Airborne transmission arises through the inhalation of aerosol droplets exhaled by an infected person and is now thought to be the primary transmission route of COVID-19. By assuming that the respiratory droplets are mixed uniformly through an indoor space, we derive a simple safety guideline for mitigating airborne transmission that would impose an upper bound on the product of the number of occupants and their time spent in a room. Our theoretical model quantifies the extent to which transmission risk is reduced in large rooms with high air exchange rates, increased for more vigorous respiratory activities, and dramatically reduced by the use of face masks. Consideration of a number of outbreaks yields self-consistent estimates for the infectiousness of the new coronavirus.
The current revival of the American economy is being predicated on social distancing, specifically the Six-Foot Rule, a guideline that offers little protection from pathogen-bearing aerosol droplets sufficiently small to be continuously mixed through an indoor space. The importance of airborne transmission of COVID-19 is now widely recognized. While tools for risk assessment have recently been developed, no safety guideline has been proposed to protect against it. We here build on models of airborne disease transmission in order to derive an indoor safety guideline that would impose an upper bound on the “cumulative exposure time,” the product of the number of occupants and their time in an enclosed space. We demonstrate how this bound depends on the rates of ventilation and air filtration, dimensions of the room, breathing rate, respiratory activity and face mask use of its occupants, and infectiousness of the respiratory aerosols. By synthesizing available data from the best-characterized indoor spreading events with respiratory drop size distributions, we estimate an infectious dose on the order of 10 aerosol-borne virions. The new virus (severe acute respiratory syndrome coronavirus 2 [SARS-CoV-2]) is thus inferred to be an order of magnitude more infectious than its forerunner (SARS-CoV), consistent with the pandemic status achieved by COVID-19. Case studies are presented for classrooms and nursing homes, and a spreadsheet and online app are provided to facilitate use of our guideline. Implications for contact tracing and quarantining are considered, and appropriate caveats enumerated. Particular consideration is given to respiratory jets, which may substantially elevate risk when face masks are not worn.
COVID-19 is an infectious pneumonia that appeared in Wuhan, Hubei Province, China, in December 2019 and has since caused a global pandemic (1, 2). The pathogen responsible for COVID-19, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), is known to be transported by respiratory droplets exhaled by an infected person (3⇓⇓⇓–7). There are thought to be three possible routes of human-to-human transmission of COVID-19: large drop transmission from the mouth of an infected person to the mouth, nose or eyes of the recipient; physical contact with droplets deposited on surfaces (fomites) and subsequent transfer to the recipient’s respiratory mucosae; and inhalation of the microdroplets ejected by an infected person and held aloft by ambient air currents (6, 8). We subsequently refer to these three modes of transmission as, respectively, “large-drop,” “contact,” and “airborne” transmission, while noting that the distinction between large-drop and airborne transmission is somewhat nebulous given the continuum of sizes of emitted droplets (11).* We here build upon the existing theoretical framework for describing airborne disease transmission (12⇓⇓⇓⇓⇓–18) in order to characterize the evolution of the concentration of pathogen-laden droplets in a well-mixed room, and the associated risk of infection to its occupants.
The Six-Foot Rule is a social distancing recommendation by the US Centers for Disease Control and Prevention, based on the assumption that the primary vector of pathogen transmission is the large drops ejected from the most vigorous exhalation events, coughing and sneezing (5, 19). Indeed, high-speed visualization of such events reveals that 6 ft corresponds roughly to the maximum range of the largest, millimeter-scale drops (20). Compliance to the Six-Foot Rule will thus substantially reduce the risk of such large-drop transmission. However, the liquid drops expelled by respiratory events are known to span a considerable range of scales, with radii varying from fractions of a micron to millimeters (11, 21).
There is now overwhelming evidence that indoor airborne transmission associated with relatively small, micron-scale aerosol droplets plays a dominant role in the spread of COVID-19 (4, 5, 7, 17⇓–19, 22), especially for so-called “superspreading events” (25⇓⇓–28), which invariably occur indoors (29). For example, at the 2.5-h-long Skagit Valley Chorale choir practice that took place in Washington State on March 10, some 53 of 61 attendees were infected, presumably not all of them within 6 ft of the initially infected individual (25). Similarly, when 23 of 68 passengers were infected on a 2-h bus journey in Ningbo, China, their seated locations were uncorrelated with distance to the index case (28). Airborne transmission was also implicated in the COVID-19 outbreak between residents of a Korean high-rise building whose apartments were linked via air ducts (30). Studies have also confirmed the presence of infectious SARS-CoV-2 virions in respiratory aerosols (31) suspended in air samples collected at distances as large as 16 ft from infected patients in a hospital room (3). Further evidence for the dominance of indoor airborne transmission has come from an analysis of 7,324 early cases outside the Hubei Province, in 320 cities across mainland China (32). The authors found that all clusters of three or more cases occurred indoors, 80% arising inside apartment homes and 34% potentially involving public transportation; only a single transmission was recorded outdoors. Finally, the fact that face mask directives have been more effective than either lockdowns or social distancing in controlling the spread of COVID-19 (22, 33) is consistent with indoor airborne transmission as the primary driver of the global pandemic.
The theoretical model developed herein informs the risk of airborne transmission resulting from the inhalation of small, aerosol droplets that remain suspended for extended periods within closed, well-mixed indoor spaces. When people cough, sneeze, sing, speak, or breathe, they expel an array of liquid droplets formed by the shear-induced or capillary destabilization of the mucosal linings of the lungs and respiratory tract (8, 34, 35) and saliva in the mouth (36, 37). When the person is infectious, these droplets of sputum are potentially pathogen bearing, and represent the principle vector of disease transmission. The range of the exhaled pathogens is determined by the radii of the carrier droplets, which typically lie in the range of 0.1 μm to 1 mm. While the majority are submicron in scale, the drop size distribution depends on the form of exhalation event (11). For normal breathing, the drop radii vary between 0.1 and 5.0 μm, with a peak around 0.5 μm (11, 38, 39). Relatively large drops are more prevalent in the case of more violent expiratory events such as coughing and sneezing (20, 40). The ultimate fate of the droplets is determined by their size and the airflows they encounter (41, 42). Exhalation events are accompanied by a time-dependent gas-phase flow emitted from the mouth that may be roughly characterized in terms of either continuous turbulent jets or discrete puffs (20, 38, 43). The precise form of the gas flow depends on the nature of the exhalation event, specifically the time dependence of the flux of air expelled. Coughs and sneezes result in violent, episodic puff releases (20), while speaking and singing result in a puff train that may be well approximated as a continuous turbulent jet (38, 43). Eventually, the small droplets settle out of such turbulent gas flows. In the presence of a quiescent ambient, they then settle to the floor; however, in the well-mixed ambient more typical of a ventilated space, sufficiently small drops may be suspended by the ambient airflow and mixed throughout the room until being removed by the ventilation outflow or inhaled (SI Appendix, section 1).
Theoretical models of airborne disease transmission in closed, well-mixed spaces are based on the seminal work of Wells (44) and Riley et al. (45), and have been applied to describe the spread of airborne pathogens including tuberculosis, measles, influenza, H1N1, coronavirus (SARS-CoV) (12⇓⇓⇓–16, 46, 47), and, most recently, the novel coronavirus (SARS-CoV-2) (17, 25). These models are all based on the premise that the space of interest is well mixed; thus, the pathogen is distributed uniformly throughout. In such well-mixed spaces, one is no safer from airborne pathogens at 60 ft than 6 ft. The Wells–Riley model (13, 15) highlights the role of the room’s ventilation outflow rate Q in the rate of infection, showing that the transmission rate is inversely proportional to Q, a trend supported by data on the spreading of airborne respiratory diseases on college campuses (48). The additional effects of viral deactivation, sedimentation dynamics, and the polydispersity of the suspended droplets were considered by Nicas et al. (14) and Stilianakis and Drossinos (16). The equations describing pathogen transport in well-mixed, closed spaces are thus well established and have recently been applied to provide risk assessments for indoor airborne COVID-19 transmission (17, 18). We use a similar mathematical framework here in order to derive a simple safety guideline.
We begin by describing the dynamics of airborne pathogen in a well-mixed room, on the basis of which we deduce an estimate for the rate of inhalation of pathogen by its occupants. We proceed by deducing the associated infection rate from a single infected individual to a susceptible person. We illustrate how the model’s epidemiological parameter, a measure of the infectiousness of COVID-19, may be estimated from available epidemiological data, including transmission rates in a number of spreading events, and expiratory drop size distributions (11). Our estimates for this parameter are consistent with the pandemic status of COVID-19 in that they exceed those of SARS-CoV (17); however, our study calls for refined estimates through consideration of more such field data. Most importantly, our study yields a safety guideline for mitigating airborne transmission via limitation of indoor occupancy and exposure time, a guideline that allows for a simple quantitative assessment of risk in various settings. Finally, we consider the additional risk associated with respiratory jets, which may be considerable when face masks are not being worn.