Flow (2021), 1 E10
doi:10.1017/ﬂo.2021.10

RESEARCH ARTICLE

Monitoring carbon dioxide to quantify the risk of indoor
airborne transmission of COVID-19
Martin Z. Bazant1, 2, * , Ousmane Kodio2
Zongyu Gu1 and John W.M. Bush2, Alexander E. Cohen1, Kasim Khan3 ,1 Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3 Independent researcher, USA*Corresponding author. E-mail: bazant@mit.edu

Received: 4 April 2021; Revised: 1 June 2021; Accepted: 12 July 2021
Keywords: Respiratory ﬂows; Ventilation; Airborne disease transmission; COVID-19; Carbon dioxide monitoring; Safety
guideline

Abstract
A new guideline for mitigating indoor airborne transmission of COVID-19 prescribes a limit on the time spent in ashared space with an infected individual (Bazant & Bush, Proceedings of the National Academy of Sciences of theUnited States of America, vol. 118, issue 17, 2021, e2018995118). Here, we rephrase this safety guideline in terms ofoccupancy time and mean exhaled carbon dioxide (CO2 ) concentration in an indoor space, thereby enabling the useof CO2 monitors in the risk assessment of airborne transmission of respiratory diseases. While CO2 concentration isrelated to airborne pathogen concentration (Rudnick & Milton, Indoor Air, vol. 13, issue 3, 2003, pp. 237–245), theguideline developed here accounts for the diﬀerent physical processes aﬀecting their evolution, such as enhanced
pathogen production from vocal activity and pathogen removal via face-mask use, ﬁltration, sedimentation anddeactivation. Critically, transmission risk depends on the total infectious dose, so necessarily depends on both thepathogen concentration and exposure time. The transmission risk is also modulated by the fractions of susceptible,
infected and immune people within a population, which evolve as the pandemic runs its course. A mathematicalmodel is developed that enables a prediction of airborne transmission risk from real-time CO2 measurements.
Illustrative examples of implementing our guideline are presented using data from CO2 monitoring in university
classrooms and oﬃce spaces.

Impact Statement
There is mounting scientiﬁc evidence that COVID-19 is primarily transmitted through indoor airborne
transmission, as arises when a susceptible person inhales virus-laden aerosol droplets exhaled by an infectious
person. A safety guideline to limit indoor airborne transmission (Bazant & Bush, 2021) has recently been
derived that complements the public health guidelines on surface cleaning and social distancing. We here
recast this safety guideline in terms of total inhaled carbon dioxide (CO2 ), as can be readily monitored inmost indoor spaces. Our approach paves the way for optimizing air handling systems by balancing healthand ﬁnancial concerns, and so informs policy for safely reopening schools and businesses as the pandemic
runs its course. Finally, our approach may be applied quite generally in the mitigation of airborne respiratory
illnesses, including inﬂuenza.

© The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative
Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction inany medium, provided the original work is properly cited.
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Martin Z. Bazant et al.

1. Introduction
Coronavirus disease 2019 (COVID-19) has caused a devastating pandemic since it was ﬁrst identiﬁed
in Wuhan, China, in December 2019 (Chen et al., 2020; Li et al., 2020). For over a year, public healthguidance has focused on disinfecting surfaces in order to limit transmission through fomites (Van
Doremalen et al., 2020) and maintaining social distance in order to limit transmission via large drops
generated by coughs and sneezes (Bourouiba, Dehandschoewercker & Bush, 2014; Rosti, Olivieri,
Cavaiola, Seminara & Mazzino, 2020). The eﬃcacy of these measures has been increasingly called into
question, however, since there is scant evidence for fomite transmission (Gandhi, Yokoe & Havlir, 2020;
Lewis, 2021), and large-drop transmission is eﬀectively eliminated by masks (Moghadas et al., 2020).
There is now overwhelming evidence that the pathogen responsible for COVID-19, severe acuterespiratory syndrome coronavirus 2 (SARS-CoV-2), is transmitted primarily through exhaled aerosol
droplets suspended in indoor air (Bazant & Bush, 2021; Greenhalgh et al., 2021; Jayaweera, Perera, Gunawardana & Manatunge, 2020; Morawska & Cao, 2020; Morawska & Milton, 2020; Prather
et al., 2020; Stadnytskyi, Bax, Bax & Anﬁnrud, 2020; Zhang, Li, Zhang, Wang & Molina, 2020b).
Notably, airborne transmission provides the only rational explanation for the so-called ‘superspreading events’, that have now been well chronicled and all took place indoors (Hamner, 2020; Hwang,
Chang, Bumjo & Heo, 2020; Kwon et al., 2020; Miller et al., 2020; Moriarty, 2020; Nishiura et al.,2020; Shen et al., 2020). The dominance of indoor airborne transmission is further supported by
the fact that face-mask directives have been more eﬀective in limiting the spread of COVID-19 thaneither social distancing directives or lockdowns (Stutt, Retkute, Bradley, Gilligan & Colvin, 2020;
Zhang et al., 2020b). Indeed, a recent analysis of spreading data from Massachusetts public schools
where masking was strictly enforced found no statistically signiﬁcant eﬀect of social distance restrictions that ranged from three feet to six feet (van den Berg et al., 2021). A recent study of schooltransmissions in Georgia found that mask use and improvements in ventilation and ﬁltration were themost eﬀective mitigation strategies, while the imposition of barriers or six-foot distancing between
desks had little eﬀect (Gettings et al., 2021). Finally, the detection of infectious SARS-CoV-2 virions suspended in hospital room air as far as 5.5 m from an infected patient provides direct evidence
for the viability of airborne transmission of COVID-19 (Lednicky et al., 2020; Santarpia et al.,2020).
With a view to informing public health policy, we proceed by developing a quantitative approachto mitigating the indoor airborne transmission of COVID-19, an approach that might be similarly
applied to other airborne respiratory diseases. The canonical theoretical framework of Wells (1955) andRiley, Murphy & Riley (1978) describes airborne transmission in an indoor space that is well-mixedby ambient air ﬂows, so that infectious aerosols are uniformly dispersed throughout the space (Beggs,
Noakes, Sleigh, Fletcher & Siddiqi, 2003; Gammaitoni & Nucci, 1997; Nicas, Nazaroﬀ & Hubbard,
2005; Noakes, Beggs, Sleigh & Kerr, 2006; Stilianakis & Drossinos, 2010). While exceptions to thewell-mixed-room approximation are known to arise (Bhagat, Wykes, Dalziel & Linden (2020); see supplementary information in Bazant & Bush, 2021), supporting evidence for the well-mixed approximation
may be found in both theoretical arguments (Bazant & Bush, 2021) and computer simulations of naturaland forced convection (Foster & Kinzel, 2021). The Wells–Riley model and its extensions have been
applied to a number of superspreading events and used to assess the risk of COVID-19 transmission in avariety of indoor settings (Augenbraun et al., 2020; Buonanno, Morawska & Stabile, 2020a; Buonanno,
Stabile & Morawska, 2020b; Evans, 2020; Miller et al., 2020; Prentiss, Chu & Berggren, 2020).
A safety guideline for mitigating indoor airborne transmission of COVID-19 has recently been derived
that indicates an upper bound on the cumulative exposure time, that is, the product of the number ofoccupants and the exposure time (Bazant & Bush, 2021). This bound may be simply expressed in termsof the relevant variables, including the room dimensions, ventilation, air ﬁltration, mask eﬃciency andrespiratory activity. The guideline has been calibrated for COVID-19 using epidemiological data fromthe best characterized superspreading events and incorporates the measured dependence of expiratory
droplet-size distributions on respiratory and vocal activity (Asadi, Bouvier, Wexler & Ristenpart, 2020a;
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Asadi et al., 2019; Morawska et al., 2009). An online app has facilitated its widespread use during the
pandemic (Khan, Bazant & Bush, 2020). The authors also considered the additional risk of turbulent
respiratory plumes and jets (Abkarian, Mendez, Xue, Yang & Stone, 2020a, 2020b), as need be considered when masks are not worn. The accuracy of the guideline is necessarily limited by uncertainties ina number of model parameters, which will presumably be reduced as more data is analysed from indoor
spreading events.
Carbon dioxide measurements have been used for decades to quantify airﬂow and zonal mixing inbuildings and so guide the design of heating, ventilation and air-conditioning (known as HVAC) systems(Fisk & De Almeida, 1998; Seppänen, Fisk & Mendell, 1999). Such measurements thus represent anatural source of data for assessing indoor air quality (Corsi, Torres, Sanders & Kinney, 2002), especiallyas they rely only on relatively inexpensive, widely available CO2 sensors. Quite generally, high CO2
levels in indoor settings are known to be associated with poor health and diminished cognitive function(Coley, Greeves & Saxby, 2007; Hung & Derossis, 1989; Salisbury, 1986; Seppänen et al., 1999).
MacNaughton et al. (2015) examined the economic and environmental costs of increasing ventilation
rates to improve indoor air quality, and concluded that the public health beneﬁts of improved ventilation
generally outweigh these costs. Statistically signiﬁcant correlations between CO2 levels and illnessrelated absenteeism in both the work place (Milton, Glencross & Walters, 2000) and classrooms (Mendell
et al., 2013; Shendell et al., 2004) have been widely reported (Li et al., 2007). Direct correlations
between CO2 levels and concentration of airborne bacteria have been found in schools (Liu et al.,2000). Correlations between outdoor air exchange rates and respiratory infections in dormitory rooms
have also been reported (Bueno de Mesquita, Noakes & Milton, 2020; Sun, Wang, Zhang & Sundell,2011). Despite the overwhelming evidence of such correlations and the numerous economic analyses
that underscore their negative societal impacts (Fisk, 2000; Milton et al., 2000), using CO2 monitors tomake quantitative assessments of the risk of indoor disease transmission is a relatively recent notion (Liet al., 2007).
Rudnick and Milton (2003) ﬁrst proposed the use of Wells–Riley models, in conjunction withmeasurements of CO2 concentration, to assess airborne transmission risk indoors. Their model treats
CO2 concentration as a proxy for infectious aerosols: the two were assumed to be produced proportionallyby the exhalation of an infected individual and removed at the same rate by ventilation. The current
pandemic has generated considerable interest in using CO2 monitoring as a tool for risk management of
COVID-19 (Bhagat et al., 2020; Hartmann & Kriegel, 2020). The Rudnick–Milton model has recently
been extended by Peng and Jimenez (2021) through consideration of the diﬀerent removal rates of CO2
and airborne pathogen. They conclude by predicting safe CO2 levels for COVID-19 transmission invarious indoor spaces, which vary by up to two orders of magnitude.
We here develop a safety guideline for limiting indoor airborne transmission of COVID-19 byexpressing the safety guideline of Bazant and Bush (2021) in terms of CO2 concentration. Doing so
makes clear that one must limit not only the CO2 concentration, but also the occupancy time. Our modelaccounts for the eﬀects of pathogen ﬁltration, sedimentation and deactivation in addition to the variableaerosol production rates associated with diﬀerent respiratory and vocal activities, all of which alter the
relative concentrations of airborne pathogen and CO2 . Our guideline thus quantiﬁes the extent to whichsafety limits may be extended by mitigation strategies such as mask mandates, air ﬁltration and theimposition of ‘quiet spaces’.
In § 2, we rephrase the indoor safety guideline of Bazant and Bush (2021) in terms of the room’s
CO2 concentration. In § 3, we present theoretical descriptions of the evolution of CO2 concentrationand infectious aerosol concentration in an indoor space, and highlight the diﬀerent physical processes
inﬂuencing the two. We then model the disease transmission dynamics, which allows for the risk ofindoor airborne transmission to be assessed from CO2 measurements taken in real time. In § 4, weapply our model to a pair of data sets tracking the evolution of CO2 concentration in speciﬁc oﬃce
and classroom settings. These examples illustrate how CO2 monitoring, when coupled with our safety
guideline, provides a means of assessing and mitigating the risk of indoor airborne transmission of
respiratory pathogens.
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Martin Z. Bazant et al.

2. Safety guideline for the time-averaged CO2 concentration
2.1. Occupancy-based safety guideline
We begin by recalling the safety guideline of Bazant and Bush (2021) for limiting indoor airborne diseasetransmission in a well-mixed space. The guideline would impose an upper bound on the cumulative
exposure time,
Nt 𝜏 <𝜖𝜆c V,
2Qb Cq sr p2m(1)

where Nt is the number of possible transmissions (pairs of infected and susceptible people) and 𝜏 is thetime in the presence of the infected person(s). The reader is referred to table 1 for a glossary of symbols
and their characteristic values. Here Qb is the mean breathing ﬂow rate and V the room volume. The
risk tolerance 𝜖 < 1 is the prescribed bound on the probability of at least one transmission, as should bechosen judiciously according to the vulnerability of the population (Garg, 2020); for example, Bazant
and Bush (2021) suggested 𝜖 = 10 % for children and 1 % for the elderly.
The only epidemiological parameter in the guideline, Cq , is the infectiousness of exhaled air, measured
in units of infection quanta per volume for a given aerosolized pathogen. The notion of ‘infection
quantum’, introduced by Wells (1955), is widely used in epidemiology to measure the expected rateof disease transmission, which may be seen as a transfer of infection quanta between pairs of infected
and susceptible individuals. For airborne transmission, a suitable concentration of infection quanta pervolume, Cq , can thus be associated with exhaled air without reference to the microscopic pathogen
concentration. Notably, Cq is known to depend on the type of respiratory and vocal activity (associatedwith an individual resting, exercising, speaking, singing, etc.), being larger for the more vigorous
activities (Bazant & Bush, 2021; Buonanno et al., 2020b). The relative susceptibility sr is introduced
as a scaling factor for Cq that accounts for diﬀerences in the transmissibility of diﬀerent respiratory
pathogens, such as bacteria or viruses (Li et al., 2008; Rudnick & Milton, 2003) with diﬀerent strains(Davies et al., 2020; Volz et al., 2021), and for diﬀerences in the susceptibility of diﬀerent populations,such as children and adults (Riediker & Morawska, 2020; Zhang et al., 2020a; Zhu et al., 2020).
The mask penetration probability, pm (r), is a function of drop size that is bounded below by 0 (forthe ideal limit of perfect mask ﬁltration) and above by 1 (as is appropriate when no mask is worn).
Standard surgical masks at low ﬂow rates allow only 0.04 %–1.5 % of the micron-scale aerosols topenetrate (Chen & Willeke, 1992), but these values should be increased by a factor of 2–10 to accountfor imperfect ﬁt (Oberg & Brosseau, 2008). Cloth masks show much greater variability (Konda et al.,2020b). The mask penetration probability may also depend on respiratory activity (Asadi et al., 2020b)
and direction of airﬂow (Pan, Harb, Leng & Marr, 2020). Here, for the sake of simplicity, we treat pm (r)as being constant over the limited aerosol size range of interest, and evaluate p̄m = pm (r̄) at the eﬀective
aerosol radius r to be deﬁned below, above which drops tend to settle to the ground faster than they areswept away by ventilation. For this eﬀective aerosol ﬁltration factor, Bazant and Bush (2021) suggestedp̄m = 1 %–5 % for surgical masks (Li et al., 2008; Oberg & Brosseau, 2008), p̄m = 10 %–40 % for hybrid
cloth face coverings and p̄m = 40 %–80 % for single-layer fabrics (Konda et al., 2020b). Notably, even
low quality masks can signiﬁcantly reduce transmission risk since the bound on cumulative exposure
time, (1), scales as p̄−2m .
Finally, we deﬁne 𝜆c = 𝜆 c (r) as an eﬀective relaxation rate of the infectious aerosol-borne pathogen
concentration, C(r, t), evaluated at the eﬀective aerosol radius r̄. The size-dependent relaxation rate ofthe droplet-borne pathogen has four distinct contributions:𝜆 c (r) = 𝜆 a + 𝜆 f (r) + 𝜆 s (r) + 𝜆 v (r).

(2)

Here, 𝜆 a is the ventilation rate, speciﬁcally the rate of exchange with outdoor air; 𝜆f (r) = pf (r)𝜆 r is
the ﬁltration rate, where pf (r) is the droplet removal eﬃciency for air ﬁltration at a rate 𝜆r (recirculated
air changes per time); 𝜆s (r) = vs (r)A/V is the net sedimentation rate for infectious droplets with the
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Table 1. Glossary of symbols arising in our theory, their units and characteristics values.
SymbolMeaning

Typical Values

Engineering parameters
N
Number of people, room occupancy𝜏Time since an infected person entered the room
V
Room volume
A
Floor surface area
H
Mean ceiling height, V/A
Ventilation outﬂow rate
Qa
𝜆a
Ventilation (outdoor air exchange) rate, Qa /V𝜆r
Recirculation air exchange rate, Qr /V
pf
Probability of droplet ﬁltration via recirculation
Filtration removal rate, pf 𝜆 r
𝜆fpm
Mask penetration probability, pm = pm (r)Physical parametersrRespiratory drop radiusDrop volume, ≈ 43 πr3Vd
ndDrop number density per radiusvsDrop settling speed
𝜆s
Drop settling rate, vs (r)/H
Qb
Mean breathing ﬂow rate
C0
Background CO2 concentration
Exhaled CO2 concentration
C2
Production rate of exhaled CO2
P2

0.1–100 μm
10−5 –106 μm3
0.01–1.0 (cm3 μm)−110−5 –102 mm s−1
10−5 –102 h−1
0.5–3.0 m3 h−1
250–450 p.p.m.
0–40 000 p.p.m.
0.02–10 m3 h−1

Epidemiological parameters
S, I
Number of susceptible and infected people
Airborne transmission rate per infected–susceptible pair𝛽a𝜆v
Pathogen (virion) deactivation rate
𝜆c
Pathogen concentration relaxation rate, 𝜆c = 𝜆 c (r)r̄Eﬀective infectious drop radiusP
Pathogen production rate / air volume / drop radiusC
Infectious pathogen concentration / air volume / radius
cv
Pathogen (virion) concentration per drop volume
ci
Pathogen infectivity, 1/(infectious dose)
Infectiousness of breath, exhaled quanta concentration
Cq𝜆q
Quanta emission rate, Qb Cq
pi
Probability a person is infected (prevalence)
Probability a person is immune (by vaccination or exposure)pim
Probability a person is susceptible, ps = 1 − pi − pimpsRelative susceptibility (or transmissibility)sr
Expected number of infected–susceptible pairs
Nt
Expected number of airborne transmissions, Nt 𝛽a 𝜏TaIndoor reproductive number, (N − 1)𝛽a 𝜏R in𝜖Risk tolerance, bound on Ta

10−6 –10 quanta h−1
0.01–10 h−1
0.1–100 h−1
0.3–5 μm
10−6 –109 (h μm)−110−8 –104 (m3 μm)−1
104 –1011 RNA copies mL−1
0.001–1.0
1–1000 quanta m−3
1–1000 quanta h−1
0–1
0–1
0–1
0.1–100–10000–100
0.001–100
0.005–0.5

1–10000–1000 h
10–104 m3
5–5000 m2
2–6 m
1–105 m3 h−1
0.1–30 h−1
0.1–30 h−1
0–1.0
0–30 h−1
0.01–0.1

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Martin Z. Bazant et al.

Stokes settling speed vs (r) sedimenting through a turbulent, well-mixed ambient in a room of height
H = V/A, volume V and ﬂoor area A (Corner & Pendlebury, 1951; Martin & Nokes, 1988); 𝜆 v (r) is
the deactivation rate of the aerosolized pathogen, which depends weakly on humidity and droplet size
(Lin & Marr, 2019; Marr, Tang, Van Mullekom & Lakdawala, 2019; Yang & Marr, 2011), and may beenhanced by other factors such as ultraviolet (UV-C) irradiation (García de Abajo et al., 2020; Hitchman,2021), chemical disinfectants (Schwartz et al., 2020) or cold plasma release (Filipić, Gutierrez-Aguirre,
Primc, Mozetič & Dobnik, 2020; Lai, Cheung, Wong & Li, 2016).
Notably, only the ﬁrst of the four removal rates enumerated in (2) is relevant in the evolution of
CO2 ; thus, the concentrations of CO2 and airborne pathogen may evolve independently. Speciﬁcally, theproportionality between the two equilibrium concentrations varies in diﬀerent indoor settings (Peng &Jimenez, 2021), for example in response to room ﬁltration (Hartmann & Kriegel, 2020). Moreover, whentransient eﬀects arise, for example, following the arrival of an infectious individual or the opening of awindow, the two concentrations adjust at diﬀerent rates. Finally, we note that there may also be sourcesof CO2 other than human respiration, such as emissions from animals, stoves, furnaces, ﬁreplaces orcarbonated drinks, as well as sinks of CO2 , such as plants, construction materials or pools of water,which we neglect for simplicity. As such, following Rudnick and Milton (2003), we assume that the
primary source of excess CO2 is exhalation by the human occupants of the indoor space.
In order to prevent the growth of an epidemic, the safety guideline should bound the indoor reproductive number, R in , which is the expected number of transmissions if an infectious person enters a room
full of susceptible people. Indeed, the safety guideline, (1), corresponds to the bound R in < 𝜖 with the
choice Nt = N − 1, and so would limit to 𝜖 the risk of an infected person entering the room of occupancy N transmitting to any other person during the exposure time 𝜏. If the epidemic is well underway
or subsiding, the guideline should take into account the prevalence of infection pi and immunity pim (asachieved by previous exposure or vaccination) in the local population. Assuming a trinomial distribution of N people who are infected, immune or susceptible, with mutually exclusive probabilities pi , pim
and ps = 1 − pi − pim , respectively, the expected number of infected–susceptible pairs is N (N − 1)pi ps .
It is natural to switch between these two limits (Nt = N − 1 and Nt = N (N − 1)pi ps ) when one infected
person is expected to be in the room, Npi = 1, and thus set
Nt = (N − 1) min{1, Npi (1 − pi − pim )}.

(3)

One may thus account for the changing infection prevalence pi and increasing immunity pim in the local
population as the pandemic evolves.
2.2. CO2 -based safety guideline
The outdoor CO2 concentration is typically in the range C0 = 350–450 p.p.m., with higher values inurban environments (Prill et al., 2000) and lower values in forests (Higuchi, Worthy, Chan & Shashkov,
2003). In the absence of other indoor CO2 sources, human occupancy in poorly ventilated spaces can lead
to CO2 levels of several thousand p.p.m. People have reported headaches, slight nausea, drowsiness anddiminished decision-making performance for levels above 1000 p.p.m. (Fisk, Satish, Mendell, Hotchi &Sullivan, 2013; Krawczyk, Rodero, Gładyszewska-Fiedoruk & Gajewski, 2016), while short exposures
to much higher levels may go unnoticed. As an example of CO2 limits in industry, the American
Conference of Governmental Industrial Hygienists recommends a limit of 5000 p.p.m. for an eighthour period and 30 000 p.p.m. for 10 min. A value of 40 000 p.p.m. is considered to be immediately
life-threatening.
We denote by C0 the background concentration that would arise in the room at zero occupancy,and by C2 the excess CO2 concentration (relative to C0 ) associated with human respiration. The total
rate of CO2 production by respiration in the room is given by P2 = NQb C2,b , where C2,b is the CO2
concentration of exhaled air, approximately C2,b = 38 000 p.p.m., although the net CO2 production
rate, Qb C2,b varies considerably with body mass and physical activity (Persily & de Jonge, 2017). If the
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production rate P2 and the ventilation ﬂow rate Q = 𝜆 a V are constant, then the steady-state value of theexcess CO2 concentration,
P2 Qb C2,b N=,(4)
C2,s =Q
𝜆a V
is simply the ratio of the individual CO2 ﬂow rate, Qb C2,b , and the ventilation ﬂow rate per person, Q/N.
The safety guideline, (1), was derived on the basis of the conservative assumption that the infectious
aerosol concentration has reached its maximum, steady-state value. If we assume, for consistency,that the CO2 concentration has done likewise, and so approached the value expressed in (4), then theguideline can be recast as a bound on the safe mean excess CO2 concentration,
C2 𝜏 =∫ 𝜏
0C2 dt <

𝜖C2,b 𝜆c N,
𝜆 q sr p2m 𝜆 a Nt

(5)

where we replace the steady excess CO2 concentration with its time average, C2  ≈ C2,s , and deﬁne
the mean quanta emission rate per infected person, 𝜆q = Qb Cq . For the early to middle stages of an
epidemic or when pi and pim are not known, we recommend setting N/Nt = 1 < N/(N − 1) ≈ 1,
for a conservative CO2 bound that limits the indoor reproductive number. In the later stages of an
epidemic, as the population approaches herd immunity ( pi → 0, pim → 1), the safe CO2 bounddiverges, N/Nt → ∞, and so may be supplanted by the limits on CO2 toxicity noted above, that lie
in the range 5000–30 000 p.p.m. for eight hour and 10 minute exposures, respectively. Our simple
CO2 -based safety guideline, (5), reveals scaling laws for exposure time, ﬁltration, mask use, infection
prevalence and immunity, factors that are not accounted for by directives that would simply impose alimit on CO2 concentration. The substantial increase in safe occupancy times, as one proceeds fromthe peak to the late stages of the pandemic, is evident in the diﬀerence between the solid and dashedlines in ﬁgure 1, which were evaluated for the case of a typical classroom in the USA (Bazant &Bush, 2021). This example shows the critical role of exposure time in determining the safe CO2 level,a limit that can be increased dramatically by eﬃcient mask use and to a lesser extent by ﬁltration.
When infection prevalence pi falls below 10 per 100 000 (an arbitrarily chosen small value), the chanceof transmission is extremely low, allowing for long occupancy times. The risk of transmission at
higher levels of prevalence, as may be deduced by interpolating between the solid and dashed lines inﬁgure 1, could also be rationally managed by monitoring the CO2 concentration and adhering to theguideline.

3. Mathematical model of CO2 monitoring to predict airborne disease transmission risk
3.1. CO2 dynamics
We follow the traditional approach of modelling gas dynamics in a well-mixed room (Shair & Heitner,
1974), as a continuous stirred tank reactor (Davis & Davis, 2012). Given the time dependence ofoccupancy, N (t), mean breathing ﬂow rate, Qb (t), and ventilation ﬂow rate, Qa (t) = 𝜆 a (t)V, one may
express the evolution of the excess CO2 concentration C2 (t) in a well-mixed room throughVdC2= P2 (t) − Qa (t)C2 ,dt(6)

P2 (t) = N (t)Qb (t)C2,b

(7)

where

is the exhaled CO2 production rate. The relaxation rate for excess CO2 in response to changes in P2 (t)is precisely equal to the ventilation rate, 𝜆a (t) = Qa (t)/V. For constant 𝜆 a , the general solution of (6)Downloaded from https://www.cambridge.org/core. 04 Oct 2021 at 20:22:51, subject to the Cambridge Core terms of use.

E10-8Martin Z. Bazant et al.
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Time (h)Figure 1. Illustration of the safety guideline, (5), which bounds the safe excess CO2 (p.p.m.) and
exposure time 𝜏 (hours). Here, we consider the case of a standard US classroom (with an area of 83.6 m2
and a ceiling height of 3.6 m) with N = 25 occupants, assumed to be children engaging in normal speech
and light activity (𝜆 q sr = 30 quanta h−1 ) with moderate risk tolerance (𝜖 = 10 %). Comparison with themost restrictive bound on the indoor reproductive number without any precautions (red line) indicatesthat the safe CO2 level or occupancy time is increased by at least an order of magnitude by the use offace masks (blue line), even with relatively inconsistent use of cloth masks (pm = 30 %). The eﬀect of air
ﬁltration (green line) is relatively small, shown here for a case of eﬃcient HEPA ﬁltration (pf = 99 %)with 17 % outdoor air fraction (𝜆f = 5𝜆 a ). All three bounds are increased by several orders of magnitude(dashed lines) during late pandemic conditions (pi ps = 10 per 100 000), when it becomes increasingly
unlikely to ﬁnd an infected–susceptible pair in the room. The other parameters satisfy (𝜆v +𝜆 s )/𝜆 a = 0.5,
as could correspond to, for example, 𝜆v = 0.3, 𝜆 s = 0.2 and 𝜆 a = 1 h−1 (1 ACH).
for C2 (0) = 0 is given by
C2 (t) =∫ t0exp(−𝜆 a (t − t ))

P2 (t ) 
dt ,
V

(8)

which can be derived by Laplace transform or using an integrating factor. The time-averaged excess
CO2 concentration can be expressed as
1C2  =𝜏∫ 𝜏
0(1 − exp(−𝜆 a (𝜏 − t)))P2 (t)
dt
Qa(9)by switching the order of time integration. If P2 (t) is slowly varying over the ventilation time scale 𝜆−1a ,the time-averaged CO2 concentration may be approximated asC2  ≈
1 − e−𝜆a 𝜏 P2 (𝜏)
P2 
−
,
Qa
𝜆a 𝜏
Qa

(10)

where the excess CO2 concentration approaches the ratio of the mean exhaled CO2 production rate to
the ventilation ﬂow rate at long times, 𝜏  𝜆−1
a , as indicated in (4).
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Flow

E10-9

3.2. Infectious aerosol dynamics
Following Bazant and Bush (2021), we assume that the radius-resolved concentration of infectious
aerosol-borne pathogen, C(r, t), evolves according to𝜕C= P(r, t) − 𝜆 c (r, t)VC,
𝜕t

(11)

P(r, t) = I (t)Qb (t)nd (r, t)Vd (r)pm (r)cv (r),(12)

Vwhere the mean production rate,depends on the number of infected people in the room, I (t), and the size distribution nd (r, t) of exhaleddroplets of volume Vd (r) containing a pathogen (i.e. virion) at microscopic concentration, cv (r). The
droplet size distribution is known to depend on expiratory and vocal activity (Asadi et al., 2019, 2020c;
Morawska et al., 2009). Quite generally, the aerosols evolve according to a dynamic sorting process(Bazant & Bush, 2021): the drop-size distribution evolves with time until an equilibrium distribution
obtains.
Given the time evolution of excess CO2 concentration, C2 (t), one may deduce the radius-resolved
pathogen concentration C(r, t) by integrating the coupled diﬀerential equation𝜕CP(r, t) dC2+ 𝜆 c (r, t)C =+ 𝜆 a (t)C2 .
(13)𝜕t
P2 (t)
dt
This integration can be done numerically or analytically via Laplace transform or integrating factors if
one assumes that 𝜆a , 𝜆 c , P and P2 all vary slowly over the ventilation (air change) time scale, 𝜆−1
a . Inthat case, the general solution takes the form


∫ t
P



exp(−𝜆 c (r)(t − t ))C2 (t ) dt ,C(r, t) ≈C2 (t) + (𝜆 a − 𝜆 c (r))
(14)
P2
0
where we consider the infectious aerosol build-up from C(r, 0) = 0.
3.3. Disease transmission dynamics
According to Markov’s inequality, the probability of at least one airborne transmission taking place
during the exposure time 𝜏 is bounded above by the expected number of airborne transmissions, Ta (𝜏),1. The expected number
and the two become equal in the (typical) limit of rare transmissions, Ta (𝜏)of transmissions to S(t) susceptible people is obtained by integrating the mask-ﬁltered inhalation rateof infection quanta over both droplet radius and time:

∫ ∞
∫ 𝜏
S(t)Qb (t)sr
C(r, t)ci (r)pm (r) dr dt,(15)
Ta (𝜏) =00

where ci (r) is the infectivity of the aerosolized pathogen. The infectivity is measured in units of infection
quanta per pathogen and generally depends on droplet size. One might expect pathogens contained insmaller aerosol droplets with r < 5 μm to be more infectious than those in larger drops, as reported by
Santarpia et al. (2020) for SARS-CoV-2, on the grounds that smaller drops more easily penetrate therespiratory tract, absorb and coalesce onto exposed tissues, and allow pathogens to escape more quickly
by diﬀusion to infect target cells. The mask penetration probability pm (r) also decreases rapidly withincreasing drop size above the aerosol range for most ﬁltration materials (Chen & Willeke, 1992; Konda
et al., 2020b; Li et al., 2008; Oberg & Brosseau, 2008), so the integration over all radii in (15) gives themost weight to the aerosol size range, roughly r < 5 μm. We note that this range includes the maxima
in exhaled droplet size distributions (Asadi et al., 2019, 2020c; Morawska et al., 2009) as deﬁned in
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E10-10

Martin Z. Bazant et al.

terms of either number or total volume in the aerosol range (Bazant & Bush, 2021); nevertheless, larger
droplets may also contribute to airborne transmission (Tang et al., 2021).
The inverse of the infectivity, c−1
i , is equal to the ‘infectious dose’ of pathogens from inhaledaerosol droplets that would cause infection with probability 1 − (1/e) = 63 %. Bazant and Bush (2021)
estimated the infectious dose for SARS-CoV-2 to be of the order of 10 aerosol-borne virions. Notably,
the corresponding infectivity, ci ∼ 0.1, is an order of magnitude larger than previous estimates for SARSCoV (Buonanno et al., 2020b; Watanabe, Bartrand, Weir, Omura & Haas, 2010), which is consistent
with only COVID-19 reaching pandemic status. The infectivity is known to vary across diﬀerent age
groups and pathogen strains, a variability that is captured by the relative susceptibility, sr . For example,
Bazant and Bush (2021) suggest assigning sr = 1 for the elderly (over 65 years old), sr = 0.68 for adults(aged 15–64) and sr = 0.23 for children (aged 0–14) for the original Wuhan strain of SARS-CoV-2,
based on a study of transmission in quarantined households in China (Zhang et al., 2020a). The authors
further suggested multiplying these values by 1.6 for the more infectious Alpha variant of concern of
the lineage B.1.1.7 (VOC 202012/01), which emerged in the UK with a reproductive number that was
60 % larger than that of the original strain (Davies et al., 2020; Volz et al., 2021). Based on the latest
epidemiological data (Abdool Karim & de Oliveira, 2021), we likewise suggest multiplying sr by 1.5
for the South African variant, 501 Y.V2, and by 1.2 for the Californian variants, B.1.427 and B.1.429.
Appropriate factors multiplying sr for more recent strains are provided on our online app (Khan, Bazant& Bush, 2020), such as 1.5 for the Beta variant B.1.351 from South Africa, 2.0 for the Gamma variant
P.1 from Brazil and 2.5 for the Delta variant B.1.617.2 from India that has become dominant at the timeof publication.
3.4. Approximate formula for the airborne transmission risk from CO2 measurements
Equations (14) and (15) provide an approximate solution to the full model that depends on the exhaleddroplet size distribution, nd (r, t), and mean breathing rate, Qb (t), of the population in the room. Since
the droplet distributions nd (r) have only been characterized in certain idealized experimental conditions(Asadi et al., 2020a, 2020c; Morawska et al., 2009), it is useful to integrate over r to obtain a simpler
model that can be directly calibrated for diﬀerent modes of respiration using epidemiological data
(Bazant & Bush, 2021). Assuming Qb (t), I (t), S(t), N (t) and nd (r, t) vary slowly over the relaxation−1
time 𝜆c , we may substitute (14) into (15) and perform the time integral of the second term to obtain
Ta (𝜏) ≈sr 𝜆 aC2,b∫ 𝜏∫ ∞
0

0





nq (r, t)pm (r) 2 Qb (t)I (t)S(t)C2 (t)𝜆 c (r)− 1 exp(−𝜆 c (r)(𝜏 − t)) dr dt,
1+
𝜆 c (r)
N (t)
𝜆a
(16)

where nq (r, t) = nd (r, t)Vd (r)cv (r)ci (r) is the radius-resolved exhaled quanta concentration.
Following Bazant and Bush (2021), we deﬁne an eﬀective radius of infectious aerosols r such that∫ ∞0

Cq (t)pm (r) 2
nq (r, t)pm (r) 2dr ≡,𝜆 c (r)𝜆 c (r)(17)∫∞
where Cq (t) = 0 nq (r, t) dr is the exhaled quanta concentration, which may vary in time with changes
in expiratory activity, for example, following a transition from nose breathing to speaking. In principle,
the eﬀective radius r can be evaluated, given a complete knowledge of the dependence on drop radiusof the mask penetration probability, pm (r), and of all the factors that determine the exhaled quantaconcentration, nq (r, t) and pathogen removal rate, 𝜆 c (r). While these dependencies are not readily
characterized, typical values of r are at the scale of several microns, based on the size dependencies ofnd (r, t), ci (r) and pm (r) noted above.
Further simpliﬁcations allow us to derive a formula relating CO2 measurements to transmission
risk. By assuming that Cq (t) varies slowly over the time scale of concentration relaxation, one may
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Flow

E10-11approximate the memory integral with the same eﬀective radius r. Thus, accounting for immunity andinfection prevalence in the population via
I (t)S(t) Nt
1
≈
= 1−
(18)
min{1, Npi ps },
N (t)
N
N
we obtain a formula for the expected number of airborne transmissions,∫ 𝜏
sr p2m 𝜆 a Nt
𝜆c
Ta (𝜏) ≈𝜆 q (t)C2 (t) 1 +− 1 exp(−𝜆c (𝜏 − t)) dt,
C2,b 𝜆c N 0
𝜆a

(19)

in terms of the excess CO2 time series, C2 (t), where 𝜆 q (t) = Cq (t)Qb (t) is the mean quanta emission
rate. It is also useful to deﬁne the expected transmission rate,∫ 𝜏
sr p2m 𝜆 a NtdTa(𝜏) = Nt 𝛽a (𝜏) =𝜆 q (t)C2 (t) exp(−𝜆c (𝜏 − t)) dt , (20)
𝜆 q (𝜏)C2 (𝜏) + (𝜆 a − 𝜆c )dt
C2,b 𝜆c N
0which allows for direct assessment of airborne transmission risk based on CO2 levels. A pair of examples
of such assessments will be presented in § 4.
Notably, the mean airborne transmission rate expected per infected–susceptible pair, 𝛽a (t), reﬂects
the environment’s memory of the recent past, which persists over the pathogen relaxation time scale,−1
𝜆c . Likewise, the CO2 concentration deﬁned in (9) has a memory of recent changes in CO2 sources orventilation, which persists over the air change time scale. Notably, the airborne pathogen concentration−1equilibrates more rapidly than CO2 ; speciﬁcally, 𝜆−1
a > 𝜆 c , since CO2 is unaﬀected by the ﬁltration,
sedimentation and deactivation rates enumerated in (2). The time delays between the production of CO2
and infectious aerosols by exhalation and their build-up in the well-mixed air of a room shows that CO2
variation and airborne transmission are inherently non-Markovian stochastic processes. As such, any
attempt to predict ﬂuctuations in airborne transmission risk would require stochastic generalizationsof the diﬀerential equations governing the mean variables, (6) and (11), and so represent a stochastic
formulation of the Wells–Riley model (Noakes & Sleigh, 2009).
3.5. Reduction to the CO2 -based safety guideline
Finally, we connect the general result, (19), with the CO2 -based safety guideline derived above, (5).
Since C2 (t) varies on the ventilation time scale 𝜆−1
a , which is necessarily longer than the relaxation time−1
scale of the infectious aerosols 𝜆c , we may assume that 𝜆q (t)C2 (t) is slowly varying and evaluate theintegral in (19). We thus arrive at the approximation𝜆 q (𝜏)C2 (𝜏) 𝜆c
sr p2m 𝜆 a Nt
− 1 + e−𝜆c 𝜏 .
(21)
Ta (𝜏) ≈𝜆 q C2 𝜏 +C2,b 𝜆c N
𝜆a
𝜆c
Since 𝜆q (t)C2 (t) is slowly varying, the second term in brackets is negligible relative to the ﬁrst for−1times longer than the ventilation time, 𝜏  𝜆−1
a > 𝜆 c . In this limit, the imposed bound on expectedtransmissions, Ta (𝜏) < 𝜖, is approximated by𝜆 q C2 𝜏 =∫ 𝜏0

𝜆 q (t)C2 (t) dt <𝜖C2,b 𝜆c N.
sr p2m 𝜆 a Nt

(22)

This formula reduces to the safety guideline, (5), in the limit of constant mean quanta emission rate, 𝜆q ,which conﬁrms the consistency of our assumptions.
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E10-12

Martin Z. Bazant et al.

4. Examples of implementation
We proceed by illustrating the process by which the guideline, (5), can be coupled to real data obtained
from CO2 monitors. Speciﬁcally, we consider time series of CO2 concentration gathered in classroom
and oﬃce settings at the Massachusetts Institute of Technology using an Atlas Scientiﬁc EZO-CO2
Embedded NDIR CO2 Sensor controlled with an Arduino Uno and an Aranet4, respectively. The
measurements were collected at desk level. Social distancing guidelines were adhered to, and masks
were worn by all participants. We assume a constant exhaled CO2 concentration of 38 000 p.p.m., anduse the global minimum of the CO2 series as the background CO2 level C0 from which the excessconcentration C2 (t) was deduced. Notably, the relatively small ﬂuctuations in the CO2 measurements
recorded in a variety of settings support the notion of a well-mixed room.
The sensors yield a time series of CO2 concentration that we use, along with (19) and (20), to
calculate the transmission rate, assuming that there is a single infected person in the room and that allothers are susceptible (Nt = N − 1). In this case, the expected number of transmissions is equal to theindoor reproductive number, Ta (𝜏) = R in (𝜏), and the transmission rate is (dTa /dt)(𝜏) = Nt 𝛽a (𝜏). The
approximations made in the derivation above are valid in these examples, so a direct numerical solutionof (13) would yield an identical result. In particular, the ‘slowly varying’ assumptions are satisﬁed,
since we keep N and I constant, and any time-dependence of Qb cancels in the ratio P/P2 . Moreover,there was no indication that the air change rate varied over the relatively short time periods considered.
Finally, the droplet distributions nd (r, t) may vary in time, but no signiﬁcant changes in mean respiratory
activity were observed or evidenced in the CO2 measurements.
We choose realistic values of the parameters that fall within the typical ranges estimated by Bazant
and Bush (2021). The mean breathing rate is set to a value, Qb = 0.5 m3 h−1 , appropriate for light
respiratory activity, and we choose an eﬀective settling speed of vs = 0.108 m h−1 appropriate forr̄ = 0.5 μm. Larger values of Qb would be appropriate for more vigorous respiratory activity, as
would accompany, for example, a workout at a gym. Our estimate for vs may be revised in response to
additional information concerning the dependence on size of the infectivity of exhaled aerosol droplets(Stadnytskyi et al., 2020). The viral deactivation rate for SARS-CoV-2 is set to 𝜆v = 0.3 h−1 , anestimate appropriate for 50 % relative humidity. We plot Nt 𝛽a (𝜏) and R in (𝜏) for cases where masks are
and are not worn, and choose a mask penetration probability of pm ≈ 0.3, as is roughly appropriate
for a cloth mask (Konda et al., 2020a). For more eﬀective masks, such as surgical masks or N95s, asmaller pm value is appropriate, as noted above. The parameters 𝜆c , 𝜆 a , H and N are chosen according
to the speciﬁc scenario presented. Exhaled COVID-19 quanta concentrations Cq for various expiratory
activities are estimated from ﬁgure 2 of Bazant and Bush (2021). In order to be conservative, we assume
that sr = 1, suitable for the most susceptible individuals exposed to the Wuhan strain of SARS-CoV-2.
When applying our model to less susceptible populations, such as children (Zhang et al., 2020a; Zhu
et al., 2020) or vaccinated individuals exposed to new variants (Sheikh et al., 2021; Nasreen et al.,2021), a smaller sr value may be used.
4.1. Small oﬃce with two workers
Figure 2a shows CO2 measurements taken in an oﬃce of length L = 4.2 m, width W = 3 m andheight H = 3 m. Initially, a single worker is present, but at 19:00, a second worker arrives at the oﬃce.
The workers exit the oﬃce at 21:09, return at 21:39 and exit again at 22:30. As the participants werespeaking, we use Cq = 72 quanta m−3 (Bazant & Bush, 2021). We compute the room’s ventilation
rate, 𝜆 a ≈ 2.3 h−1 , from the exponential relaxation that follows the occupants’ exit from the oﬃce (asindicated by the orange curve in ﬁgure 2). The oﬃce was equipped with moderate ventilation, which we
characterize with pf = 0.99 and 𝜆 r = 6 h−1 . The background concentration C0 = 420 p.p.m. is evident
both before and after occupancy.

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Flow
(a)
One person in office
Two people
in officeCO2
600(p.p.m.)

0.10
0.05

500
400

(b)

Ntβa

0.20
Empty officerelaxation of CO2 0.15

800
700

E10-13

Mar 25, 16:00

Mar 25, 20:00

Mar 26, 00:00
2021

0

0.4
0.3

asks
out m
h
t
i
W

Rin 0.2
0.1

ter
With fil

Safety guideline

With masks
0

19:00

19:30

20:00
Time

20:30

21:00
Mar 25, 2021

Figure 2. Measured CO2 concentration and calculated transmission rate in a two-person oﬃce.
(a) Black dots represent the concentration of CO2 . The solid blue, dashed magenta and dash–dot
green curves represents the transmission rate, as calculated from (20) for three diﬀerent scenarios, twoof which were hypothetical: (blue) the pair are not wearing masks and there is no ﬁltration present;(magenta) the pair are not wearing masks and there is ﬁltration present; (green) the pair are wearing
masks and there is no ﬁltration present. The orange solid curve denotes the period of exponential relaxation following the exit of the room’s occupants, from which one may infer both the room’s ventilation
rate, 𝜆 a = 2.3 h−1 , and the background CO2 concentration, C0 = 420 p.p.m. (b) Corresponding blue,
magenta and green curves, deduced by integrating (20), indicate the total risk of transmission over thetime of shared occupancy. If the pair were not wearing masks, the safety limit R in < 0.1 would be violated after approximately an hour.
As shown in ﬁgure 2b, if the oﬃce workers were not wearing masks, the safety guideline of expectedtransmissions Ta < 10 % would be violated after approximately an hour together. However, oﬃce workers wearing cloth masks would not violate the safety guideline during the 2.5 h spent together. Filtration
without masks also extends the occupancy time limits; however, R in approaches the safety limit after
approximately 2 h. While this example should not be taken as a deﬁnitive statement of danger or safety inthis setting, it does serves to illustrate how our CO2 guideline can be implemented in a real-life situation.
4.2. University classroom adhering to social distancing guidelines
We next monitor CO2 levels during a university lecture. There were N = 12 participants in a lecture hall of length L = 13 m, width W = 12 m and height H = 3 m. The lecture started at 13:05
and ﬁnished at 13:50. Four people remained in the room for 30 min after the class. The classroom
has mechanical ventilation, which is characterized in terms of 𝜆a ≈ 5.2 h−1 (corresponding to a
12 min outdoor air change time; here, we estimate 𝜆a by ﬁtting the build-up of CO2 , see ﬁgure 3).
During the lecture, the professor spoke while the students were quiet and sedentary; thus, we assume
Cq = 30 quanta m−3 (Bazant & Bush, 2021). The maximum CO2 concentration reached in the classroom
was ≈550 p.p.m., the excess level no more than 100 p.p.m. Thus, the safety limit was never exceeded,
and would not have been even if masks had not been worn. We note that this lecture hall was particularly large, well-ventilated and sparsely populated, and so should not be taken as being representativeof a classroom setting. A more complete assessment of safety in schools would require integrating CO2
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E10-14

Martin Z. Bazant et al.
(a)
0.04

550
12 people inclassroom

0.03
0.02

Fit of CO2 build-up

Ntβa

CO2
(p.p.m.) 500

0.01
450

13:10

13:20

13:30

13:40
13:50
Mar 18, 2021

0(b) 0.10
Safety guideline

Rin 0.05

ks
Without mas
With masks

0

13:10

13:20

13:30

Time

13:40
13:50
Mar 18, 2021

Figure 3. Measured CO2 concentration and calculated transmission rate for 12 masked students in auniversity lecture hall. (a) Black dots represent the concentration of CO2 . The dash–dot green and solid
blue curves represent the transmission rate, as calculated from (20), when the occupants are wearing
masks, and in the hypothetical case where they are not, respectively. The gold curve indicates a ﬁt to
the transient build-up of CO2 from which we infer an air change rate of 𝜆 a = 5.2/h. (b) The dash–dot
green and solid blue curves indicate the total risk of transmission with and without masks, respectively,
as deduced by integrating (20) over time. Even had masks not been worn, the safety guideline would nothave been violated during the lecture.
concentrations over a considerably longer time interval. For example, if students are tested weekly for
COVID-19, then one should assess the mean CO2 concentration in class during the course of an entire
school week. Nevertheless, this second example further illustrates the manner in which our model may
be applied to real-world settings and suggests that precautions such as ventilation, ﬁltration and maskuse, can substantially increase safe occupancy times.
5. Conclusion
Mounting evidence suggests that COVID-19 is spread primarily via indoor airborne transmission. Suchan inference is no surprise, as such is also the case for many other respiratory illnesses, including
inﬂuenza, tuberculosis, measles and severe acute respiratory syndrome (as is caused by a precursor to
SARS-CoV-2, the coronavirus SARS-CoV). More than a year into the pandemic, public health guidancecontinues to emphasize the importance of social distancing and surface cleaning, despite evidence thatmask directives are much more eﬀective than either in limiting airborne transmission. We have here
illustrated the manner in which CO2 monitoring may be used in conjunction with the safety guideline ofBazant and Bush (2021) to assess the risk of indoor airborne respiratory disease transmission, includingthat of COVID-19. Morawska et al. (2021) emphasize the need for improved air quality standards tocombat airborne respiratory diseases, similar in spirit to food safety standards for restaurants. We hopethat our study might prove useful in deﬁning such standards, and in informing personal and policy
decisions about closing and reopening indoor spaces, such as schools and businesses.
We have here reformulated the COVID-19 indoor safety guideline of Bazant and Bush (2021),
expressing it in terms of cumulative exposure to CO2 , which can be readily monitored in real time formost indoor spaces. In so doing, we have built upon the seminal work of Rudnick and Milton (2003), as
was recently extended and applied to COVID-19 by Peng and Jimenez (2021). The guideline of Bazant
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Flow

E10-15and Bush (2021) makes clear that, since the risk of indoor airborne infection is determined by the total
volume of pathogen inhaled, safety limits intended to protect against it must be expressed in terms ofoccupancy time. Likewise, in the context of CO2 measurements, safety limits cannot be expressed solely
in terms of a limit on CO2 levels, but must also depend on occupancy time. As we have demonstrated
with our two case studies, the safety guideline, (5), when coupled with CO2 monitors, allows for realtime assessment of risk of airborne disease transmission R in in indoor spaces. Moreover, this approachhas the distinct advantage that one can assess certain key model parameters, including the backgroundconcentration of CO2 and the room’s ventilation rate, directly from the CO2 measurements.
Within a well-mixed space, CO2 is eﬀectively a passive scalar that tracks the ambient ﬂow, and isremoved only through exchange with outdoor air. Aerosol-borne pathogens are subject to additionalremoval mechanisms, including ﬁltration (by face masks and internal circulation), sedimentation anddeactivation. Thus, the concentration of CO2 cannot be taken as a proxy for those of an aerosolborne pathogen without resolving the proportionality constant between the two that results from these
additional removal mechanisms. We stress that the eﬀects of face mask use are dramatic in reducing theratio of aerosol-borne pathogens to CO2 concentration, and in reducing the risk of indoor transmission.
Finally, we note that the additional removal mechanisms acting on the droplet-borne pathogen alter
not only its equilibrium concentrations relative to that of CO2 , but their relaxation times in transient
situations, as may be treated using the mathematical formalism of Bazant and Bush (2021).
We emphasize that the caveats enumerated by Bazant and Bush (2021) concerning the limitations of
their safety guideline apply similarly here. First, there is considerable uncertainty in a number of model
parameters, including the critical viral load and relative susceptibility. While our inferences are consistent
with available data, we hope that these uncertainties will be reduced as more COVID-19 spreading eventsare characterized and analysed. Second, when masks are not worn, there is substantial additional risk ofshort-range airborne transmission from respiratory jets and plumes, as accompany breathing, speaking(Abkarian et al., 2020a; Abkarian & Stone, 2020), coughing and sneezing (Bourouiba et al., 2014).
Third, while the assumption of the well-mixed room is widely applied and represents a reasonable ﬁrst
approximation, it is known to have limitations (Bhagat et al., 2020; Linden, Lane-Serﬀ & Smeed, 1990;
Linden, 1999). Notably, measured ﬂuctuations in CO2 levels provide a direct means of assessing thevalidity of the well-mixed-room hypothesis, especially when several sensors are used simultaneously at
diﬀerent locations in the same space. Indeed, CO2 monitoring has been used for decades to assess thequality of air handling and zonal mixing in buildings (Cheng et al., 2011; Fisk & De Almeida, 1998;
Hung & Derossis, 1989; Salisbury, 1986; Seppänen et al., 1999), and can now be repurposed to assess
the risk of indoor airborne disease transmission.
Our transmission theory and safety guideline provide a quantitative basis for the use of CO2 monitors
in assessing the risk of indoor airborne disease transmission. Speciﬁcally, the simple guideline, (5),and mathematical formulae connecting CO2 data to the evolving transmission risk, (19)–(20), pave theway for real-time assessment of personal risk in indoor spaces. Our safety guideline might also be
implemented in spaces where not only CO2 is monitored, but also other relevant model parameters. For
example, changes in occupancy could be monitored at entrances and exits, while monitoring decibellevels and type of vocalization would serve to inform the infectious aerosol production rate (Asadi et al.,2019, 2020c; Morawska et al., 2009). One could further envision feeding all such data into air regulation
controls in order to ensure that our CO2 -based indoor safety limit is never violated. We note that sucha prospect would be most easily achieved in quasi-steady circumstances in which a ﬁxed population is
behaving in a predictable fashion over the course of an event of known duration, for example, students
in a lecture hall or passengers on a charter bus. In more complicated situations, our model provides
a framework for optimizing sensor-based, demand-controlled ventilation (Fisk & De Almeida, 1998),
with a view to limiting transmission risk while reducing energy consumption and system costs (Risbeck
et al. 2021a, 2021b). Finally, our model provides a general framework for using CO2 monitors to mitigate
the indoor airborne transmission of other respiratory illnesses, including the seasonal ﬂu.
In order to facilitate the application of our safety guideline, in the supplementary material we provide
a link to an online app that computes the safety guideline in terms of both room occupancy and CO2
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Martin Z. Bazant et al.

levels (Khan et al., 2020). The CO2 -based guideline, available in the app’s advanced mode, may be usedin conjunction with CO2 monitors to formulate safe reopening strategies for indoor spaces in the laterstages of the pandemic.
Acknowledgements. We thank L. Champion for valuable discussions and for sharing preliminary data and Roberto Rondanelli
for references.
Funding Statement. The authors received no funding for this work.
Declaration of Interests. The authors report no conﬂict of interest.
Data Availability Statement. Raw data are available from M.Z.B. and O.K.
Ethical Standards. The research meets all ethical guidelines, including adherence to the legal requirements of the study country.
Supplementary Material. Supplementary material are available at https://indoor-covid-safety.herokuapp.com.

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