2.1 Occupancy-based safety guideline

We begin by recalling the safety guideline of Reference Bazant and BushBazant and Bush (Reference Bazant and Bush2021) for limiting indoor airborne disease transmission in a well-mixed space. The guideline would impose an upper bound on the cumulative exposure time,

(1)\begin{equation} N_t \tau < \frac{\epsilon \overline{\lambda}_c V }{Q_b^{2} C_q s_r \overline{p}_m^{2}}, \end{equation}

where $N_t$ is the number of possible transmissions (pairs of infected and susceptible people) and $\tau$ is the time 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 $Q_b$ is the mean breathing flow rate and $V$ the room volume. The risk tolerance $\epsilon <1$ is the prescribed bound on the probability of at least one transmission, as should be chosen judiciously according to the vulnerability of the population (Reference GargGarg, 2020); for example, Reference Bazant and BushBazant and Bush (Reference Bazant and Bush2021) suggested $\epsilon =10\,\%$ for children and 1 % for the elderly.

Table 1.

Glossary of symbols arising in our theory, their units and characteristics values.

The only epidemiological parameter in the guideline, $C_q$, 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 Reference WellsWells (Reference Wells1955), is widely used in epidemiology to measure the expected rate of 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 per volume, $C_q$, can thus be associated with exhaled air without reference to the microscopic pathogen concentration. Notably, $C_q$ is known to depend on the type of respiratory and vocal activity (associated with an individual resting, exercising, speaking, singing, etc.), being larger for the more vigorous activities (Reference Bazant and BushBazant & Bush, 2021; Reference Buonanno, Stabile and MorawskaBuonanno et al., 2020b). The relative susceptibility $s_r$ is introduced as a scaling factor for $C_q$ that accounts for differences in the transmissibility of different respiratory pathogens, such as bacteria or viruses (Reference Li, Guo, Wong, Chung, Gohel and LeungLi et al., 2008; Reference Rudnick and MiltonRudnick & Milton, 2003) with different strains (Reference Davies, Barnard, Jarvis, Kucharski, Munday, Pearson and EdmundsDavies et al., 2020; Reference Volz, Mishra, Chand, Barrett, Johnson, Geidelberg and FergusonVolz et al., 2021), and for differences in the susceptibility of different populations, such as children and adults (Reference Riediker and MorawskaRiediker & Morawska, 2020; Reference Zhang, Litvinova, Liang, Wang, Wang, Zhao and YuZhang et al., 2020a; Reference Zhu, Bloxham, Hulme, Sinclair, Tong, Steele and ShortZhu et al., 2020).

The mask penetration probability, $p_m(r)$, is a function of drop size that is bounded below by 0 (for the 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 to penetrate (Reference Chen and WillekeChen & Willeke, 1992), but these values should be increased by a factor of 2–10 to account for imperfect fit (Reference Oberg and BrosseauOberg & Brosseau, 2008). Cloth masks show much greater variability (Reference Konda, Prakash, Moss, Schmoldt, Grant and GuhaKonda et al., 2020b). The mask penetration probability may also depend on respiratory activity (Reference Asadi, Cappa, Barreda, Wexler, Bouvier and RistenpartAsadi et al., 2020b) and direction of airflow (Reference Pan, Harb, Leng and MarrPan, Harb, Leng & Marr, 2020). Here, for the sake of simplicity, we treat $p_m(r)$ as being constant over the limited aerosol size range of interest, and evaluate $\bar {p}_m=p_m(\bar {r})$ at the effective aerosol radius $\overline {r}$ to be defined below, above which drops tend to settle to the ground faster than they are swept away by ventilation. For this effective aerosol filtration factor, Reference Bazant and BushBazant and Bush (Reference Bazant and Bush2021) suggested $\bar {p}_m=1\,\%\text {--}5\,\%$ for surgical masks (Reference Li, Guo, Wong, Chung, Gohel and LeungLi et al., 2008; Reference Oberg and BrosseauOberg & Brosseau, 2008), $\bar {p}_m=10\,\%\text {--}40\,\%$ for hybrid cloth face coverings and $\bar {p}_m=40\,\%\text {--}80\,\%$ for single-layer fabrics (Reference Konda, Prakash, Moss, Schmoldt, Grant and GuhaKonda et al., 2020b). Notably, even low quality masks can significantly reduce transmission risk since the bound on cumulative exposure time, (1), scales as $\bar {p}_m^{-2}$.

Finally, we define $\overline {\lambda }_c=\lambda _c(\overline {r})$ as an effective relaxation rate of the infectious aerosol-borne pathogen concentration, $C(r,t)$, evaluated at the effective aerosol radius $\bar {r}$. The size-dependent relaxation rate of the droplet-borne pathogen has four distinct contributions:

(2)\begin{equation} \lambda_c(r) = \lambda_a + \lambda_f(r) + \lambda_s(r) + \lambda_v(r). \end{equation}

Here, $\lambda _a$ is the ventilation rate, specifically the rate of exchange with outdoor air; $\lambda _f(r)=p_f(r)\lambda _r$ is the filtration rate, where $p_f(r)$ is the droplet removal efficiency for air filtration at a rate $\lambda _r$ (recirculated air changes per time); $\lambda _s(r)=v_s(r)A/V$ is the net sedimentation rate for infectious droplets with the Stokes settling speed $v_s(r)$ sedimenting through a turbulent, well-mixed ambient in a room of height $H = V/A$, volume $V$ and floor area $A$ (Reference Corner and PendleburyCorner & Pendlebury, 1951; Reference Martin and NokesMartin & Nokes, 1988); $\lambda _v(r)$ is the deactivation rate of the aerosolized pathogen, which depends weakly on humidity and droplet size (Reference Lin and MarrLin & Marr, 2019; Reference Marr, Tang, Van Mullekom and LakdawalaMarr, Tang, Van Mullekom & Lakdawala, 2019; Reference Yang and MarrYang & Marr, 2011), and may be enhanced by other factors such as ultraviolet (UV-C) irradiation (Reference García de Abajo, Hernández, Kaminer, Meyerhans, Rosell-Llompart and Sanchez-ElsnerGarcía de Abajo et al., 2020; Reference HitchmanHitchman, 2021), chemical disinfectants (Reference Schwartz, Stiegel, Greeson, Vogel, Thomann, Brown and LewisSchwartz et al., 2020) or cold plasma release (Reference Filipić, Gutierrez-Aguirre, Primc, Mozetič and DobnikFilipić, Gutierrez-Aguirre, Primc, Mozetič & Dobnik, 2020; Reference Lai, Cheung, Wong and LiLai, Cheung, Wong & Li, 2016).

Notably, only the first of the four removal rates enumerated in (2) is relevant in the evolution of ${\rm CO}_{2}$; thus, the concentrations of ${\rm CO}_{2}$ and airborne pathogen may evolve independently. Specifically, the proportionality between the two equilibrium concentrations varies in different indoor settings (Reference Peng and JimenezPeng & Jimenez, 2021), for example in response to room filtration (Reference Hartmann and KriegelHartmann & Kriegel, 2020). Moreover, when transient effects arise, for example, following the arrival of an infectious individual or the opening of a window, the two concentrations adjust at different 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 drinks, as well as sinks of CO$_2$, such as plants, construction materials or pools of water, which we neglect for simplicity. As such, following Reference Rudnick and MiltonRudnick and Milton (Reference Rudnick and Milton2003), 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, $\mathcal {R}_{\textrm {in}}$, 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 $\mathcal {R}_{\textrm {in}}<\epsilon$ with the choice $N_t=N-1$, and so would limit to $\epsilon$ the risk of an infected person entering the room of occupancy $N$ transmitting to any other person during the exposure time $\tau$. If the epidemic is well underway or subsiding, the guideline should take into account the prevalence of infection $p_i$ and immunity $p_{\textrm {im}}$ (as achieved 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 $p_i$, $p_{\textrm {im}}$ and $p_s = 1 - p_i - p_{\textrm {im}}$, respectively, the expected number of infected–susceptible pairs is $N(N-1)p_ip_s$. It is natural to switch between these two limits ($N_t=N-1$ and $N_t=N(N-1)p_ip_s$) when one infected person is expected to be in the room, $Np_i=1$, and thus set

(3)\begin{equation} N_t = (N-1) \min\{1, N p_i (1-p_i - p_{\textrm{im}})\}.\end{equation}

One may thus account for the changing infection prevalence $p_i$ and increasing immunity $p_{\textrm {im}}$ in the local population as the pandemic evolves.

2.2 ${CO}_{\textit{2}}$-based safety guideline

The outdoor ${\rm CO}_{2}$ concentration is typically in the range $C_0=350\text {--}450$ p.p.m., with higher values in urban environments (Reference PrillPrill et al., 2000) and lower values in forests (Reference Higuchi, Worthy, Chan and ShashkovHiguchi, Worthy, Chan & Shashkov, 2003). In the absence of other indoor ${\rm CO}_{2}$ sources, human occupancy in poorly ventilated spaces can lead to ${\rm CO}_{2}$ levels of several thousand p.p.m. People have reported headaches, slight nausea, drowsiness and diminished decision-making performance for levels above 1000 p.p.m. (Reference Fisk, Satish, Mendell, Hotchi and SullivanFisk, Satish, Mendell, Hotchi & Sullivan, 2013; Reference Krawczyk, Rodero, Gładyszewska-Fiedoruk and GajewskiKrawczyk, Rodero, Gładyszewska-Fiedoruk & Gajewski, 2016), while short exposures to much higher levels may go unnoticed. As an example of ${\rm CO}_{2}$ limits in industry, the American Conference of Governmental Industrial Hygienists recommends a limit of 5000 p.p.m. for an eight-hour 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 $C_0$ the background concentration that would arise in the room at zero occupancy, and by $C_2$ the excess CO$_2$ concentration (relative to $C_0$) associated with human respiration. The total rate of ${\rm CO}_{2}$ production by respiration in the room is given by $P_2=N Q_b C_{2,b}$, where $C_{2,b}$ is the ${\rm CO}_{2}$ concentration of exhaled air, approximately $C_{2,b} = 38\,000$ p.p.m., although the net ${\rm CO}_{2}$ production rate, $Q_b C_{2,b}$ varies considerably with body mass and physical activity (Reference Persily and de JongePersily & de Jonge, 2017). If the production rate $P_2$ and the ventilation flow rate $Q = \lambda _a V$ are constant, then the steady-state value of the excess ${\rm CO}_{2}$ concentration,

(4)\begin{equation} C_{2,s} = \frac{P_2}{Q} = \frac{Q_b C_{2,b} N}{\lambda_a V},\end{equation}

is simply the ratio of the individual ${\rm CO}_{2}$ flow rate, $Q_b C_{2,b}$, 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 ${\rm CO}_{2}$ concentration has done likewise, and so approached the value expressed in (4), then the guideline can be recast as a bound on the safe mean excess ${\rm CO}_{2}$ concentration,

(5)\begin{equation} \langle C_2 \rangle \tau = \int_0^{\tau} C_2\,\textrm{d} t< \frac{\epsilon C_{2,b}}{\lambda_q s_r \overline{p}_m^{2}} \frac{\overline{\lambda}_c}{\lambda_a} \frac{N}{N_t},\end{equation}

where we replace the steady excess ${\rm CO}_{2}$ concentration with its time average, $\langle C_2 \rangle \approx C_{2,s}$, and define the mean quanta emission rate per infected person, $\lambda _q=Q_bC_q$. For the early to middle stages of an epidemic or when $p_i$ and $p_{\textrm {im}}$ are not known, we recommend setting $N/N_t = 1 < N/(N-1) \approx 1$, for a conservative ${\rm CO}_{2}$ bound that limits the indoor reproductive number. In the later stages of an epidemic, as the population approaches herd immunity ($p_i\to 0$, $p_{\textrm {im}}\to 1$), the safe ${\rm CO}_{2}$ bound diverges, $N/N_t \to \infty$, and so may be supplanted by the limits on ${\rm CO}_{2}$ toxicity noted above, that lie in the range 5000–30 000 p.p.m. for eight hour and 10 minute exposures, respectively. Our simple ${\rm CO}_{2}$-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 a limit on ${\rm CO}_{2}$ 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 difference between the solid and dashed lines in figure 1, which were evaluated for the case of a typical classroom in the USA (Reference Bazant and BushBazant & Bush, 2021). This example shows the critical role of exposure time in determining the safe ${\rm CO}_{2}$ level, a limit that can be increased dramatically by efficient mask use and to a lesser extent by filtration. When infection prevalence $p_i$ 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 ${\rm CO}_{2}$ concentration and adhering to the guideline.

Figure 1.

Illustration of the safety guideline, (5), which bounds the safe excess $\textit{CO}_{\textit{2}}$ (p.p.m.) and exposure time $\tau$ (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 $N=\textit{25}$ occupants, assumed to be children engaging in normal speech and light activity ($\lambda _q s_r = \textit{30}$ quanta h$^{-1}$) with moderate risk tolerance ($\epsilon =\textit{10 %}$). Comparison with the most restrictive bound on the indoor reproductive number without any precautions (red line) indicates that the safe $\textit{CO}_{\textit{2}}$ level or occupancy time is increased by at least an order of magnitude by the use of face masks (blue line), even with relatively inconsistent use of cloth masks ($p_m=\textit{30 %}$). The effect of air filtration (green line) is relatively small, shown here for a case of efficient HEPA filtration ($p_f=\textit{99 %}$) with 17 % outdoor air fraction ($\lambda _f=\textit{5} \lambda _a$). All three bounds are increased by several orders of magnitude (dashed lines) during late pandemic conditions ($p_ip_s=\textit{10}$ per 100 000), when it becomes increasingly unlikely to find an infected–susceptible pair in the room. The other parameters satisfy $(\lambda _v+\lambda _s)/\lambda _a=\textit{0.5}$, as could correspond to, for example, $\lambda _v=\textit{0.3}$, $\lambda _s=\textit{0.2}$ and $\lambda _a=\textit{1}$ h$^{-1}$ (1 ACH).

3.1 $CO_{\textit{2}}$ dynamics

We follow the traditional approach of modelling gas dynamics in a well-mixed room (Reference Shair and HeitnerShair & Heitner, 1974), as a continuous stirred tank reactor (Reference Davis and DavisDavis & Davis, 2012). Given the time dependence of occupancy, $N(t)$, mean breathing flow rate, $Q_b(t)$, and ventilation flow rate, $Q_a(t)= \lambda _a(t) V$, one may express the evolution of the excess ${\rm CO}_{2}$ concentration $C_2(t)$ in a well-mixed room through

(6)\begin{equation} V \frac{\textrm{d} C_2}{\textrm{d} t} = P_2(t) - Q_a(t) C_2, \end{equation}

where

(7)\begin{equation} P_2(t) = N(t) Q_b(t) C_{2,b} \end{equation}

is the exhaled ${\rm CO}_{2}$ production rate. The relaxation rate for excess ${\rm CO}_{2}$ in response to changes in $P_2(t)$ is precisely equal to the ventilation rate, $\lambda _a(t) = Q_a(t)/V$. For constant $\lambda _a$, the general solution of (6) for $C_2(0)=0$ is given by

(8)\begin{equation} C_2(t) = \int_0^{t} \exp({-\lambda_a(t-t')}) \frac{P_2(t')}{V} \textrm{d} t', \end{equation}

which can be derived by Laplace transform or using an integrating factor. The time-averaged excess ${\rm CO}_{2}$ concentration can be expressed as

(9)\begin{equation} \langle C_2 \rangle = \frac{1}{\tau}\int_0^{\tau} \left(1 - \exp({-\lambda_a(\tau-t)})\right)\frac{P_2(t)}{Q_a} \textrm{d} t \end{equation}

by switching the order of time integration. If $P_2(t)$ is slowly varying over the ventilation time scale $\lambda _a^{-1}$, the time-averaged ${\rm CO}_{2}$ concentration may be approximated as

(10)\begin{equation} \langle C_2 \rangle \approx \frac{\langle P_2 \rangle}{Q_a} - \left(\frac{1-\textrm{e}^{-\lambda_a\tau}}{\lambda_a\tau}\right) \frac{P_2(\tau)}{Q_a},\end{equation}

where the excess ${\rm CO}_{2}$ concentration approaches the ratio of the mean exhaled ${\rm CO}_{2}$ production rate to the ventilation flow rate at long times, $\tau \gg \lambda _a^{-1}$, as indicated in (4).

3.2 Infectious aerosol dynamics

Following Reference Bazant and BushBazant and Bush (Reference Bazant and Bush2021), we assume that the radius-resolved concentration of infectious aerosol-borne pathogen, $C(r,t)$, evolves according to

(11)\begin{equation} V \frac{\partial C}{\partial t} = P(r,t) - \lambda_c(r,t) V C, \end{equation}

where the mean production rate,

(12)\begin{equation} P(r,t) = I(t)Q_b(t)n_d(r,t)V_d(r)p_m(r)c_v(r), \end{equation}

depends on the number of infected people in the room, $I(t)$, and the size distribution $n_d(r,t)$ of exhaled droplets of volume $V_d(r)$ containing a pathogen (i.e. virion) at microscopic concentration, $c_v(r)$. The droplet size distribution is known to depend on expiratory and vocal activity (Reference Asadi, Wexler, Cappa, Barreda, Bouvier and RistenpartAsadi et al., 2019, Reference Asadi, Wexler, Cappa, Barreda, Bouvier and Ristenpart2020c; Reference Morawska, Johnson, Ristovski, Hargreaves, Mengersen, Corbett and KatoshevskiMorawska et al., 2009). Quite generally, the aerosols evolve according to a dynamic sorting process (Reference Bazant and BushBazant & Bush, 2021): the drop-size distribution evolves with time until an equilibrium distribution obtains.

Given the time evolution of excess ${\rm CO}_{2}$ concentration, $C_2(t)$, one may deduce the radius-resolved pathogen concentration $C(r,t)$ by integrating the coupled differential equation

(13)\begin{equation} \frac{\partial C}{\partial t} + \lambda_c(r,t) C = \frac{P(r,t)}{P_2(t)} \left(\frac{\textrm{d} C_2}{\textrm{d} t} + \lambda_a(t) C_2 \right). \end{equation}

This integration can be done numerically or analytically via Laplace transform or integrating factors if one assumes that $\lambda _a$, $\lambda _c$, $P$ and P ₂ all vary slowly over the ventilation (air change) time scale, $\lambda _a^{-1}$. In that case, the general solution takes the form

(14)\begin{equation} C(r,t) \approx \frac{P}{P_2}\left(C_2(t) + (\lambda_a-\lambda_c(r)) \int_0^{t} \exp({-\lambda_c(r)(t-t')})C_2(t')\,\textrm{d} t'\right), \end{equation}

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 $\tau$ is bounded above by the expected number of airborne transmissions, $T_a(\tau )$, and the two become equal in the (typical) limit of rare transmissions, $T_a(\tau )\ll 1$. The expected number of transmissions to $S(t)$ susceptible people is obtained by integrating the mask-filtered inhalation rate of infection quanta over both droplet radius and time:

(15)\begin{equation} T_a(\tau) = \int_0^{\tau} S(t) Q_b(t) s_r \left( \int_0^{\infty} C(r,t) c_i(r) p_m(r)\,\textrm{d} r \right)\textrm{d} t , \end{equation}

where $c_i(r)$ 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 in smaller aerosol droplets with $r < 5\ \mathrm {\mu }$m to be more infectious than those in larger drops, as reported by Reference Santarpia, Herrera, Rivera, Ratnesar-Shumate, Reid, Denton and LoweSantarpia et al. (Reference Santarpia, Herrera, Rivera, Ratnesar-Shumate, Reid, Denton and Lowe2020) for SARS-CoV-2, on the grounds that smaller drops more easily penetrate the respiratory tract, absorb and coalesce onto exposed tissues, and allow pathogens to escape more quickly by diffusion to infect target cells. The mask penetration probability $p_m(r)$ also decreases rapidly with increasing drop size above the aerosol range for most filtration materials (Reference Chen and WillekeChen & Willeke, 1992; Reference Konda, Prakash, Moss, Schmoldt, Grant and GuhaKonda et al., 2020b; Reference Li, Guo, Wong, Chung, Gohel and LeungLi et al., 2008; Reference Oberg and BrosseauOberg & Brosseau, 2008), so the integration over all radii in (15) gives the most weight to the aerosol size range, roughly $r < 5\ \mathrm {\mu }$m. We note that this range includes the maxima in exhaled droplet size distributions (Reference Asadi, Wexler, Cappa, Barreda, Bouvier and RistenpartAsadi et al., 2019, Reference Asadi, Wexler, Cappa, Barreda, Bouvier and Ristenpart2020c; Reference Morawska, Johnson, Ristovski, Hargreaves, Mengersen, Corbett and KatoshevskiMorawska et al., 2009) as defined in terms of either number or total volume in the aerosol range (Reference Bazant and BushBazant & Bush, 2021); nevertheless, larger droplets may also contribute to airborne transmission (Reference Tang, Bahnfleth, Bluyssen, Buonanno, Jimenez, Kurnitski and DancerTang et al., 2021).

The inverse of the infectivity, $c_i^{-1}$, is equal to the ‘infectious dose’ of pathogens from inhaled aerosol droplets that would cause infection with probability $1-(1/e)=63\,\%$. Reference Bazant and BushBazant and Bush (Reference Bazant and Bush2021) estimated the infectious dose for SARS-CoV-2 to be of the order of 10 aerosol-borne virions. Notably, the corresponding infectivity, $c_i \sim 0.1$, is an order of magnitude larger than previous estimates for SARS-CoV (Reference Buonanno, Stabile and MorawskaBuonanno et al., 2020b; Reference Watanabe, Bartrand, Weir, Omura and HaasWatanabe, 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, $s_r$. For example, Reference Bazant and BushBazant and Bush (Reference Bazant and Bush2021) suggest assigning $s_r=1$ for the elderly (over 65 years old), $s_r=0.68$ for adults (aged 15–64) and $s_r=0.23$ 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 (Reference Zhang, Litvinova, Liang, Wang, Wang, Zhao and YuZhang 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 (Reference Davies, Barnard, Jarvis, Kucharski, Munday, Pearson and EdmundsDavies et al., 2020; Reference Volz, Mishra, Chand, Barrett, Johnson, Geidelberg and FergusonVolz et al., 2021). Based on the latest epidemiological data (Reference Abdool Karim and de OliveiraAbdool Karim & de Oliveira, 2021), we likewise suggest multiplying $s_r$ 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 s_r for more recent strains are provided on our online app (Reference Khan, Bazant and BushKhan, 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 time of publication.

3.4 Approximate formula for the airborne transmission risk from $CO_{\textit{2}}$ measurements

Equations (14) and (15) provide an approximate solution to the full model that depends on the exhaled droplet size distribution, $n_d(r,t)$, and mean breathing rate, $Q_b(t)$, of the population in the room. Since the droplet distributions $n_d(r)$ have only been characterized in certain idealized experimental conditions (Reference Asadi, Bouvier, Wexler and RistenpartAsadi et al., 2020a, Reference Asadi, Wexler, Cappa, Barreda, Bouvier and Ristenpart2020c; Reference Morawska, Johnson, Ristovski, Hargreaves, Mengersen, Corbett and KatoshevskiMorawska 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 (Reference Bazant and BushBazant & Bush, 2021). Assuming $Q_b(t)$, $I(t)$, $S(t)$, $N(t)$ and $n_d(r,t)$ vary slowly over the relaxation time $\overline {\lambda }_c^{-1}$, we may substitute (14) into (15) and perform the time integral of the second term to obtain

(16)\begin{align} T_a(\tau) \approx \frac{s_r \lambda_a}{C_{2,b}} \int_0^{\tau} \int_0^{\infty} \frac{n_q(r,t)p_m(r)^{2}}{\lambda_c(r)} \frac{Q_b(t) I(t) S(t) C_2(t)}{N(t)} \left[ 1 + \left(\frac{\lambda_c(r)}{\lambda_a} - 1 \right) \exp({-\lambda_c(r)(\tau-t)})\right]\textrm{d} r\,\textrm{d} t, \end{align}

where $n_q(r,t)=n_d(r,t)V_d(r)c_v(r)c_i(r)$ is the radius-resolved exhaled quanta concentration.

Following Reference Bazant and BushBazant and Bush (Reference Bazant and Bush2021), we define an effective radius of infectious aerosols $\overline {r}$ such that

(17)\begin{equation} \int_0^{\infty} \frac{n_q(r,t) p_m(r)^{2}}{\lambda_c(r)} \textrm{d} r \equiv \frac{C_q(t) p_m(\overline{r})^{2}}{\lambda_c(\overline{r})},\end{equation}

where $C_q(t) = \int _0^{\infty } n_q(r,t)\,\textrm {d} r$ 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 $\overline {r}$ can be evaluated, given a complete knowledge of the dependence on drop radius of the mask penetration probability, $p_m(r)$, and of all the factors that determine the exhaled quanta concentration, $n_q(r,t)$ and pathogen removal rate, $\lambda _c(r)$. While these dependencies are not readily characterized, typical values of $\overline {r}$ are at the scale of several microns, based on the size dependencies of $n_d(r,t)$, $c_i(r)$ and $p_m(r)$ noted above.

Further simplifications allow us to derive a formula relating ${\rm CO}_{2}$ measurements to transmission risk. By assuming that $C_q(t)$ varies slowly over the time scale of concentration relaxation, one may approximate the memory integral with the same effective radius $\overline {r}$. Thus, accounting for immunity and infection prevalence in the population via

(18)\begin{equation} \frac{I(t) S(t)}{N(t)} \approx \frac{N_t}{N} = \left(1 - \frac{1}{N}\right) \min\{1,Np_ip_s\}, \end{equation}

we obtain a formula for the expected number of airborne transmissions,

(19)\begin{equation} T_a(\tau) \approx \frac{s_r \overline{p}_m^{2}}{C_{2,b}} \frac{\lambda_a}{\overline{\lambda}_c} \frac{N_t}{N} \int_0^{\tau} \lambda_q(t) C_2(t)\left[ 1 + \left(\frac{\overline{\lambda}_c}{\lambda_a} - 1\right) \exp({-\overline{\lambda}_c(\tau-t)})\right]\textrm{d} t, \end{equation}

in terms of the excess ${\rm CO}_{2}$ time series, $C_2(t)$, where $\lambda _q(t)=C_q(t)Q_b(t)$ is the mean quanta emission rate. It is also useful to define the expected transmission rate,

(20)\begin{align} \frac{\textrm{d} T_a}{\textrm{d} t}(\tau) = N_t \beta_a(\tau) = \frac{s_r \overline{p}_m^{2}}{C_{2,b}} \frac{\lambda_a}{\overline{\lambda}_c} \frac{N_t}{N} \left[\lambda_q(\tau) C_2(\tau) + (\lambda_a - \overline{\lambda}_c) \int_0^{\tau} \lambda_q(t)C_2(t) \exp({-\overline{\lambda}_c(\tau-t)})\,\textrm{d} t\right], \end{align}

which allows for direct assessment of airborne transmission risk based on ${\rm CO}_{2}$ levels. A pair of examples of such assessments will be presented in § 4.

Notably, the mean airborne transmission rate expected per infected–susceptible pair, $\beta _a(t)$, reflects the environment's memory of the recent past, which persists over the pathogen relaxation time scale, $\overline {\lambda }_c^{-1}$. Likewise, the ${\rm CO}_{2}$ concentration defined in (9) has a memory of recent changes in ${\rm CO}_{2}$ sources or ventilation, which persists over the air change time scale. Notably, the airborne pathogen concentration equilibrates more rapidly than CO$_2$; specifically, $\lambda _a^{-1} > \overline {\lambda }_c^{-1}$, since ${\rm CO}_{2}$ is unaffected by the filtration, sedimentation and deactivation rates enumerated in (2). The time delays between the production of ${\rm CO}_{2}$ and infectious aerosols by exhalation and their build-up in the well-mixed air of a room shows that ${\rm CO}_{2}$ 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 differential equations governing the mean variables, (6) and (11), and so represent a stochastic formulation of the Wells–Riley model (Reference Noakes and SleighNoakes & Sleigh, 2009).

3.5 Reduction to the $CO_{\textit{2}}$-based safety guideline

Finally, we connect the general result, (19), with the ${\rm CO}_{2}$-based safety guideline derived above, (5). Since $C_2(t)$ varies on the ventilation time scale $\lambda _a^{-1}$, which is necessarily longer than the relaxation time scale of the infectious aerosols $\overline {\lambda }_c^{-1}$, we may assume that $\lambda _q(t)C_2(t)$ is slowly varying and evaluate the integral in (19). We thus arrive at the approximation

(21)\begin{equation} T_a(\tau) \approx \frac{s_r \overline{p}_m^{2}}{C_{2,b}} \frac{\lambda_a}{\overline{\lambda}_c} \frac{N_t}{N} \left[\langle \lambda_q C_2 \rangle \tau + \frac{\lambda_q(\tau)C_2(\tau)}{\overline{\lambda}_c} \left(\frac{\overline{\lambda}_c}{\lambda_a}-1 + \textrm{e}^{-\overline{\lambda}_c\tau} \right) \right]. \end{equation}

Since $\lambda _q(t)C_2(t)$ is slowly varying, the second term in brackets is negligible relative to the first for times longer than the ventilation time, $\tau \gg \lambda _a^{-1} > \overline {\lambda }_c^{-1}$. In this limit, the imposed bound on expected transmissions, $T_a(\tau ) < \epsilon$, is approximated by

(22)\begin{equation} \langle \lambda_q C_2 \rangle \tau = \int_0^{\tau} \lambda_q(t)C_2(t)\,\textrm{d} t < \frac{\epsilon C_{2,b}}{s_r \overline{p}_m^{2}} \frac{\overline{\lambda}_c}{\lambda_a} \frac{N}{N_t}.\end{equation}

This formula reduces to the safety guideline, (5), in the limit of constant mean quanta emission rate, $\lambda _q$, which confirms the consistency of our assumptions.

4.1 Small office with two workers

Figure 2a shows $\text{CO}_{2}$ measurements taken in an office of length $L = 4.2$ m, width $W = 3$ m and height $H = 3$ 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 were speaking, we use $C_q=72$ quanta m$^{-3}$ (Reference Bazant and BushBazant & Bush, 2021). We compute the room's ventilation rate, $\lambda _a \approx 2.3\ {\rm h}^{-1}$, from the exponential relaxation that follows the occupants’ exit from the office (as indicated by the orange curve in figure 2). The office was equipped with moderate ventilation, which we characterize with $p_f=0.99$ and $\lambda _r = 6$ h$^{-1}$. The background concentration $C_0 = 420$ p.p.m. is evident both before and after occupancy.

Figure 2.

Measured $\textit{CO}_{2}$ concentration and calculated transmission rate in a two-person office. (a) Black dots represent the concentration of $\textit{CO}_{2}$. The solid blue, dashed magenta and dash–dot green curves represents the transmission rate, as calculated from (20) for three different scenarios, two of 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, $\lambda _a = \textit{2.3}$ h$^{-1}$, and the background CO$_2$ concentration, $C_0 = \textit{420}$ p.p.m. (b) Corresponding blue, magenta and green curves, deduced by integrating (20), indicate the total risk of transmission over the time of shared occupancy. If the pair were not wearing masks, the safety limit $\mathcal {R}_{\textrm {in}} < \textit{0.1}$ would be violated after approximately an hour.

As shown in figure 2b, if the office workers were not wearing masks, the safety guideline of expected transmissions $T_a < 10\,\%$ 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, $\mathcal {R}_{\textrm {in}}$ approaches the safety limit after approximately 2 h. While this example should not be taken as a definitive statement of danger or safety in this setting, it does serves to illustrate how our ${\rm CO}_{2}$ guideline can be implemented in a real-life situation.

4.2 University classroom adhering to social distancing guidelines

We next monitor ${\rm CO}_{2}$ 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 $\lambda _a \approx 5.2$ h$^{-1}$ (corresponding to a 12 min outdoor air change time; here, we estimate $\lambda _a$ by fitting the build-up of ${\rm CO}_{2}$, see figure 3). During the lecture, the professor spoke while the students were quiet and sedentary; thus, we assume $C_q = 30$ quanta m$^{-3}$ (Reference Bazant and BushBazant & Bush, 2021). The maximum ${\rm CO}_{2}$ concentration reached in the classroom was ${\approx }550$ 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 representative of a classroom setting. A more complete assessment of safety in schools would require integrating ${\rm CO}_{2}$ concentrations over a considerably longer time interval. For example, if students are tested weekly for COVID-19, then one should assess the mean ${\rm CO}_{2}$ 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 mask use, can substantially increase safe occupancy times.

Figure 3.

Measured $\textit{CO}_{\textit{2}}$ concentration and calculated transmission rate for 12 masked students in a university lecture hall. (a) Black dots represent the concentration of $\textit{CO}_{\textit{2}}$. 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 $\textit{CO}_{\textit{2}}$ from which we infer an air change rate of $\lambda _a = \textit{5.2}/h$. (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 not have been violated during the lecture.