The Well-Mixed Room
where is the particle settling speed and is, again, the probability of drop filtration in the recirculation flow . Owing to the dependence of the settling speed on particle radius, the population of each drop size evolves, according to Eq. 1, at different rates. Two limiting cases of Eq. 1 are of interest. For the case of , drops of infinitesimal size that are neither deactivated nor removed by filtration, it reduces to the Wells–Riley model (44, 45). For the case of , a nonreacting suspension with no ventilation, it corresponds to established models of sedimentation from a well-mixed ambient (51, 54). For the sake of notational simplicity, we define a size-dependent sedimentation rate as the inverse of the time taken for a drop of radius r to sediment from ceiling to floor in a quiescent room.
Model predictions for the steady-state, droplet radius-resolved aerosol volume fraction, , produced by a single infectious person in a well-mixed room. The model accounts for the effects of ventilation, pathogen deactivation, and droplet settling for several different types of respiration in the absence of face masks (). The ambient conditions are taken to be those of the Skagit Valley Chorale superspreading incident (25, 27) ( m, , , m, , and ). The expiratory droplet size distributions are computed from the data of Morawska et al. (ref. 11, figure 3) at for aerosol concentration per log-diameter, using . The breathing flow rate is assumed to be /h for nose and mouth breathing, /h for whispering and speaking, and /h for singing.
thereby accounting for the protective properties of masks, and allowing for the possibility that the infectivity depends on droplet size. Different droplet sizes may emerge from, and penetrate into, different regions of the respiratory tract (34, 37, 79), and so have different ; moreover, virions in relatively small droplets may diffuse to surfaces more rapidly and so exchange with bodily fluids more effectively. Such a size dependence in infectivity, , is also consistent with reports of enhanced viral shedding in micron-scale aerosols compared to larger drops for both influenza virus (60) and SARS-CoV-2 (31). Finally, we introduce a relative transmissibility (or susceptibility), , to rescale the transmission rate for different subpopulations or viral strains.
Indoor Safety Guideline
The number of susceptibles, , may include all others in the room (), or be reduced by the susceptible probability , the fraction of the local population not yet exposed or immunized. In the limit of , one may interpret as the probability of the first transmission, which is approximately equal to the sum of the independent probabilities of transmission to any particular susceptible individual in a well-mixed room.‡In SI Appendix, section 3, we show that this guideline follows from standard epidemiological models, including the Wells–Riley model, but note that it has broader generality. The exact transient safety bound appropriate for the time-dependent situation arising directly after an infected index case enters a room is evaluated in SI Appendix, section 2.
where, for the sake of simplicity, we assume constant mask filtration over the entire range of aerosol drop sizes. We define the microscopic concentration of infection quanta per liquid volume as , and the concentration of infection quanta or “infectiousness” of exhaled air, . The latter is the key disease-specific parameter in our model, which can also be expressed as the rate of quanta emission by an infected person, . The second equality in Eq. 4 defines the effective infectious drop radius , given in SI Appendix, Eq. S7. The third equality defines the dilution factor, , the ratio of the concentration of infection quanta in the well-mixed room to that in the unfiltered breath of an infected person. As we shall see in what follows, provides a valuable diagnostic in assessing the relative risk of various forms of exposure.
where , and is the air purification rate associated with air exchange, air filtration, and viral deactivation. The effect of relative humidity on the droplet size distribution can be captured by multiplying by , since the droplet distributions used in our analysis were measured at (11).
the interpretation of which is immediately clear. To minimize risk of infection, one should avoid spending extended periods in highly populated areas. One is safer in rooms with large volume and high ventilation rates. One is at greater risk in rooms where people are exerting themselves in such a way as to increase their respiration rate and pathogen output, for example, by exercising, singing, or shouting. Since the rate of inhalation of contagion depends on the volume flux of both the exhalation of the infected individual and the inhalation of the susceptible person, the risk of infection increases as . Likewise, masks worn by both infected and susceptible persons will reduce the risk of transmission by a factor , a dramatic effect given that for moderately high-quality masks (74, 75).
Application to COVID-19
Estimates of the “infectiousness” of exhaled air, , defined as the peak concentration of COVID-19 infection quanta in the breath of an infected person, for various respiratory activities. Values are deduced from the drop size distributions reported by Morawska et al. (11) (blue bars) and Asadi et al. (39) (orange bars). The only value reported in the epidemiological literature, quanta/, was estimated (25) for the Skagit Valley Chorale superspreading event (27), which we take as a baseline case () of elderly individuals exposed to the original strain of SARS-CoV-2. This value is rescaled by the predicted infectious aerosol volume fractions, , obtained by integrating the steady-state size distributions reported in Fig. 1 for different expiratory activities (11). Aerosol volume fractions calculated for various respiratory activities from figure 5 of Asadi et al. (39) are rescaled so that the value quanta/ for “intermediate speaking” matches that inferred from Morawska et al.’s (11) for “voiced counting.” Estimates of for the outbreaks during the quarantine period of the Diamond Princess (26) and the Ningbo bus journey (28), as well as the initial outbreak in Wuhan City (2, 81), are also shown (see SI Appendix for details).
The COVID-19 indoor safety guideline would limit the cumulative exposure time (CET) in a room with an infected individual to lie beneath the curves shown. Solid curves are deduced from the pseudo-steady formula, Eq. 5, for both natural ventilation (/h; blue curve) and mechanical ventilation (/h; red curve). Horizontal axes denote occupancy times with and without masks. Evidently, the Six-Foot Rule (which limits occupancy to ) becomes inadequate after a critical time, and the Fifteen-Minute Rule becomes inadequate above a critical occupancy. (A) A typical school classroom: 20 persons share a room with an area of 900 ft2 and a ceiling height of 12 ft (, ). We assume low relative transmissibility (), cloth masks (), and moderate risk tolerance () suitable for children. (B) A nursing home shared room (, ) with a maximum occupancy of three elderly persons (), disposable surgical or hybrid-fabric masks (), and a lower risk tolerance () to reflect the vulnerability of the community. The transient formula, SI Appendix, Eq. S8, is shown with dotted curves. Other parameters are quanta/, /h, /h, and m.
Beyond the Well-Mixed Room
In certain instances, meaningful estimates may be made for both and x. For example, if a couple dines at a restaurant, x would correspond roughly to the distance across a table, and would correspond to the fraction of the time they face each another. If N occupants are arranged randomly in an indoor space, then one expects and . When strict social distancing is imposed, one may further set x to the minimum allowed interperson distance, such as 6 ft. Substitution from Eq. 5 reveals that the second term in Eq. 7 corresponds to the risk of transmission from respiratory jets, as deduced by Yang et al. (106), aside from the factor . We note that any such guideline intended to mitigate against short-range airborne transmission by respiratory plumes will be, as is , dependent on geometry, flow, and human behavior, while our guideline for the mitigation of long-range airborne transmission  is universal.
Discussion and Caveats
that may be evaluated using appropriate and values (listed in SI Appendix, Table S2). One’s risk increases linearly with the number of people in a room and duration of the event. Relative risk decreases for large, well-ventilated rooms and increases when the room’s occupants are exerting themselves or speaking loudly. While these results are intuitive, the approach taken here provides a physical framework for understanding them quantitatively. It also provides a quantitative measure of the relative risk of certain environments, for example, a well-ventilated, sparsely occupied laboratory and a poorly ventilated, crowded, noisy bar. Along similar lines, the weighted average of , provides a quantitative assessment of one’s risk of airborne infection over an extended period. It thus allows for a quantitative assessment of what constitutes an exposure, a valuable notion in defining the scope of contact tracing, testing, and quarantining.