Supporting Information
Beyond Six Feet: A Guideline to Limit Indoor
Airborne Transmission of COVID-19
Martin Z. Bazanta,b,1 and John W. M. BushbaDepartment of Chemical Engineering; b Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139

This manuscript was compiled on January 20, 2021

1. Validity of the model assumptions
We proceed by providing rationale for the two key assumptionsof our model, and discussing their range of validity.
A. The well-mixed room. Two flow types typically contributeto vigorous mixing of indoor air: buoyancy-driven and forced
convection. Buoyancy-driven convection results from the action of gravity on differences in air density, typically due totemperature gradients. Forced-air heating (from below) or
cooling (from above) of a room typically lead to localized flowsthat are subject to hydrodynamic instabilities that prompt
mixing on a relatively larger scale. Air-conditioning units generate relatively cold, dense air that sinks as a turbulent plumeand entrains ambient air as it sinks to the floor (1). Heatingvents and radiators create buoyant counterparts, turbulent
thermal plumes that rise to the ceiling. In relatively coolrooms, temperature gradients associated with body heat leadto turbulent thermal plumes rising from individuals. Respiratory jets, plumes and puffs also typically have some buoyancy,
leading to updrafts from these turbulent respiratory flows (2).
When masks are worn, the horizontal momentum of respiratory
flows is greatly suppressed, but the exhaled air typically leads
to a rising turbulent buoyant plume (3). Additional large-scale
flows, either laminar or turbulent, are driven by horizontal
temperature gradients in the vicinity of closed windows. The
net effect of such buoyancy-driven flows is to promote mixingof the indoor air.
Room-scale forced convection may also be driven by themechanical air circulation system or the respiration and movement of its occupants. The hydrodynamic stability of an air
flow with a characteristic speed v and length scale L is prescribed by the Reynolds number, Re = vL/‹a , where ‹a is the
kinematic viscosity of air. For example, laminar flow aroundobstacles typically becomes unstable to vortex shedding for
Re > 100 and turbulence for Re > 2000. For an indoor air flow
forced by mechanical ventilation with a characteristic speedv = (Q + Qr )/A = (⁄a + ⁄r )H and length scale corresponding
to that of a room, L = H = 3m, the characteristic Reynolds
number Re ¥ 110 for ⁄a = 0.3/h and Re ¥ 2000 for ⁄a = 8/h,even in the absence of recirculation flow (⁄r = 0). Thus, forced
convection will typically lead to high-Reynolds-number flowsthat may be characterized by some combination of vortex
Bazantshedding and turbulence. Use of fans, air-filtration units oropen windows will further increase airflow and mixing. Humanmovement may also generate vigorous, high-Reynolds-number
flows. Human respiration without masks leads to turbulent
puff trains and jets (4), coughing and sneezing to turbulent,buoyant puff clouds (2).
In turbulent flows, the largest eddies are at the scale ofthe confining geometry and control the convective transport.
Turbulent mixing by forced convection at the scale of the roomis thus characterized by the eddy diffusivity, De ¥ 12 vH, as has
been verified for the transport of tracer gas in instrumented
homes (5). The characteristic mixing time, ·mix = H 2 /2De =≠1 ≠1
H/v = (⁄≠1, is thus prescribed by the time scalea + ⁄r )of the faster of the two processes, outdoor air exchange andinternal air recirculation. In either case, the room should bewell mixed by the time the bioaerosol concentration reaches its
steady state, since the concentration relaxation rate necessarily
exceeds the air exchange rate due the influence of droplet
settling, filtration and viral deactivation.
In summary, indoor air is typically well mixed by either
laminar or turbulent flows driven by some combination of
mechanical air circulation, buoyancy-driven flows and humanactivity (6). The success of turbulent deposition models (7–10) in rationalizing measured aerosol mass transfer to indoor
surfaces (7, 11, 12) provides further support for the validityof the well-mixed-room approximation. While variations fromwell-mixedness may arise, for example through the development of stratification (6), the approximation of the well-mixed
room has been widely applied in describing indoor settings,
specifically in the context of long-range airborne disease transmission (13–17).

B. Droplet transfer to surfaces . In the continuous stirredtank reactor (CSTR)(18), a canonical system in chemicalengineering, the relative rate of chemical reaction to convection
is prescribed by the dimensionless Damköller number. In
the context of airborne disease transmission, we consider the‘reaction’ corresponding to aerosol-borne virion removal froma well-mixed room, and so define Da = ⁄c /⁄a in terms of theconcentration relaxation rate ⁄c (r) and the air change rate
⁄a . To distinguish between the distinct contributions to this1

To whom correspondence should be addressed. E-mail: bazantmit.edu

1–11

reaction from bulk and surface removal processes, we write
Da = Dab + Das =(⁄v + ⁄f )V
hA
+
.
Q
Q

[S1]

Bulk disinfection of the air may arise through viral deactivation
at a rate ⁄v (r) or air filtration at a rate ⁄f (r) = pf (r)⁄r . We
denote by h the virion mass-transfer coefficient from the bulk
to the surfaces, which necessarily has the units of velocity.
In the field of aerosol science (7, 8, 19), it is customary
to refer to the dimensionless droplet mass-transfer coefficientas the ‘deposition efficiency’, ÷ = hA/Q, and to relate it tovarious particle transport resistances. Such resistances are
expressed in dimensionless form in terms Stokes numbers (12),each of which represents the ratio of two particle speeds. There
are three potentially relevant resistances to drop mass-transferfrom the bulk air to surfaces: gravitational settling, diffusionand surface deposition. We thus express the efficiency ofdroplet deposition from the bulk in terms of three Stokes
numbers:÷ ≠1 = (Stg + Std )≠1 + St≠1
[S2]
dep .
We note that the droplet deposition efficiency ÷ is equivalent
to the "particle loss-rate coefficient", —, customarily definedfor particulate aerosols in indoor air (7))
The gravitational Stokes number, Stg = vs /va , prescribes
the relative magnitudes of the Stokes settling speed, vs =2ﬂd gr2 /9µa and the ambient air flow, va = Q/A = ⁄a L. We
note that inertial corrections to Stokes law (20) only become
important for droplets with r > 100µm, and so are not relevant
for aerosols. For the largest respirable droplets (r = 5µm)with poor, natural ventilation (0.35 ACH) in a typical room
(H = 2.7m), settling dominates convection since Stg = 12 ∫ 1.
Conversely, for smaller droplets (r = 0.5µm) with hospitalgrade ventilation (25 ACH), settling can be neglected since
Stg = 0.0017
π 1. Drops with radius exceeding a criticalvalues, rc = 9µa ⁄a H/(2ﬂd g) corresponding to Stg = 1, tend
to settle to the surfaces before being removed by ventilation.
For a given droplet size r, we write Stg (r) = (r/rc )2 . The
droplets of interest in our model of airborne transmission arethose with Stg < 1.
For submicron particles suspended in a well-mixed ambient,diffusion may influence the transfer of the particle through thelower viscous boundary layer, either by Brownian motion or
eddy diffusion for the case of a turbulent boundary layer (7,12, 19). The importance of such particle diffusion is prescribedby the diffusional Stokes number (12), Std = D/(”va ), where
D is the particle diffusivity and ” is the viscous boundary layer
thickness near the depositing surface, which sets the diffusional
mass-transfer coefficient, hd = D/”. We may alternatively
write Std = Sh(Re, Sc)/Pe, where the Péclet number, Pe =va H/D, prescribes the relative magnitudes of convection anddiffusion, and the Sherwood number, Sh = hd H/D, is thedimensionless diffusion mass-transfer coefficient, averaged overthe surface.
We first consider Brownian droplet diffusion through a laminar boundary layer, as is typical of natural ventilation overhorizontal surfaces. For a given flow field, kinematic viscosity‹a = µ/ﬂa and Brownian diffusivity D = kB T /6ﬁµa rd (as follows from the Stokes-Einstein relation), the Sherwood number
depends on the Schmidt number Sc = ‹a /D and Reynolds number Re = Pe/Sc. Using the relation, Sh = 0.664 Re1/2 Sc1/3 ,relevant for a laminar boundary layer on a flat plate (21), we
2estimate, for the case of bioaerosols with radius (r = 0.5µm)
with poor ventilation (0.35 ACH), that Std = 7 ◊ 10≠6 π 1(Sc = 5 ◊ 105 , Re = 47, Sh = 180, Pe = 2.5 ◊ 107 ), whichindicates that particle diffusion can be safely neglected. Indeed, the removal of droplets by Brownian diffusion becomescomparable to that by gravitational settling, Std > Stg , only
for r < 13 nm at 0.35 ACH, a critical radius much smaller
than that of a single SARS-CoV-2 virion (rv = 60nm).
Next, we consider droplet diffusion through a turbulent
boundary layer, as might arise for high ventilation rates. In
their seminar work on aerosol mass transfer, Corner andPendleton (8) augmented the Brownian diffusivity, D, withthe turbulent eddy diffusivity, De = Ke y 2 , where Ke is a turbulence intensity parameter, proportional to the local mean
velocity gradient, and y is the distance from the surface. A
better fit to experimental data was achieved by expressing theeddy diffusion as De = Ke ” 2 (y/”)n (9, 10), where ” is theboundary layer thickness ” and n = 2.6 ≠ 2.8 is an empiricalscaling exponent (7). Using this approach, Lai & Nazaroff (7)developed a comprehensive, experimentally validated model ofaerosol mass transfer to surfaces and showed that, even for thehighest building ventilation rates, turbulent diffusion competes
with gravitational settling only for particles with r < 0.2µm.
This critical size is much smaller than that of stable respiratory
droplets (22–24), and only a few times larger than a single
virion. We thus conclude that gravitational settling dominates
diffusion, both Brownian and turbulent, in respiratory aerosol
removal from indoor air.
Finally, the role of droplet deposition is determined by
Stdep = vdep /va = Stg vdep /vs , where vdep is the net liquid
deposition velocity, specifically the surface adsorption flux
per bulk liquid concentration, as accounts for the probabilityof droplets sticking to the surface rather than bouncing off.
Experiments have shown that vdep /vs is approximately constant for water droplets with characteristic diameter of 1-10
µm in a variety of settings, including droplets settling fromclouds onto the forest canopy(25) and from humid air onto
acrylic pipes(26, 27)). For droplets of radius r = 5µm settlingat vs = 0.306 cm/s, the deposition velocity vdep ¥ 5 cm/s.
We thus conclude that deposition kinetics are fast comparedto droplet transport, since Stdep ¥ 20 Stg , and can be safely
neglected as a small mass-transfer resistance in series.
In summary, for the respiratory droplets of interest, thedominant contribution to droplet mass transfer to surfaces
for the relevant range of respiratory aerosols, r = 0.1 ≠ 10µm
comes from gravitational settling, h ¥ vs . We are thus justified
in writing
1 22r
÷ ¥ Stg =[S3]
rc
as assumed in the main text. This deduction allows us toconveniently express the droplet mass transfer from complex
indoor airflows in terms of the Stokes settling velocity. The
effects of boundary-layer or surface mass-transfer resistance are
negligible, and the timescale of settling of respiratory dropletsfrom a well-mixed ambient is prescribed by the Stokes settlingtime.

2. Exact solution for the transient guideline
The general safety guideline, Eq. [3], can be written as,
Rin (· ) = Ns sr È—a Í· , in terms of the time-averaged airborne
transmission rate, which can be broken into steady-state andBazanttransient terms:⁄ ·È—a Í(· )

=

1
·

—a

=

Q2b
V

—a (· )=0—a (t)dt = — a ≠

⁄ Œ
0

⁄ Œ
2

Qb
V·

0

—a (· )[S4]

nq (r)dr⁄c (r)[S5]"
nq (r) !
1 ≠ e≠⁄c (r)· dr⁄c (r)2[S6]

where nq (r) = nd (r)Vd (r)cv (r)ci (r) and ⁄c (r) = ⁄a (1 +(r/rc )2 ) + ⁄v (r) + ⁄f (r). The steady-state term — a can be
expressed as Eq. [4], where the mean sedimentation speedvs (r) and mean suspended droplet size, r, are defined byvs (r)=‹(r)2 v3r‹(r)rc

42

= Cq

3⁄ Œ
0

nq (r)dr
1 + (r/(‹(r)rc ))2

4≠1

≠1

[S7]

with v = Q/A and ‹(r) =1 + (⁄v (r) + ⁄f (r))/⁄a , which
gives most weight in the average to the suspended aerosol
droplets with r < rc .
For short exposures or poor ventilation, the transient correction can reduce the indoor reproductive number, resulting
in a more permissive guideline. In the case of monodisperse
droplets of size r = r̄, the transient term can be approximatedas, —a /— a = (1 ≠ e≠⁄c · )/(⁄c · ) ¥ 1/(1 + ⁄c · ). We may thus
recast the general safety guideline in the form,Ns sr —· < ‘(1 + (⁄c (r)· )≠1 )[S8]which reduces to the pseudo-steady criterion, Ns sr · — < ‘ forexposure times long relative to the concentration relaxation
time, ⁄c (r)· ∫ 1. In the limit of short exposures, ⁄c (r)· π 1,
we obtain a refined safety guideline, Ns sr (—· )(⁄c (r)· ) < ‘,which is less strict because it reflects the leeway associatedwith the time taken for the build-up of the airborne pathogen
following the arrival of an infected person. During this period,≠1/2
the safe exposure time scales as ·max ≥ Ns
, and the CET≠1bound diverges as Ns ·max ≥ ·in the · æ 0 limit. This
divergence simply reflects the fact that transmission will notoccur if people do not spend sufficient time together.

3. Deduction of the indoor safety guideline from epidemiological models
We here demonstrate that our indoor safety guideline, Eq. [3],may be deduced from standard epidemiological models.
While our theoretical model of the well-mixed room wasdeveloped to describe airborne transmission from a single individual to a fixed number of others in a well-mixed room,
it bears noting that this scenario is but one of a broader
class of transmission events. First, there are many situations
where air conditioning or forced ventilation mixes air between
rooms, in which case the compound room is effectively awell-mixed space. Second, there are incidents when several
individuals are initially infected. Third, the population may
be together for a period long with respect to the incubation
time (approximately 5.5 days (28)), in which case the number of infectious individuals necessarily increases with time.
Examples of this broader class of transmission events might
include cruise ships (29–33), meat and poultry processing facilities (34, 35) and prisons (36, 37). We proceed by detailingthe epidemiological models describing this more general classof events.
BazantFollowing Noakes et al.(38), we adopt the standard SEIR
model from epidemiology (39–41), an extension of the original
SIR model of Kermack and McKendrick(42) of susceptible(S), infected (I) and recovered (R) populations to include an
exposed population (E) that has acquired the pathogen andso become infected but not yet infectious:
dS
dE
dI
dR
= ≠—SI,= —SI ≠ –E,= –E ≠ “I,
= “I
dt
dt
dt
dt
[S9]
where – is the incubation rate, —(t) is the transmission rate
(setting sr = 1), and “ is the recovery rate of the infected.
To describe a spreading event in a closed environment, weconsider I0 infected people entering, at time t = 0, a room
full of Ns = S0 = N ≠ I0 susceptible individuals (setting
ps = 1), none of which were previously exposed, E0 = 0.
The reproductive number, R0 = S0 —/“, describes the initial
exponential rate of increase (R0 > 1) or decrease (R0 < 1)of the infected fraction of the population (39). In the casewhere the population is being tested and infected individuals
removed from the population via isolation, “ may alternatively
be considered as a proxy for testing frequency. Noting that the
combined incubation time and time to recovery for COVID-19
is thought to exceed 2 weeks (43), we here consider spreading
events of duration · that are short relative to the recovery
time, · π “ ≠1 , and so eliminate R from consideration.
The Wells-Riley model is effectively a reduction of the SIR
model based on the assumption of slow incubation over thetimescale of the event, · π –≠1 , in which case the numberof infected persons in the room remains fixed, I(t) = I0 . One
expects such to be the case for events of relatively short duration, such as the Skagit Choir incident. However, such an
approximation is not likely to be valid in the case of indoor
super-spreading events of longer duration, · > –≠1 . To adequately model such events, we consider the possibility thatthe number of infectors increases with time, which requires a
description of the nonlinear dynamics of disease transmission.
The resulting SEI model to be considered here is a system of three nonlinear ordinary differential equations. While
the system may be solved numerically, deduction of a safety
guideline requires analytical results. Our goal is thus to solve
this SEI model for the infection (or “secondary attack") rate,iS (· ) = (E(· ) + I(· ) ≠ I0 )/S0 . Since exact solutions may
be cumbersome (as for the SIR model (44)), we simplify the
analysis by considering two limits, where the incubation time–≠1 is short or long relative to the event timescale, · , andthe transmission timescale — ≠1 . While the incubation timeof COVID-19 is not precisely known, it is bounded above by
the time to develop symptoms, which spans 2-14 days with anaverage 5.5 days (28).
For slow incubation (– π —, · ≠1 ) (38), the number of
infected persons remains nearly constant, I(t) ¥ I0 , and theresulting SE model is easily solved to derive the secondary
attack rate (45),≠I0iS (· ) = 1 ≠ es·0—(t)dt,[S10]

In the opposite limit of fast incubation (– ∫ —, · ≠1 ), exposed
individuals are rapidly infected, so the exposed, non-infected
population need not be considered, E(t) = 0. Solving the3resulting SI model yields
iS (· ) =C

1 ≠ i≠1
0

3

N

e

s·
0

—(t)dt4≠1 D≠1

≠1

[S11]

,

where i0 = I0 /N is the initial percentage of infected individuals.
These simple solutions, (S10) and (S11), cover the two limitingcases of fast and slow incubation relative to transmission.
The safety criterion Rin < ‘, which limits the probabilityof the first transmission to lie below a small tolerance, isuniversal, in that it does not depend on the choice of model
to describe the subsequent spreading dynamics. In particular,
the safety criterion, I(· ) < 1 + ‘, with I0 = 1 corresponds to
Rin
=
N ≠1

⁄ ·
0

—(t)dt <;!

"≠ ln 1 ≠ N ‘≠1#$1ln (NN≠1)(1+‘)
N≠1≠‘(slow)(fast)[S12]
according to Eqs. (S10) and (S11) for slow and fast incubation,
respectively. In the small-tolerance limit, ‘ π 1, these bounds
both reduce to, Rin < ‘, and are thus independent of the
incubation rate. More generally, the linear response of any
Markovian, mean-field theory (with neither memory nor manybody correlations) to any perturbation, such as the arrivalof a new infector, must begin at the mean transmission rate.
Indeed, the same universal bound may be simply derived
s ·fromthe infinitesimal cumulative probability of transmission 0 —(t)from one infector to a set of N ≠ 1 independent susceptibles.

4. Inference of infection quanta from disease spreading data
It is an important point that the concentration of “infection
quanta" per exhaled air volume, Cq , as may be inferred fromspreading data, is necessarily model dependent. By s
definition,·
the time integral of the transmission rate, p(· ) = 0 —(t)dt,
is equal to the expected number of quanta transferred from
one infector to one susceptible. In models of indoor airborne
transmission (16, 38, 45), infection quanta are further related
to the evolving pathogen-laden droplet concentration by Eq.
[4]. While the mathematical definition of infection quantais unambiguous in terms of the instantaneous transmissionrate, —(t), between two isolated individuals, the inference ofquanta concentrations from field data for the spreading rateis inevitably model dependent because it involves interactions
among an evolving population. Specifically, the long-time
growth in the number of infections is influenced by nonlinearities in the transmission dynamics, as governed by the chosen
epidemiological model, for example, the SEIR model, Eq. [S1].
In the original Wells-Riley model (38, 45–48), one infection
quantum is defined as the amount of pathogen required to
infect an average of 1 ≠ 1/e ¥ 63.2% of susceptible people
in an enclosed space. This inference is based on fitting fielddata to Eq. [S2], which relates one net quantum transferred,
I0 q(t) = 1, to the secondary infection rate, iS (t) = 1 ≠ (1/e).
This approach is valid for times short relative to the incubation
time and has been successfully applied to extract quantaemission rates for various viral diseases (49–52), including
SARS-CoV-2 (16, 53). However, the Wells-Riley model cannot
be reliably used to infer infection quanta from super-spreading
events evolving over time scales that exceed the incubationtime.
4

For such situations, here we take an approach appropriate for the case of fast incubation that necessarily implies a
different relation between released infection quanta and theobserved secondary infection rate. If we adopt the SI modelwith fast incubation, then the long-term behavior of Eq. [S3]indicates that one quantum (I0 q(t) = 1) will infect a fraction
iS = [1 + (N/I0 )/(eN/I0 ≠ 1)]≠1 of N people in an enclosed
space, I0 of which were initially infected, as a result of transmission amplification by the growing number of infectors. In
this model, a single quantum from one person infects anotherwith probability iS = [1 + 2/(e2 ≠ 1)]≠1 ¥ 76.2% for N = 2,
but manages to infect everyone in a large group, iS æ 1 asN æ Œ. This dependence of infection quanta on epidemiological model parameters, such as the incubation rate, reflects
the fact that the fitted “infection quantum" is a measure ofcontagiousness at the scale of a group that is not necessarily
proportional to the microscopic pathogen concentration. Notably, infection quanta are well defined in the limit of shorttimes and slow transmission, where all models reduce to theWells-Riley model with a constant number of infectors. Fromthe modeling perspective, the notion of infection quanta is
thus unambiguous only in this limit. Finally, we recall that,regardless of the model used to infer it, the actual value of
infection quanta may still vary considerably between spreading events, owing to its dependence on activity level of the
population and other physiological factors (16, 54).

5. Fitting to super-spreading events
Table S1 shows the data used to infer the concentration of
infection quanta Cq exhaled by an infected individual for fourwell-known indoor super-spreading events of COVID-19, forwhich physical parameters can be estimated with reasonable accuracy from published data: the Skagit Valley Chorale (43, 53),
the Diamond Princess cruise ship (30, 31), the Ningbo tourbus (55, 56), and the initial Wuhan outbreak (64), interpreted
as mainly indoor spreading at home (60). The best characterized case of the Skagit Valley Chorale is used in the main text
to calibrate the safety guideline. Here we consider the otherthree spreading events, for which only relatively rough estimates of certain model parameters can be made. Despite theresulting uncertainty, these events provide additional support
for the inferences made in the main text.
In all situations, we assume an effective particle radius r =2.0µm at the upper limit of the infectious range of suspended
droplets (65), corresponding to an effective settling speed ofvs (r) = 1.78 m/h. Note that the effective radius would besomewhat larger, r = 3.0µm, if we were to neglect the size
dependence of infectivity and set nq =constant, and use ourprediction of the steady-state airborne droplet distribution
Cs (r)/cv for singing in the choir room (Fig. 1). We also assume
no use of face masks, pm = 1 and an airborne virus deactivation
rate, ⁄v = 0.3/h (53) that lies between the existing estimatesof zero (66) and 0.63/h (67)
A. The Diamond Princess. The data reported for the Diamond

Princess is particularly useful in that it captures the time
evolution of the infected persons among a fixed population
over an extended period, corresponding to the 12-day quarantine (30, 31), after which passengers and crew began todisembark. The resulting data reported for I(t) is best described in terms of the fast-incubation limit (Figure S1). FitBazant

Skagit Church Choir
Ningbo Tour Bus
Diamond Princess
Wuhan City Outbreak

N

· (h)

I0

I(· )

⁄a (1/h)A (m2 )H (m)

Qb (m3 /h)

Cq (q/m3 )⁄q (q/h)

61
68
3711
3.03

2.5
1.7
288
132

1
1
20
1

53
21
354
1.63

0.65
1.25
8
0.34

180
25
139,000
90

4.5
1.8
2.1
2.4

1.0
0.5
0.5
0.5

870
90
30
29

870
45
15
14

Table S1. Data from four COVID-19 spreading events used to infer the concentration, Cq , or emission rate, ⁄q , of exhaled infection quanta,on the basis of the assumption of indoor airborne transmission. 1. The Skagit Valley Choir event (43). We use existing estimates of relevant
physical parameters (53) and the Wells-Riley model (16, 45–47, 53), Eq. [S10], appropriate for slow incubation. 2. A tour bus transported 68
people (including the driver) on a 100 minute round-trip journey to a Buddhist ceremony in Ningbo, China (55, 56). One index case infected
23 fellow passengers, three of which are assumed to have been infected at the ceremony, where the infection rate was 7/172 = 4% forother attendees. The interior bus dimensions are estimated from a Dongfeng 67-seat luxury tour bus made in Hubei, China, which has outerdimensions of 10.49m◊2.5m◊3.2m and matches the floor plan of the Ningbo tour bus (56). The air change rate, ⁄a = 1.25 ACH is estimatedfrom previous studies of air quality in transit buses with closed windows (57). 3. The Diamond Princess cruise ship during its 12-day port
quarantine in Yokohama, Japan (30, 31). We infer I0 = 20 and N — = 0.25/day by fitting the confirmed case history to our fast-incubation
solution, Eq. [S11], as shown in Fig. S1. We estimate relevant volume and area from floor plans of the Diamond Princess (33, 58), where
passengers and crew mainly occupy 14 floors of living space of beam width 38 m, an average length equal to 90% of the ship’s length 290 m,
and mean ceiling height 2.1 m. We also assume a standard cruise-ship ventilation rate (8 ACH) for partially recirculated air conditioning (59).
4. Initial outbreak in Wuhan City, Hubei Province, China. We assume that the population-level spreading is dominated by indoor aerosol
transmission with slow incubation (Eq. [S10]) in single-family apartments (60) with a mean family size of 3.03 (61), mean apartment area of
315 SF/person (62) and mean ventilation rate of 0.34 ACH (63). We estimate a mean exposure time of · = 5.5 days until symptoms (andpatient isolation), and use the average R0 = 3.3 estimated for the population during the initial outbreak (64) in place of Rin (· ).

B. Ningbo tour bus. We proceed by considering another super-

spreading event, involving a Buddhist blessing ceremony heldat the Tiantong Temple in Ningbo, Zhejiang Province, China
on January 19, 2020 (55, 56). The confirmed index case, Ms. S,
was thought to have contracted the virus two days earlier from
dinner guests who had traveled to Wuhan during the initialoutbreak. Ms. S traveled to the ceremony on a tour bus (Bus2) for 50 minutes with 68 people, including the driver, andreturned on the same bus with the same seating arrangement.
Since it was winter, the windows were likely closed. No outside
BazantCOVID-19 transmission aboard the
Diamond Princess cruise ship

1200

1000

Number of Infections

ting to the available data allows us to infer that the initial
number of passengers infected on February 3, 2020 was approximately I0 = 20, and that the concentration of infection quantacharacterizing this particular spreading event was Cq = 30
quanta/m3 . Furthermore, it suggests that the incubation time
is significantly less than 12 days, as is consistent with current
estimates for the average time between exposure and the onsetof symptoms being 2-5 days (54).
Our fitting of I0 and Cq for the Diamond Princess is basedon the hypothesis of a “well-mixed ship". While such an approach is not traditional, and would be contested by those who
do not believe that airborne transmission was prevalent on the
Diamond Princess (68), it is consistent with a growing bodyof evidence (69). Analysis of the ship’s floor plans and ventilation system has indicated recirculation of heated interior air
throughout the entirety of the ship during cold weather (-5¶C), without adequate filtration for virus-containing aerosol
droplets (32, 33). Moreover, a detailed statistical analysisof the SARS-CoV-2 transmission history between passenger
cabins revealed no significant correlation between new cases
and the sharing of rooms with previously confirmed cases.
Airborne transmission was further suggested by several examples of new cases emerging in single-occupancy cabins despite
no known contact with other cases (32). Like many otherindoor COVID-19 spreading events (53, 56, 70) in which position in an enclosed space was uncorrelated with likelihood oftransmission (67), the cruise line outbreaks present evidence
that strongly supports the notion of airborne transmission of
SARS-CoV-2 through well-mixed indoor spaces (60, 71, 72).

Fast incubation theory
I(0)=20, Nb = 0.25/day
800

600

Port quarantine
Yokohama, JapanN = 3711

Positive test cases
for SARS-CoV-2

400

200

1000+ passengers depart

0
0

5

10

15

20

Time (days since February 3, 2020)
Fig. S1. Validation and calibration of the fast-incubation model for the super-spreading
event on board the Diamond Princess cruise ship, during its twelve day quarantine
at Yokohama, Japan in February 2020 (30). Fitting the number of SARS-CoV-2
positive cases versus time (blue data points) during the quarantine period, to thefast-incubation solution of the SEI model, Eq. [S11], (red curve) yields estimates forthe initial number infected (I0 = 20) and the transmission rate (N — = 0.25/day),from which we infer the concentration of infection quanta from breathing (Cq = 30
quanta/m3 ). The relevant physical parameters are listed in Table S1.

5

Activity
breathing at rest
breathing heavily
whispering
speaking
singing softly
singing

Experiment (74)nose in, nose out (b-nn)
nose in, mouth out (b-nm)
whispered counting (c-w-p)
voiced counting (c-v-p)
whispered “aahs" (aah-w-p)
voiced “aahs" (aah-v-p)

Qb (m3 /h)„1 (10≠16 )Cq (q/m3 )⁄q (q/h)

0.5
0.5
0.75
0.75
1.0
1.0

0.35
1.3
1.5
2.9
4.1
39

8.8
33
37
72
103
970 (53)

4.2
16
28
54
103
970

Table S2. Activity dependence of airborne transmission of COVID-19. Expiratory droplet size distributions for different activities are takenfrom the experiments reported by Morawska et al. (74), and reasonable estimates are made for the exhaled air volume per time Qb from3
3
3
breathings rc (0.5 m /h), speaking (0.75 m /h) and singing (1.0 m /h). Our model is then used to predict the steady-state aerosol volume fraction,„1 =„s (r) dr, that results from exhalation of a single infectious individual in a setting corresponding to the Skagit Valley Choir room0
(Table S1), by integrating the distributions shown in Fig. 1 up to the critical radius rc = 2.5µm. The concentration of COVID-19 infection
quanta in the breath of an infected individual is assumed to be Cq = 970 q/m3 for the singing case, as estimated for the Skagit Valley Chorale
incident by Miller et al. (53), and values of Cq for other activities are calculated by rescaling with the appropriate ratio of aerosol volume
fractions, „1 . The quanta emission rate for each activity is then given by ⁄q = Qb Cq .

air was supplied by mechanical ventilation, although the air
was recirculated continuously. No masks were worn by thepassengers, as there had not previously been any cases of
COVID-19 in Zhejiang Province. Of the 68 passengers, 23 newcases of COVID-19 were confirmed, with their locations being
evenly distributed across the bus, largely uncorrelated withproximity to Ms. S (56). Most had no close contact with theindex case. However, some spatial patterns of transmission
could be discerned on the bus, such as fewer cases among thewindow seats, possibly due to recirculation air flows. The
epidemiological evidence for indoor airborne transmission is
overwhelming (56).
Of the 172 others that attended the ceremony, only 7 individuals (4.1%) became infected at the 150-minute worship
event, which was mostly held outside. Assuming that the bus
passengers were infected at the temple at the same rate as theothers, we may estimate that 3 of 23 transmissions occurred
there, so only 20 on the round-trip bus ride. The publishedfloorplan of the bus (56) resembles that of a Dongfeng 67seat luxury tour bus made in Hubei Province, China, whose
specifications we use to estimate interior dimensions. The
most uncertain model parameter is the air exchange rate, ⁄a ,which we presume to be associated with natural ventilation.
We further assume that windows were closed and that air
filtration was limited. Although ⁄a = 1.8 ≠ 3.7/h has beenmeasured in passenger cars with closed windows and recirculating fans (73), transit buses tend to have lower values, suchas ⁄a = 1.25/h (57), a value that we adopt here. We also
assume a typical seated breathing flow rate, Qb = 0.5 m3 /h.
Using these approximations for the model parameters, weapply the slow-incubation model (S10) to infer Cq = 90 q/m3
and ⁄q = 45 q/h. This value of Cq falls in the range ofintermediate speaking obtained from other inferences in themain text, which would be consistent with passengers speaking
amongst themselves on the voyage. Given the uncertainty inthe air exchange rate, the inferred value of Cq could also be
consistent with normal breathing with only occasional speech.
Finally, we note that the average age of the bus passengers
was 59 (56), which suggests a relatively vulnerable population,
for which one expects to infer a relatively high Cq .
C. Wuhan outbreak. Table S1 also shows how the theory

can be used to interpret the population-level reproductive
number in terms of indoor airborne transmission, assumingthat the primary disease transmission arises in family apartments, as shown in a recent analysis of COVID-19 spreading in
6

China (60). We consider the initial outbreak in Wuhan, Hubei
Province and assume physical parameters appropriate for typical apartments and families. We then use the average population reproductive number (64) to assert Rin (· ) = ÈR0 Í = 3.3,
where · = 5.5 days is the mean time for an newly infectedfamily member to show symptoms and be isolated or removed.
As shown in the main text (Fig.2), the inferred valuesof Cq = 29 quanta/m3 for family apartments in the Wuhan
outbreak and Cq = 30 quanta/m3 for the Diamond Princess
cruise ship are consistent with the light respiratory activities
expected to be most prevalent in those settings, such as sleeping and quiet speech. Both inferred Cq values are greatly
exceeded by the inference of Cq = 870 (made here, usingr̄ = 2.0µm and ⁄a = 0.65/h; see Table S1) or 970 quanta/m3(made by Miller et al (53), averaging over simulations withvariable ⁄a = 0.3 ≠ 1.0/h and ⁄s = 0.3 ≠ 1.5/h) for singing, but
in a manner consistent with the increased pathogen output
associated with more vigorous speaking or singing (23, 75).
These inferences build confidence in our estimates of exhaledquanta concentrations, Cq , for various respiratory activities,
as shown in Figure 2 of the main text.

6. The dependence of airborne disease transmission
on respiratory activity
It is well established that aerosol droplet production varies
strongly with the form of expiratory activity (74, 76). For example, vocalization greatly increases aerosol emission relative
to quiet breathing, roughly in proportion to the amplitude of
the sound produced (23, 77). In Table S2, the aerosol volume
fraction produced by different activities, obtained by integrating the distributions in Fig. 1 up to rc = 2.5µm, are used torescale the inferred quanta concentration for singing (53) inthe Skagit Valley Choir room into those appropriate for otheractivities. The resulting values are consistent with the valuesof Cq inferred for the Diamond Princess quarantine and theinitial Wuhan City outbreak (see Table S1 and Fig. 2).

7. Application of the Safety Guideline
In order to illustrate how to implement our safety guideline,
we provide a spreadsheet as Supplementary Data, which has
also been implemented for general use as an online app, translated into many languages (78). We note that, following
the submission of our manuscript, it has come to our attention that the groups of José-Luiz Jimenez (79) and Lidia
BazantMorawska (80) have produced spreadsheets with extensive
documentation (17, 81) similar in spirit to ours, designed toprovide quantitative risk assessments for safe management of
indoor spaces during the COVID-19 pandemic. Since theiranalyses and others (82, 83) are based on the same physicalpicture of airborne transmission in a well-mixed space, theirpredictions are consistent with ours. However, their recommendations rely on calculations of pathogen concentration
and risk in different settings, while ours are based entirely on
our Safety Guideline, as succinctly expressed by the bound on
cumulative exposure time in Eq. [5].
Our spreadsheet and online app enable the calculation of
the suggested maximum cumulative exposure times for specificindoor spaces. In this section, we offer guidance on how to
use the spreadsheet for both safety assessment and contact
tracing, specifically, how to select suitable parameters andproperly interpret the results. The input parameters, coloredin pink in the spreadsheet and in italics below, are divided
into the following four categories.
A. Physical Parameters.. The geometry of the indoor space is

specified by its floor area, A, and mean ceiling height, H, fromwhich the volume V = AH is calculated using appropriate
unit conversions. The ventilation outflow rate, Q = V ⁄a , iscalculated from the air exchange rate, ⁄a , typically expressedin terms of air changes per hour (ACH). This critical input parameter is governed by national or local standards for different
types of indoor spaces, such as the ASHRAE standards (62.1)
in the United States (84). As noted in the main text, naturalventilation may be approximated as ⁄a = 0.34/h, which has
been measured in bedrooms with closed windows (63) and isconsidered to be the minimum standard (84), although thisvalue will vary with both location and quality of construction.
For residences, classrooms, businesses, and public spaces, ⁄ausually falls in the range 4≠8/h. Crowded spaces, such as bars,
nightclubs and restaurants, typically require more vigorous
ventilation, ⁄a = 15 ≠ 20/h. Minimum ventilation standards
for American hospitals have increased from 12 to 18 ACH (85).
Most chemical and biological laboratories have ⁄a in the rangeof 6 ≠ 12/h, but those handling toxic or infectious materials
may have ⁄a as high as 20 ≠ 30/h. Revised ASHRAE standards, intended to mitigate the spread of airborne infectious
diseases, recommend a minimum of ⁄a = 6/h for all indoor
spaces (86).
Forced air filtration and droplet settling in ducts may
also be included through the effective air filtration rate,⁄f (r) = pf (r)⁄r , where ⁄r = Qr /V is the recirculation air
exchange rate passing through the filter and pf (r) is droplet filtration probability, taken to be a constant over the aerosol size
range. The United States Environmental Protection Agency
defines high-efficiency particulate air (HEPA) filtration (87)as removing pf = 99.97% of aerosol particles, while ordinary air filters are assigned Minimum Efficiency Reporting
Value (MERV) ratings corresponding to pf > 20%–90% in
specified aerosol size ranges (88). For ventilation systems
with indoor air recirculation, the primary outdoor air fraction,
Zp = Q/(Q + Qr ), is usually specified by indoor air quality(IAQ) standards. For example, Zd = 20% for classrooms in
the United States (89)), where Q + Qr is the total flow rate
and v = (Q + Qr )/A is the mean air velocity. Alternatively, airfiltration may be accomplished by indoor free-standing units
with a specified recirculation flow rate of Qr .
Bazant

The relative humidity RH of the indoor air is another physical input parameter, which affects both respiratory droplet
evaporation and viral deactivation (90, 91).
B. Physiological Parameters. The first physiological parame-

ter is the volumetric breathing flow rate, Qb , which is approximately 0.5 m3 /h for resting and light activity. Average values
for healthy males and females have been reported as 0.49, 0.54,
1.38, 2.35, and 3.30 m3 /h for resting, standing, light exercise,moderate exercise and heavy exercise, respectively (92), andused in simulations of airborne transmission of COVID-19 (16).
The second physiological parameter is the mean respiratory
aerosol droplet size r̄ for the suspended infectious droplets
responsible for airborne disease transmission. The precise
definition of r̄ is given in Eq. (S7), in terms of the distributionof droplet sizes for different types of respiration (23, 74, 77),
the size-dependent infectivity of aerosol droplets (65), and thesettling and ventilation rates. As illustrated in the main text,
a typical value for r̄ is 2-3 µm. We note that these values
are roughly consistent with the standard definition of aerosol
droplets, as those having r < 5µm (16). The effect of relative
humidity on the size of stable droplets
after evaporation (93)can be estimated by rescaling r̄ by 3 0.4/(1 ≠ RH), since
the droplet distributions used to calibrate the guideline were
measured at RH = 60% (74).
C. Disease Parameters. The most important disease parame-

ter is the infectiousness of exhaled air, Cq , the infection quanta
per volume. Using all of the limited information currently
available, we estimate Cq = 30 q/m3 for normal breathing
and light activity and provide our best estimates of Cq for
different respiratory activities in Fig. 2. Our analysis indicatesthat Cq can be an order of magnitude larger for singing orother vigorous respiratory activities, or an order of magnitude
smaller for sleeping and light nose breathing.
The second disease parameter, sr , is the relative transmissibility (or susceptibility) of the virus. This transmissibility
necessarily depends not only on the susceptibility of the population, which is known to be age-dependent (54, 94–98), buton the viral strain (99, 100). For example, a study of transmission in quarantined households during lockdowns in China
reported sr = 0.23 for children (aged 0–14) and sr = 0.68 for
adults (aged 15-64) relative to sr = 1 for the elderly (over65 years old) (101). Initial estimates of R0 from the United
Kingdom for the new variant of concern (VOC 202012/01,
lineage B.1.1.7) suggest that sr is increased by 60% relative tothe original strain of SARS-CoV-2 and show signs of elevatedrisk of infection among children (99, 100).
The third disease parameter is the viral deactivation rate,⁄v , at which the aerosol-bound virus loses infectiousness, which
for SARS-CoV-2 has been estimated to lie in the range ofzero (66) to 0.63/h (67). Taking into account results forother aerosolized viruses (90, 91, 102), the viral deactivation
rate is approximated as linear in relative humidity, ⁄v =⁄v,50 RH, where the value ⁄v,50 is specified. The effective viraldeactivation rate may also be enhanced by ultraviolet radiation(UV-C) (103) or airborne dispersal of chemical disinfectants(e.g. H2 O2 , O3 ) (104).

D. Precautionary Parameters. The first precautionary param-

eter is the mask penetration factor, pm , defined as the fraction7

of infectious aerosol droplets that pass through the mask during exhalation or inhalation, averaging over any dependencies
on respiratory activity (105) or direction of airflow (106). Alternatively, the filtration efficiency is defined as 1 ≠ pm .
Many studies are available to help assign this value for different types of face coverings, ranging from cloth coverings
to surgical masks (107–110). Although filtration efficiency
depends on drop size, it is typically nearly constant in theaerosol range, up to several microns in diameter (107, 108).
Typical values for disposable medical masks are in the range
pm = 1≠5% and vary with fit and applied pressure (108, 109),
while for simple cloth face masks, values of pm = 10 ≠ 40%
are typical for hybrid, multi-layer fabrics and pm = 40 ≠ 80%
for single-layer or coarse fabrics (110).
The second precautionary parameter is the disease transmission tolerance, ‘. We note that ‘ = 1 corresponds to thebaseline of one expected transmission during the occupancy period. The choice of ‘ should take into account the vulnerabilityof the population, which for COVID-19 is a strong function ofage and pre-existing medical conditions (54, 95, 111). Relative
to the median age of 69 in the Skagit Valley Chorale spreading
incident used to calibrate our model, the relative rate of hospitalization with COVID-19 (95) can be calculated as 2.5% (ages0-4), 0.8% (ages 5-17), 20% (ages 18-49), 61% (ages 50-64),
130% (ages 75-84), and 145% (ages > 85). For the elderly,
especially those with preexisting conditions or co-morbidity,‘ π 1 should be chosen. For the young and healthy (in regions
where hospitals are not overwhelmed and vulnerable groups
are protected), larger values of ‘ could be considered (112). As
noted in the main text, choosing a sufficiently small ‘ will also
serve to mitigate against prolonged exposure to respiratory
jets, whose contribution to pathogen transport may dominate
in the absence of face-mask use (3).

of a room visited by the index case for a time · should beconsidered as contacts for the purpose of tracing the infection
network. If the guideline is violated, then all occupants of theroom must be considered contacts, regardless of their distancefrom the index case. Compared to the current CDC definitionof contact (114) – spending more than 15 minutes less than 6feet apart from an infected person – this definition, based on
our consideration of airborne transmission, may thus identify
significantly more contacts to be traced and quarantined.
G. Disclaimer. Our Indoor Safety Guideline calculator is an

evolving tool intended to familiarize the interested user withthe factors influencing the risk of indoor airborne transmissionof COVID-19, and to assist in the quantitative assessmentof risk in various settings. We note that uncertainty in andintrinsic variability of model parameters may lead to errors aslarge as an order of magnitude, which may be compensated forby choosing a sufficiently small risk tolerance. Our guideline
does not take into account short-range transmission throughrespiratory jets, which may substantially elevate risk whenface masks are not being worn, in a manner discussed in themain text. Use of the Indoor Safety Guideline is the sole
responsibility of the user. It is being made available without
guarantee or warranty of any kind. The authors do not accept
any liability from its use.

8. Online course
The scientific principles underlying this work are also taught ina free, self-paced massive, open online course (MOOC) (115).

9. Glossary of Mathematical Symbols
See Table S3.

E. Results. The spreadsheet first computes properties of theinfectious aerosol per infected individual in the room, which
are primarily of technical interest: the effective droplet settling speed vs (r̄), the concentration relaxation rate ⁄c (r̄), thedilution factor, fd , and the infectiousness of the ambient air,fd Cq , in steady state per infected person in the room.
The spreadsheet computes the safety guideline in two ways
with the key results highlighted in green. First, the occupancy
limit Nmax can be calculated for a given exposure time · , asis set by the typical residence time of people in the indoor
space. The corresponding minimum outdoor airflow per person,
Q/Nmax , may be compared with local standards, such as 3.8
L/s/person for retail spaces and classrooms and 10 L/s/personfor gyms and sports facilities in Europe (113), or the ASHRAE
Standards in the United States, typically 5-20 cfm/person
depending on the type of space (84). Second, the time limit·max is calculated for a given occupancy N . These bounds
are plotted and may be compared to the Six-Foot Rule and
15-Minute Rule, both of which invariably violate our guideline.
For the bounds on both Nmax and ·max , two results arereported: the transient bound, which accounts for the buildup
of infectious aerosols in the air after the entrance of an infected
person, and the more conservative steady-state bound, whichis relevant after the relaxation time ⁄c · ∫ 1.
F. Contact Tracing. The spreadsheet can also be used as a toolfor contract tracing. With a conservative tolerance, such as‘ = 0.01, the guideline defines whether or not the N occupants
8

BazantSymbolEngineering Parameters

Meaning

Characteristic values

N
·
V
A
H
Q
⁄a
Qr
⁄r
Zp
pf
⁄f (r)

Number of persons, room occupancy
Time since an infected person entered the room
Room volume
Floor surface area
Mean ceiling height, V /A
Ventilation outflow rate
Ventilation (outdoor air exchange) rate, Q/V
Recirculation flow rate
Recirculation air exchange rate, Qr /VPrimary outdoor air fraction, Q/(Q + Qr )Probability of droplet filtration via recirculation
Filtration removal rate, pf ⁄r

1 – 1000
0-1000 h
10–104 m3
5–5,000 m2
2–6 m
1–105 m3 /h
0.1–30 h≠1
1–105 m3 /h
0.1–30 h≠1
0.1-1.0
0–1.0
0–30 h≠1

Respiratory drop radius
Critical airborne drop radiusDrop volume, ¥ 4
ﬁr3
3
Drop number density per radius
Air density
Drop mass density
Drop relative density, ﬂ = ﬂd ≠ ﬂa
Air viscosity
Gravitational acceleration
Drop settling speed
Drop settling rate, vs (r)/H
Drop initial solute volume fraction
Relative humidity
Equilibrium drop radius after evaporation
Turbulent entrainment coefficient of respiratory jet
Momentum flux of a respiratory jet
Average area of the mouth opening
Turbulent pathogen concentration in a jet relative to the mouth
Distance from infected person
Probability of being in the respiratory jet of an infected person

0.1–100 µm
0.2–20µm
10≠5 –106 µm3
0.01–1.0 drops/(cm3 · µm)
1.2 kg/m3
1000–1500 kg/m3
1000–1500 kg/m3
18 µPa·s
9.8 m/s2
10≠5 –102 mm/s
10≠5 –102 h≠1
0.01–10%
10–100%10≠2 -10µm
0.1–0.15
103 - 104 cm4 /s2
1-10 cm2
0–1
0.1-10 m
0–1

Number of susceptible, exposed, infected and recovered persons
Incubation, transmission and removal rates
Mean airborne transmission rate per infected/susceptible pair
Steady-state and time-averaged mean transmission rate
Mean breathing flow rate
Pathogen (virion) deactivation rate
Pathogen concentration relaxation rate
Effective infectious drop radius
Pathogen dilution factor, Qb /(⁄c (r̄)V )Pathogen production rate per air volume per drop radius
Mean pathogen concentration per air volume per drop radius
Steady-state airborne pathogen concentration per drop radius, P (r)/(⁄c (r)V )Steady-state infectious aerosol volume fraction distribution, Css(r)/cv (r)rcSteady-state aerosol volume fraction (from one infected person),„s (r)dr10≠6 –10 quanta/h≠1
10≠6 –10 quanta/h≠1
0.5–3.0 m3 /h
0.01–10 h≠1
0.1-100 h≠1
0.3–5µm
10≠5 –10≠2
10≠6 –109 virions/(h·µm)
10≠8 –104 virions/(m3 · µm)
10≠8 –104 virions/(m3 · µm)
10≠19 –10≠13 /µm
10≠19 –10≠12

Physical Parametersrrc
Vdnd (r)
ﬂa
ﬂd
ﬂd
µa
g
vs (r)⁄s (r)
„s
RH
req
–t
M
Am
Cj (x)/C0x
pj
Epidemiological Parameters
S(t), E(t), I(t), R(t)–, — , “—a (t)—¯a , È—a ÍQb⁄v⁄c (r)r̄
fd
P (r)C(r, t)Cs (r)„s (r)„1
cv (r)ci (r)
cq (r)Cq⁄q
pm (r)
pi
ps
sr
Ns
R0
Rin
‘
Nmax

0

Pathogen (virion) concentration per drop volume
Pathogen infectivity, 1/(infectious dose)
Concentration of infection quanta per drop volume, ci cv
Infectiousness of breath, exhaled quanta concentration
Quanta emission rate, Qb Cq
Probability of drop transmission through mask, 1≠(filtration efficiency)
Probability a person is infected, prevalence of infection
Probability a person is susceptible (not exposed, immune, or vaccinated)
Relative transmissibility (or susceptibility)
Expected number of susceptible persons Æ N
Reproductive number, S(0)—/“Indoor reproductive number, Ns sr È—a Í·
Risk tolerance, bound on expected number of transmissions
Maximum safe room occupancy

104 –1011 RNA copies/mL
0.001–1.0
10–1011 quanta/mL
1–1000 quanta/m3
1–1000 quanta/h
0.01–0.1
0–1
0–1
0.1–100–10000.01–100.001–100
0.005–0.5
1–1000

Table S3. Terms arising in our theoretical formulation, their units and characteristics values.

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