A guideline to limit indoor airborne transmission of
COVID-19
Martin Z. Bazanta,b,1

and John W. M. BushbaDepartment of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139; and b Department of Mathematics, Massachusetts
Institute of Technology, Cambridge, MA 02139

The current revival of the American economy is being predicatedon social distancing, specifically the Six-Foot Rule, a guideline thatoffers little protection from pathogen-bearing aerosol droplets
sufficiently small to be continuously mixed through an indoor
space. The importance of airborne transmission of COVID-19 is
now widely recognized. While tools for risk assessment haverecently been developed, no safety guideline has been proposed
to protect against it. We here build on models of airborne disease transmission in order to derive an indoor safety guidelinethat would impose an upper bound on the “cumulative exposure
time,” the product of the number of occupants and their timein an enclosed space. We demonstrate how this bound depends
on the rates of ventilation and air filtration, dimensions of theroom, breathing rate, respiratory activity and face mask use of
its occupants, and infectiousness of the respiratory aerosols. By
synthesizing available data from the best-characterized indoor
spreading events with respiratory drop size distributions, we estimate an infectious dose on the order of 10 aerosol-borne virions.
The new virus (severe acute respiratory syndrome coronavirus
2 [SARS-CoV-2]) is thus inferred to be an order of magnitude
more infectious than its forerunner (SARS-CoV), consistent withthe pandemic status achieved by COVID-19. Case studies are presented for classrooms and nursing homes, and a spreadsheet andonline app are provided to facilitate use of our guideline. Implications for contact tracing and quarantining are considered, andappropriate caveats enumerated. Particular consideration is given
to respiratory jets, which may substantially elevate risk when facemasks are not worn.

the assumption that the primary vector of pathogen transmission is the large drops ejected from the most vigorous exhalation
events, coughing and sneezing (5, 19). Indeed, high-speed visualization of such events reveals that 6 ft corresponds roughly to the
maximum range of the largest, millimeter-scale drops (20). Compliance to the Six-Foot Rule will thus substantially reduce therisk of such large-drop transmission. However, the liquid drops
expelled by respiratory events are known to span a considerable
range of scales, with radii varying from fractions of a micron tomillimeters (11, 21).
There is now overwhelming evidence that indoor airborne
transmission associated with relatively small, micron-scale
aerosol droplets plays a dominant role in the spread of COVID19 (4, 5, 7, 17–19, 22), especially for so-called “superspreading
events” (25–28), which invariably occur indoors (29). For example, at the 2.5-h-long Skagit Valley Chorale choir practice thattook place in Washington State on March 10, some 53 of 61
attendees were infected, presumably not all of them within
6 ft of the initially infected individual (25). Similarly, when 23
of 68 passengers were infected on a 2-h bus journey in Ningbo,
China, their seated locations were uncorrelated with distance tothe index case (28). Airborne transmission was also implicated in
the COVID-19 outbreak between residents of a Korean high-risebuilding whose apartments were linked via air ducts (30). Studies have also confirmed the presence of infectious SARS-CoV-2
SignificanceAirborne transmission arises through the inhalation of aerosol
droplets exhaled by an infected person and is now thought
to be the primary transmission route of COVID-19. By assuming that the respiratory droplets are mixed uniformly throughan indoor space, we derive a simple safety guideline for mitigating airborne transmission that would impose an upperbound on the product of the number of occupants and theirtime spent in a room. Our theoretical model quantifies theextent to which transmission risk is reduced in large rooms
with high air exchange rates, increased for more vigorous
respiratory activities, and dramatically reduced by the use offace masks. Consideration of a number of outbreaks yields
self-consistent estimates for the infectiousness of the newcoronavirus.

COVID-19 | infectious aerosol | airborne transmission | SARS-CoV-2
coronavirus | indoor safety guideline

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C

OVID-19 is an infectious pneumonia that appeared in
Wuhan, Hubei Province, China, in December 2019 and hassince caused a global pandemic (1, 2). The pathogen responsible
for COVID-19, severe acute respiratory syndrome coronavirus
2 (SARS-CoV-2), is known to be transported by respiratory
droplets exhaled by an infected person (3–7). There are thought
to be three possible routes of human-to-human transmissionof COVID-19: large drop transmission from the mouth of an
infected person to the mouth, nose or eyes of the recipient; physical contact with droplets deposited on surfaces (fomites) andsubsequent transfer to the recipient’s respiratory mucosae; andinhalation of the microdroplets ejected by an infected person andheld aloft by ambient air currents (6, 8). We subsequently refer tothese three modes of transmission as, respectively, “large-drop,”“contact,” and “airborne” transmission, while noting that the distinction between large-drop and airborne transmission is somewhat nebulous given the continuum of sizes of emitted droplets(11).∗ We here build upon the existing theoretical frameworkfor describing airborne disease transmission (12–18) in orderto characterize the evolution of the concentration of pathogenladen droplets in a well-mixed room, and the associated risk ofinfection to its occupants.
The Six-Foot Rule is a social distancing recommendation by
the US Centers for Disease Control and Prevention, based on
PNAS 2021 Vol. 118 No. 17 e2018995118

Author contributions: M.Z.B. and J.W.M.B. designed research; M.Z.B. and J.W.M.B.
performed research; M.Z.B. analyzed data; and M.Z.B. and J.W.M.B. wrote the paper.y
The authors declare no competing interest.yThis article is a PNAS Direct Submission. R.Z. is a guest editor invited by the Editorial
Board.yThis open access article is distributed under Creative Commons Attribution License 4.0 (CCBY).y
1To whom correspondence may be addressed. Email: bazant@mit.edu.yThis article contains supporting information online at https://www.pnas.org/lookup/suppl/
doi:10.1073/pnas.2018995118/-/DCSupplemental.y
Published April 13, 2021.
* The possibility of pathogen resuspension from contaminated surfaces has also recently

been explored (9, 10).

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Edited by Renyi Zhang, Texas A&M University, College Station, TX, and accepted by Editorial Board Member John H. Seinfeld March 3, 2021 (received forreview September 9, 2020)

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virions in respiratory aerosols (31) suspended in air samples collected at distances as large as 16 ft from infected patients in a
hospital room (3). Further evidence for the dominance of indoor
airborne transmission has come from an analysis of 7,324 earlycases outside the Hubei Province, in 320 cities across mainland
China (32). The authors found that all clusters of three or morecases occurred indoors, 80% arising inside apartment homes and
34% potentially involving public transportation; only a single
transmission was recorded outdoors. Finally, the fact that facemask directives have been more effective than either lockdowns
or social distancing in controlling the spread of COVID-19 (22,33) is consistent with indoor airborne transmission as the primary
driver of the global pandemic.
The theoretical model developed herein informs the risk ofairborne transmission resulting from the inhalation of small,aerosol droplets that remain suspended for extended periods
within closed, well-mixed indoor spaces. When people cough,sneeze, sing, speak, or breathe, they expel an array of liquid
droplets formed by the shear-induced or capillary destabilization of the mucosal linings of the lungs and respiratory tract(8, 34, 35) and saliva in the mouth (36, 37). When the person
is infectious, these droplets of sputum are potentially pathogen
bearing, and represent the principle vector of disease transmission. The range of the exhaled pathogens is determined by theradii of the carrier droplets, which typically lie in the range of
0.1 µm to 1 mm. While the majority are submicron in scale,the drop size distribution depends on the form of exhalation
event (11). For normal breathing, the drop radii vary between
0.1 and 5.0 µm, with a peak around 0.5 µm (11, 38, 39). Relatively large drops are more prevalent in the case of more violent
expiratory events such as coughing and sneezing (20, 40). The
ultimate fate of the droplets is determined by their size and theairflows they encounter (41, 42). Exhalation events are accompanied by a time-dependent gas-phase flow emitted from themouth that may be roughly characterized in terms of either continuous turbulent jets or discrete puffs (20, 38, 43). The precise
form of the gas flow depends on the nature of the exhalation
event, specifically the time dependence of the flux of air expelled.
Coughs and sneezes result in violent, episodic puff releases (20),while speaking and singing result in a puff train that may bewell approximated as a continuous turbulent jet (38, 43). Eventually, the small droplets settle out of such turbulent gas flows.
In the presence of a quiescent ambient, they then settle to thefloor; however, in the well-mixed ambient more typical of a ventilated space, sufficiently small drops may be suspended by theambient airflow and mixed throughout the room until being
removed by the ventilation outflow or inhaled (SI Appendix,
section 1).
Theoretical models of airborne disease transmission in closed,
well-mixed spaces are based on the seminal work of Wells (44)and Riley et al. (45), and have been applied to describe thespread of airborne pathogens including tuberculosis, measles,
influenza, H1N1, coronavirus (SARS-CoV) (12–16, 46, 47), and,most recently, the novel coronavirus (SARS-CoV-2) (17, 25).
These models are all based on the premise that the space ofinterest is well mixed; thus, the pathogen is distributed uniformly
throughout. In such well-mixed spaces, one is no safer fromairborne pathogens at 60 ft than 6 ft. The Wells–Riley model(13, 15) highlights the role of the room’s ventilation outflow
rate Q in the rate of infection, showing that the transmission
rate is inversely proportional to Q, a trend supported by dataon the spreading of airborne respiratory diseases on college
campuses (48). The additional effects of viral deactivation, sedimentation dynamics, and the polydispersity of the suspended
droplets were considered by Nicas et al. (14) and Stilianakis
and Drossinos (16). The equations describing pathogen transport in well-mixed, closed spaces are thus well established andhave recently been applied to provide risk assessments for indoor
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airborne COVID-19 transmission (17, 18). We use a similarmathematical framework here in order to derive a simple safety
guideline.
We begin by describing the dynamics of airborne pathogen ina well-mixed room, on the basis of which we deduce an estimate for the rate of inhalation of pathogen by its occupants.
We proceed by deducing the associated infection rate from asingle infected individual to a susceptible person. We illustrate
how the model’s epidemiological parameter, a measure of theinfectiousness of COVID-19, may be estimated from available
epidemiological data, including transmission rates in a number of
spreading events, and expiratory drop size distributions (11). Ourestimates for this parameter are consistent with the pandemic
status of COVID-19 in that they exceed those of SARS-CoV(17); however, our study calls for refined estimates through consideration of more such field data. Most importantly, our study
yields a safety guideline for mitigating airborne transmission via
limitation of indoor occupancy and exposure time, a guidelinethat allows for a simple quantitative assessment of risk in various
settings. Finally, we consider the additional risk associated withrespiratory jets, which may be considerable when face masks are
not being worn.
The Well-Mixed Room
We first characterize the evolution of the pathogen concentration in a well-mixed room. The assumption of well mixedness
is widely applied in the theoretical modeling of indoor airborne
transmission (14, 16, 17), and its range of validity is discussed in
SI Appendix, section 1. We describe the evolution of the airborne
pathogen by adapting standard methods developed in chemicalengineering to describe the “continuously stirred tank reactor”(49), as detailed in SI Appendix, section 1. We assume that the
droplet-borne pathogen remains airborne for some time before
being extracted by the room’s ventilation system, inhaled, orsedimenting out. The fate of ejected droplets in a well-mixed
ambient is determined by the relative magnitudes of two speeds:
the settling speed of the drop in quiescent air, vs , and the ambient air circulation speed within the room, va . Drops of radiusr ≤ 100 µm and density ρd descend through quiescent air ofdensity ρa and dynamic viscosity µa at the Stokes settling speedvs (r ) = 2∆ρgr 2 /(9µa ), prescribed by the balance between gravity and viscous drag (50), where g is the gravitational accelerationand ∆ρ = ρd − ρa .
We consider a well-mixed room of area A, depth H , andvolume V = HA with ventilation outflow rate Q and outdoor
air change rate (typically reported as air changes per hour,or ACH) λa = Q/V . Mechanical ventilation imposes an additional recirculation flow rate Qr that further contributes to thewell-mixed state of the room, but alters the emergent drop
size distributions only if accompanied by filtration. The mean
air velocity, va = (Q + Qr )/A, prescribes the air mixing time,τa = H /va = H 2 /(2Da ), where Da = va H /2 is the turbulent diffusivity defined in terms of the largest eddies (51, 52), those on
the scale of the room (53). The timescale of the droplet settling from a well-mixed ambient corresponds to that througha quiescent ambient (51, 52, 54), as justified in SI Appendix,
section 1. Equating the characteristic times of droplet settling,H
p/vs , and removal, V /Q, indicates a critical drop radius rc =9λa H µa /(2g∆ρ) above which drops generally sediment out,
and below which they remain largely suspended within the roomprior to removal by ventilation outflow. We here define airborne transmission as that associated with droplets with radiusr < rc . The relevant physical picture, of particles settling from awell-mixed environment, is commonly invoked in the contexts ofstirred aerosols (51) and sedimentation in geophysics (54). The
additional effects of ventilation, particle dispersity, and pathogen
deactivation in the context of airborne disease transmission wereconsidered by Nicas et al. (14), Stilianakis and Drossinos (16)Bazant and Bush
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A guideline to limit indoor airborne transmission of COVID-19

the dependence λv (RH ) is not yet well characterized experimentally for SARS-CoV-2, we follow Miller et al. (25) andtreat the deactivation rate as bounded by existing data, specifically, λv = 0 [no deactivation measured in 16 h at 22 ± 1 ◦ C
and RH = 53 ± 11% (66)] and λv = 0.63/h [corresponding toa half life of 1.1 h at 23 ± 2 ◦ C and RH = 65% (67)]. Pending further data for SARS-CoV-2, we assume λv = 0.6RH h−1 ,and note the rough consistency of this estimate with that for
MERS-CoV (Middle East Respiratory Syndrome coronavirus)
at 25 ◦ C and RH = 79% (68), λv = 1.0/h. Finally, we note that
effective viral deactivation rates may be enhanced using either
ultraviolet radiation (UV-C) (69) or chemical disinfectants(e.g. H2 O2 , O3 ) (70).
The influence of air filtration and droplet settling in ventilation ducts may be incorporated by augmenting λv (r ) byan amount λf (r ) = pf (r )λr , where pf (r ) is the probability ofdroplet filtration and λr = Qr /V . The recirculation flow rate,
Qr , is commonly expressed in terms of the primary outdoor air
fraction, Zp = Q/(Qr + Q), where Q + Qr is the total airflow
rate. We note that the United States Environmental Protection
Agency defines high-efficiency particulate air (HEPA) filtration(71) as that characterized by pf > 99.97% for aerosol particles.
Ordinary air filters have required Minimum Efficiency Reporting
Value (MERV) ratings of pf = 20 to 90% in specific size ranges.
Other types of filtration devices (22), such as electrostatic precipitators (72) with characteristic pf values of 45 to 70%, can also be
included in this framework.
We seek to characterize the concentration C (r , t) (specifically, number/volume per radius) of pathogen transported bydrops of radius r . We assume that each of I (t) infectious
individuals exhales pathogen-laden droplets of radius r at a constant rate P (r ) = Qb nd (r )Vd (r )pm (r )cv (r ) (number/time per
radius), where Qb is the breathing flow rate (exhaled volume pertime). We introduce a mask penetration factor, 0 < pm (r ) < 1,that roughly accounts for the ability of masks to filter droplets
as a function of drop size (73–76).† The concentration, C (r , t),of pathogen suspended within drops of radius r then evolves
according to
V∂C= I P − (Q + pf Qr + vs A + λv V ) C∂t[1]

Rate of Production rate Loss rate from ventilation, filtration=−,
change from exhalation
sedimentation, and deactivation
where vs (r ) is the particle settling speed and pf (r ) is, again, theprobability of drop filtration in the recirculation flow Qr . Owing
to the dependence of the settling speed on particle radius, thepopulation of each drop size evolves, according to Eq. 1, at different rates. Two limiting cases of Eq. 1 are of interest. For the caseof λv = vs = Qr = 0, drops of infinitesimal size that are neither
deactivated nor removed by filtration, it reduces to the Wells–
Riley model (44, 45). For the case of λv = P = Q = Qr = 0, a
nonreacting suspension with no ventilation, it corresponds toestablished models of sedimentation from a well-mixed ambient(51, 54). For the sake of notational simplicity, we define a sizedependent sedimentation rate λs (r ) = vs (r )/H = λa (r /rc )2 as
the inverse of the time taken for a drop of radius r to sedimentfrom ceiling to floor in a quiescent room.
When one infected individual enters a room at time t = 0,
so that I (0) = 1, the radius-resolved
pathogenconcentration

increases as C (r , t) = Cs (r ) 1 − e −λc (r )t , relaxing to a steadyvalue, Cs (r ) = P (r )/(λc (r )V ), at a rate λc (r ) = λa + λf (r ) +†For the sake of simplicity, we do not consider here the dependence of pm on respiratory
activity (77) or direction of airflow (78), but note that, once reliably characterized, thesedependencies might be included in a straightforward fashion.

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and Buonanno et al. (17, 18), whose models will be built uponhere.
In SI Appendix, section 1, we provide justification for ourassumption of the well-mixed room. It is noteworthy that, evenin the absence of forced ventilation, there will generally be somemixing in an enclosed space: Natural ventilation will lead toflows through windows and doors, as well as leakage throughconstruction materials and joints. Moreover, occupants serve toenhance airflow through their motion and respiration. Traditionally, ventilation standards for American homes (American Society of Heating, Refrigerating and Air-Conditioning Engineers[ASHRAE]) recommend a minimal outdoor air exchange rate of
λa = 0.35/h, a value comparable to the average of 0.34/h reported
for Chinese apartments, including those in winter in Wuhan(55). Even with such minimal ventilation rates, for a room ofheight H = 2.1 m, there is an associated critical drop size ofradius rc = 1.3 µm. In order to guard against infectious aerosols,
ASHRAE now recommends ventilation rates greater than λa =6/h, which corresponds to rc = 5.5 µm. The “airborne” dropletsof interest here, those of radius r < rc , thus constitute a significant fraction of those emitted in most respiratory events(11, 23, 38).
Wells (56) argued that exhaled drops with diameter lessthan approximately 100 µm will evaporate before settling. The
resulting “droplet nuclei” consist of residual solutes, including
dissolved salts, carbohydrates, proteins, and pathogens, which
are typically hygroscopic and retain significant quantities ofbound water (57, 58). For apdroplet with initial radius r0 ,the equilibrium size, req = r0 3 φs /(1 − RH ), is reached overan evaporation timescale, τe = r02 /(θ(1 − RH )), where φs isthe initial solute volume fraction, RH is the relative humidity, and θ = 4.2 × 10−10 m2 /s at 25 ◦ C (58). In dry air (RH 1), saliva droplets, which typically contain 0.5% solutes and asimilar√ volume of bound water (φs ≈ 1%), can thus lose up to1 − 3 0.01 ≈ 80% of their initial size (58). Conversely,
droplets√of airway mucus shrink by as little as 1 − 3 0.2 ≈ 40%, since
they typically contain 5 to 10% gel-forming mucins (glycosylated proteins) and comparable amounts of bound water (59).
The evaporation time at 50% RH ranges from τe = 1.2 ms for
r0 = 0.5µm to 12 s at 50 µm. These inferences are consistent
with experiments demonstrating that stable respiratory aerosol
distributions in the range req < 10 µm are reached within 0.8 sof exhalation (11). While we note that the drop size distributions will, in general, depend on the relative humidity, we proceed by employing the equilibrium drop distributions measured
directly (11, 38).
We consider a polydisperse suspension of exhaled dropletscharacterized by the number density nd (r ) (per volume of air,per radius) of drops of radius r and volume Vd (r ) = 4/3πr 3 .
The drop size distribution nd (r ) is known to vary strongly withrespiratory activity and various physiological factors (11, 17, 39).
The drops contain a microscopic pathogen concentration cv (r ),a drop size-dependent probability of finding individual virions (3,31, 60), usually taken to be that in the sputum (RNA copies per
milliliter) (17, 61).
The virions become deactivated (noninfectious) at a rate λv (r )that generally depends on droplet radius, temperature, andhumidity (62). Using data for human influenza viruses (63), aroughly linear relationship between λv and RH can be inferred(62, 64), which provides some rationale for the seasonal variationof flu outbreaks, specifically, the decrease from humid summers
to dry winters. Recent experiments on the aerosol viability ofmodel viruses (bacteriophages) by Lin and Marr (65) have further revealed a nonmonotonic dependence of λv on relative
humidity. Specifically, the deactivation rate peaks at intermediate values of relative humidity, where the cumulative exposureof virions to disinfecting salts and solutes is maximized. SinceDownloaded at MIT LIBRARIES on April 15, 2021

λs (r ) + λv (r ). Note that both the equilibrium concentration andthe timescale to approach it are decreased by the combined
effects of ventilation, air filtration, particle settling, and deactivation (14, 64). Owing to the dependence of this adjustment
process on the drop size, one may understand it as a dynamic
sifting process wherein larger droplets settle out and reach theirequilibrium concentration relatively quickly. However, we note
that, in the absence of filtration and deactivation (λf = λv = 0),
the adjustment time, λ−1c , depends only weakly on drop size,
varying from V /(2Q) for the largest airborne drops (with radiusrc ) to V /Q for infinitesimal drops. The sedimentation rate of
the “airborne” droplets of radius r ≤ rc is thus bounded aboveby the air exchange rate, λs (r ) ≤ λa . The exhaled drop size distribution depends strongly on respiratory activity (11, 17, 38, 39);thus, so too must the radius-resolved concentration of airborne
pathogen. The predicted dependence on respiratory activity (11)of the steady-state volume fraction of airborne droplets, φs (r ) =Cs (r )/cv (r ), is illustrated in Fig. 1.

We define the airborne disease transmission rate, βa (t), as themean number of transmissions per time per infectious individual
per susceptible individual. One expects βa (t) to be proportional
to the quantity of pathogen exhaled by the infected person, andto that inhaled by the susceptible person. Gammaitoni and Nucci(12) defined the airborne transmission rate as βa (t) = Qb ci Cs (t)for the case of a population evolving according to the Wells–
Riley model and inhaling a monodisperse suspension. Here, ci
is the viral infectivity, the parameter that connects the fluid
physics to the epidemiology, specifically, the concentration ofsuspended pathogen to the infection rate. We note its relation to the notion of “infection quanta” in the epidemiologicalliterature (44). Specifically, ci < 1 is the infection quanta perpathogen, while ci−1 > 1 is the “infectious dose,” the number ofaerosol-borne virions required to cause infection with probability
1 − e −1 = 63%.
For the polydisperse suspension of interest here, we define theairborne transmission rate as

Fig. 1. Model predictions for the steady-state, droplet radius-resolved aerosol volume fraction, φs (r), produced by a single infectious person in a well-mixed
room. The model accounts for the effects of ventilation, pathogen deactivation, and droplet settling for several different types of respiration in the absenceof face masks (pm = 1). The ambient conditions are taken to be those of the Skagit Valley Chorale superspreading incident (25, 27) (H = 4.5 m, A = 180 m2 ,λa = 0.65 h−1 , rc = 2.6 µm, λv = 0.3 h−1 , and RH = 50%). The expiratory droplet size distributions are computed from the data of Morawska et al. (ref. 11,
figure 3) at RH = 59.4% for aerosol concentration per log-diameter, using nd (r) = (dC/d log D)/(r ln 10). The breathing flow rate is assumed to be 0.5 m3 /h
for nose and mouth breathing, 0.75 m3 /h for whispering and speaking, and 1.0 m3 /h for singing.

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C (r , t)pm (r )ci (r )dr ,[2]0

thereby accounting for the protective properties of masks, andallowing for the possibility that the infectivity ci (r ) depends on
droplet size. Different droplet sizes may emerge from, and penetrate into, different regions of the respiratory tract (34, 37, 79),
and so have different ci (r ); moreover, virions in relatively small
droplets may diffuse to surfaces more rapidly and so exchangewith bodily fluids more effectively. Such a size dependence ininfectivity, ci (r ), is also consistent with reports of enhanced
viral shedding in micron-scale aerosols compared to larger drops
for both influenza virus (60) and SARS-CoV-2 (31). Finally,
we introduce a relative transmissibility (or susceptibility), sr , torescale the transmission rate for different subpopulations or viralstrains.
Indoor Safety Guideline
The reproduction number of an epidemic, R0 , is defined as
the mean number of transmissions per infected individual. Provided R0 < 1, a disease will not spread at the population level(80). Estimates of R0 for COVID-19 have been used to compare its rate of spread in different regions and its dependence
on different control strategies (33, 81–83) and, most recently,
viral variants (84, 85). We here define an analogous reproductive number for indoor, airborne transmission, Rin (τ ), as the
expected number of transmissions in a room of total occupancy
N over a time τ from a single infected person entering at t = 0.
Our safety guideline sets a small risk tolerance  (typically 1 to10%) for the indoor reproductive number, defined asZ τRin (τ ) = Nsβa (t)dt < .
[3]0

The number of susceptibles, Ns = ps (N − 1), may include allothers in the room (ps = 1), or be reduced by the susceptible probability ps < 1, the fraction of the local population not
yet exposed or immunized. In the limit of   1, one mayinterpret Rin (τ ) as the probability of the first transmission,
which is approximately equal to the sum of the Ns independent probabilities of transmission to any particular susceptible
individual in a well-mixed room.‡ In SI Appendix, section 3, weshow that this guideline follows from standard epidemiologicalmodels, including the Wells–Riley model, but note that it has
broader generality. The exact transient safety bound appropriate for the time-dependent situation arising directly after an
infected index case enters a room is evaluated in SI Appendix,
section 2.
We here focus on a simpler, more conservative guideline that
follows for long times relative to the air residence time, τ  λ−1a(which may vary from minutes to hours, and is necessarily greaterthan λc (r )−1 ), when the airborne pathogen has attained its equilibrium concentration C (r , t) → Cs (r ). In this equilibrium case,the transmission rate (2) becomes constant,Z ∞βanq (r )Q 2p2
Q 2 p 2 Cq
2= b m
dr = b m= pm
fd λ q ,[4]srV
λc (r )
V λc (r )0

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where, for the sake of simplicity, we assume constant mask
filtration pm over the entire range of aerosol drop sizes. We
define the microscopic concentration of infection quanta perliquid volume as nq (r ) = nd (r )Vd (r )cv (r )ci (r ), and the concentration of infection quanta or “infectiousness” of exhaled air,‡Markov’s inequality ensures that the probability of at least one transmission, P1 ,is bounded above by the expected number of transmissions, P1 ≤ Rin . In the limit,
Rin <   1, these quantities are asymptotically equal, since P1 = 1 − (1 − p(τ ))Ns ∼R
Ns p(τ ) = Rin for Ns independent transmissions of probability, p(τ ) = 0τ βa (t)dt  1.

Bazant and Bush
A guideline to limit indoor airborne transmission of COVID-19

R∞
Cq = 0 nq (r )dr . The latter is the key disease-specific parameter in our model, which can also be expressed as the rate ofquanta emission by an infected person, λq = Qb Cq . The second
equality in Eq. 4 defines the effective infectious drop radius r ,given in SI Appendix, Eq. S7. The third equality defines the dilution factor, fd = Qb /(λc (r̄ )V ), the ratio of the concentration of
infection quanta in the well-mixed room to that in the unfilteredbreath of an infected person. As we shall see in what follows,
fd provides a valuable diagnostic in assessing the relative risk ofvarious forms of exposure.
We thus arrive at a simple guideline, appropriate for steadystate situations, that bounds the cumulative exposure time(CET),
λc V + v s A(N − 1)τ <  2 2.
[5]
Qb pm Cq srwhere v s = vs (r ), and λc = λa + λf (r̄ ) + λv (r̄ ) is the air purification rate associated with air exchange, air filtration, andviral deactivation. The effect of relative humidity on thedropletsize distribution can be captured by multiplying r̄ byp3
0.4/(1 − RH ), since the droplet distributions used in our
analysis were measured at RH = 60% (11).
By noting that the sedimentation rate of aerosols is usuallyless than the air exchange rate, λs (r ) < λa , and by neglecting
the influence of both air filtration and pathogen deactivation, we
deduce, from Eq. 5, a more conservative bound on the CET,
Nτ <

λa V
,
2 C s
Qb2 pmq r[6]

the interpretation of which is immediately clear. To minimize
risk of infection, one should avoid spending extended periods inhighly populated areas. One is safer in rooms with large volume
and high ventilation rates. One is at greater risk in rooms where
people are exerting themselves in such a way as to increase their
respiration rate and pathogen output, for example, by exercising, singing, or shouting. Since the rate of inhalation of contagion
depends on the volume flux of both the exhalation of the infectedindividual and the inhalation of the susceptible person, the risk ofinfection increases as Qb2 . Likewise, masks worn by both infected
and susceptible persons will reduce the risk of transmission by a2
factor pm, a dramatic effect given that pm ≤ 0.1 for moderately
high-quality masks (74, 75).
Application to COVID-19
The only poorly constrained quantity in our guideline is the epidemiological parameter, Cq sr , the product of the concentrationof exhaled infection quanta by an infectious individual, Cq , andthe relative transmissibility, sr . We emphasize that Cq and sr
are expected to vary widely between different populations (86–91), among individuals during progression of the disease (92,93), and between different viral strains (84, 85). Nevertheless,
we proceed by making rough estimates for Cq for different respiratory activities on the basis of existing epidemiological data
gathered from early superspreading events of COVID-19. Ourinferences provide a baseline value for Cq , relevant for elderly
individuals exposed to the original strain of SARS-CoV-2, that
we may rescale by the relative transmissibility sr in order toconsider different populations and viral strains. We make these
inferences with the hope that such an attempt will motivate
the acquisition of more such data, and so lead to improved
estimates for Cq and sr for different populations in various
settings.
An inference of Cq = 970 quanta/m3 was made by Miller
et al. (25) in their recent analysis of the Skagit Valley Chorale
superspreading incident (27), on the basis of the assumptionthat the transmission was described in terms of the Wells–Riley
model (12, 13, 17, 45). To be precise, they inferred a quanta
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emission rate of λq = Cq Q̄b = 970 quanta/h for a mean breathing
rate of Q̄b = 1.0 m3 /h appropriate for singing (25). This inferenceis roughly consistent with studies of other related viral diseases.
For example, Liao et al. (46) estimated Cq = 28 quanta/m3 fromthe rate of indoor spreading of SARS-CoV, in a hospital and anelementary school. Estimates of Cq for H1N1 influenza fall in therange 15 to 128 quanta/m3 (47). For SARS-CoV-2, Buonanno
et al. (17) estimate a Cq range of 10.5 to 1,030 quanta/m3 , on
the basis of the estimated infectivity ci = 0.01 to 0.1 of SARSCoV (94) and the reported viral loads in sputum (92, 93, 95),
and note that the precise value depends strongly on the infected
person’s respiratory activity. Notably, their range spans the highvalue inferred for the Skagit Valley Chorale (25), and all of ourinferences to follow.
We proceed by estimating quanta concentrations, Cq , or,equivalently, quanta emission rates, λq = Qb Cq , for different
forms of respiration. First, we solve Eq. 1 to obtain the steadystate radius-resolved droplet volume fraction φs (r ) for various
hypothetical expiratory activities in the room of the Skagit Valley Chorale, using the drop size distributions of Morawska et al.
(11). Our results are shown in Fig. 1. Integrating each curve up
to the critical radius rc , we then obtain an activity-dependent
volRr
ume fraction of infectious airborne droplets φ1 = 0 c φs (r )dr inthe choir room (see SI Appendix). Finally, we assume the inferredvalue, Cq = 970 quanta/m3 , for the superspreading incident (25)that resulted from the expiratory activity most resembling singing[voiced “aahs” with pauses for recovery (11)], and deduce values of Cq for other forms of respiration by rescaling with theappropriate φ1 values. Our predictions for the dependence of
Cq on respiratory activity are shown in Fig. 2. For validation, we
also show estimates for Cq based on the recent measurementsof activity-dependent aerosol concentrations reported by Asadi
et al. (38, 39). Specifically, we calculated the aerosol volume fractions from the reported drop-size distributions (from figure 5 ofref. 39) for a different set of expiratory activities that included
various breathing patterns and speaking aloud at different volumes. We then used these volume fractions to rescale the value
Cq = 72 quanta/m3 for speaking at intermediate volume (39),which we chose to match the value inferred for the most similar
respiratory activity considered by Morawska et al. (11), specifically, voiced counting with pauses (11). Notably, the quanta concentrations so inferred, Cq , are consistent across the full range ofactivities, from nasal breathing at rest (1 to 10 quanta/m3 ) to oral
breathing and whispering (5 to 40 quanta/m3 ), to loud speaking
and singing (100 to 1,000 quanta/m3 ).
Our inferences for Cq from a number of superspreading events
are also roughly consistent with physiological measurements ofviral RNA in the bodily fluids of COVID-19 patients at peakviral load. Specifically, our estimate of Cq = 72 quanta/m3 forvoiced counting (11) and intermediate-volume speech (39) withintegrated aerosol volume fractions φ1 = 0.36 and 0.11 (µm/cm)3
corresponds, respectively, to microscopic concentrations of cq =ci cv = 2 × 108 and 7 × 108 quanta/mL (see SI Appendix). Respiratory aerosols mainly consist of sputum produced by thefragmentation (96) of mucous plugs and films in the bronchioles and larynx (34–36). Larger droplets are thought to form
by fragmentation of saliva in the mouth (36, 37). Airborne viralloads are usually estimated from that of saliva or sputum (61,92, 93, 95, 97). After incubation, viral loads, cv , in sputum tend
to peak in the range 108 to 1011 RNA copies per milliliter (92,93, 95), while much lower values have been reported for otherbodily fluids (92, 93, 98). Virus shedding in the pharynx remains

Fig. 2. Estimates of the “infectiousness” of exhaled air, Cq , defined as the peak concentration of COVID-19 infection quanta in the breath of an infected
person, for various respiratory activities. Values are deduced from the drop size distributions reported by Morawska et al. (11) (blue bars) and Asadi etal. (39) (orange bars). The only value reported in the epidemiological literature, Cq = 970 quanta/m3 , was estimated (25) for the Skagit Valley Chorale
superspreading event (27), which we take as a baseline case (sRr = 1) of elderly individuals exposed to the original strain of SARS-CoV-2. This value is rescaledby the predicted infectious aerosol volume fractions, φ1 = 0rc φs (r)dr, obtained by integrating the steady-state size distributions reported in Fig. 1 fordifferent expiratory activities (11). Aerosol volume fractions calculated for various respiratory activities from figure 5 of Asadi et al. (39) are rescaled so thatthe value Cq = 72 quanta/m3 for “intermediate speaking” matches that inferred from Morawska et al.’s (11) for “voiced counting.” Estimates of Cq for theoutbreaks during the quarantine period of the Diamond Princess (26) and the Ningbo bus journey (28), as well as the initial outbreak in Wuhan City (2, 81),are also shown (see SI Appendix for details).

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three different age groups and the original strain of SARS-CoV2 in our case studies. However, we anticipate the need to revise
these sr values for new viral variants, such as the lineage B.1.1.7(VOC 202012/01) (84, 85), which recently emerged in the United
Kingdom with 60% greater transmissibility and elevated risk ofinfection among children.
In summary, our inferences of Cq and sr from a diverse
set of indoor spreading events and from independent physiological data are sufficiently self-consistent to indicate that thevalues reported in Fig. 2 may prove to be sufficient to apply thesafety guideline in a quantitative fashion. Our hope is that ourattempts to infer Cq will motivate the collection of more suchdata from spreading events, which might then be used to refine
our necessarily crude initial estimates.
Case Studies
We proceed by illustrating the value of our guideline in estimating the maximum occupancy or exposure time in two settingsof particular interest, the classroom and an elder care facility. Considering our inferences from the data and the existing
literature, it would appear reasonable to illustrate our guideline for COVID-19 with the conservative choice of Cq = 30
quanta/m3 . However, we emphasize that this value is expected
to vary strongly with different demographics and respiratory
activity levels (17). In taking the value of Cq = 30 quanta/m3 ,we are assuming that, in both settings considered, occupants
are engaged in relatively mild respiratory activities consistent
with quiet speech or rest. In assessing critical CETs for given
populations, we stress that the tolerance  is a parameter thatshould be chosen judiciously according to the vulnerability of the
population, which varies dramatically with age and preexisting
conditions (86–89).
We first apply our guideline to a typical American classroom,designed for an occupancy of 19 students and their teacher, andchoose a modest risk tolerance,  = 10% (Fig. 3A). The importance of adequate ventilation and mask use is made clear by ourguideline. For normal occupancy and without masks, the safe
time after an infected individual enters the classroom is 1.2 h fornatural ventilation and 7.2 h with mechanical ventilation, according to the transient bound, SI Appendix, Eq. S8. Even with clothmask use (pm = 0.3), these bounds are increased dramatically, to
8 and 80 h, respectively. Assuming 6 h of indoor time per day,a school group wearing masks with adequate ventilation would
thus be safe for longer than the recovery time for COVID-19(7 d to 14 d), and school transmissions would be rare. We stress,
however, that our predictions are based on the assumption of a“quiet classroom” (38, 77), where resting respiration (Cq = 30)
is the norm. Extended periods of physical activity, collective
speech, or singing would lower the time limit by an order ofmagnitude (Fig. 2).
Our analysis sounds the alarm for elderly homes and long-termcare facilities, which account for a large fraction of COVID-19
hospitalizations and deaths (86–88). In nursing homes in New
York City, law requires a maximum occupancy of three andrecommends a minimum area of 80 ft2 per person. In Fig. 3B,
we plot the guideline for a tolerance of  = 0.01 transmission
probability, chosen to reflect the vulnerability of the community. Once again, the effect of ventilation is striking. For naturalventilation (0.34 ACH), the Six-Foot Rule fails after only 3 minunder quasi-steady conditions, or after 17 min for the transient
response to the arrival of an infected person, in which case theFifteen-Minute Rule is only marginally safe. With mechanicalventilation (at 8 ACH) in steady state, three occupants could
safely remain in the room for no more than 18 min. This example provides insight into the devastating toll of the COVID-19
pandemic on the elderly (86, 88). Furthermore, it underscores
the need to minimize the sharing of indoor space, maintain

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high during the first week of symptoms and reaches 7 × 108 RNA
copies per throat swab (92) (typically 1 mL to 3 mL). Since viralloads are 20 to 50% greater in sputum than in throat swabs (93),the most infectious aerosols are likely to contain cv ≈ 109 RNA
copies per milliliter. Using this viral load and assuming ci = 2%
based on previous inferences for SARS-CoV (94), Buonanno
et al. (17) estimated cq = 2 × 107 quanta/mL for SARS-CoV-2,
an order of magnitude below our inferences obtained directlyfrom spreading data for COVID-19 (11, 39). The inference that
SARS-CoV-2 is 10 times more infectious than SARS-CoV, withci ≈ 10% (an infectious dose on the order of 10 aerosol-borne
virions), is consistent with the fact that only the former caused apandemic.
Our findings are consistent with emerging virological (3, 31,
66, 67) and epidemiological (5, 19, 23, 28, 29) evidence that
SARS-CoV-2 is present and extremely infectious in respiratory
aerosols and that indoor airborne transmission is the dominant
driver of the COVID-19 pandemic (4, 22). Further support forthis hypothesis is provided by crudely applying our indoor transmission model to a number of slightly less well characterized
spreading events, as detailed in SI Appendix, all of which yieldroughly consistent values of Cq (shown in Fig. 2). For the initial outbreak of COVID-19 in Wuhan City (2, 81), we assume
that spreading occurred predominantly in family apartments,
as is consistent with the inference that 80% of transmission
clusters arose in people’s homes (32). We may then tentatively equate the average reproduction number estimated for
the Wuhan outbreak (81), R0 = 3.3, with the indoor reproduction number, Rin (τ ). We use τ = 5.5 d as the exposure
time, assuming that it corresponds to mean time before theonset of symptoms and patient isolation. We consider the mean
household size of three persons in a typical apartment witharea 30 m2 per person and a winter bedroom ventilation rateof 0.34 ACH (55), and assume that λv = 0.3/h and r̄ = 2 µm.
We thus infer Cq = 30 quanta/m3 , a value expected for normal
breathing (Fig. 2).
For the Ningbo bus incident, all model parameters are known
except for the air exchange rate. We estimate λa = 1.25/h for amoving bus with closed windows, based on studies of pollutantsin British transit buses (99). We thus infer Cq = 90 quanta/m3 , avalue that lies in the range of intermediate speaking, as might be
expected onboard a bus filled to capacity. Considering the uncertainty in λa , one might also infer a value consistent with resting
on a quiet bus; in particular, choosing λa = 0.34/h yields Cq = 57
quanta/m3 . Finally, we infer a value of Cq = 30 quanta/m3 fromthe spreading event onboard the quarantined Diamond Princess
cruise ship (26), a value consistent with the passengers being
primarily at rest. However, we note that the extent to which
the Diamond Princess can be adequately described in terms ofa well-mixed space remains the subject of some debate (see SI
Appendix, section 5).
We proceed by making the simplifying assumption that thedependence of Cq on expiratory activity illustrated in Fig. 2
is universal, but retain the freedom to rescale these values
by the relative transmissibility sr for different age groups andviral strains. It is well established that children have considerably lower hospitalization and death rates (86–88), but there
is growing evidence that they also have lower transmissibility(89–91, 100, 101). A recent study of household clusters suggeststhat children are rarely index cases or involved in secondarytransmissions (89). The best controlled comparison comes fromquarantined households in China, where social contacts werereduced sevenfold to eightfold during lockdowns (101). Compared to the elderly (over 65 y old) for which we have assigned
sr = 1, the relative susceptibility of adults (aged 15 y to 64 y) was
found to be sr = 68%, while that of children (aged 0 y to 14 y)
was sr = 23%. We proceed by using these values of sr for these

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Fig. 3. The COVID-19 indoor safety guideline would limit the cumulative exposure time (CET) in a room with an infected individual to lie beneath thecurves shown. Solid curves are deduced from the pseudo-steady formula, Eq. 5, for both natural ventilation (λa = 0.34/h; blue curve) and mechanicalventilation√ (λa = 8.0/h; red curve). Horizontal axes denote occupancy times with and without masks. Evidently, the Six-Foot Rule (which limits occupancy toNmax = A/(6 ft)) becomes inadequate after a critical time, and the Fifteen-Minute Rule becomes inadequate above a critical occupancy. (A) A typical schoolclassroom: 20 persons share a room with an area of 900 ft2 and a ceiling height of 12 ft (A = 83.6 m2 , V = 301 m3 ). We assume low relative transmissibility(sr = 25%), cloth masks (pm = 30%), and moderate risk tolerance ( = 10%) suitable for children. (B) A nursing home shared room (A = 22.3 m2 , V = 53.5 m3 )with a maximum occupancy of three elderly persons (sr = 100%), disposable surgical or hybrid-fabric masks (pm = 10%), and a lower risk tolerance ( = 1%)
to reflect the vulnerability of the community. The transient formula, SI Appendix, Eq. S8, is shown with dotted curves. Other parameters are Cq = 30
quanta/m3 , λv = 0.3/h, Qb = 0.5 m3 /h, and r = 0.5 µm.

adequate, once-through ventilation, and encourage the use offace masks.
In both examples, the benefit of face masks is immediately−2
apparent, since the CET limit is enhanced by a factor pm, theinverse square of the mask penetration factor. Standard surgical masks are characterized by pm = 1 to 5% (73, 74), and so
allow the CET to be extended by 400 to 10,000 times. Even cloth
face coverings would extend the CET limit by 6 to 100 times forhybrid fabrics (pm = 10 to 40%) or 1.5 to 6 times for single-layer
fabrics (pm = 40 to 80%) (75). Our inference of the efficacy offace masks in mitigating airborne transmission is roughly consistent with studies showing the benefits of mask use on COVID-19
transmission at the scales of both cities and countries (22, 33, 83).
Air filtration has a less dramatic effect than face mask use inincreasing the CET bound. Nevertheless, it does offer a meansof mitigating indoor transmission with greater comfort, albeit atgreater cost (22, 72). Eq. 5 indicates that even perfect air filtration, pf = 1, will only have a significant effect in the limit of highly
recirculated air, Zp  1. The corresponding minimum outdoor
airflow per person, Q/Nmax , should be compared with local standards, such as 3.8 L/s per person for retail spaces and classrooms
and 10 L/s per person for gyms and sports facilities (72). In theabove classroom example with a typical primary outdoor air fraction of Zp = 20% (22), the air change rate λa could effectively
be increased by a factor of 4.6 by installing a MERV-13 filter,
pm = 90%, or a factor of 5.0 with a HEPA filter, pm = 99.97%.
At high air exchange rates, the same factors would multiply the
CET bound.
Next, we illustrate the value of our guideline in contact tracing (82), specifically, in prescribing the scope of the testing ofpeople with whom an infected index case has had close contact.
The CDC presently defines a COVID-19 “close contact” as any
encounter in which an individual is within 6 ft of an infected person for more than 15 min. Fig. 3 makes clear that this definition
may grossly underestimate the number of individuals exposed
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to a substantial risk of airborne infection in indoor spaces. Our
study suggests that, whenever our CET bound (5) is violated
during an indoor event with an infected person, at least one
transmission is likely, with probability . When the tolerance 
exceeds a critical value, all occupants of the room should be considered close contacts and so warrant testing. For relatively short
exposures (λa τ  1) initiated when the index case enters theroom, the transient bound should be considered (SI Appendix,
section 2).
We proceed by considering the implications of our guideline for the implementation of quarantining and testing. While
official quarantine guidelines emphasize the importance of isolating infected persons, our study makes clear the importance ofisolating and clearing infected indoor air. In cases of home quarantine of an infected individual with healthy family members,
our guideline provides specific recommendations for mitigating
indoor airborne transmission. For a group sharing an indoor
space intermittently, for example, office coworkers or classmates,
regular testing should be done with a frequency that ensuresthat the CET between tests is less than the limit set by theguideline. Such testing would become unnecessary if the time
limit set by the CET bound greatly exceeds the time taken for
an infected person to be removed from the population. For
the case of a symptomatic infected person, this removal timeshould correspond to the time taken for the onset of symptoms (∼5.5 d). To safeguard against asymptomatic individuals,
one should use the recovery time (∼14 d) in place of theremoval time.
Finally, we briefly discuss how the prevalence of infection in
the population affects our safety guideline. Our guideline sets a
limit on the indoor reproductive number, the risk of transmission from a single infected person in the room. It thus implicitly
assumes that the prevalence of infection in the population, pi ,is relatively low. In this low-pi limit, the risk of transmission
increases with the expected number of infected persons in the
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Beyond the Well-Mixed Room
The model developed herein describes the risk of small respiratory drops (r < rc ) in the case where the entirety of theroom is well mixed. There are undoubtedly circumstances where
there are substantial spatial and temporal variations of thepathogen concentration from the mean (7, 42). For example,
it is presumably the spatial variations from well mixedness that
result in the inhomogeneous infection patterns reported fora number of well-documented transmission events in closed
spaces, including a COVID outbreak in a Chinese restaurant(4), and SARS outbreaks on airliners (103). Circumstances
have also been reported where air conditioner-induced flows
appear to have enhanced direct pathogen transport between
infected and susceptible individuals (104). In the vicinity of
an infected person, the turbulent respiratory jet or puff will
have a pathogen concentration that is substantially higher than
the ambient (20, 43). Chen et al. (42) referred to infection
via respiratory plumes as “short-range airborne transmission,”and demonstrated that it poses a substantially greater risk thanlarge-drop transmission. In order to distinguish short-range
airborne transmission from that considered in our study, we
proceed by referring to the latter as “long-range airborne”
transmission.
On the basis of the relatively simple geometric form of turbulent jet and puff flows, one may make estimates of the form of
the mixing that respiratory outflows induce, the spatial distribution of their pathogen concentration, and so the resulting risk
they pose to the room’s occupants. For the case of the turbulent
jet associated with relatively continuous speaking or breathing,turbulent entrainment of the ambient air leads to the jet radiusr = αt x increasing linearly with distance x from the source,
where αt ≈ 0.1 to 0.15 is the typical jet entrainment coefficient(20, 42, 43). The conservation of momentum flux M = πρa r 2 v 2then indicates that the jet speed decreases with distance from√the source according to v (x ) = M 1/2 /(αt x πρa ). Concurrently,
turbulent entrainment results in the pathogen concentration
1/2
within the jet decreasing according to Cj (x )/C0 = Am /(αt x ),where Am ≈ 2 cm2 denotes the cross-sectional area of themouth, and C0 = Cq /cv is the exhaled pathogen concentration.§
Abkarian et al. (43) thus deduce that, for the respiratory jet
generated by typical speaking, the concentration of pathogen isdiminished to approximately 3% of its initial value at a distanceof 2 m.
In a well-mixed room, the mean concentration of pathogen
produced by a single infected person is fd C0 . For example, in thelarge, poorly ventilated room of the Skagit Valley Chorale, we
compute a dilution factor, fd = Qb /(λc (r̄ )V ), of approximately
0.001. We note that, since λc (r ) > λa = Q/V , the dilution factor satisfies the bound, fd ≤ Qb /Q. For typical rooms and air
exchange rates, fd lies in the range of 0.0001 to 0.01. With§

These expressions for v(x) and C(x) are valid in the limit of x > xv , where xv is thevirtual origin of the jet, typically on the order of 10 cm (20, 105). Near-field expressions
well behaved at x = 0 are given by replacing x with x + xv , and normalizing such that
C(0) = C0 .

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the dilution factor of the well-mixed room and the dilution
rate of respiratory jets, we may now assess the relative risk toa susceptible person of a close encounter (either episodic or
prolonged) with an infected individual’s respiratory jet, and an
exposure associated with sharing a room with an infected person for an extended period. Since the infected jet concentration
Cj (x ) decreases with distance from its source, one may assess its
pathogen concentration relative to that of the well-mixed room,
1/2
Cj (x )/(fd C0 ) = Am /(αt fd x ). There is thus a critical distance,
1/2
Am /(αt fd ), beyond which the pathogen concentration in the jet
is reduced to that of the ambient. This distance exceeds 10 m forfd in the aforementioned range and so is typically much greaterthan the characteristic room dimension. Thus, in the absence ofmasks, respiratory jets may pose a substantially greater risk than
the well-mixed ambient.
We first consider a worst-case, close-contact scenario in which
a person directly ingests a lung full of air exhaled by an infected
person. An equivalent amount of pathogen would be inhaledfrom the ambient by anyone within the room after a time τ =Vb /(Qb fd ), where Vb ≈ 500 mL is the volume per breath. For
the geometry of the Skagit choir room, for which fd = 0.001, thecritical time beyond which airborne transmission is a greater risk
than this worst-case close encounter with a respiratory plume isτ = 1.0 h. We next consider the worst-case scenario governedby the Six-Foot Rule, in which a susceptible person is directlyin the path of an infected turbulent jet at a distance of 6 ft,
over which the jet is diluted by a factor of 3% (43). The associated concentration in the jet is still roughly 30 times higher than
the steady-state concentration in the well-mixed ambient (whenfd = 0.001), and so would result in a commensurate amplification of the transmission probability. Our guideline could thus
be adopted to safeguard against the risk of respiratory jets ina socially distanced environment by reducing  by a factor of
C (6ft)/(fd C0 ), which is 3 to 300 for fd in the range of 0.0001
to 0.01. We note that the latter worst-case scenario describes a
static situation where a susceptible individual is seated directlyin the respiratory plume of an infected individual, as may arise ina classroom or airplane (103). More generally, with a circulating
population in an indoor setting, one would expect to encounter
an infected respiratory plume only for some small fraction of thetime, consideration of which would allow for a less conservative
choice of .
We may thus make a relatively crude estimate for the additional risk of short-range plume transmission, appropriate whenmasks are not being worn (pm = 1), by adding a correction to
our safety guideline [5]. We denote by pj the probability that asusceptible neighbor lies in the respiratory plume of the infected
person, and denote by x > 0 the distance between nearest neighbors, between which the risk of infection is necessarily greatest.
We thus deduce
"
#
1/2
pj Am
Rin (τ ) 1 +< .
[7]
Ns fd αt x
In certain instances, meaningful estimates may be made for bothpj and x . For example, if a couple dines at a restaurant, x would
correspond roughly to the distance across a table, and pj would
correspond to the fraction of the time they face each another.
If N occupants are arranged randomly
pin an indoor space, thenone expects pj = tan−1 αt /π and x = A/N . When strict social
distancing is imposed, one may further set x to the minimum
allowed interperson distance, such as 6 ft. Substitution from Eq.
5 reveals that the second term in Eq. 7 corresponds to the riskof transmission from respiratory jets, as deduced by Yang et al.
(106), aside from the factor pj . We note that any such guideline
intended to mitigate against short-range airborne transmission
by respiratory plumes will be, as is [7], dependent on geometry,
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room, Npi , and the tolerance should be lowered in proportion to Npi if it exceeds one. Conversely, when Npi → 0, thetolerance might be increased proportionally until the recommended restrictions are deemed unnecessary.
For instructions on how to apply our guideline to other situations, we refer the interested reader to the spreadsheet provided
in SI Appendix. There, by specifying a given room geometry,
ventilation rate, and respiratory activity, one may deduce themaximum CET in a particular indoor setting, and so define precisely what constitutes an exposure in that setting. An online app
based on our guideline has also been developed (102).

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flow, and human behavior, while our guideline for the mitigationof long-range airborne transmission [5] is universal.
We note that the use of face masks will have a marked effecton respiratory jets, with the fluxes of both exhaled pathogen andmomentum being reduced substantially at their source. Indeed,
Chen et al. (42) note that, when masks are worn, the primary
respiratory flow may be described in terms of a rising thermal
plume, which is of significantly less risk to neighbors. With apopulation of individuals wearing face masks, the risk posed byrespiratory jets will thus be largely eliminated, while that of thewell-mixed ambient will remain.
Finally, we stress that our guideline is based on the average
concentration of aerosols within the room. For every region ofenhanced airborne pathogen concentration, there is necessarilya region of reduced concentration and lower transmission risk
elsewhere in the room. The ensemble average of the transmission risk over a number of similar events, and the time-averagedtransmission risk in a single event, are both expected to approachthat in the well-mixed steady state, as in ergodic processes instatistical mechanics. This feature of the system provides rationale for the self-consistency of our inferences of Cq , based on
the hypothesis of the well-mixed room, from the diverse set of
spreading events considered herein.
Discussion and Caveats
We have focused here primarily on airborne transmission, forwhich infection arises through inhalation of a critical quantity ofairborne pathogen, and neglected the roles of both contact andlarge-drop transmission (6). While motivated by the COVID-19
pandemic, our theoretical framework applies quite generally toairborne respiratory illnesses, including influenza. Moreover, wenote that the approach taken, coupling the droplet dynamics tothe transmission dynamics, allows for a more complete description. For example, consideration of conservation of pathogen
allows one to calculate the rate of pathogen sedimentation andassociated surface contamination, consideration of which would
allow for quantitative models of contact transmission and so
inform cleaning protocols.
Typical values for the parameters arising in our model are
listed in SI Appendix, Table S1. Respiration rates Qb have beenmeasured to be ∼ 0.5 m3 /h for normal breathing, and may
increase by a factor of 3 for more strenuous activities (17).
Other parameters, including room geometry, ventilation, andfiltration rates, will obviously be room dependent. The mostpoorly constrained parameter appearing in our guideline is Cq sr ,the product of the concentration of pathogen in the breath of
an infected person and the relative transmissibility. The latter,
sr , was introduced in order to account for the dependence oftransmissibility on the mean age of the population (86–88, 91)
and the viral strain (84, 85). The value of Cq sr was inferredfrom the best characterized superspreading event, the Skagit
Valley Chorale incident (25), as arose among an elderly population with a median age of 69 y (27), for which we assign
sr = 1. The Cq value so inferred was rescaled using reported
drop size distributions (11, 23, 38) allowing us to estimate Cq
for several respiratory activities, as listed in Fig. 3. Further comparison with inferences based on other spreading events of newviral strains among different populations would allow for refinement of our estimates of Cq and sr . We thus appeal to the
public health community to document the physical conditions
enumerated in SI Appendix, Table S1 for more indoor spreading
events.
Adherence to the Six-Foot Rule would limit large-drop transmission, and adherence to our guideline, Eq. 5, would limit
long-range airborne transmission. We have also shown how thesizable variations in pathogen concentration associated with respiratory flows, arising in a population not wearing face masks,
might be taken into account. Consideration of both short-range
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and long-range airborne transmission leads to a guideline of theform of Eq. 7 that would bound both the distance between occupants and the CET. Circumstances may also arise where a room
is only partially mixed, owing to the absence or deficiency of air
conditioning and ventilation flows, or the influence of irregularities in the room geometry (107). For example, in a poorly
ventilated space, contaminated warm air may develop beneaththe ceiling, leading to the slow descent of a front between
relatively clean and contaminated air, a process described by“filling-box” models (107). In the context of reducing COVID19 transmission in indoor spaces, such variations from well
mixedness need be assessed on a room-by-room basis. Nevertheless, the criterion [5] represents a minimal requirement
for safety from long-range airborne infection in well-mixed,indoor spaces.
We emphasize that our guideline was developed specificallywith a view to mitigating the risk of long-range airborne transmission. We note, however, that our inferences of Cq camefrom a number of superspreading events, where other modesof transmission, such as respiratory jets, are also likely to havecontributed. Thus, our estimates for Cq are necessarily overestimates, expected to be higher than those that would have arisen
from purely long-range airborne transmission. Consequently,
our safety guideline for airborne transmission necessarily provides a conservative upper bound on CET. We note that theadditional bounds required to mitigate other transmission modes
will not be universal; for example, we see, in Eq. 7, that the danger of respiratory jets will depend explicitly on the arrangementof the room’s occupants. Finally, we reiterate that the wearing ofmasks largely eliminates the risk of respiratory jets, and so makes
the well-mixed room approximation considered here all the more
relevant.
Our theoretical model of the well-mixed room was developed
specifically to describe airborne transmission between a fixed
number of individuals in a single well-mixed room. Nevertheless,we note that it is likely to inform a broader class of transmission
events. For example, there are situations where forced ventilation mixes air between rooms, in which case the compound room
becomes, effectively, a well-mixed space. Examples consideredhere are the outbreaks on the Diamond Princess and in apartments in Wuhan City (see SI Appendix); others would include
prisons. There are many other settings, including classrooms andfactories, where people come and go, interacting intermittently
with the space, with infected people exhaling into it, and susceptible people inhaling from it, for limited periods. Such settings arealso informed by our model, provided one considers the mean
population dynamics, and so identifies N with the mean numberof occupants.
The guideline [5] depends on the tolerance , whose value in aparticular setting should be set by the appropriate policy makers,
informed by the latest epidemiological evidence. Likewise, theguideline includes the relative transmissibility sr of a given viralstrain within a particular subpopulation. These two factors may
be eliminated from consideration by using [6] to assess the relative behavioral risk posed to a particular individual by attending
a specific event of duration τ with N other participants. We thus
define a relative risk index,
IR =2
N τ Cq Qb2 pm
,
λa V

[8]

that may be evaluated using appropriate Cq and Qb values (listed
in SI Appendix, Table S2). One’s risk increases linearly with thenumber of people in a room and duration of the event. Relative risk decreases for large, well-ventilated rooms and increases
when the room’s occupants are exerting themselves or speaking loudly. While these results are intuitive, the approach taken
here provides a physical framework for understanding themBazant and Bush
A guideline to limit indoor airborne transmission of COVID-19

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quantitatively. It also provides a quantitative measure of the relative risk of certain environments, for example, a well-ventilated,sparsely occupied laboratory and a poorly ventilated, crowded,
noisy bar. Along similar lines, the weighted average of [8], provides a quantitative assessment of one’s risk of airborne infection
over an extended period. It thus allows for a quantitative assessment of what constitutes an exposure, a valuable notion in
defining the scope of contact tracing, testing, and quarantining.
Above all, our study makes clear the inadequacy of the SixFoot Rule in mitigating indoor airborne disease transmission,and offers a rational, physically informed alternative for managing life in the time of COVID-19. If implemented, our safety
guideline would impose a limit on the CET in indoor settings,
violation of which constitutes an exposure for all of the room’s
occupants. Finally, while our study has allowed for an estimate of
the infectiousness of COVID-19, it also indicates how new datacharacterizing indoor spreading events may lead to improved

estimates thereof and so to quantitative refinements of our safety
guideline.
The spreadsheet included in Dataset S1 provides a simple
means of evaluating the CET limit for any particular indoor setting. A convenient online app based on our safety guideline isalso available (102). The app and spreadsheet also enable theuse of data from CO2 sensors (47) to improve the accuracy of thesafety guideline (108). A glossary of terms arising in our study ispresented in SI Appendix, Table S3.

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Data Availability. All study data are included in the article andsupporting information.

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ENGINEERINGACKNOWLEDGMENTS. We thank William Ristenpart and Sima Asadi forsharing experimental data, and Lesley Bazant, Lydia Bourouiba, Daniel
Cogswell, Mark Hampden-Smith, Kyle Hofmann, David Keating, Lidia
Morawska, Nels Olson, Monona Rossol, and Renyi Zhang for importantreferences.

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