key: cord-0680984-jhnq4wbm
authors: Mittal, Rajat
title: A Simple Mathematical Framework for Estimating Risk of Airborne Transmission of COVID-19 with Application to Facemask Use and Social Distancing
date: 2020-08-03
journal: nan
DOI: nan
sha: 81f6d1e51518ac6d40ad090f4e997bfadae69375
doc_id: 680984
cord_uid: jhnq4wbm

A simple mathematical model for estimating the risk of airborne transmission of a respiratory infection such as COVID-19, is presented. The model employs basic concepts from fluid dynamics and incorporates the known scope of factors involved in the airborne transmission of such diseases. Simplicity in the mathematical form of the model is by design, so that it can serve not only as a common basis for scientific inquiry across disciplinary boundaries, but, also be understandable by a broad audience outside science and academia. The model is used to assess the protection from transmission afforded by face coverings made from a variety of fabrics. In addition, the reduction in transmission associated with physical distancing between the host and susceptible as well as the level of physical activity of the host and/or the susceptible in enhancing transmission risk, is also examined.

COVID-19 spread across the world with a speed and intensity that laid bare the limits of our understanding of the transmission pathways and the associated factors that are key to the spread of such diseases. There is however an emerging consensus that "airborne transmission," where virion bearing respiratory droplets and droplet nuclei (also called respiratory aerosols) expelled by an infected person (the ``host'') are inhaled by a "susceptible" individual, constitutes an important mode for the spread of COVID-19 [1] [2] [3] [4] [5] . Questions regarding the size of the droplets involved [6] [7] [8] [9] and the range of such transmission 10 can be bypassed by noting that the key element that differentiates airborne transmission from the droplet and contact routes of transmission 11 is the essential role of inhalation by the susceptible in this pathway for transmission. Generally, it is the small (<10 μm) particles that are likely to be entrained into the inhalation current of a person, but environmental conditions as well as the proximity between the host and the susceptible could allow larger particles/droplets to play a role in airborne transmission.

Irrespective of the size of droplets or the range involved, airborne transmission of COVID-19 and other respiratory infections involve the following sequence of events (see Fig. 1 ):

1. generation, expulsion and aerosolization of virus-containing droplets from the mouth and nose of an infected host; 2. dispersion and transport via ambient air currents of this respiratory aerosol to a susceptible; and 3. inhalation of droplets/aerosols, and deposition of virus in the respiratory mucosa of the susceptible.

Each phase in this sequence has complex dependencies on a variety of factors that may include the morphological properties and pathogenicity of the virus, the health status of the host and/or the susceptible, environmental conditions, and the presence/effectiveness of face coverings being used by the host and/or susceptible. Given this complexity of phenomenology and the many factors involved, it is not surprising that even after more than 8 months of the world dealing with the COVID-19 pandemic, there are fundamental questions that continue to confound scientists, policy makers and the members of the public at-large. These include questions such as: what factors have enabled the SARS-CoV-2 to spread so much faster and more extensively than other similar viruses in the recent past 12, 13 ? Why is the rate of infection so different in different regions/countries of the world 14 ? How much lower is the likelihood of transmission in an outdoor environment compared to an indoor environment 10, 15 ? How do policies and societal behavior such as compliance with mask wearing and social distancing affect the rate of transmission 16, 17 ?

Scientists spanning fields such as biomedicine, epidemiology, virology, public health, fluid dynamics, aerosol physics, public policy, behavioral psychology, and others, are tackling these as well as other important questions. However, what is lacking is a simple conceptual framework (or model) that encapsulates the complex, multifactorial scope of this problem in a manner that not only serves as a common basis for scientific inquiry across disciplinary boundaries, but also as a tool to more easily communicate the factors associated with the spread of this disease, to a wide range of stakeholders including non-scientists such as policy-makers, public media, and the public at-large. Given the rapidly evolving nature of the pandemic and the resurgence of infections in many commmunities 18 , the importance of clear communication of infection risk across scientific disciplines, as well as to policy/decision makers and other segments of society, is more important than ever.

In 1961, Dr. Frank Drake, an astronomer and astrophysicist involved in the search for extraterrestrial intelligence, conceived a conceptual framework to predict the number of technological civilizations that may exist in our galaxy. The Drake Equation 19 ,20 , as it has become known, involves a number of probabilistic factors, which when multiplied together, result in the number of civilizations within our galaxy, at any given moment, that humanity could communicate with. The power of this equation is not in that it actually allows us to predict this number with a known level of certainty, but in the fact that it provides an easy to understand framework for grasping the key factors involved in something that seems inestimable: the number of advanced lifeforms that exist elsewhere in our galaxy.

Motivated by the Drake Equation, and based on the idea that airborne transmission is successful if a susceptible inhales a viral dose that exceeds the minimum infectious dose 21, 22 , I propose the following inequality that predicts the possibility of airborne transmission of SARS-CoV-2 from an infected host to a susceptible:

In the above expression:

̇ℎ : rate of expulsion of respiratory droplets from the nose and mouth of the host.

ℎ : fractional viral emission load -average number of virions contained in expelled droplets ℎ : fraction of expelled droplets that make it past the face-covering of the host ℎ : fraction of expelled droplets that aerosolize (i.e. become suspended in the air) :

fraction of aerosolized droplets that transport to the vicinity of the susceptible :

fraction of aerosolized droplets transported to the vicinity of the susceptible that contain viable virions :

fraction of aerosols in the vicinity of the susceptible that would be inhaled by a susceptible not wearing a face covering :

fraction of inhaled aerosols that are filtered by the face covering of the susceptible. :

duration of exposure of the susceptible to the aerosols from the host. : minimum number of inhaled virions required to initiate infection in the susceptible.

The use of mathematical models to predict infection rates is well-established in epidemiology 21,23,24 and the above inequality belongs among such models (see Sec. 8 for discussion on this). As with any model, the above model has a number of underlying assumptions (see Sec. 9) but the potential advantage of the above model is that it presents transmission risk via a simple mathematical expression that on one hand, captures the entire scope of factors that may be involved in airborne transmission, and on the other, is easy to convey to scientists from a wide range of fields, non-scientists such as policy makers, public officials, and public media, as well as even members of the general public. 

As laid out in a previous publication, 11 each stage in the airborne transmission process is mediated by complex flow phenomena, ranging from air-mucous interaction and liquid sheet fragmentation inside the respiratory tract, to turbulence in the expiratory jet/ambient flow and flow-induced droplet evaporation and particle dispersion, to inhalation and deposition of aerosols in the lungs. Furthermore, nonpharmaceutical approaches employed to mitigate respiratory infections such as social distancing and the wearing of face masks, are also rooted in the principles of fluid dynamics. Thus, fluid dynamics is central to all important physical aspects of the airborne transmission of respiratory infections such as COVID-19, and it therefore stands to reason that this connection to flow physics will appear in any model of airborne transmission. So is the case in the current model.

In the sections that follow, we provide additional details about the key variables involved in the CAT inequality and describe the caveats associated with this model.

The CAT inequality (Eq. 1) naturally segregates into three sets of variables: the first set depend primarily on the host, the second on the environment, and the third, on the susceptible. We now describe the factors that each of these variables depend as well as our state of knowledge regarding each variable.

̇ℎ is the rate of expulsion of respiratory droplets from the nose and mouth of the host, and is one of the most extensively studied parameters within the arena of airborne transmission 6, 7, 9, [25] [26] [27] [28] . Droplets are formed from the mucus and saliva that lines our respiratory and oropharyngeal tracts, and these droplets are expelled with the air that is exhaled out of our mouth and nose. Thus, the droplet expulsion rate may itself be considered as the product of the rate of droplet formation in the respiratory tract and the fraction of these formed droplets that are expelled during the expiratory event. Much of the work on respiratory droplet expulsion, including some of the earliest work 9 focused on the rate (or total count) of droplets expelled during spasmodic events such as sneezing and coughing. However, recent attention has focused on droplet generation during talking and breathing 7, 8, 25 due to the recognition that viral shedding from asymptomatic/presymptomatic individuals (who are not coughing or sneezing) may be an important differentiator in the high spreading rate of SARS-CoV-2 infections compared to earlier coronavirus outbreaks 2, 29 .

While the conventional notion is that sneezing has the highest rate of droplet generation followed by coughing, talking 7,30 and breathing 6 (in that order), the very large scatter in measured data 6, 25, 31 makes it difficult to validate this notion. Indeed, a recent study 26 even showed that a single cough from a person exhibiting symptoms of the common cold can generate (on average) nearly 5 million droplets, and this is many orders of magnitude higher than some previous measurements 6, 9 . Indeed it could be useful to express ̇ℎ as the product of the volume exhalation rate of the host, (̇ℎ) and the number density of droplets (i.e. droplets per unit volume) in the exhaled gas ( ℎ ). This is because for a given individual, ℎ might not vary significantly during activities such as breathing, and ̇ℎ would therefore increase linearly with the exhalation rate ̇ℎ . Exhalation rate for an adult can range from about 100 ml.s -1 at rest to to 2000 ml.s -1 during intense exercise 32 . Measured values of ℎ for breathing 33 are about 0.2 ml -1 and this suggests that ̇ℎ for breathing could range from about 20 to 400 droplets per second depending on the ventilation rate. Values of ℎ during normal speech in the same experiment were found to be about four times higher and other studies have found that droplet emission increased with the loudness of speech 7 . Finally, recent attention has focused on "super producers,": individuals who according to some studies, generate droplets at rates that are 10 or more times higher than others 34 . Thus, even for normal breathing, ̇ℎ could range from about 10 to 5000 s -1 depending on the exhalation rate and the emission phenotype of the individual, and speech could increase the upper range by another order of magnitude. Thus, phenotype and expiratory activity of the host alone could increase the transmission risk by a factor of 1000 or more. ℎ is the fractional viral load of a respiratory droplet and there is currently no data on this variable for SARS-CoV-2. Indirect measures based on volume concentration of viral load in oral fluid samples collected from COVID-19 patients combined with simple statistical models have been used to suggest that 37% of 50 μm size droplets and 0.37% of 10 μm size droplets would contain virions 30 . No confirmation of these estimates from direct measurement of respiratory aerosols is available so far, and there is evidence that suggests that these simple volume-fraction based estimates might significantly underestimate the viral load of the small (<5 μm) droplets 27 . Furthermore, the fractional viral load also likely depends on the location in the respiratory tract from where the droplet originates because pathogens tend to colonize specific regions of the respiratory tract, and because the surface area density of the mucus volume varies throughout the respiratory tract 35 . There is however, no quantification of this effect. However, even a low-end estimate of say a 0.5% fractional viral load, combined with 200,000 droplets/cough, would result in the shedding of 1000 virions in each cough.

ℎ is the fraction of expelled droplets that aerosolize, i.e. get suspended in the air. It is generally found that droplets smaller than about 10 μm can remain suspended in the air whereas droplets larger than 50 μm fall to the ground rapidly 33, 36, 37 . Thus, the size distribution of the expelled droplets is a key determinant of ℎ . A number of studies have examined the distribution of droplet size expelled during various expiratory activities 6, 7, 9, [25] [26] [27] [28] and these studies show that droplet size can vary from 0.01 to 1000 microns. The consensus is that breathing generates the smallest particles, with talking, coughing, and sneezing generating increasingly larger droplets (in that order) 38 . There is however a large scatter in this data, and this might be due to individual differences 39 as well as the changes in mucosal fluid induced by the pathogen 38 . Finally, the fluid in the expelled droplets also evaporates rapidly resulting in a reduction in size, and this rate of evaporation depends on environmental conditions (temperature and humidity) as well as the velocity of the droplets. These dependencies can, however, be determined, for the most part, from first principles 37,40 .

represents the fraction of aersolized respiratory aerosol droplets/droplet nuclei from the infected host that are transported to the immediate vicinity of the susceptible, and this is one variable where environmental factors play a dominant role. These include air currents, temperature and humidity. Ambient air currents determine the "time-of-flight" as well as the dilution in concentration of the bioaerosol that arrives near the susceptible.

Even though we know the dependencies of the variable , it is still a difficult variable to estimate since environmental factors can be so highly variable. For example, even for a host and susceptible in the same room, this variable could change significantly given the relative location of the two individuals, the operational status of the air conditioning, and the location of the individuals relative to the air conditioning diffusers and vents 15, 41, 42 . The estimation of this parameter becomes even more difficult in indoor spaces such as buildings where rooms share a high-volume air conditioning (HVAC) system. In highdensity indoor spaces such as classrooms, aircraft cabins, gyms, buses, trains, etc., anthropogenic effects generated due to human movement and body heat generated thermal plumes 43, 44 could also have a significant effect on this variable.

Estimation of in outdoor environments presents a different challenge. While these outdoor environments do not have confining boundaries and localized inflow/outflow regions that dominate the flow patterns, effects due to atmospheric turbulence 45, 46 , local wind and weather conditions, convection effects due to thermal gradients, and other environmental factors have to be taken into account. Furthermore, even in outdoor settings, the presence of high human density (such as at sporting events, social gatherings etc.) could introduce significant anthropogenic effects on the dispersion and transport of respiratory aerosols.

As the aerosol plume from the host travels downstream, it expands due to diffusion, entrainment, and turbulence-induced mixing. This results in a drop in concentration (aerosol particles per unit volume) with distance from the host. To further understand how this enters the estimation of the variable , we introduce the variable , which is the volume of air surrounding the face of the susceptible that would be inhaled by the susceptible (see Fig 2) . If the concentration near the host is and the mean concentration in the volume at a distance of ℎ is ̅ , then due to this dilution in concentration can be expressed as ̅ / . The volume can be estimated given the inspiratory status of the susceptible (see discussion of ) but in the current model, a choice for that eliminates dependence of this variable on the susceptible, is the maximum possible volume of air that can be inhaled by a human per second, which is about 2 liters 32 . Thus, can be estimated under these assumptions that ̅ / is known or can be estimated. It is noted that could be expressed in term of the ratio of the particle expulsion rate to the exhalation rate of the host as =̇ℎ/̇ℎ.

A number of studies have measured the spreading rate of the exhalation jets formed from various expiratory activities [47] [48] [49] and this spreading rate with downstream distance can vary significantly for breathing, talking and coughing. External flow currents and thermal convection 32 effects, will however deform the shape of the expiratory plume and may enhance non-uniform concentration of the respiratory Notwithstanding these complexities, the general decay of aerosol concentration with distance underscores the reduction in transmission risk with physical distance. In Sec. 8, we will employ canonical models of scalar dispersion to provide some estimates of the protection factor associated with physical distancing.

represents the fraction of respiratory aerosol particles from the infected host that arrive in the immedite vicinity of the susceptible with viable virions. Ambient air currents determine the "time-offlight", which combined with temperature, humidity and UV exposure determine the viability of the virions carried in the aerosols. A study has shown that the SARS-CoV-2 virion can stay viable in aerosol form for 3 or more hours 13 , but high temperature 50 and sunlight/UV exposure 51, 52 are both detrimental to virion viability. Humidity, on the other hand, has a more complex effect on the viability of airborne viruses 53 , and this has made it difficult to correlate transmission risk with regional and seasonal variations in environmental conditions 54 . In general could be modeled as − / where T is the time-of-flight and is the half-life of virions in aerosolized form, which depends on the virus, as well as the temperature, humidity and UV exposure.

is the fraction of bioaerosols from the host in the vicinity of the susceptible that would be inhaled by a susceptible not wearing a face covering. This variable primarily depends on the inspiratory status of the susceptible. At rest, an adult human inhales about 100 ml of air per second 55 but this value can go up 20 fold during intense exercise 32 . Within the context of the current model, could be estimated as the ratio of the susceptible's inspiratory rate to the maximum possible inspiratory rate for a human, which can be assumed to be 2000 ml/second 32 (the volume here is the same as in the previous section). With this prescription, for the average healthy adult male, could vary from 0.05 during rest to 1.0 during intense exercise. Beyond the exercise state of the individual, tidal volume (volume inhaled per breath) and minute ventilation (volume inhaled per minute) also depend on age 56 , gender, body weight 57,58 and the respiratory health of the person, and these factors can be easily accounted for in . For instance, measured values of minute ventilation for women are about 20% lower than for men 57 and this would translate to a 20% reduction in for women. Similarly, short adults can have resting inspiratory rates that are about 20% lower than tall adults 57 and this would result in a proportionate reduction in . Inspiration rates for preteens can be three-fold lower than adults 56 and this would also reduce proportionately. Thus, the above discussion suggests that differential inspiration rates could play a role in the age and body-weight associated COVID-19 prevalence disparities noted in recent studies 59, 60 . The effect of physical activityassociated changed in ventilation rates on transmission risk is examined in Sec. 8.

is the infectious dose for airborne transmission. In the arena of infectious diseases, infectious dose is often expressed as HID50 22 which is the minimum infectious dose required to initiate infection in 50% of inoculated humans. This number is usually obtained via controlled studies where human volunteers are exposed to different viral loads. However, such studies are not available for potentially lethal viruses such as SARS-CoV-2. Studies on the infectious dose for Influenza A indicate an HID50 of O(1000) virus particles 22, 61 . Studies of MERS-CoV in mice found a similar infectious dose 62 , so the HID50 for humans accounting for the larger body weight, could be two or more orders of magnitudes higher. It is important to note that for Influenza A, infectivity via aerosols has been found to be O(10 5 ) higher than via a nasopharyngeal (i.e. nasal swab) route, 63 highlighting the exceptional effectiveness of the airborne route for transmission of respiratory infections. Infectivity of airborne viruses also depends on the carrier droplet size. Small (~2 μm) droplets deposit deeper in the lungs and have been shown to be two or more orders of magnitude more infective than larger (>10 μm) droplets 64 . Finally, the infectious dose might also depend on the age and health status of the susceptible. Determination of for SARS-CoV-2 remains one of the most important tasks for scientists working in this arena.

The remaining variable is the duration of exposure of the susceptible to the aerosols from the host and based on the CAT inequality, transmission risk is directly proportional to this duration of exposure.

Face coverings appear in the two factors ℎ and as fractions of aerosols/droplets that pass through the face coverings of the host and susceptible, respectively, and there is much data available to estimate these variables. These face covering-related variables depend on two factors -the material of the face covering and the fit of the face covering on the face of the individual. A perfectly fit N95 face mask for instance, stops 95% or more of the particles that go through it and would therefore be equal to 0.05 for such a mask. Thus, the wearing of a well-fit N95 mask by either the host or the susceptible reduces the transmission risk by a factor or 20. Furthermore, if both individuals are wearing such masks, the transmission risk, according to Eq. 1, reduces by a factor or 400.

Surgical masks have been measured to block 30% to 60% of respiratory aerosols 65, 66 and another study a surgical mask reduced aerosol shedding if Influenza A virions from infected hosts by a factor of 3.4. 28 This suggests that even surgical masks worn by both the host and the susceptible could reduce transmission risk by factors ranging from 2 to about 10. Another recent study of viral shedding with and without surgical face masks from patients with influenza, coronavirus (SARS) and rhinovirus provides clear evidence of the ability of such face covering to reduce transmissibility of the virus 67 . Finally, we point out that even homemade cloth masks provide significant protection against airborne infections. In fact, a recent study showed that a cloth mask with three cloth layers of two easily obtainable fabrics (1 layer of 600 threads-per-inch cotton and 2 layers of silk), provides nearly 90% filtration of breath aerosol droplets 65 .

The fitment of the mask is important for overall protection since a loose-fitting mask with perimeter leaks allows unfiltered aerosols to bypass the mask. Leaks are a particular problem for outward protection (i,e, reducing emission of respiratory aerosols by the infected host) since the process of expiration pushes the mask outwards and enhances perimeter leaks 11 . Nevertheless, it has been shown that even an imperfectly fit cloth mask with perimeter gaps can filter out 30% of the aerosols 65 and would halve the risk of transmission if worn both by the host and the susceptible.

Finally, in addition to filtering aerosol particles, face covering also reduce the velocity of exhalation jet 68, 69 This could increase the expansion angle of respiratory jet and reduce the initial penetration distance of the respiratory droplets, 70 thereby altering . A recent study found that the neck gaiters worn as face covering might facilitate the breakup of large droplets into smaller one 71 , and in doing so increase ., thereby increasing transmission risk.

The model now applied to address three distinct questions: what protection is afforded by different face coverings; how does risk decrease with increased physical distance between the host and susceptible, and finally, to what degree does the level of physical activity, as manifested in the ventilation rates of the host and/or the susceptible, affect transmission risk.

Protection Afforded by Face Coverings: We start with the effect of facemasks and employ data from Konda et al. 65 on the filtration efficiency (FE) of common fabrics used in respiratory face masks. These authors examined 16 different fabrics and face masks. For most of these, tests no leaks were allowed but for three fabrics/samples, and they prescribed leaks that ranged from 0.5-2% of the active sample area in an attempt to mimic the perimeter leaks typical for worn face masks. Table 1 in the paper of Konda et al 65 provides separate average filtration efficiencies for particles less than and greater than 0.3 μm. In the current analysis, we take an average of these two values as the aggregate filtration efficiency of the mask and estimate that the fraction of aerosols/droplets that passed through the face coverings is given by = 1 − /100. Given these values and the assumption that the filtration efficiency is the same for inward as well as outward protection (i.e. ℎ = = ) we can now estimate the unilateral protection factor if either the host or the susceptible wears this mask as PF= −1 . The corresponding bilateral protection factor, i.e. when both individuals wear masks is then given by PF= −2 . These PFs normalized by the corresponding situation where neither individual is wearing a mask, are plotted in Fig.  3 for the various cases from Konda et al. 65

The plot shows that the 1-layer 80-TPI cotton mask provides the lowest PF (1.1 for unilateral and 1.3 for bilateral mask wearing). Surgical and N95 masks provide PFs of 8 and 13 respectively for unilateral mask wearing, and 67 and 175 respectively for bilateral mask wearing. Interestingly, based on this data, a number of multi-layers masks made from common fabrics outperform the N95 masks by significant factors, and the top performing mask is the cotton-chiffon mask, that provide a unilateral(bilateral) protection factor of 53(2770). The presence of leaks around the mask significantly diminishes the performance of the mask; for surgical masks, the unilateral(bilateral) PFs drop to 1.9(3.6) and for the cotton-silk mask, it drops from 27(711) to 1.5(2.3) due to leaks. Thus, the prevention or minimization of leaks is the biggest hurdle in achieving high protection factors with face masks. Nevertheless, even bilateral protection factors of 2 could significantly diminish infection rates and reduce the reproductive number (R0) of COVID-19 72 to values that could help bring the pandemic under control 73 . Two caveats in the above analysis are worth noting: first, for most face coverings, outward protection is lower than inward protection 11 ; and second, filtration efficiencies measurements for face covering show significant scatter and the numbers in Fig. 3 might therefore also have large uncertainties.

Protection due to Physical Distancing: The model is used next to examine the protection from transmission afforded by physical distance between the host and the susceptible. As mentioned earlier, the distance between the host and the susceptible is a dominant factor in the variable associated with the transport of virion bearing aerosols. Here, we employ simple models of the aerosol dispersion to estimate the protection factors associated with physical distancing. Two extreme conditions for an outdoor transmission scenario are considered: in the first, the host expels the aerosols with a velocity ( ) that far exceeds the velocity of the ambient flow velocity ( ∞ ), and in the second, the ambient velocity is significantly larger than the aerosol jet velocity (see Fig. 4a ). The former can be approximated as a jet in quiescent flow and the latter, to a horizontal plume from a point source in a crossflow. Assuming that the flow is turbulent, we can employ data from studies of such flows in canonical configurations that shows that the peak concentration for the first case decays as ( ) ~ −1 (see Zarruk & Cowen 74 ) and that for the second as ( ) ~ ( −3/2 ) (see Sykes and Henn 75,76 and Vinkovic et al. 76 ). Thus, assuming that the mean concentration in the inhaled volume of the susceptible is equal to � ℎ �, dispersion induced dilution at a distance of ℎ would result in ~ ( ℎ )/ , and a corresponding protection factor due to physical distancing of −1 .

(a) (b)

are at a unit distance. Fig. 4b shows the protection factors due to physical distancing for these two scenarios and we note that since the y-intercepts of the two curves have been individually normalized to unity at a unit distance, direct numerical comparison between the two conditions is not appropriate. The plot does however indicate that in the absence of a crossflow, the protection factor increases linearly with distance, whereas when the crossflow velocity is significantly larger than the exhaled jet velocity, the protection factor increases at a faster rate of ℎ 1.5 . Given that wind conditions can be highly variable, the more conservative linear increase in the protection factor with distance would be appropriate. Thus, an easy to convey message from this analysis based on known flow physics is that physical distancing affords at least a linear increase in protection from transmission.

Level of Physical Activity and Transmission Risk: The final application of the model is to examine the potential effect of physical/exercise intensity on the risk of transmission. This would be relevant to settings such as gyms, sports/exercise facilities, and even gatherings/events, schools, and workplace situations where levels of physical activity might exceed levels that are considered sedentary. Exercise intensity enters the CAT inequality through the ventilation rates of the host and the susceptible. As pointed out earlier, in expiratory activities such as breathing and talking, the particle expulsion rate (̇ℎ) may be proportional to the minute ventilation rate of the host (̇ℎ l/min). For the susceptible, the variable is directly proportional to the inhalation rate of the susceptible (̇ l/min). Employing established definitions 77 that relate exercise intensity to oxygen consumption rates, and further assuming proportionality between oxygen consumption rates and minute ventilation, and that the maximum minute ventilation rate for an adult is 120 l/min 78 we can estimate the increased transmission risk with exercise intensity over rest condition as ̇ℎ ×̇/ 0 2 , where 0 is the minute ventilation rate of a adult in a sedentary condition. In the current estimation procedure, 0 is set at 6 liters/min, which is 5% of the max minute ventilation rate 77 .

Fig. 5 shows this increased transmission risk for the five exercise intensity levels and it can be seen that even for a susceptible in a sedentary state, the transmission risk goes up by a factor of 8 if the host is at a moderate intensity of exercise. In a setting such as a shopping mall, an outdoor market, a warehouse or a high-school, where activity levels of hosts and susceptibles could be in the light to moderate range, increase in transmission risk due just to increased ventilation rates, would, according to the current model, be up to 64 times higher. In a facility such as a gym or for instance a basketball practice, where intensity levels could be in "vigorous" range, transmission risk could be nearly 200 times higher due to increased exhalation and inhalation rates of the individuals involved.

Existing models for estimating infection risk via airborne transmission can be classified into Wells-Riley type and dose-response models 79 . Within this context, the current model should be considered a doseresponse model since it explicitly makes use of the infectious dose ( ) to predict infection risk. The current model, as proposed, can further be characterized as a deterministic (as opposed to stochastic) dose-response model since it assumes that a dose larger than the threshold dose of results in an infection 79 .

The Wells-Riley type models are based on the notion of an infectious "quantum" which is defined as the quantity of expelled aerosol required to cause an infection in a susceptible 24, 21 . Within the context of the CAT inequality, the quanta emission rate can be expressed as �ḣ ℎ / � and Eq. 1 could easily be reformulated to express risk in terms of this quantity. However, the quanta emission rate combines a host-dependent variable (the rate of viral shedding) with a susceptible dependent variable (the infectious dose) and makes it difficult to delineate the effects of the distinct states (health, inspiratory state, mask wearing, etc.) of the two individuals involved. Furthermore, the vast majority of such models assume a "well-mixed" state for the aerosols in the environments and this does not allow for "local" effects 79 that have direct bearing on practices such as social distancing.

Dose-response models of varying degree of complexity have been developed 79 . Many of these models allow for spatial and temporal inhomogeneities in the aerosol concentrations and are well-suited for detailed modeling of infection risk in a variety of scenarios. Some of the recent models that have been developed can incorporate data on ambient flow conditions from computational models or experimental measurements 80 . However, such models are expressed in mathematically complex forms, which diminishes comprehensibility outside disciplinary expertise, and makes it particularly difficult to communicate the underlying ideas to non-scientists. As shown in the previous sections, the simplicity of the current model and its phenomenology-based compartmentalization into host, environment and susceptible dependent variables, not only allows for easier comprehension by a wide range of audiences, but also provides quick estimates of factors including but not limited to the wearing of a mask (and the type of mask worn), physical distancing, inspiratory status of the host and susceptible and expiratory activities (breathing, talking, coughing sneezing etc.).

The notion that "a model is a lie that helps you see the truth 2 ," certainly applies to the current model as well. The CAT inequality is an attempt to model the highly complex, multifactorial process of airborne transmission of a respiratory infection such as COVID-19, and the following caveats and limitations of this model are worth pointing out:

1. The choice of the variables in the CAT inequality is not unique and other combinations of the variables are possible. In particular, the variables shown in the CAT inequality could be decomposed further; for instance, ̇ℎ can be expressed as the rate of droplet generation in the respiratory tract and the fraction of generated droplets that are expelled from the mouth. Such a variable separation might be appropriate, for instance, to isolate the effect of therapies that attempt to diminish the droplet generation rate via alteration of the mucous properties 34 . 2. The inequality assumes that the rate of arrival of virion bearing aerosols in the vicinity of the susceptible is constant. The CAT inquality can be modified to include a time-dependent emission and arrival rate 21 but this would increase the complexity of the mathematical expression. 3. The CAT inequality could be missing important, but as yet unknown effects. For instance, the use of in the model assumes that it is the accumulated dose of virus that determines transmission. While this assumption is quite standard in the arena of infectious diseases [21] [22] [23] it is plausible that the rate at which this infectious dose is delivered to the respiratory tract of the susceptible is also important in initiating an infection. For instance, 1000 virions inhaled over a short duration (say minutes) might overwhelm the immune system whereas the same viral dose delivered over a much longer duration (say hours) might allow the immune system to mount an effective response and avoid infection. 4. The variables in the CAT inequality are more accurately represented as probablity density functions (PDFs) given the stochastic nature of the processes involved 81 . For instance, respiratory droplets of different sizes are expelled at different rates 9,28,31,35 during an expiratory event and the rate of droplet emission ̇ℎ could therefore be expressed as a droplet size-dependent PDF. Similarly, the viral loading of respiratory droplets ( ℎ ) as well as the infectious dose are expected to be functions of droplet size, and could therefore be represented by droplet size dependent PDFs. 5. Variables in the CAT inequality are not necessarily mutually independent and this is not suprising given the fact that many variables in the expression have common dependencies. The dependency of many variables on particle size is described above. Other examples include the face covering on the host which modifies ℎ . However, the alteration of the expiratory jet due to the mask could also affect the aerosolization variable ℎ of the expelled droplets as well as entrainment into the ambient air current, which would could affect ℎ . 6. The model assumes a single host but the CAT inequality can easily account for multiple host by summing the left hand side for multiple infected hosts.

The CAT Inequality is a simple mathematical model for estimating the risk of airborne transmission of infectious diseases such as COVID-19, that is designed to convey the key factors involved to a wide range of stakeholders ranging from scientists in various disciplines, to policy makers, public media and even the general public. As shown though specific examples, the model provides a framework for interpreting the effect of behaviors such as wearing masks, physical distancing, and the intensity of physical activity/exercise on infection risk in terms that are easy to convey to a range of audiences. In closing, we point out that while the transmission model presented here is inspired by the Drake Equation, the current model is not a speculative model but a deterministic one. This is because we understand much more about the factors involved in this transmission model than we do about the factors in the Drake equation. Indeed, as discussed in the paper, estimates for many of the variables in the CAT inequality can be obtained from existing data or from basic principles of fluid dynamics, physiology, and virology. Even for the variables for which we currently do not have good estimates, we understand the underlying dependencies as well as the procedures/methods required to estimate these variables, and it is expected that ongoing studies will close these gaps in our understanding, and provide better quantification of all the variables involved in this model.

I would like to acknowledge Drs. Sanjay E. Sarma, Howard Stone and Jae Ho Lee for providing feedback on a draft of this article, and Shantanu Bailoor for help in preparing some figures for this paper.

There are no disclosures to report.

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