be confused with
the conical shape of high speed jets in a fluid at rest Abkarian and Stone
(2020); Abkarian  _et al._ (2020); Yang  _et al._ (2020). In an indoor space,
viral particles are stored, which translates into a background homogeneous
concentration $\bar{C}$, controlled by the ventilation rate $Q$.

##  II Airborne transmission mechanism

The main entry of SARS-CoV-2 virus is through the upper respiratory
epithelium. To colonize a cell, an inhaled viral particle, interacts through
the spike protein – which is cleaved by a host cell protease, mostly the
TMPRSS2 protease for the wild Wuhan-1 strain – with a host cell membrane
protein, the ACE2 receptor. Cleavage of the spike protein is necessary for a
conformational change so that it can effectively interact with the ACE2
receptor. This interaction leads to the formation of a virus-ACE2 complex
which triggers the internalization of the virus inside the cell. It replicates
its RNA molecule, produce the proteins required for self-assembly of new viral
particles, which are released, leading to the colonization of neighboring
cells Snijder  _et al._ (2020).

An organism may become infected if a sufficient amount of viral particles
interact with cells expressing both the TMPRSS2 protease and the ACE2 receptor
and if the virus is able to hack into cellular mechanisms to produce and
disseminate new virions. From the upper respiratory epithelium, which is the
first tissue to be infected (nasal epithelium for viral strains before
Omicron, but also throat epithelium for Omicron, due to a weaker dependency to
the TMPRSS2 protease), the virus, embedded in mucus, is carried to the
trachea, to the lungs or the esophagus, and finally to deeper organs. It has
also the possibility to reach the brain and to colonize certain cells of the
cortex.

When the virus is concentrated in the nasal cavity, it is disseminated via a
mist of fine droplets of mucus or saliva dispersed by breathing, talking or
singing. A sneeze or cough produces larger droplets containing viral particles
(Figure S3). The evaporation of mucosalivary liquid droplets in the air is
controlled by the ambient relative humidity $\mathrm{RH}$ and by its content
in surfactants, proteins and electrolytes. Initially, transport of water
molecules from the droplets to the surrounding air is diffusive so that the
drop squared radius decreases linearly in time. After a very short time, the
droplet stabilizes at an equilibrium radius at which the viral particle is
surrounded by proteins and water Mikhailov  _et al._ (2004); Pöhlker  _et al._
(2021); Seyfert  _et al._ (2022); Chong  _et al._ (2021); Merhi  _et al._
(2022). The influence of droplet chemical composition on the equilibrium
radius is still poorly understood Huynh  _et al._ (2022); Vejerano and Marr
(2018); Oswin  _et al._ (2022). The radius shrinks by a factor $2$ to $5$ (see
Supplementary Information): droplets emitted during breathing, below
$20\;{\rm\mu m}$, are therefore sub-micronic or micronic and stay suspended in
the air.

The transmission risk increases with the intake viral dose $d$, defined as the
amount of infectious viral particles inhaled by a person, cumulated over time.
$d$ increases with the time of exposure to the virus, with the inhalation rate
$q_{e}$, i.e. the product of the breathing rate by the tidal volume (for light
exercise, $q_{e}\simeq 0.5\;{\rm m^{3}/hour}$), and with the concentration
$C_{i}$ of infectious viral particles in the inhaled air. As a dose is a
quantity of virus, it can be measured using quantitative RT-PCR and is then
expressed in genome units (GU). However, it is better adapted to measure a
dose by infecting a culture cell monolayer, in plaque-forming units (PFU).
Plaque-forming units measure the ability of viral particles to replicate and
to be secreted by the chosen cell type, while qRT-PCR measures the number of
RNA molecules and is not sensitive to replication potential. On generic Vero
cells, the amount of virus needed on average to form one cell lysis has
decreased from $1400\leavevmode\nobreak\ {\rm GU/PFU}$ for the wild strain to
$240\leavevmode\nobreak\ {\rm GU/PFU}$ for the Delta variant Poydenot  _et
al._ (2022).

The inhaled dose $d$ is the product of two factors:

  * •

a purely biological factor reflecting the exhalation flux of viral particles
and the ability of the virus to infect a person

  * •

a purely physical factor $\epsilon$ reflecting the dilution of viral particles
between exhalation and inhalation.

We will investigate these two factors independently in the next sections.

##  III Dispersion and transport of CO2 and viral particles

Average concentration – In a public space where many people are gathered, let
us single out two people: one is infectious and exhales viral particles and
the second one inhales the air. The air exhaled by the infectious person is
concentrated in viral particles which are gradually diluted in the ambient
air, in a way similar to the smoke of a cigarette. Outdoors, far from an
infected person, the viral particle concentration vanishes. However, indoors,
viral particles accumulate: the concentration in the wake of an infected
person decays from its exhalation concentration $C_{e}$, which is high, to a
constant concentration $\bar{C}$ far away. The dilution factor $\epsilon$ is
defined as the ratio of the viral concentration $C_{i}$ in the inhaled air and
the viral concentration in exhaled air $C_{e}$ (Figure 1). Consider first a
closed room of volume $V$ where there is a single person exhaling viral
particles at a concentration $C_{e}$. A ventilation of flow rate $Q$ replaces
exhaust air at the average concentration $\bar{C}$ by fresh air. The average
concentration $\bar{C}$ obeys the conservation equation:
$V\mathrm{d}\bar{C}/{\mathrm{d}t}=q_{e}C_{e}-Q\bar{C}$. It is a linear
relaxation equation whose exponential relaxation time is $V/Q$. The average
dilution factor tends towards the steady state solution
$\bar{\epsilon}=\bar{C}/C_{e}=q_{e}/Q$ Rudnick and Milton (2003).

Dispersion – We have investigated experimentally and theoretically the
turbulent dispersion of viral particles and CO2 in a turbulent flow of mean
velocity $\bar{v}$, characterized by a root mean square velocity $\sigma_{V}$.
This corresponds to a generic situation where horizontal drafts (indoors) or
wind (outdoors) dominate over thermal plumes, so that natural convection can
be neglected. We have for instance shown in a recent paper that this was the
case in the corridors of commercial malls Poydenot  _et al._ (2021). The
dispersion of CO2 and of oil droplets of typical size $10\leavevmode\nobreak\
{\rm\mu m}$ is studied separately, but in the same conditions, in the wind
tunnel schematized in Figure 2 (b). The smoke concentration field is measured
using high resolution pictures ($7380\times 4920$) with a $2\;{\rm s}$
exposure time (Figure S1). The CO2 concentration profile $C^{{\rm CO}_{2}}(x)$
along the axis is measured after careful sampling of air using a syringe: the
air is transferred into a chamber where a vacuum has been set, equipped with a
CO2 non-dispersive infrared sensor (Figure S2).

As dispersion is slow compared to the axial convection, the axial coordinate
$x$ is equivalent to time $t=x/\bar{v}$. Transverse diffusion is controlled by
random motion and leads, according to the central limit theorem, to a quasi-
Gaussian radial profile of concentration (Poydenot  _et al._ , 2021). As the
flux across a transverse section is equal to the source emission rate
$q_{e}C_{e}$, the concentration field takes the form:

$C=C_{e}\;\frac{q_{e}}{\pi\sigma_{R}^{2}\bar{v}}\;\exp\left(-\frac{r^{2}}{2\sigma_{R}^{2}}\right)+\bar{C}$
(1)

The measurement of the dispersion radius $\sigma_{R}$ as a function of $x$
gives the law of spatial decay of the concentration.

Figure 2: Rescaled concentration profile of $10\;{\rm\mu m}$ size oil droplets
(black solid line) and CO2 (Red circles) in a wind tunnel experiment. The CO2
concentration along the axis is measured after sampling the gaz with a
syringe. The oil droplet concentration is measured using the optical
absorption of light. CO2 and oil aerosol are introduced successively at a rate
$Q$, $20\leavevmode\nobreak\ {\rm cm}$ downstream of the turbulence generating
grid. The working velocity is $\bar{v}=9\;{\rm m/s}$. The green line is the
best fit by Eqs (1-S8).

Because of turbulence, the velocity of an elementary volume of fluid is
correlated along its Lagrangian trajectory. This drives transported particle
velocities to be correlated too. We model the dispersion as a Langevin process
characterized by an exponentially decaying Lagrangian correlation function:
$\overline{\mathbf{v^{\prime}}(t)\mathbf{v^{\prime}}(t+\tau)}=\sigma_{V}^{2}\exp(-\tau/{\mathcal{T}})$,
where ${\mathcal{T}}$ is the Lagrangian integral time scale. The dispersion of
fluid particles injected at a source point at time $t=0$ is given by Taylor’s
theorem, which gives at time $t=x/\overline{v}$ Poydenot  _et al._ (2021):

$\sigma_{R}^{2}=\frac{2}{3}\sigma_{V}^{2}{\mathcal{T}}^{2}\left[\exp\left(-\frac{x}{{\overline{v}}{\mathcal{T}}}\right)+\frac{x}{{\overline{v}}{\mathcal{T}}}-1\right]$
(2)

At large distances $x\gg\overline{v}\mathcal{T}$, it predicts a diffusive
regime
$\sigma_{R}\sim\sigma_{V}\left({\mathcal{T}}x/{\overline{v}}\right)^{1/2}$,
the concentration along the axis ($r=0$) decaying as $1/x$. The most important
dispersion takes place at short distances $x\ll\overline{v}\mathcal{T}$.
Transport obeys a ballistic-like regime of the form
$\sigma_{R}\sim\sigma_{V}x/\bar{v}$, which corresponds to a fast decay of the
concentration along the axis ($r=0$) as:

$C-\bar{C}=C_{e}\;\frac{q_{e}\bar{v}}{\pi(\sigma_{V}x)^{2}}.$ (3)

Face masks and respirators – In public places where face masks are mandatory,
the aerodynamical contribution to $\epsilon$ must be multiplied by the product
of the exhalation and inhalation mask filtration factors. All types of masks
totally filter droplets of a fraction of millimeter. However, they have very
different efficiencies between $0.1\;{\rm\mu m}$ and $0.5\;{\rm\mu m}$, which
is the range of equilibrium sizes of the smallest mucus droplets after
evaporation. Mask efficiency presents a minimum around $0.3\;{\rm\mu m}$,
where the particles are large enough to make Brownian collection inefficient,
and tiny enough to have small inertia, insufficient to make them leave the
streamlines wrapping around the fibers Hinds (1999). Filtration factors are
determined by the material properties and the respiratory activity at play,
but most importantly by proper mask fit. Cloth and surgical masks tend to be
much looser fitted than respirators, thereby greatly reducing their filtration
factor compared to what their fabric could achieve alone Cappa  _et al._
(2021). Cloth masks have an effective filtration factor around $\lambda=0.70$,
on average Pan  _et al._ (2021); Hill  _et al._ (2020). Assuming they are
well-fitted, surgical masks have an effective filtration factor $\lambda=0.28$
Lindsley  _et al._ (2021a); Oberg and Brosseau (2008) and N95/FFP2 respirators
$\lambda=0.10$ Qian  _et al._ (1998); Asadi  _et al._ (2020).

##  IV Modeling the transmission risk

Intake viral dose – Transmission risk assessment requires the estimate of the
probability of infection under a given intake viral dose $d$. The simplest
hypothesis is to assume an independent action of all inhaled replicable viral
particles, which means that a single virus can initiate the infection.
However, more than one is statistically needed, as the probability that a
single infectious viral particle overwhelms the host immunity defences
successfully is small, typically between $10^{-3}$ and $10^{-2}$ Poydenot  _et
al._ (2022). For a person having inhaled an intake dose $d$, the probability
law of infection $p(d)$ takes the form:

$p(d)=1-e^{-ad}$ (4)

where the susceptibility $a$ is the inverse of the infection dose defined, for
each individual, as the intake dose for which the probability of infection is
$1-1/e\simeq 63\;\%$. $a$ is widely distributed across individuals, according
to a probability distribution $f(a)$. We denote by $\bar{a}=\int
af(a)\mathrm{d}a$ the population average of $a$. $\bar{a}^{-1}$ is the
infectious dose, called the quantum of infection when used as a convenient
unit for a quantity of infectious viral particles; the product $\bar{a}d$ is
then the dose expressed in “quanta”. For the wild strain Wuhan-1, the
infectious quantum has been estimated around $5.6\times 10^{5}{\rm GU}$ and
$400\;{\rm PFU}$ Poydenot  _et al._ (2022). Importantly, the distribution
$f(a)$ includes the effect of vaccinal or infection induced immunisation. The
infectious quantum is indeed a population average over different immunity
conditions.

Relation between inhaled dose and viral exhalation rate – In the literature
Peng and Jimenez (2021); Vouriot  _et al._ (2021); Kriegel  _et al._ (2020);
Lelieveld  _et al._ (2020); Bazant and Bush (2021); Pöhlker  _et al._ (2021);
Vuorinen  _et al._ (2020) devoted to airborne transmission of SARS-CoV-2, most
authors have considered that the viral emission rate, i.e. the number of virus
exhaled per unit time, is the product of the volume of mucus droplets emitted
per unit time by the viral load in the nasal cavity for mucus droplets, in the
throat for saliva droplets Buonanno  _et al._ (2020) or deeper in the lungs
for respiratory aerosols. This excludes the possibility of a viral enrichment
at the interface between mucosalivary fluid and air, which could lead to a
viral content of droplets which is not proportional to their initial volume.

Infectivity is proportional to the concentration of infectious viral particles
in the exhaled air, noted $C_{e}$. It is assumed here to follow the same
kinetics as the infectious viral load. It also depends on biological factors
which differ from one viral strain to the other. The viral kinetics results
from the competition between viral replication and immune response. For
simplicity, $C_{e}$ can be assumed to present the same temporal profile
amongst patients:

$C_{e}=C_{m}\;\psi\left(t-t_{c}\right)$ (5)

where $C_{m}$ is a characteristic concentration and $t_{c}$ the infection
time. Both $C_{e}$ and $C_{m}$ are expressed in ${\rm PFU/m^{3}}$. The
rescaled viral load curve $\psi(t)$ is dimensionless and can be considered as
the average over the sub-population considered. Typically, $\psi$ increases
exponentially in the presymptomatic period and decreases exponentially at long
times, as shown in Figure 3 Jang  _et al._ (2021); Jones  _et al._ (2021);
Sethuraman  _et al._ (2020). We choose here the following normalization for
$\psi(t)$, which provides an unambiguous definition of the mean contagious
time $T$:

$\displaystyle\int_{0}^{\infty}\psi(t)\mathrm{d}t=T,\quad\int_{0}^{\infty}t\;\psi(t)\mathrm{d}t=T^{2}$
(6)

The characteristic viral concentration $C_{m}$ is highly variable among
infected people. Moreover, it depends on the vaccination status, on former
infections, on the quality of the immune system, etc. In order to consider an
average over all statistical realizations, we introduce the probability
density function $g(C_{m})$, which is typically log-normal. Depending on the
hypotheses used to extract $C_{m}$ from sparse RT-qPCR measurements, its
dispersion is found between $0.1$ to $1$ $\log_{10}\;{\rm copies/mL}$ above
and below average Jones  _et al._ (2021); Monel  _et al._ (2021); Elie  _et
al._ (2021).

Figure 3: Viral kinetics of SARS-CoV-2. Average viral load in the nasal cavity
as a function of time since infection from Killingley, et al. Killingley  _et
al._ (2022), using a strain close to Wuhan-1. Blue squares are obtained by
quantitative RT-PCR and expressed in genom units (GU). Orange squares are
obtained by counting lysis plaques and expressed plaque forming units (PFU).
Right axis: Model dimensionless viral load $\psi(t)$ as a function of time
$t$, in days, after infection.

The inhaled dose can be expressed as:

$d=\int dt\;\lambda^{2}\;q\;C_{i}=\int dt\;\lambda^{2}\;\epsilon\;q\;C_{m}$
(7)

The key quantity is the mean integrated viral emission $\bar{h}$, which
accounts for all the biological part of the risk. By definition, it is the
dose in “quanta” exhaled by a patient during the entire infectious period:

$\bar{h}=\int dt\bar{q}\bar{a}C_{e}=\bar{q}\bar{a}\bar{C}_{m}T$ (8)

It is defined as an average over the sub-population attending the public space
considered and may depend on the particular activity taking place in the
public space through the mean inhalation rate $\bar{q}$. The integrated viral
emission $\bar{h}$ is between $450$ and $500$ quanta for Wuhan-1, $800$ for
Alpha, $1500$ for Delta, $2800$ for Omicron BA.1 and $4700$ for Omicron BA.2
Poydenot  _et al._ (2022).

Environmental risk – We define the transmission risk of a public space as the
average number of infections $r$ that an infected person staying in for an
infinitely long time would cause, on average. This public space contributes to
epidemic decay if $r<1$ and to its growth otherwise. $r$ characterizes the
environmental conditions and behaviors in the public space. We consider a
public place hosting $N$ people during a given period of time, among which $M$
are infectious. The epidemic incidence $I(t)$ is the fraction of a given
population infected per unit time. In practice, the fraction of the population
$I(t)\mathrm{\delta}t$ infected between $t$ and $\delta t$ is measured with
$\delta t=1\;{\rm day}$, although conceptually $\delta t$ is infinitesimal.
The prevalence $P$ is the fraction of this population currently infectious and
is related to $I$ by the equation:

$P(t)=\int_{-\infty}^{0}I(t^{\prime})\psi(t-t^{\prime})\;\mathrm{d}t^{\prime}$
(9)

On average, the number $M$ of infectious people among $N$ is $\bar{M}=PN$.
Under the Poissonian hypothesis, the mean number $rM$ of people infected reads

$rM=\sum_{i=1}^{N-M}\left(1-e^{-a_{i}d_{i}}\right)$ (10)

where $d_{i}$ is the intake dose of the individual labelled $i$ while
$1/a_{i}$ is their infection dose. In the case where all $N$ people are
statistically subjected to the same intake dose $d_{i}=d$, the risk $r$ reads:

$r=(N-M)\int_{0}^{\infty}f(a)\left(1-e^{-ad}\right)\;\mathrm{d}a$ (11)

The quantity $\rho(d)=\int f(a)\left(1-e^{-ad}\right)\;\mathrm{d}a$ is the
probability to get infected when an intake dose $d$ is inhaled and is called
the dose response function. We consider now the limit where the intake dose,
expressed in quanta $\bar{a}d$, has a very low probability of being larger
than $1$. This excludes super-spreading events, which occur when one
infectious person (or several) with a large exhaled concentration $C_{m}$
attends an under-ventilated place, leading to multiple simultaneous
infections. Then, performing the linearization $1-\exp(-ad)\simeq ad$, the
equation simplifies into
$\bar{Z}=(N-M)\bar{a}d=M(N-M)\bar{a}\epsilon\bar{q}\bar{C}_{m}T$. The average
number of secondary infections $r=\bar{Z}/M$ is therefore proportional to the
biological factor $\bar{h}$, to the dilution ratio $\epsilon$:

$\bar{r}=(N-M)\epsilon\bar{h}$ (12)

For the same concentration of viral particles in the air, the transmission
risk increases with the number of people $(N-M)$ susceptible to be infected as
they all have an equal individual risk of being infected.

##  V Results

Transport of respiratory aerosols – Does the dispersion of CO2 and virus laden
droplets obey the same law? To answer this question, we compare experimentally
and theoretically the turbulent dispersion of an aerosol of oil droplets and
of CO2 in a controlled turbulent flow. Figure 2 (a) shows that the
concentration decays of the gaz and of the aerosol superimpose without any
adjustable parameter. This remarkable result is highly non-trivial. Inertial
particles disperse at the same rate as massless particles if the particle
velocity correlation is close enough to the fluid velocity correlation
function. Inertia acts as a low-pass filter of the fluid velocity, with a cut-
off time controlled by the Stokes times $\tau_{S}=\rho_{p}d^{2}/18\eta$, which
is the particle response time to a change in the fluid velocity Bec  _et al._
(2010).

The quantitative criterion for the aerosolization of particles of different
sizes size must compare the particle inertia to the dispersion by turbulent
fluctuations. The stopping distance after which a droplet exiting the upper
airways with a large initial velocity stops in a fluid at rest Hinds (1999)
includes inertia but ignores turbulent particle diffusion. Inertia acts all
along the particle trajectory: the effect of particle size on turbulent
dispersion is to make large enough particles slow-paced at changing their
velocities, so that they leave the gas streamlines and decorrelate from the
flow. We have shown here that the relevant turbulent property to which inertia
must be compared is the Lagrangian time-scale $\mathcal{T}$. This experimental
result is in agreement with correlations functions found from direct numerical
simulations of particles in turbulence Biferale  _et al._ (2008). Particles
form an aerosol phase whenever turbulence can homogenize their concentration.
Settling gradually depletes the phase, but does not determine whether droplets
should be regarded as individual ballistic particles or as an Eulerian phase.
Settling times incorrectly assume that the air is still. Our experiments show
that gravity is not the relevant physical mechanism that separates small
particles from large particles, but the balance of inertia and turbulence is,
and the dimensionless ratio $\mathrm{St}\equiv\tau_{S}/\mathcal{T}$ is the
correct way of defining the Stokes number for the dispersion problem (see
Supplementary Information for a derivation). If the Stokes number
$\mathrm{St}$ is much smaller than $1$, particles presents a negligible
inertia, and are therefore aerosolized and dispersed according to Equation
(S8).

In the wind tunnel used in our experiments, we measured $\mathcal{T}=9\;{\rm
ms}$ Poydenot  _et al._ (2021). The crossover above which inertia becomes
important ($\tau_{S}=\mathcal{T}$) is therefore expected for a droplet
diameter $\sim 50\;{\rm\mu m}$ much larger than the oil droplet diameter. For
a meter scale ventilated room, the Lagrangian timescale $\mathcal{T}$ is in
the range of $10^{2}\;{\rm ms}$ and the cross-over droplet diameter is larger
than $\sim 100\;{\rm\mu m}$. To conclude, CO2 and viral particles are
dispersed at the same rate so that the dimensionless factor $\epsilon$ can
indeed be determined quantitatively from the measurement of the CO2
concentration.

Figure 2 (a) shows that Equations (1-S8), with a single relaxation time-scale
$\mathcal{T}$, perfectly fit the data. This result is particularly subtle and
surprising. At such high Reynold numbers (between $10^{5}$ and $10^{6}$),
turbulence is fully developed in space. According to the energy cascade
picture of turbulence, kinetic energy is injected at large scale and
dissipated by viscosity at small scale. At intermediate length scales, called
the inertial range, energy is transferred by inertial effects Lesieur (2008),
which cause non-trivial velocity correlations between two points separated in
space. The typical flow evolution time scale at the dissipation scale is
called the Kolmogorov time $\tau_{K}$ and is used in the turbulence community
to define a different particle Stokes number $\mathrm{St}$ Biferale  _et al._
(2008). Indeed, $\tau_{S}/\tau_{K}$ is relevant for the pair dispersion of
particles Richardson and Walker (1926); Bourgoin (2015). The transport of a
single particle is a Lagrangian problem Poydenot  _et al._ (2021), and
therefore does not involve the Kolmogorov time, but rather the Lagrangian
decorrelation time $\mathcal{T}$. Surprisingly, for the Reynolds numbers
accessible numerically and experimentally, Lagrangian two-points correlation
functions in time do not feature any inertial range, while Eulerian two-points
space correlation functions do Biferale  _et al._ (2008). Figure 2 shows that
we do not find any signature of an inertial range on the Lagrangian
correlation function in time either. We do not observe any signature of a
power-law regime, but instead, a simple ballistic-like regime. Simple
arguments à la Kolmogorov would rather predict a much faster power-law decay
of the velocity correlation function, leading quickly to a diffusive law and
therefore of a decay of $C-\bar{C}$ as $x^{-1}$.

Experiments with a volunteer breathing through the mouth Poydenot  _et al._
(2021) have shown that viral particles are injected at a length scale
$a\approx 0.3\;{\rm m}$, set by the head size, which is significantly smaller
than the turbulent integral scale. Equation (3) can directly be used for
exhalations, by changing $x$ to $x+a$, so that:

$\epsilon=\bar{\epsilon}+\frac{q_{e}\bar{v}}{\pi\sigma_{V}^{2}(x+a)^{2}}$ (13)

The aerosol concentration therefore present a fast decay as the inverse
squared distance $x$ to the infected person. It is inversely proportional to
the wind speed, at constant fluctuation rate $\sigma_{V}/\bar{v}\sim 10^{-1}$.
Equation 13 constitutes a central result of this paper, as it controls the
spatial structure of the transmission risk outdoors.

Transmission risk and CO2 concentration – The $N$ people present exhale CO2
while only the $M$ infected people emit viral particles. As a consequence, the
expression of $\epsilon$ in terms of the average CO2 concentration $\langle
C\rangle$ also contains a factor $N$:

$\langle\epsilon\rangle=\frac{\langle C^{{\rm CO}_{2}}\rangle-C^{{\rm
CO}_{2}}_{\infty}}{N\;C^{{\rm CO}_{2}}_{e}}$ (14)

where $C_{e}\simeq 37500\;{\rm ppm}$ is the average CO2 concentration in
exhaled air. It must be emphasized that Equation (14) does not assume any
relationship between CO2 and virus emission rates and is equally valid for
breathing, talking or singing. It only hypothesizes that airborne viral
particles are inhaled. The two factors balance each others in the transmission
risk, which reads, after averaging over $M$:
$\bar{r}=\lambda^{2}(1-P)N\langle\epsilon\rangle\bar{h}$. The prevalence is in
general much smaller than $1$ so that we obtain the final formula relating the
viral transmission risk to the CO2 concentration:

$\boxed{\bar{r}=\lambda^{2}\frac{\langle C^{{\rm CO}_{2}}\rangle-C^{{\rm
CO}_{2}}_{\infty}}{C^{{\rm CO}_{2}}_{e}}\bar{h}}$ (15)

This equation is remarkable: the dependence of the environmental risk, which
characterizes the creation of transmission chains, with respect to the volume
of the room, the ventilation or the occupancy number are all encoded in the
space averaged CO2 concentration $\langle C\rangle$. The result is more subtle
than it seems at first sight. Indeed, the $N$ occupants of a public space all
exhale CO2 but only the $M$ infected ones exhale virions. Let us compare the
risk of a well-ventilated lecture hall (say, at $750\;{\rm ppm}$ of CO2), with
$50$ students, to the risk if the same students are spread out in two
conventional rooms with the same CO2 concentration. Obviously, all other
things being equal, the probability of a student being infectious is twice
lower when $25$ students are grouped together, rather than $50$. However,
since the ventilation must be proportional to the room occupancy to keep the
CO2 level, the viral particles are twice more concentrated in the small room
and therefore the inhaled dose doubles. Under the above assumption, the
average number of people infected is the same, although the risk is
distributed differently.

The average CO2 concentration $\langle C^{{\rm CO}_{2}}\rangle$ can be defined
to take into account both the long-range dilution $\bar{\epsilon}$ and the
short-range contribution, present both indoor and outdoor, due to the higher
aerosol concentration in the wake of infectious people. To achieve this, the
CO2 concentration must be averaged over the surface where one more person
would stand in the public place. The reasoning is the same as before. Each
time a CO2 molecule reaches the measurement point, there is a probability
$M/N$ that it has been exhaled by one of the $M$ infectious people. In that
case, viral particles follow the same path as CO2 molecules. The dilution
between inhaled and exhaled viral particle concentrations is still valid, but
only if averaged over the possible permutations of the $M$ infectious people
amongst the $N$ people. Again, the transmission risk is distributed very
differently in the wake of infectious and non-infected people but the average
remains correct. The quantitative determination of the aerosol transmission
risk is directly applicable to outdoor spaces, where it is limited to this
dispersion cone in the wake of infected people. The short range transmission
risk outdoors is generally smaller than indoors, due to larger air flow
velocities.

Let us illustrate on two examples the consequences of equations (3) and (13).
Consider a shopping mall corridor in which we have measured the air draft
velocity to be typically $0.2\;{\rm m/s}$. This corresponds to an extra CO2
concentration of $160\;{\rm ppm}$ at $1\;{\rm m}$, of $50\;{\rm ppm}$ at
$2\;{\rm m}$, of $10\;{\rm ppm}$ at $5\;{\rm m}$. Consider now an outdoor
situation with a $1\;{\rm m/s}$ wind. This corresponds to an extra CO2
concentration of $30\;{\rm ppm}$ at $1\;{\rm m}$, of $10\;{\rm ppm}$ at
$2\;{\rm m}$, of $2\;{\rm ppm}$ at $5\;{\rm m}$ (Figure S7).

Figure 4 shows the linear relationship between the aerosol transmission risk
and the CO2 average concentration for different face masks and viral strains.
The natural risk threshold is $r<1$, below which an infected person infects on
average less than one other person, making the epidemic recede. With the
current level of mask-wearing in public (Figure S4, Figure S5), which leads to
a small filtration factor $\lambda^{2}=0.2$, the maximum excess CO2
concentration should be $70\;{\rm ppm}$. This corresponds to a CO2
concentration of at most $C=490\;{\rm ppm}$. If the risk is large but
ventilation cannot be changed, N95/FFP2 respirators could be mandated. Social
acceptability of such mandates could be managed by targeting first high-risk
spaces, such as public transportation Bertone  _et al._ (2022), then gradually
expanding the scope of non-pharmaceutical interventions based on reproduction
number and prevalence thresholds, following transparent and planned rules. If
respirators were mandated, the factor $\lambda^{2}$ would be decreased by $50$
and the maximum excess CO2 concentration should be $670\;{\rm ppm}$. The risk
becomes smaller than $1$ for standard ventilation conditions. Without
mandatory face masks, the maximum excess CO2 concentration should be
$C=13\;{\rm ppm}$. This is far too small to be achievable in practice.
Ventilation with fresh air is energy consuming, both in winter and during heat
waves. It is therefore important to complement ventilation with HEPA filters
or UV-C neon lights air purifiers (Figure S6), which are already used reliably
and regularly Raeiszadeh and Adeli (2020), are cost-effective techniques for
destroying nucleic acids, DNA or RNA from bacteria, viruses or other micro-
organisms present in the air.

Figure 4: Airborne transmission risk $r$ as a function of the concentration
excess $\langle C^{\mathrm{CO}_{2}}\rangle-C^{\mathrm{CO}_{2}}_{\infty}$ for
the Omicron BA.1 strain ($\bar{h}=2800\;{\rm quanta}$). Solid line: no face
mask. Dotted line: current level of mask-wearing in public, when mandatory
($\lambda^{2}=0.2$, Figure S4). Dashed line: FFP2 respirator
($\lambda^{2}=0.02$). Inset: schematic showing the spatial distribution of
short-range transmission risk in a public space. $\langle
C^{\mathrm{CO}_{2}}\rangle$ is by definition the space average of the CO2
concentration over all places where a further person would spontaneously
stand.

##  VI Discussion

We have obtained, using controlled approximations, a very simple expression
showing a linear relationship between airborne transmission risk and CO2
concentration. Several effects have been neglected, which we discuss here.

The effect of gravity is to transport droplets vertically at the settling
velocity. It can be neglected when the latter is small in front of turbulent
velocities. Typically, the $10\;{\rm\mu m}$ oil droplets settle at $3\;{\rm
mm.s^{-1}}$. For the wind tunnel used here, the fluctuating velocity is
typically $\sigma_{V}\sim 0.7\leavevmode\nobreak\ {\rm m.s^{-1}}$ which
corresponds to a cross-over droplet diameter, above which settling is
dominant, of $150\leavevmode\nobreak\ {\rm\mu m}$. For a meter scale
ventilated room, the cross-over diameter remains of order
$10^{2}\leavevmode\nobreak\ {\rm\mu m}$. For an aerosol of viral particles,
deposition is therefore negligible, as can be checked from models taking it
into account Bazant and Bush (2021); Peng and Jimenez (2021); Bazant  _et al._
(2021). As evaporation is very fast, the effect of gravity is at worst
transient and limited to the largest droplets.

We assume that dispersion is controlled only by the ambient air flow, and not
by expiratory air flows. Unmasked coughs and sneezes can create large
velocities that dominate over ambient transport, in which aerosols are exhaled
in a buoyant jet that gets mixed at its boundaries Bourouiba (2021); Bourouiba
_et al._ (2014). Face masks greatly reduce expiratory velocities, so that
transport by the ambient flow is recovered $20\;{\rm cm}$ away from the source
Deng and Chen (2021); Verma  _et al._ (2020); Simha and Rao (2020). Our work
is particularly applicable outdoor, where there is no mixing but rather a
horizontal turbulent flow, and wind velocities easily dominate over exhalation
(Figure S7). Measurements of crowds taking into account head orientation,
crowd density and walking velocity show that horizontal velocities completely
control the transmission risk Garcia  _et al._ (2021); Mendez  _et al._
(2022).

Virions transported in an aerosol phase are gradually degraded by the damage
done by antiviral proteins in mucosalivary fluid Malamud  _et al._ (2011) and
by UV-B radiation in sunlight or by UV-C. The inactivation rate of enveloped,
airborne viruses such as SARS-CoV-2 depends on the physicochemical composition
of the aerosol droplets, relative humidity $\mathrm{RH}$ and temperature in
ways that are still unclear. Relative humidity plays an important role: the
inactivation rate has a maximum at high $\mathrm{RH}$, reaches a minimum at
intermediate $\mathrm{RH}$ then increases again for lower $\mathrm{RH}$ Yang
and Marr (2011); Tang (2009); Smither  _et al._ (2020); Huynh  _et al._
(2022). This suggests that virions can associate with proteins which protects
them both from dessication and antivirals Vejerano and Marr (2018). Ignoring
viral inactivation is our main simplifying assumption outdoors, as sunlight
may make it much faster. Inactivation can be modeled as a relaxation process
of time-scale $\tau$, once $C$ is defined as the concentration of replicable
viral particles:
$\mathrm{d}C/{\mathrm{d}t}=(q_{e}C_{e}-Q\bar{C})/V-\bar{C}/\tau$. The dilution
factor $\epsilon$ and the transmission risk are therefore divided by
$1+V/(Q\tau)$. The assumption made before corresponds to the regime where
$Q\gg V/\tau$. As $\tau$ is on the order of an hour, this condition requires
at least several “air changes per hour” (ACH) to be satisfied, which is the
case in correctly ventilated public spaces. In steady state, measuring
$\epsilon$ is therefore equivalent to measure both occupancy and ventilation
rate. In transient conditions, often found in large public spaces, this
approximation fails.

The mean integrated viral emission $\bar{h}$ has three sources of variability.
The most important one is the variation of this biological factor between
different public spaces, depending on the type of respiratory activity taking
place there and on the characteristics of the population attending it. For
breathing, the respiratory rate $q_{e}$ can vary by a factor $2$-$3$,
depending on activity level and physiological characteristics. Here, it should
be understood as the average value for the level of activity inside the public
space, eg. its typical value at rest for a theater, light exercise for a
shopping mall where people walk, and exercise for a sport facility.

The viral exhalation rate dependence on respiratory activity, talking and
singing is scarcely known. It has been estimated that talking increases it by
a factor $\sim 10$ Coleman  _et al._ (2022); Shen  _et al._ (2022). However,
few speech samples were culture-positive, either due to protocol limitations,
or because the virus may have lost some of its ability to replicate when
exiting through the mouth Coleman  _et al._ (2022); Adenaiye  _et al._ (2021).
It is therefore difficult to define $\bar{h}$ for each public space, and it
must be averaged over the whole society. Individual exhalation rates, peak
viral load and individual infection doses are also broadly distributed, which
creates another source of variability. There is also a variability originating
from the number of contacts between individuals. Individuals who have many
contacts get infected earlier and are thus removed from the pool of
susceptible individuals, which decreases $\bar{h}$ over time.

The last simplifying assumption is a small intake dose $\bar{a}d$ (in quanta),
which corresponds to a small dilution factor $\epsilon$. Equation (14) shows
that this hypothesis corresponds to a situation which is either well
ventilated or with a large number of people. This is generally the case for
public spaces. By contrast, domestic spaces in winter generically gather few
people in bad ventilation conditions, during long periods of time. This
assumption excludes super-spreading events in which almost everyone gets
infected because of these large deviations and poor ventilation.

##  VII Conclusion

In this article, we have introduced an effective definition of the airborne
transmission risk $r$ associated with a public space, defined as the average
secondary infections per initially infected person. Under the commonly
accepted hypothesis of no cooperation between virions, the risk is computed in
the limit where it is low. It is related to the integrated quantum emission
$\bar{h}$, to the mask filtration factor $\lambda^{2}$ and to the dilution
factor $\epsilon$ between exhaled and inhaled air. $\bar{h}$ accounts for all
biological aspects and depends on the viral strain considered but also on the
characteristics of the sub-population attending the public space considered.
In that sense, $\bar{h}$ takes into account the average respiratory activity
in the public space in question. $\epsilon$ accounts for all hydrodynamical
aspects and is decomposed into an average contribution controlled by
ventilation and a spatially dependent contribution, localized in the
dispersion cone of infected people, controlled by turbulent air flows. Indoor
and outdoor spaces both present a risk of airborne transmission at short-
range, in the dilution cone of the exhaled breath. But indoor, enclosed spaces
only, present a risk of airborne transmission at long-range. Equation (13)
provides the answer to our initial question (iii) as it provides the law
governing the short range contribution to the transmission, both indoor and
outdoor.

Equation (15) constitutes a central result of the article, as it provides an
answer to our initial question (i). Indeed, it incorporates the ventilation
flow rate, the room volume and the occupancy number into a single measurable
quantity: the CO2 concentration. The disappearance of the occupancy $N$ from
Equation (15) at large $N$ is non-trivial and comes from two factors balancing
each other: on the one hand, CO2 is exhaled by all individuals present, and
not only by people infected by the virus; on the other hand, the transmission
risk increases linearly with the number of people susceptible to be infected.
As the risk is proportional to the CO2 concentration, the acceptable CO2
concentration below which $r<1$ can be readily adjusted as new variants with
higher $\bar{h}$ keep on appearing, based on a feedback using the measured
reproduction number.

The simplicity of Equation (15) is based on the third important result derived
here, both experimentally and theoretically: the dispersion of CO2 and
airborne viral particles are governed by the same law, for usual Reynolds
numbers (question ii). The quantitative criterion is a small enough Stokes
number $\mathrm{St}\equiv\tau_{S}/\mathcal{T}$ defined as the ratio of the
Stokes time to the Lagrangian integral time. This result, albeit simple in
appearance, results from a subtle effect of Lagrangian turbulence. This has
important implications in determining infectivity times: as settling is not
the relevant physical mechanism for aerosol transport, the infectivity of
particles between $20\;{\rm\mu m}$ and $100\;{\rm\mu m}$ could be
underestimated by models in which the air is still Merhi  _et al._ (2022). We
are not aware of previous simultaneous measurements of gas and particles in
the same controlled flow.

The transmission of SARS-CoV-2 in public spaces is predominantly airborne. In
complement to vaccination, treatments for the vulnerable patient population
and the test-trace-isolate strategy, the infection risk can be reduced by a
combination of four collective actions:

  * •

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

  * •

Monitoring CO2 to measure the risk and adjust practices.

  * •

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.

Ventilation guidelines or regulations in public spaces are centered around
balancing thermal comfort with energy consumption and perceived air quality,
not infection prevention Morawska  _et al._ (2021), in part because the
physical basis of airborne transmission was poorly understood Randall  _et
al._ (2021); Jimenez  _et al._ (2022). The approach presented here defines an
unambiguous transmission risk, encoding most hydrodynamical aspects into a
single measurable quantity. It is applicable to all airborne pathogens and to
many flow configurations. It also defines a maximum acceptable risk: $r=1$ is
the threshold above which the epidemic continues to propagate. In that sense,
it is a step towards ventilation regulation for infection prevention.

The transmission risk model outdoors is extremely easy to implement in
practice. Combining Equations (12) and (13), we find

$r=\frac{\bar{h}q_{e}\bar{v}}{\pi\sigma_{V}^{2}(x+a)^{2}}.$ (16)

At the request of municipal services, we have applied it to the ”Canal Saint-
Martin” in Paris, France, the popular banks of a canal along which many people
eat and drink. Measuring the mean distance between visitors (typically
$1\;{\rm m}$), we find that when the wind is aligned with the banks, $r$ is
below $1$ for the Wuhan-1 strain and for wind speeds above $1.5\;{\rm m/s}$.
However, the wind is not always perfectly aligned, and fluctuations in the
wind direction lead to dispersion over the canal, and therefore reduced risk.
Recommendation to local policymakers was to emphasize the lower risk outdoors
but simultaneously to mandate face masks during the very few days where there
is not enough wind to disperse viral particles. Using a slightly revised
dispersion formula, adapted to a cycling group, we have similarly been able to
establish that the transmission risk is low in the Tour de France peloton, but
not indoors, in the hotels hosting the cyclists.

We propose that the paradigm of airborne infection prevention policies could
shift from a set of independent nonpharmaceutical interventions (mask
mandates, occupancy limits, social distancing) to a standard of risk,
translated to a maximum acceptable CO2 concentration. Each public place has
several options to meet this imposed standard, itself depending on the local
community transmission, by lowering the maximum occupancy or by investing in
better ventilation, by mandating masks or respirators, or by purifying the air
with HEPA filters or UV-C flash lights (Figure S6). Restrictions can be
gradually added at different reproduction rates and prevalence thresholds by
targeting high-risk spaces first to balance social acceptability. Education on
good quality masks and their wider availability could greatly reduce risk in
public transportation Bertone  _et al._ (2022) and shopping malls. If, after
targeting first high-risk spaces, the global reproduction rate is still above
$1$, restrictions can be expanded to lower-risk spaces, until the epidemic
recedes. It would also be possible, for example, to mandate that mechanical
ventilation in public buildings have two flow rates: one that allows for
ordinary air renewal, keeping the CO2 concentration below, say,
$1000\leavevmode\nobreak\ {\rm ppm}$, and another much more powerful one
designed for a drastic lowering of pathogen concentrations in the air, despite
energy or comfort costs. Based on the results presented here, French shopping
malls have successfully used smoke extractor fans, renewing the air in five
minutes ($12\;{\rm ACH}$), to that effect since May 2021. Insofar as smoke
extractors are compulsory in this type of public spaces, this has not resulted
in any additional costs.

## Acknowledgments

The authors thank Joël Pothier and Alice Lebreton for the fruitful
discussions.

## Funding

Unibail-Rodamco-Westfield on behalf of Conseil National des Centres
Commerciaux (CNCC) has funded this work under the CNRS contract 217977, who
asked the authors to make recommendations for a health protocol aiming to
reduce and quantify the transmission risk in shopping centers. The conclusions
of the present article are therefore of direct interest for the funding
company. The authors declare no financial competing interest. 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 responsibility in the decision to submit it for publication.

## Author contributions statement

F.P., I.A., E.C., S.D., A.J., I.K., A.L., E.M. and B.A. conceived and
conducted the experiments, all authors analyzed the results. B.A., J.H. and
F.P. wrote and reviewed the manuscript.

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Risk assessment for long and short range airborne transmission of SARS-CoV-2,
indoors and outdoors
Supplementary Information
Florian Poydenota, Ismael Abdourahamanea, Elsa Caplaina, Samuel Dera, Jacques
Haiechb, Antoine Jallona, Inés Khoutamia, Amir Loucifa, Emil Marinova, and
Bruno Andreottia* aLaboratoire de Physique de l’École normale supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université Paris Cité, F-75005
Paris, France.
bCogitamus Laboratory and CNRS UMR 7242 BSC, 300 Bd Sébastien Brant, CS 10413,
67412 Illkirch Cedex.
∗Electronic address<EMAIL_ADDRESS>
(Dated: August 27, 2024)

##  VIII Experimental methods

###  VIII.1 Wind tunnel

Dispersion of aerosols and CO2 is studied in a suction-flow wind tunnel with a
$120\times 23\times 23\;{\rm cm}$ test section. A turbulence generating grid
($3\times 3$ grid of $7\times 7\;{\rm cm}$ squares separated by $1\;{\rm cm}$
wide bars) is place after the air inlet contraction section, $30\;{\rm cm}$
upstream of the test section. Wind velocity is measured with a hot-wire
anemometer (test 405i). The flow Reynolds number is between $10^{5}$ and
$10^{6}$.

###  VIII.2 Aerosol measurements

Aerosols are produced from heated oil vapor injected at a controlled rate and
power through a $6\;{\rm mm}$ nozzle. A cloud of droplets nucleates at a few
millimeters downstream. The test section is illuminated from above by a LED
array with a diffuser. The cloud is made dilute enough so that double
scattering is negligible and the light intensity scattered is linear with
respect to the local aerosol concentration. High resolution pictures with a
$1$ or $2\;{\rm s}$ exposure time are taken with a Digital Single-Lens Reflex
camera. In order to keep the image data linear with respect to the light
intensity, the raw images files are debayered using a bilinear approximation
and no further image processing is performed. We keep only the green channel
to measure the light intensity. We average sets of $10$ pictures taken in the
same conditions to achieve a satisfying statistical convergence. Residual
background light is removed by substracting the image of the tunnel without
droplets. The illumination intensity profile along the tunnel axis $I_{0}(x)$
is calibrated using a diffusive object. $I_{0}(x)$ is flat, except at the
start and the end of the measurement section where it decreases because of
greater distance to the illumination source.

The intensity ${\mathcal{I}}(x,y)$ of a single pixel is an integral over the
$z$ axis of the scattered intensity, which is Gaussian as the concentration is
Gaussian along a transverse section. Integrating it over $z$, the image
intensity field ${\mathcal{I}}(x,y)$ is Gaussian along $y$:

${\mathcal{I}}(x,y)=I_{0}(x){\mathcal{C}}(x)\sigma_{R}(x)\exp{\left(-\frac{\left(y-y_{0}(x)\right)^{2}}{2\sigma_{R}^{2}(x)}\right)}$
(S1)

As shown on Figure S1, we fit each transverse pixel line over $y$ by this
corrected Gaussian profile to extract ${\mathcal{C}}(x)$, which is
proportional to the concentration on the axis, and the radius $\sigma_{R}(x)$.
${\mathcal{C}}(x)\sigma_{R}^{2}(x)$ is a constant independent of $x$:
conservation of mass is respected by our measurement technique.

Figure S1: Transverse intensity profiles at various distances from the oil
outlet. The best fit by a Gaussian gives ${\mathcal{C}}(x)$, $\sigma_{R}(x)$
and $y_{0}(x)$.

###  VIII.3 CO2 measurements

Solid CO2 pellets are sublimated inside a pressurized container heated with a
controlled power. A variable area flow meter controls the gas injection rate.
The gas is injected at room temperature into the test section. CO2 is then
sampled along the dispersion cone axis by slowly suctioning air into a
$60\;{\rm mL}$ syringe with a long needle. Three replicates are taken at the
same point. A laser beam pointing along the flow axis helps position precisely
the needle tip. After sampling, syringes are sealed until analysis.

The content of the syringe is transferred into a $300\;{\rm mL}$ chamber
equipped with a non-dispersive CO2 infrared sensor with a range of
$0$–$50\,000\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ and with an absolute
pressure sensor. Using a vacuum pump, a primary vacuum is made in the sealed
chamber. Then, the $60\;{\rm mL}$ sample is sucked into the chamber. Finally,
nitrogen is immediately added to the chamber to reach the ambient pressure.
The CO2 concentration is recorded as a function of time. After a few minutes,
the sensor equilibrates and the concentration reaches a steady state. Figure
S2 shows a typical temporal evolution of the measured CO2 concentration. It
rises rapidly as the sensor is near the entrance of the chamber, then the
concentration homogenizes inside and the sensor relaxes to equilibrium with an
exponential decay. There remains a small amount of leakage and adsorption that
we fit with a linear decrease in time.

Since we measure its volumetric injection rate $Q$, the absolute volumic
concentration of CO2 is therefore known along the axis. This is not the case
for the oil droplet aerosol, where only the relative concentration is known.
The measured concentration on the axis $C_{M}(x)$ can be related to the
transverse concentration profile:

$C(r,x)=C_{M}(x)e^{-r^{2}/2\sigma_{R}^{2}}$ (S2)

Conservation of mass imposes that

$\int_{0}^{\infty}2\pi
r\mathrm{d}rC_{M}e^{-r^{2}/2\sigma_{R}^{2}}=2\pi\sigma_{R}^{2}(x)C_{M}(x)$
(S3)

is set by the injection rate, i.e.

$2\pi\sigma_{R}^{2}(x)C_{M}(x)=\frac{Q}{\overline{v}}$ (S4)

We therefore plot $\overline{v}C_{M}/Q$ for CO2 and $1/2\pi\sigma_{R}^{2}$ for
oil droplets.

Figure S2: Red dots: typical measured CO2 concentration inside the vacuum
chamber. The concentration rises rapidly as CO2 is introduced and passes by
the sensor, then the sensor equilibrates with an exponential decay. There
remains a very slow decay, linear in time, due to leakage and adsorption.
Solid black line: best fit by an exponential decay superimposed with a linear
decrease in time.

##  IX Turbulent dispersion: Lagrangian approach

Turbulent diffusion, like many diffusive processes (Risken and Frank, 1989),
can be seen through two complementary lenses: a Langevin equation describing
particle dynamics and a Fokker-Planck equation describing the spacial
evolution of the concentration field. We develop these two approaches here. As
turbulence acts as a fluctuating noise, in both cases we use the Reynolds
decomposition (Lesieur, 2008) to split any field or dynamical variable $A$
into its ensemble average over many realizations of the flow $\overline{A}$
and its fluctuating part with zero mean $A^{\prime}$:
$A=\overline{A}+A^{\prime}$.

We first consider a single particle of diameter $d$, density $\rho_{p}$, in a
fluid of density $\rho_{f}$ and viscosity $\eta$. Turbulent diffusion for
particles without inertia is a correlated Brownian motion (Batchelor, 1949,
1950, 1952). The simplest form of a Langevin equation that leads to a finite
correlation time gives an exponentially decaying velocity correlation function
(Risken and Frank, 1989; Sawford, 1991, 2001; Salazar and Collins, 2009):

$\overline{\mathbf{v^{\prime}}(t)\mathbf{v^{\prime}}(t+\tau)}=\sigma_{V}^{2}\exp(-\tau/{\mathcal{T}})$
(S5)

The correlation time ${\mathcal{T}}$ is called the Lagrangian integral time
scale, defined in the general case as:

${\mathcal{T}}=\frac{1}{\sigma_{V}^{2}}\;\int_{0}^{\infty}\overline{\mathbf{v^{\prime}}(t)\mathbf{v^{\prime}}(t+\tau)}\mathrm{d}\tau$
(S6)

The dispersion of fluid particles injected at a source point at time $t=0$ is
given by Taylor’s theorem (Taylor, 1922; Sawford, 2001; Salazar and Collins,
2009):

$\frac{d\overline{{\mathbf{r}}^{2}(t)}}{dt}=2\overline{{\mathbf{r}}(t){\mathbf{v}}(t)}=2\int_{0}^{t}\overline{{\mathbf{v}}\left(t^{\prime}\right){\mathbf{v}}(t)}\mathrm{d}t^{\prime}.$
(S7)

Applied to equation (S5) at time $t=x/\overline{v}$, the transverse spread is
(Sawford, 2001; Du  _et al._ , 1995; Salazar and Collins, 2009; Poydenot  _et
al._ , 2021):

$\sigma_{R}^{2}=\frac{2}{3}\sigma_{V}^{2}{\mathcal{T}}^{2}\left[\exp\left(-\frac{x}{{\overline{v}}{\mathcal{T}}}\right)+\frac{x}{{\overline{v}}{\mathcal{T}}}-1\right]$
(S8)

At short and long distances compared to $\overline{v}\mathcal{T}$,
$\sigma_{R}$ can be rewritten as the mean square displacement at time
$t=x/\overline{v}$ of a diffusive process with an effective diffusion
coefficient $D$:

$\sigma_{R}^{2}\sim D\frac{x}{\overline{v}}$ (S9)

The effective diffusion coefficient is constant at long distances
$x\gg\overline{v}\mathcal{T}$, as expected for an uncorrelated random walk of
velocity $\sigma_{V}$ and mean free path $\sigma_{V}\mathcal{T}$:

$D\sim\sigma_{V}^{2}\mathcal{T}$ (S10)

At short distances $x\ll\overline{v}\mathcal{T}$, particle motion is
correlated so that the mean free path is rather the distance traveled
$\sigma_{R}$:

$D\sim\sigma_{V}\sigma_{R}$ (S11)

This gives the two scalings reported in the article.

Unlike passive scalars, particles are subjected to gravity and inertia. The
equation of motion of a single particle reads (Maxey and Riley, 1983)

$\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{r}=\mathbf{v},\quad\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}=-\frac{1}{\tau_{S}}(\mathbf{v}-\mathbf{u}[\mathbf{r}])+\left(1-\frac{\rho_{f}}{\rho_{p}}\right)\mathbf{g}$
(S12)

where $\mathbf{u}=\overline{\mathbf{u}}+\mathbf{u^{\prime}}$ is the fluid
velocity and $\tau_{S}=\rho_{p}d^{2}/18\eta$ the Stokes times, which is the
particle response time to a change in the fluid velocity given by the Stokes
force $-3\pi\eta d(\mathbf{v}-\mathbf{u})$.
$\left(1-\rho_{f}/\rho_{p}\right)\mathbf{g}$ is a buoyancy term. In the
absence of turbulence, particles fall with a velocity

$\mathbf{v}_{\rm fall}=\frac{(\rho_{p}-\rho_{f})\mathbf{g}d^{2}}{18\eta}.$
(S13)

If the fluid presents a constant mean velocity, particles move on average at a
velocity $\overline{\mathbf{v}}=\overline{\mathbf{u}}+\mathbf{v}_{\rm fall}$.
We consider again for simplicity that the fluid velocity correlation function
(but not the particle velocity correlation function
$\overline{\mathbf{v^{\prime}}(t)\mathbf{v^{\prime}}(t+\tau)}$) decays
exponentially over a time ${\mathcal{T}}$:

$\overline{\mathbf{u^{\prime}}(t)\mathbf{u^{\prime}}(t+\tau)}=\sigma_{U}^{2}\exp(-\tau/{\mathcal{T}})$
(S14)

The aerosol phase forms when turbulent velocity fluctuations are large enough
to counteract particle settling (Friedlander, 2000) i.e. when
$\sigma_{U}>v_{\rm fall}$.

The fluctuating velocity obeys:

$\mathbf{v^{\prime}}(t)=\frac{1}{\tau_{S}}\int_{-\infty}^{t}\mathbf{u^{\prime}}(t^{\prime})\exp{\left(-\frac{t-t^{\prime}}{\tau_{S}}\right)}\mathrm{d}t^{\prime}$
(S15)

The particle velocity correlation function is the low-passed fluid velocity
correlation:

$\overline{\mathbf{v^{\prime}}(t)\mathbf{v^{\prime}}(t+\tau)}=\frac{1}{2\tau_{S}}\int_{-\infty}^{\tau}\overline{\mathbf{u^{\prime}}(t_{2})\mathbf{u^{\prime}}(t_{2}+t_{1})}\exp{\left(-\frac{\tau-
t_{1}}{\tau_{S}}\right)}\mathrm{d}t_{1}$ (S16)

Performing the Reynolds decomposition
$\mathbf{v}=\overline{\mathbf{v}}+{\mathbf{v^{\prime}}}$ and using the Fourier
transform of the fluid velocity autocorrelation function, we find:

$\overline{\mathbf{v^{\prime}}(t)\mathbf{v^{\prime}}(t+\tau)}=\frac{\sigma_{U}^{2}}{1-\frac{\tau_{S}^{2}}{{\mathcal{T}}^{2}}}\left(\exp{(-\tau/\mathcal{T})}-\frac{\tau_{S}}{{\mathcal{T}}}\exp{(-\tau/\tau_{S})}\right)$
(S17)

Inertia acts as a low-pass filter of the fluid velocity (Bec  _et al._ ,
2010). The dimensionless ratio $\mathrm{St}=\tau_{S}/\mathcal{T}$ is called
the Stokes number and characterizes the relative influence of particle inertia
and hydrodynamic drag. If the Stokes number $\mathrm{St}$ is much smaller than
$1$, particles presents a negligible inertia and the particle velocity
correlation function reduces to the fluid velocity correlation function
$\overline{\mathbf{v^{\prime}}(t)\mathbf{v^{\prime}}(t+\tau)}=\overline{\mathbf{u^{\prime}}(t)\mathbf{u^{\prime}}(t+\tau)}$.
Conversely, in the limit where the Stokes number $\mathrm{St}$ is much larger
than $1$, the correlation function decays exponentially as
$\sigma_{U}^{2}\mathrm{St}\exp{(-\tau/\tau_{S})}$.

##  X Turbulent dispersion: Reynolds-averaged Eulerian approach

In the continuum approximation, we introduce the concentration
$C(\mathbf{r},t)=\overline{C}(\mathbf{r},t)+C^{\prime}(\mathbf{r},t)$ of
particles and its flux $\mathbf{j}$. The conservation equation reads:

$\frac{\partial C}{\partial t}+\nabla\cdot\mathbf{j}=0$ (S18)

Transport by the the average flow leads to a flux equal to the concentration
times velocity $\mathbf{j}=C\mathbf{v}$. Consider now thermal diffusion
induced by random microscopic velocity fluctuations at
$\mathbf{v}=\mathbf{0}$. Would the concentration be homogeneous i.e. constant
in space, the diffusive flux would vanish. At leading order, random exchanges
between neighboring layers of fluid lead to a flux proportional to the
gradient of concentration, oriented from high to low concentration. The
phenomenological relation is known as Fick’s law:

$\mathbf{j}=-D_{m}\mathbf{\nabla}C$ (S19)

and the molecular diffusion coefficient $D_{m}$ is assumed to be independent
of $C$, which must be true in the dilute limit $C\to 0$. Plugging this
relation into the continuity equation, we get the linear diffusion equation:

$\frac{\partial C}{\partial t}=D_{m}\nabla^{2}C$ (S20)

With turbulence, the relevant transport equation is obtained by Reynolds
averaging the concentration conservation equation:

$\frac{\partial\overline{C}}{\partial
t}+\nabla\cdot(\overline{\mathbf{v}}\overline{C})=\nabla\cdot\left(D_{m}\nabla\overline{C}-\overline{\mathbf{v^{\prime}}C^{\prime}}\right)$
(S21)

For the reasons invoked for molecular diffusion, the flux
$\overline{\mathbf{v^{\prime}}C^{\prime}}$ can be expressed using a gradient
diffusion assumption:

$\overline{\mathbf{v^{\prime}}C^{\prime}}=-D_{t}\nabla\overline{C}$ (S22)

where $D_{t}$ is the turbulent diffusion coefficient, also called eddy
diffusivity. The averaged concentration therefore obeys:

$\frac{\partial\overline{C}}{\partial
t}+\nabla\cdot(\overline{\mathbf{v}}\overline{C})=\nabla\cdot\left(D\nabla\overline{C}\right)$
(S23)

where $D=D_{m}+D_{t}$ is the effective diffusion coefficient. In the
Lagrangian approach, we have introduced the decorrelation time $\mathcal{T}$.
For the Eulerian framework, we must introduce the inertial length scale
$\mathcal{L}$; above this space length, fluid velocities are uncorrelated.

For a point source at the origin in an average flow along the $x$ direction in
cylindrical $(r,\theta,x)$ coordinates and neglecting longitudinal diffusion,
the average concentration $\overline{C}$ obeys a convection-diffusion equation
which reduces in the steady state to:

$\overline{v}\frac{\partial\overline{C}}{\partial
x}=r^{-1}\frac{\partial}{\partial
r}\left(r\;D\;\frac{\partial\overline{C}}{\partial r}\right)$ (S24)

This equation has the same structure as the diffusion equation, except that
time is replaced by the space coordinate $x$. In the regime described by
Taylor, the turbulent diffusion coefficient $D$ does not depend on $r$, the
equation admits an exact solution:

$\overline{C}=\frac{q_{e}C_{e}}{\pi\sigma_{R}^{2}\bar{v}}\;\exp\left(-\frac{r^{2}}{2\sigma_{R}^{2}}\right)$
(S25)

where the multiplicative factor is obtained by identifying the mass flow rate
across any section to the source injection rate $q_{e}C_{e}$. The dispersion
radius $\sigma_{R}$ obeys the equation:

$\bar{v}\frac{\mathrm{d}\sigma_{R}^{2}}{\mathrm{d}x}=2D$ (S26)

We now 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 homogeneous and isotropic and write the diffusion equation
in spherical coordinates:

$\frac{\partial\overline{C}}{\partial t}=r^{-2}\frac{\partial}{\partial
r}\left(r^{2}\;D\;\frac{\partial\overline{C}}{\partial r}\right)$ (S27)

Again, considering a constant source of flow rate $q_{e}$ and concentration
$C_{e}$, a steady state solution gradually appears, which obeys (Cheng  _et
al._ , 2011):

$\frac{\mathrm{d}\overline{C}}{\mathrm{d}r}=-\frac{q_{e}C_{e}}{4\pi r^{2}\;D}$
(S28)

At large scale, $D\sim\sigma_{V}\mathcal{L}$ can be considered as constant so
that $\overline{C}$ decreases as $r^{-1}$. In the intermediate range of
scales, using again the ballistic approximation $D\sim\sigma_{V}r$,
$\overline{C}$ decreases as $r^{-2}$. The scaling laws derived before
therefore still hold, but up to a geometrically determined multiplicative
constant $\alpha$.

##  XI Droplet production

Coughing, sneezing, singing, speaking, laughing or breathing produce droplets
of mucosalivary fluid in two range of sizes. Droplets above $100\;{\rm\mu m}$
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 (Abkarian  _et al._ , 2020; Abkarian and
Stone, 2020)), with an average around $500\;{\rm\mu m}$ (Morawska  _et al._ ,
2009; Johnson  _et al._ , 2011; Bourouiba, 2021). 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 (Kooij  _et
al._ , 2018; Bourouiba  _et al._ , 2014). The latter form droplets by a
capillary instability referred to as the beads-on-a-string (Scharfman  _et
al._ , 2016). In the second case, a film forms between the lips, which
destabilizes into filaments, themselves exhibiting the beads-on-a-string
instability.

Droplets below $20\;{\rm\mu m}$ form either by bubble bursting events in the
lungs alveoli or by turbulent destabilization of liquid films covering the
lower and upper airways. Between $20\;{\rm\mu m}$ and $100\;{\rm\mu m}$,
almost no droplets form, indicating 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 (Villermaux, 2007). The
average droplet size, around $4\;{\rm\mu m}$ 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 reduce the
droplet size, and contributes to prevent accumulation of fluid in airways. The
main entry zone is through the nasal epithelium for viral strains before
Omicron and more specifically a subset of cells of the nasal epithelium
expressing both the ACE2 receptor and the TMPRSS2 protease, and the throat
epithelium for Omicron, due to a weaker dependency to the TMPRSS2 protease.
Other entry zones exist as well as different receptors and proteases (Singh
_et al._ , 2020). In first approximation, they can be considered as minor
routes in the dissemination of the epidemic and in the risk of infection. The
emission of mucus droplets containing viral particles in the nasal cavity is
dominant, but has not been much investigated so far (Li  _et al._ , 2021b; Ma
_et al._ , 2021).

Figure S3: (a) Sketch of the aerosol generation mechanism inside the
bronchioles. The walls are elastic and collapse during exhalation; when air
flows back into the bronchioles, it spreads them apart, leading to the
formation of a fluid film which then bursts into submicronic droplets
(Malashenko  _et al._ , 2009; Johnson and Morawska, 2009; Mittal  _et al._ ,
2020; Halpern and Grotberg, 2003; Haslbeck  _et al._ , 2010). (b) Sketch of
the aerosol generation mechanism inside the upper respiratory tract. Shear
destabilizes the respiratory fluid layer lining the walls (Kataoka  _et al._ ,
1983; Moriarty and Grotberg, 1999; Mittal  _et al._ , 2020), and droplets
detach from the waves. In both cases, if virions are present, they get
entrained when the film forms and are found in the subsequent aerosol
droplets.

###  XI.1 Evaporation process and virus inactivation

We discuss here a basic model of evaporation of droplets, before
efflorescence, in humid air. The evaporation of liquid droplets in the air is
controlled by the ambient relative humidity $\mathrm{RH}$. 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 (Pruppacher and Klett, 2010; Smith  _et al._ , 2020)

$a^{2}(t)=a_{0}^{2}-2D_{\mathrm{eff}}\left(1-\mathrm{RH}\right)t$ (S29)

where $D_{\mathrm{eff}}=1.3\,10^{-10}\;{\rm m^{2}/s}$ is an effective
diffusion coefficient taking into account both diffusive transport of mass and
its slowdown due to evaporative cooling. Evaporation takes places at the
surface of the drop, which explains the linear behavior of $a^{2}$ with $t$.
Since the drops are small, evaporation is very fast compared to the time they
spend aloft: a $4\text{\,}\mu\mathrm{m}$ drop at $70\text{\,}\%$ RH completely
evaporates in $0.2\text{\,}\mathrm{s}$, while it takes
$20\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ to fall a distance of
$2\text{\,}\mathrm{m}$ under its own weight.

The classical picture (Wells, 1934) considers evaporating droplets as
independent. This is true for small droplets dispersed inside a room, but not
of droplets inside a cough or sneeze spray. In that case, $\mathrm{RH}$ is
roughly uniform and close to $100\text{\,}\%$ inside the aerosol jet, meaning
that no evaporation takes places except at the spray boundaries (Villermaux
_et al._ , 2017; Ng  _et al._ , 2021). This makes these drops extremely long-
lived, up to a hundred times the isolated drop lifetime (Chong  _et al._ ,
2021; de Oliveira  _et al._ , 2021; Smith  _et al._ , 2020).

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 $\sim$99\text{\,}\%$$ water in volume. The solutes stabilize drops at a
finite radius $a_{\mathrm{eq}}$, at which they still contain water (Mikhailov
_et al._ , 2004):

$a_{\mathrm{eq}}=\left(\frac{M_{w}}{\rho_{w}}\sum_{i}\frac{\nu_{i}c_{i}}{M_{i}}\right)^{1/3}\;\frac{a_{0}}{\left(1-\mathrm{RH}\right)^{1/3}}$
(S30)

The sum is done over all solutes $i$. $c_{i}$ is the mass concentration of
solute $i$, $M_{i}$ its molar mass, $\nu_{i}$ its degree of dissociation ($2$
for NaCl). We model respiratory fluid by a mixture of NaCl and the total
average protein content (Nicas  _et al._ , 2005; Vejerano and Marr, 2018):
$c_{\text{electrolytes}}=$9\text{\,}\mathrm{g}\mathrm{/}\mathrm{L}$$
(physiological NaCl concentration),
$c_{\text{proteins}}=$70\text{\,}\mathrm{g}\mathrm{/}\mathrm{L}$$,
$M_{\text{proteins}}=$70\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{o}\mathrm{l}$$.
This gives $a_{\mathrm{eq}}\approx
10^{-1}a_{0}\;\left(1-\mathrm{RH}\right)^{-1/3}$: typically, at medium
relative humidity, aerosol drops formed at $5\;{\rm\mu m}$ remain at
$1\;{\rm\mu m}$, which is significantly larger than the virus itself. By
stabilizing the droplet at a finite radius, solutes reduce the evaporation
time to

$t_{\mathrm{ev}}=\frac{a_{0}^{2}}{2D_{\mathrm{eff}}(1-\mathrm{RH})}\left(1-\left(\frac{a_{\mathrm{eq}}}{a_{0}}\right)^{2}\right)$
(S31)

The solute effect on the evaporation time is small at low $\mathrm{RH}$ since
the drop shrinks by a lot; at $99\;\%$ RH, a $4\text{\,}\mu\mathrm{m}$ drop
has its evaporation time increased by $80\;\%$.

In aerosol droplets at equilibrium, virions are gradually degraded by the
damage done by antiviral proteins in saliva (Malamud  _et al._ , 2011) and by
UV-B radiation in sunlight or by UV-C. The inactivation rate of enveloped,
airborne viruses increases (Yang and Marr, 2011; Tang, 2009; Chan  _et al._ ,
2011; Smither  _et al._ , 2020) with $\mathrm{RH}$: this suggests that virions
can associate with proteins which protects them both from dessication and
antivirals (Vejerano and Marr, 2018). Another possible effect of ambient
conditions results the intrinsic temperature preference of the coronavirus
spike protein (Laporte  _et al._ , 2021), as the temperature in the nose
depends on both temperature and humidity.

##  XII Masks and respirators

Figure S4: Mask wearing in public spaces of the Paris suburban region (Ile-de-
France) where face masks are mandatory, mid April 2021. $N=1708$. Green:
correct fitting. Orange: centimeter-scale leakage along the nose or the
cheeks. Red: No fitting at all.

$\lambda_{i}$ and $\lambda_{e}$ are respectively the inhalation and exhalation
filtration factor of the face coverings worn by the two people. Filtration
factors are determined by the material properties, the fitness of the face
coverings and the respiratory activity at play. Cloth masks have a material
filtration efficiency of $20\%$–$40\%$ (Pan  _et al._ , 2021; Fischer  _et
al._ , 2020; Hill  _et al._ , 2020; Zangmeister  _et al._ , 2020; Rengasamy
_et al._ , 2010; Shakya  _et al._ , 2017; Drewnick  _et al._ , 2021), tightly
woven material being more efficient. Surgical masks have a $50\%$–$95\%$
material filtration efficiency (Hill  _et al._ , 2020; Lindsley  _et al._ ,
2021a; Pan  _et al._ , 2021; Oberg and Brosseau, 2008), and N95/FFP2
respirators $95\%$–$99\%$ (Hill  _et al._ , 2020; Zangmeister  _et al._ ,
2020; Qian  _et al._ , 1998; Rengasamy  _et al._ , 2010). Inhalation (i.e.
exposure reduction) and exhalation (i.e. aerosol source control) are not
symmetrical as they induce different airflows around the face covering.
Moreover, the flow is different for instance for coughs and tidal breathing,
which is associated with a $20$ times smaller typical flow rate (Pöhlker  _et
al._ , 2021). Surgical masks block $56\%$–$86\%$ of cough aerosols and
$42\%$–$91\%$ of exhaled aerosols (Blachere  _et al._ , 2021; Lindsley  _et
al._ , 2021a, b; Asadi  _et al._ , 2020); N95 respirators block $73\%$–$99\%$
of cough aerosols (Lindsley  _et al._ , 2021a, b; Asadi  _et al._ , 2020) and
$95\%$–$99\%$ of exhaled aerosols (Lindsley  _et al._ , 2021a; Asadi  _et al._
, 2020). Efficiencies for exposure reduction are roughly similar at the most
penetrating particle size in well-mixed conditions (Pan  _et al._ , 2021;
Lindsley  _et al._ , 2021c). Filtration efficiency strongly depends on proper
mask fit: cloth and surgical masks tend to be much looser fitted than
respirators, thereby greatly reducing their filtration factor compared to
their material efficiency (Cappa  _et al._ , 2021). Improved surgical mask fit
by double masking or knotting and tucking can bring their filtration
efficiency up to $60\%$–$99\%$ (Sickbert-Bennett  _et al._ , 2021; Brooks  _et
al._ , 2021; Clapp  _et al._ , 2021; Blachere  _et al._ , 2021). We take as an
average $\lambda=\sqrt{\lambda_{i}\lambda_{e}}$ for both inhalation and
exhalation. $\lambda=1$ when no mask is worn; $\lambda=0.70$ for cloth masks
($30\%$ efficiency); $\lambda=0.28$ for well-fitted surgical masks ($72\%$
efficiency); $\lambda=0.10$ for N95 respirators ($90\%$ efficiency). For
respirators, fit checks are needed to reach high efficiencies, protective
enough for workplace aerosol exposure. However, most untrained users achieve
on their own $90\%$ and higher filtration, making respirators a valuable
choice for the general public (Rembialkowski  _et al._ , 2017; Brosseau,
2010).

We have performed a quantitative study of face mask wearing in public spaces
around Paris. Investigators pair up in a public place and observe the people
around them without taking notes, keeping their count. One of the two
investigators is familiar with the public place and the other not. They decide
on the type of mask they wear and the most appropriate memorization technique,
after a reconnaissance visit. Every 10 people, they discreetly write down the
quantitative measurements on the type of mask they wear and on the mask
fitting. Every twenty minutes, in a calm environment, without witnesses, they
discuss the social characteristics of the place under study, at this time of
day and this day of the week, in order to prevent perception biases. Ideally,
the investigators fill, for each person, the following categories: gender,
age, occupational characteristics, visible religious attributes, spoken
language, alone or accompanied, behavior. Special attention is paid to the
difference between externally imposed rules vs internalized rules, when noting
the behaviors in the presence or not of a control (Goffman, 1988; Hall, 1990;
Winkin, 2000).

Social aspects will be discussed in a separate paper. Here, it is used to
ensure that face mask wearing statistics reflect social activities at
different times of the week, in the Paris region. Figure S4 shows the
resulting histogram. Only indoor public spaces where masks are mandatory have
been included. FFP2 respirators represent $3\;\%$ only of face barriers and
correctly worn protective coverings (surgical or FFP2) only $42\;\%$. We can
estimate that the mean filtration factor $\lambda_{i}\lambda_{e}$ is around
$0.25$ only, which gives a dose reduction by a factor of $4$ in French public
spaces where masks are mandatory. As even minute differences in respirator fit
can lead to large variations in filtration, this visual method of estimating
mask fit could lead to overestimating the average filtration efficiency. In
Georgia, USA during November-December 2020, mask wearing for students and
staff members was associated with a $2.7$-fold reduction in COVID-19 cases
(Gettings, 2021). In the UK, between February and April 2021, schoolchildren
aged 10-14 did not wear masks while those aged 15-19 did. Figure S5 compares
the cases in these two populations. The reproduction rate that can be
attributed to schools is roughly halved by masks, which gives a mean
filtration factor $\lambda^{2}=0.5$. We have observed that young people poorly
fit their masks, which could explain the discrepancy with the expected
filtration factor. Another explanation could be that an important part of the
transmission takes places during unmasked lunch time.

Figure S5: Cases per million people in the United Kingdom, from 1 February to
12 April 2021. Top panel: schoolchildren aged 10-14, with no mandatory masks.
The best fit by an exponential provides the reproduction number: $R=1.34\pm
0.04$ during the school period vs $R=0.81\pm 0.03$ before and $R=0.70\pm 0.03$
after. Bottom panel: schoolchildren aged 15-19, with mandatory masks. The best
fit by an exponential provides the reproduction number: $R=1.07\pm 0.04$
during the school period vs $R=0.82\pm 0.03$ before and $R=0.70\pm 0.03$
after.

##  XIII Risk reduction techniques

Figure S6: Hydrodynamic solutions to mitigate short-range transmission by
dispersing aerosols away from individuals. (a) Fans force the circulation of
air through HEPA filters at a cafeteria table, with an Upper-Room Ultraviolet
Germicidal Irradiation (UVGI). (b) A large fan with a filter is slightly
tilted upward and aimed at a line of static people waiting. The cone of
aerosols they emit cannot reach anyone in the line. (c) Ventilation controlled
by the occupancy number, to reduce energy consumption. Recycled air inside
ventilation ducts is decontaminated by UV-C light. (d) An air flow inside the
floor of a club pushes air up through small holes, preventing aerosols from
spreading laterally and dispersing them towards the ceiling where the
ventilation system can remove them. (e) A fan at a café table pulls air,
filters it and expels it upwards. (f) Fresh air is injected at the bottom of
the room and supplied where needed to create air flows that protect each
occupant of the room individually (Morawska  _et al._ , 2021).

The transmission risk outdoors, without masks, is real at short distances and
for long periods of time. Figure S7 shows the risk as a function of distance.
The risk is above $1$ at low wind speeds below $2.5\;{\rm m}$. 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 S6 panel
b). 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 outdoor dance floors, injection of air at high flow rate, say,
$1-10\;{\rm m^{3}/hour/person}$ may be sufficient to reduce the risk.

The absence of masks when eating and drinking poses 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 (Morawska  _et al._ , 2021) (Figure S6 a, e).

Disposable respiratory personal protection equipment is expensive, but various
protocols involving UV-C, heat and hydrogen peroxide vapor have been designed
to extend its lifetime cycle and use, and could be considered for the general
public (CDC, 2020; John  _et al._ , 2021).

Figure S7: Transmission risk $r$ for Omicron BA.1 ($\bar{h}=2800$) as a
function of the distance between people inside the public space, for the
observed mask wearing shown in Figure S4 ($\lambda^{2}=0.2$) and different
wind speeds. Green area: acceptable risk, ie. $r<1$.

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