key: cord-0706131-qnyj7no3 authors: Bazant, M. Z.; Kodio, O.; Cohen, A. E.; Khan, K.; Gu, Z.; Bush, J. W. M. title: Monitoring carbon dioxide to quantify the risk of indoor airborne transmission of COVID-19 date: 2021-04-07 journal: nan DOI: 10.1101/2021.04.04.21254903 sha: 54a6136c40812a32c1cad58dcf395ded1c7df8ab doc_id: 706131 cord_uid: qnyj7no3 A new guideline for mitigating indoor airborne transmission of COVID-19 prescribes a limit on the time spent in a shared space with an infected individual (Bazant & Bush, 2021). Here, we rephrase this safety guideline in terms of occupancy time and mean exhaled carbon dioxide concentration in an indoor space, thereby enabling the use of $mathrm{CO_2}$ monitors in the risk assessment of airborne transmission of respiratory diseases. While CO2 concentration is related to airborne pathogen concentration (Rudnick & Milton, 2003), the guideline 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 and deactivation. Critically, transmission risk depends on the total infectious dose, so necessarily depends on both the pathogen concentration and exposure time. The transmission risk is also modulated by the relative magnitudes of susceptible, infected and immune persons within a population, which evolve as the pandemic runs its course. A general mathematical theory is developed to predict airborne transmission risk from CO2 time series in real time and clarify various approximations that lead to the guideline. Illustrative examples of assessing transmission risk and implementing the guideline are presented using data from CO2 monitoring in university classrooms and office spaces. Coronavirus disease 2019 has caused a devastating global pandemic since it was first identified in Wuhan, China in December 2019 Li et al., 2020) . For over a year, public health guidance 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 et al., 2014) . The e cacy of these measures has been increasingly called into question, however, since there is scant evidence for fomite transmission (Lewis, 2021) and large-drop transmission is e ectively 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 (Prather et al., 2020; Morawska and Milton, 2020; Morawska and Cao, 2020; Jayaweera et al., 2020; Zhang et al., 2020b; Bazant and Bush, 2021) . Notably, airborne transmission provides the only rational explanation for the so-called "super-spreading events", which have now been well chronicled and all took place indoors (Miller et al., 2020; Moriarty, 2020; Hamner, 2020; Shen et al., 2020; Nishiura et al., 2020; Kwon et al., 2020; Hwang et al., 2020) . The dominance of indoor airborne transmission is further supported by the fact that face-mask directives have been more e ective in limiting the spread of COVID-19 than either social distancing directives or lockdowns (Zhang et al., 2020b; Stutt et al., 2020) . Indeed, a recent analysis of spreading data from Massachusetts public schools where masking was strictly enforced found no statistically significant e ect of social distance restrictions that ranged from 3 feet to 6 feet (van den Berg et al., 2021) . Finally, the detection of infectious SARS-CoV-2 virions suspended in hospital room air as far as 18 feet 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 approach to 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) and Riley et al. (1978) describes airborne transmission in an indoor space that is well-mixed by ambient air flows, so that infectious aerosols are uniformly dispersed throughout the space (Gammaitoni and Nucci, 1997; Beggs et al., 2003; Nicas et al., 2005; Noakes et al., 2006; Stilianakis and Drossinos, 2010) . While exceptions to the well-mixed room are known to arise (Bhagat et al., 2020) , supporting evidence for the well-mixed approximation may be found in both theoretical arguments (Bazant and Bush, 2021) and computer simulations of natural and forced convection (Foster and Kinzel, 2021) . The Wells-Riley model and its extensions have been applied to a number of super-spreading events and used to assess the risk of COVID-19 transmission in a variety of indoor settings (Miller et al., 2020; Buonanno et al., 2020b,a; Prentiss et al., 2020; Evans, 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 of occupants and the exposure time (Bazant and Bush, 2021) . This bound may be simply expressed in terms of the relevant variables, including the room dimensions, ventilation, air filtration, mask e ciency and respiratory activity. The guideline has been calibrated for COVID-19 using epidemiological data from the best characterized super-spreading events and incorporates the measured dependence of expiratory droplet-size distributions on respiratory and vocal activity (Morawska et al., 2009; Asadi et al., 2019 Asadi et al., , 2020a ). An online app has facilitated its widespread use during the pandemic (Khan et al., 2020) . The authors also considered the additional risk of turbulent respiratory plumes and jets (Abkarian et al., 2020a,b) , as need be considered when masks are not worn. The accuracy of the guideline is necessarily limited by uncertainties in a number of model parameters, which will presumably be reduced as more data is analyzed from indoor spreading events. Carbon dioxide measurements have been used for decades to quantify airflow and zonal mixing in buildings and so guide the design of heating, ventilation, and air-conditioning (HVAC) systems (Fisk and De Almeida, 1998; Seppänen et al., 1999) . Such measurements thus represent a natural source of data for assessing indoor air quality, especially as they rely only on relatively inexpensive, widely available CO 2 sensors. Quite generally, high carbon dioxide levels in indoor settings are known to be associated with poor health (Salisbury, 1986; Hung and Derossis, 1989; Seppänen et al., 1999) . Statistically significant correlations between CO 2 levels and illness-related absenteeism in both the work place (Milton et al., 2000) and classrooms Shendell et al., 2004) have been widely reported (Li et al., 2007) . Direct correlations between CO 2 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 dorm rooms have also been reported (Sun et al., 2011; Bueno de Mesquita et al., 2020) . Despite the overwhelming evidence of such correlations and the numerous economic analyses that underscore their negative societal impacts (Milton et al., 2000; Fisk, 2000) , using CO 2 monitors to make quantitative assessments of the risk of indoor disease transmission is a relatively recent notion (Li et al., 2007) . Rudnick and Milton (2003) first proposed the use of Wells-Riley models, in conjunction with measurements of CO 2 concentration, to assess airborne transmission risk indoors. Their model treats CO 2 concentration as a proxy for infectious aerosols: the two were assumed to be produced proportionally by the exhalation of an infected individual and removed at the same rate by ventilation. The current pandemic has generated considerable interest in using CO 2 monitoring as a tool for risk management of COVID-19 (Bhagat et al., 2020; Hartmann and Kriegel, 2020) . The Rudnick-Milton model has recently been extended by Peng and Jimenez (2020) through consideration of the di erent removal rates of CO 2 and airborne pathogen. They conclude by predicting safe CO 2 levels for COVID-19 transmission in various 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 by expressing the safety guideline of Bazant and Bush (2021) in terms of CO 2 concentration. Doing so makes clear that one must limit not only the CO 2 concentration, but the occupancy time. Our model accounts for the e ects of pathogen filtration, sedimentation and deactivation in addition to the variable aerosol production rates associated with di erent respiratory and vocal activities, all of which alter the relative concentrations of airborne pathogen and CO 2 . Our guideline thus quantifies the extent to which safety limits may be extended by mitigation strategies such as mask directives, air filtration and the imposition of 'quiet spaces'. In §2, we rephrase the indoor safety guideline of Bazant and Bush (2021) in terms of the room's carbon dioxide concentration. In §3, we present theoretical descriptions of the evolution of CO 2 concentration and infectious aerosol concentration in an indoor space, and highlight the di erent physical processes influencing the two. We then model the disease transmission dynamics, which allows for the risk of indoor airborne transmission to be assessed from CO 2 measurements taken in real time. In §4, we apply our model to a pair of data sets tracking the evolution of CO 2 concentration in specific o ce and classroom settings. These examples illustrate how CO 2 monitoring, when coupled with our safety guideline, provides a means of assessing and mitigating the risk of indoor airborne transmission of respiratory pathogens. We begin by recalling the safety guideline Bazant and Bush (2021) for limiting indoor airborne disease transmission in a well-mixed space. The guideline would impose an upper bound on the cumulative exposure time: where # C is the number of possible transmissions (pairs of infected and susceptible persons) and g is the time in the presence of the infected person(s). The risk tolerance n is chosen to bound the probability of one transmission. & 1 is the mean breathing flow rate and + the room volume. ⇠ @ is the infectiousness of exhaled air in units of "infection quanta" (Wells, 1955; Noakes and Sleigh, 2009 ) per volume for a given aerosolized pathogen, and is known to depend on the type of respiratory and vocal activity (resting, exercising, speaking, singing, etc.) (Buonanno et al., 2020b; Bazant and Bush, 2021) . The relative susceptibility B A acts as a scaling factor for ⇠ @ that accounts for di erences in the transmissibility of di erent respiratory pathogens, such as bacteria or viruses (Rudnick and Milton, 2003; Li et al., 2008) . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 7, 2021. ; https://doi.org/10.1101/2021.04.04.21254903 doi: medRxiv preprint with di erent strains (Volz et al., 2021; Davies et al., 2020) , and for di erences in the susceptibility of di erent populations, such as children and adults (Riediker and Morawska, 2020; Zhang et al., 2020a; Zhu et al., 2020) . The mask transmission probability, ? < = ? < (A), (Chen and Willeke, 1992; Oberg and Brosseau, 2008; Konda et al., 2020; Li et al., 2008) may depend on respiratory activity (Asadi et al., 2020b) and direction of airflow (Pan et al., 2020) , and is evaluated at the e ective aerosol radius A to be defined below. Finally, _ 2 = _ 2 (A) is the relaxation rate of the infectious aerosol-borne pathogen concentration, ⇠ (A, C). The reader is referred to Table 1 for a glossary of symbols. The size-dependent relaxation rate of the droplet-borne pathogen has four distinct contributions, where _ 0 is the ventilation rate (outdoor air exchanges per time). and _ 5 (A) = ? 5 (A)_ A is the filtration rate, where ? 5 (A) is the droplet removal e ciency for air filtration at a rate _ A (recirculated air changes per time). _ B (A) = E B (A) /+ is the net sedimentation rate for infectious droplets with the Stokes settling velocity E B (A) sedimenting through a well-mixed ambient to a floor of area (Corner and Pendlebury, 1951; Martin and Nokes, 1988) . Finally, _ E (A) is the deactivation rate of the aerosolized pathogen, which depends weakly on humidity and droplet size (Yang and Marr, 2011; Lin and Marr, 2019; Marr et al., 2019) , and may be enhanced by other factors such as ultraviolet (UV-C) irradiation (Hitchman, 2021; García de Abajo et al., 2020) , chemical disinfectants (Schwartz et al., 2020) , or cold plasma release (Filipi∆ et al., 2020; Lai et al., 2016) . Notably, only the first of the four removal rates apparent in Eq. ( 2) is relevant in the evolution of CO 2 ; thus, the concentrations of CO 2 and airborne pathogen are not strictly slaved to one another. Specifically, the proportionality between the two equilibrium concentrations varies in di erent indoor settings Peng and Jimenez (2020) , for example in response to room filtration Hartmann and Kriegel (2020) . Moreover, when transient e ects arise, for example, following the arrival of an infectious individual or the opening of a window, the two concentrations adjust at di erent rates. Finally, we note that there may also be sources of CO 2 other than human respiration, such as emissions from animals, stoves, furnaces, fireplaces, or carbonated beverages, as well as sinks of CO 2 , 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 CO 2 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 8= , which is the expected number of transmissions if an infectious person enters a room full of susceptible persons. Indeed, the safety guideline, Eq. (1), corresponds to the bound R 8= < n with the choice # C = # 1, and so would limit the risk of an infected person entering the room of occupancy # transmitting to any other during the exposure time g. If the epidemic is well underway or subsiding, the guideline should take into account the prevalence of infection ? 8 and immunity ? 8< (as achieved by previous exposure or vaccination) in the local population. Assuming a trinomial distribution of # persons who are infected, immune or susceptible, with mutually exclusive probabilities ? 8 , ? 8< and ? B = 1 ? 8 ? 8< , respectively, the expected number of infected-susceptible pairs is # (# 1) ? 8 ? B . It is natural to switch between these two limits (# C = # 1 and # C = # (# 1) ? 8 ? B ) when one infected person is expected to be in the room, # ? 8 = 1, and thus set One may thus account for the changing infection prevalence ? 8 and increasing immunity ? 8< in the local population as the pandemic evolves. The total rate of CO 2 production by respiration in the room is given by % 2 = #& 1 ⇠ 2,1 , where ⇠ 2,1 is the CO 2 concentration of exhaled air, approximately ⇠ 2,1 = 38, 000 ppm, although the net CO 2 production rate, & 1 ⇠ 2,1 varies considerably with body mass and physical activity (Persily and de Jonge, 2017) . If the production rate % 2 and the ventilation flow rate & = _ 0 + are constant, then the steadystate value of the excess CO 2 concentration, relative to the steady background concentration ⇠ 0 at zero . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 7, 2021. ; https://doi.org/10.1101/2021.04.04.21254903 doi: medRxiv preprint occupancy, is given by which is simply the ratio of the individual CO 2 flow rate, & 1 ⇠ 2,1 , to the ventilation flow rate per person, &/#. We note that the outdoor CO 2 concentration is typically in the range ⇠ 0 = 250 450 ppm, with higher values in urban environments (Prill et al., 2000) . In the absence of other indoor CO 2 sources, human occupancy in poorly ventilated spaces can easily lead to CO 2 levels of several thousand ppm. People have reported headaches, slight nausea, drowsiness, and decreased decision-making performance for levels above 1000 ppm Krawczyk et al., 2016) , while short exposures to much higher levels may go unnoticed. As an example of CO 2 limits in industry, the American Conference of Governmental Industrial Hygienists recommends a limit of 5000 ppm for an 8-hour period and 30,000 ppm for 10 minutes. A value of 40, 000 ppm is considered to be immediately life-threatening. The safety guideline, Eq. (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 CO 2 concentration has done likewise, and so approached the value expressed in Eq. (4), then the guideline can be recast as a bound on the safe mean excess CO 2 concentration, where we replace the steady excess CO 2 concentration with its time average, h⇠ 2 i ⇡ ⇠ 2,B , and define the mean quanta emission rate per infected person, _ @ = & 1 ⇠ @ . For the early to middle stages of an epidemic or when ? 8 and ? 8< are not known, we recommend setting #/# C = 1 < #/(# 1) ⇡ 1, for a conservative CO 2 bound that limits the indoor reproductive number. In the later stages of an epidemic, as the population approaches herd immunity (? 8 ! 0, ? 8< ! 1), the safe CO 2 bound diverges, #/# C ! 1, and so may be supplanted by the limits on carbon dioxide toxicity noted above, that lie in the range 5, 000 30, 000 ppm for 8-hour and 10-minute exposures. Our simple CO 2 -based safety guideline, Eq. (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 a limit on CO 2 concentration. The substantial increase in safe occupancy times, as one proceeds from the peak to the late stages of the pandemic, is evident in the di erence between the solid and dashed lines in Figure 1 , which were evaluated for the case of a typical classroom in the United States (Bazant and Bush, 2021) . This example shows the critical role of exposure time in determining the safe CO 2 level, a limit that can be increased dramatically by mask use and to a lesser extent by filtration. When infection prevalence ? 8 falls below 10 per 100,000 (an arbitrarily chosen small value), the chance of 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 in Figure 1 , could also be rationally managed by monitoring the CO 2 concentration and adhering to the guideline. We follow the traditional approach of modeling gas dynamics in a well mixed room (Shair and Heitner, 1974) , as a continuous stirred tank reactor (Davis and Davis, 2012) . Given the time dependence of occupancy, # (C), mean breathing flow rate, & 1 (C), and ventilation flow rate, & 0 (C) = _ 0 (C)+, one may express the evolution of the excess CO 2 concentration ⇠ 2 (C) in a well-mixed room through . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 7, 2021. where is the exhaled CO 2 production rate. The relaxation rate for excess CO 2 in response to changes in % 2 (C) is precisely equal to the ventilation rate, _ 0 (C) = & 0 (C)/+. For constant _ 0 , the general solution of Eq. (6) for ⇠ 2 (0) = 0 is given by which can be derived by Laplace transform or using an integrating factor. The time-averaged excess CO 2 concentration can be expressed as by switching the order of time integration. If % 2 (C) is slowly varying over the ventilation time scale _ 1 0 , the time-averaged CO 2 concentration may be approximated as where the excess CO 2 concentration approaches the ratio of the mean exhaled CO 2 production rate to the ventilation flow rate at long times, g _ 1 0 , as indicated in Eq. (4). Following Bazant and Bush (2021) , we assume that the radius-resolved concentration of infectious aerosol-borne pathogen, ⇠ (A, C), evolves according to where the mean production rate, depends on the number of infected persons in the room, (C), and the size distribution = 3 (A, C) of exhaled droplets of volume + 3 (A) containing pathogen (i.e. virions) at microscopic concentration, 2 E (A). The droplet size distribution is known to depend on expiratory and vocal activity (Morawska et al., 2009; Asadi et al., 2019 Asadi et al., , 2020c . Quite generally, the aerosols evolve according to a dynamic sorting process (Bazant and Bush, 2021) : the drop-size distribution evolves with time until an equilibrium distribution is obtained. Given the time evolution of excess CO 2 concentration, ⇠ 2 (C), one may deduce the radius-resolved pathogen concentration ⇠ (A, C) by integrating the coupled di erential equation, This integration can be done numerically or analytically via Laplace transform or integrating factors if one assumes that _ 0 , _ 2 , % and % 2 all vary slowly over the ventilation (air change) time scale, _ 1 0 . In that case, the general solution takes the form where we consider the infectious aerosol build-up from ⇠ (A, 0) = 0. According to Markov's inequality, the probability of at least one transmission taking place during the exposure time g is bounded above by the expected number of airborne transmissions, ) 0 (g), and the two become equal in the (typical) limit of rare transmissions, ) 0 (g) ⌧ 1. The expected number of transmissions to ((C) susceptible persons is obtained by integrating the mask-filtered inhalation rate of infection quanta over both droplet radius and time, where 2 8 (A) 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 in smaller aerosol droplets with A < 5`m 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 traverse the respiratory tract, absorb and coalesce onto exposed tissues, and allow pathogens to escape more quickly by di usion to infect target cells. The mask transmission probability ? < (A) also decreases rapidly with increasing drop size above the aerosol range for most filtration materials (Chen and Willeke, 1992; Oberg and Brosseau, 2008; Konda et al., 2020; Li et al., 2008) , so the integration over radius in Eq. (15) gives the most weight to the aerosol size range, roughly A < 5`m, which also coincides with the maxima in exhaled droplet size distributions (Morawska et al., 2009; Asadi et al., 2019 Asadi et al., , 2020c . The inverse of the infectivity, 2 1 8 , is equal to the "infectious dose" of pathogens from inhaled aerosol droplets that would cause infection with probability 1 (1/4) = 63%. Bazant and Bush (2021) estimated the infectious dose for SARS-CoV-2 to be on the order of ten aerosol-borne virions. Notably, the corresponding infectivity, 2 8 ⇠ 0.1, is an order of magnitude larger than previous estimates for SARS-CoV (Watanabe et al., 2010; Buonanno et al., 2020b) , which is consistent with only COVID-19 reaching pandemic status. The infectivity is known to vary across di erent age groups and pathogen strains, a variability that is captured by the relative susceptibility, B A . For example, Bazant and Bush (2021) suggest assigning B A = 1 for the elderly (over 65 years old), B A = 0.68 for adults (aged 15-64) and B A = 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 variant of concern of the lineage B.1.1.7 (VOC 202012/01), which recently emerged in the United Kingdom with a reproductive number that was 60% larger than that of the original strain (Volz et al., 2021; Davies et al., 2020) . Equations (14) and (15) provide an approximate solution to the full model that depends on the exhaled droplet size distribution, = 3 (A, C), and mean breathing rate, & 1 (C), of the population in the room. Since the droplet distributions = 3 (A) have only been characterized in certain idealized experimental conditions (Morawska et al., 2009; Asadi et al., 2020a,c) , it is useful to integrate over A to obtain a simpler model that can be directly calibrated for di erent modes of respiration using epidemiological data (Bazant and Bush, 2021) . Assuming & 1 (C), (C), ((C), # (C) and = 3 (A, C) vary slowly over the relaxation time _ 1 2 , we may substitute Eq. (14) into Eq. (15) and perform the time integral of the second term to obtain where = @ (A, C) = = 3 (A, C)+ 3 (A)2 E (A)2 8 (A) is the radius-resolved exhaled quanta concentration. Following Bazant and Bush (2021) , we define an e ective radius of infectious aerosols A such that where ⇠ @ (C) = Ø 1 0 = @ (A, C)3A 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 e ective radius A can be evaluated, given a complete knowledge of the dependence on drop radius of the mask filtration e ciency, ? < (A), and of all the factors that determine the exhaled quanta concentration, = @ (A, C) and pathogen removal rate, _ 2 (A). While these dependencies are not readily characterized, typical values of A are at the scale of several microns, based on the size dependencies of = 3 (A, C), 2 8 (A) and ? < (A) noted above. Further simplifications allow us to derive a formula relating CO 2 measurements to transmission risk. By assuming that ⇠ @ (C) varies slowly over the timescale of concentration relaxation, one may approximate the memory integral with the same e ective radius A. Thus, accounting for immunity and infection prevalence in the population via we obtain a formula for the expected number of airborne transmissions, in terms of the excess CO 2 time series, ⇠ 2 (C), where _ @ (C) = ⇠ @ (C)& 1 (C) is the mean quanta emission rate. It is also useful to define the expected transmission rate, which allows for direct assessment of airborne transmission risk based on CO 2 levels. A pair of examples of such assessments will be presented in §4. Notably, the mean airborne transmission rate per expected infected-susceptible pair, V 0 (C), reflects the environment's memory of the recent past, which persists over the pathogen relaxation time scale, _ 1 2 . The CO 2 concentration in Eq. (9) has a longer memory of past changes in CO 2 sources or ventilation, which persists over the air change time scale, _ 1 0 > _ 1 2 , since CO 2 is una ected by the filtration, sedimentation and deactivation rates evident in Eq. (2). The time delays between the production of CO 2 and infectious aerosols by exhalation and their buildup in the well mixed air of a room shows that CO 2 variation and airborne transmission are inherently non-Markovian stochastic processes. As such, any attempt to predict fluctuations in airborne transmission risk would require stochastic generalizations of the di erential equations governing the mean variables, Eqs. (6) and (11), and so represent a stochastic formulation of the Wells-Riley model (Noakes and Sleigh, 2009 ). Finally, we connect the general result, Eq. (19), with the CO 2 based safety guideline derived above, Eq. (5). Since ⇠ 2 (C) varies on the ventilation time scale, _ 1 0 , which is necessarily longer than the relaxation time scale of the infectious aerosols, _ 1 2 , we may assume that _ @ (C)⇠ 2 (C) is slowly varying and evaluate the integral in Eq. (19). We thus arrive at the approximation Since _ @ (C)⇠ 2 (C) is slowly varying, the second term in brackets is negligible relative to the first for times longer than the ventilation time, g _ 1 0 > _ 1 2 . In this limit, the imposed bound on expected transmissions, ) 0 (g) < n, is approximated by This formula reduces to the safety guideline, Eq. (5), in the limit of constant mean quanta emission rate, _ @ , which confirms the consistency of our assumptions. We proceed by illustrating the process by which the guideline, Eq. (5), can be coupled to real data obtained from CO 2 monitors. Specifically, we consider time series of CO 2 concentration gathered in classroom and in o ce 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. Social distancing guidelines were adhered to, and masks were worn by all participants. We assume a constant exhaled CO 2 concentration of 38, 000 ppm, and use the global minimum of the CO 2 series as the background CO 2 level ⇠ 0 from which the excess concentration ⇠ 2 (C) was deduced. Notably, the relatively small fluctuations in the CO 2 measurements recorded in a variety of settings support the notion of a well-mixed room. From Equations (19) and (20), we calculate the expected number of transmissions and transmission rate, assuming that there is one infected person in the room (# C = # 1). In this case, the expected number of transmissions is equal to the indoor reproductive number, ) 0 (g) = R 8= (g), and the transmission rate is 3) 0 3C (g) = # C V 0 (g). We choose realistic values of the parameters that fall within the typical ranges estimated by Bazant and Bush (2021) . 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