key: cord-0714873-lppge7zq
authors: Ho, Clifford K.
title: Modeling Airborne Pathogen Transport and Transmission Risks of SARS-CoV-2
date: 2021-02-24
journal: Appl Math Model
DOI: 10.1016/j.apm.2021.02.018
sha: 1a8dbc4f498f45b4ba38237d250d50b974429641
doc_id: 714873
cord_uid: lppge7zq

An integrated modeling approach has been developed to better understand the relative impacts of different expiratory and environmental factors on airborne pathogen transport and transmission, motivated by the recent COVID-19 pandemic. Computational fluid dynamics (CFD) modeling was used to simulate spatial-temporal aerosol concentrations and quantified risks of exposure as a function of separation distance, exposure duration, environmental conditions (e.g., airflow/ventilation), and face coverings. The CFD results were combined with infectivity models to determine probability of infection, which is a function of the spatial-temporal aerosol concentrations, viral load, infectivity rate, viral viability, lung-deposition probability, and inhalation rate. Uncertainty distributions were determined for these parameters from the literature. Probabilistic analyses were performed to determine cumulative distributions of infection probabilities and to determine the most important parameters impacting transmission. This modeling approach has relevance to both pathogen and pollutant dispersion from expelled aerosol plumes.

In ~March 2020, social distancing was introduced throughout the United States to mitigate transmission of the SARS-CoV-2 virus, which causes COVID-19. Six feet (~2 m) was proposed as a safe separation distance to prevent the spread of COVID-19. The basis for six feet of separation stems from early studies [1] , which determined that larger respiratory droplets (> ~100 microns) that may carry infectious pathogens are likely to "fall out" due to gravity within ~1 -2 m. However, more recent studies of COVID-19 suggest that smaller respiratory droplets (~ 5 microns or less) can remain airborne for long periods, and SARS-CoV-2 can remain viable in these smaller aerosolized droplets for up to several hours [2, 3] . In addition, these smaller droplets have been identified as a significant contributor to infectious aerosols [4, 5] . Figure 1 illustrates the potential airborne transmission pathways from expelled respiratory droplets. Several COVID-19 outbreaks including a restaurant in Guangzhou, China [6] , a call center in South Korea [7] , and a choir practice in the state of Washington [8, 9] also indicate a strong potential for airborne transmission of COVID-19. Fomite transmission can also occur through contact of contaminated surfaces, but this work focuses on transmission via inhalation of expelled aerosols during coughing/sneezing and breathing/talking and the impact of environmental factors. yields a Reynolds number on the order of 1000, which is 30 -40 times less than the Reynolds number for coughing or sneezing. Thus, the exhaled aerosols during breathing and talking have much lower momentum than coughing or sneezing and do not propagate as far.

However, because the size of the droplets that are emitted during tidal breathing are small, the exhaled aerosol plume can remain suspended for long periods. Thus, despite the lower viral load per exhalation event relative to coughs or sneezes, the persistence of the small aerosolized droplets and continuous nature of breathing and/or talking can increase the potential for transmission, especially in enclosed spaces with low fresh-air exchange.

Aerosol transport during various expiratory events has been simulated in previous studies using computational models to determine pathogen concentrations in various settings [24] [25] [26] [27] [28] [29] [30] [31] [32] . The models range from simple well-mixed compartment models to high-fidelity computational fluid dynamics (CFD) models. Previous CFD models were used to interrogate transient concentration distributions in environments ranging from offices to airplane cabins, and a few have quantified the probability of infection risks. You et al. [33] and Yan et al. [34] 

The general modeling approach in this study is to use CFD models to simulate expelled aerosol plume dispersion and perform comparative studies of exposure risks of expiratory events under various scenarios. Spatial and temporal simulations of the relative concentrations of the expelled pathogen (assumed to be uniformly distributed in the vapor plume) are compared and used to determine risks of exposure and probability of infection. High-fidelity turbulence models are available to simulate time-varying turbulent processes initiated by violent expiratory events (e.g.,

Large Eddy Simulation (LES) [35] ). However, in these studies, a time-averaged turbulence model (e.g., Reynolds-Averaged or Favre-Averaged Navier Stokes equations using a k- turbulence model) was implemented to reduce the computational expense and evaluate a large number of scenarios.

Relative trends in time-integrated concentrations and exposure risks as a function of time and location are assumed to be adequately captured by the time-averaged turbulence models; the objective is to perform comparative risk analyses of different configurations and scenarios rather than to make absolute predictions. In addition, we assume that the relative distribution and concentration of pathogens (droplet nuclei) can be represented by the transient dispersion of the expelled vapor plume. We neglect potential transmission from large droplets, which we assume will fall out due to gravity. Thus, the analyses in this work focuses on small droplets (aerosols) that remain aloft in the air for long periods and have been identified as a significant contributor to airborne transmission [4] .

Several modeling and experimental studies have shown that these small particles (a few microns or smaller) follow the bulk airflow and can be accurately represented by a tracer gas [17, 30, 31, 36] . In the experimental study of Bivolarova et al. [36] , particles of three sizes (0.07, 0.7, and 3.5 microns) and nitrous oxide tracer gas were generated in a room simultaneously at the same location with various ventilation rates and configurations. Sampling at different locations within the room showed that "tracer gas can be used to evaluate the distribution of aerosol particles in ventilated rooms." Gupta et al. [30] and Zhang et al. [31] also concluded that small particles behaved like a tracer gas and followed the bulk airflow during testing and modeling of particle transport in an airplane cabin, and Gupta et al. [30] CAD package Solidworks, which makes geometry and mesh creation seamless and efficient for various scenarios and configurations.

The simulated mouth was 2 cm x 2 cm with a 24° curvature in the vertical direction to account for vertical spread. The thermodynamic boundary conditions that were used for the exhaled vapor and ambient conditions are summarized in Table 1 A representative computational domain used to simulate a cough is shown in Figure 2 . Lateral symmetry is assumed in the z-direction under quiescent conditions or when airflow is in the xdirection such that a vertical symmetry plane is applied at z=0 to reduce the size of the computational domain. When airflow is in the z-direction (crossflow), symmetry is not applied, and the lateral boundary is extended in the z-direction. Vertical symmetry is not assumed due to buoyancy effects. Pressure-opening boundary conditions are applied at all external boundaries to allow for flow into and out of the domain. Air flowing into the domain due to local pressure gradients is assumed to be at ambient conditions with properties shown in Table 1 . respectively, was therefore used in the comparative simulations for computational expediency.

A grid convergence study was also performed and is described in Section 2.1.3.

The number of elements that were generated ranged from ~10 5 -10 6 hexahedral cells Figure 3 shows a sensitivity study comparing the simulated results of different mouth shapes (square arc vs. circular) and exhaled velocity profiles (step function vs. triangular) following a cough. The circular mouth has a diameter of 0.0226 m, which is slightly larger than the sides of the square-arc mouth (0.02 m) to yield the same total area. All velocity profiles yield the same mass and total volume of exhaled air and water vapor. Results show that slight changes in the shape of the mouth and exhaled velocity profile can yield differences in the overall profile of the transient exhaled water vapor concentrations. The square-shaped velocity profile, while shorter in duration than the triangular velocity profile, yields higher initial sustained velocities resulting in plumes that propagate further. The square-arc shape of the mouth and triangular velocity profile appear to produce a more bimodal plume shape relative to the circular mouth that eventually results in separation and a "pinch-off" of the leading edge (see Section 2.1 for more details). For consistency, the square-arc mouth shape and triangular velocity profile were used as a baseline to simulate coughs under varying environmental and face-covering scenarios. and water vapor from the mouth. The relative humidity of the exhaled and ambient air did not have a significant impact on buoyancy due to the relatively low concentration of water vapor in the exhaled air. Previous studies have shown that increased relative humidity can also reduce evaporation of respiratory droplets, leading to larger droplet sizes and increased deposition [17] .

A grid convergence study was performed on simulations of a cough under quiescent conditions with different numbers of elements. Figure 4 shows simulated relative humidities as a function of distance from the source for different times (t = 0.05 s, 0.2 s, and 1 s) following the initiation of a cough for different numbers of elements. Results show that for coarse meshes with a total number of elements less than ~50,000, the results display evidence of numerical dispersion. At 0.05 s, which is the time of peak velocity during the cough, the meshes with greater than ~50,000 elements yielded a sharp profile of the relative humidity, indicating a propagating wave-like pulse while the expelled velocity remained high. The coarser meshes exhibited a more diffuse profile, indicative of numerical dispersion. At 0.2 s, the cough had ended and the simulated relative humidity decreased from 100% near the source to values near 50% (ambient) further away. The simulated relative humidities in the finer meshes decreased to 50% at ~0.7 m, indicating that the expelled plume had not yet reached that location. The simulated relative humidities in the coarser meshes were above 50% at these longer distances, another indication of numerical dispersion. Finally, at a time of 1.0 s, the simulated relative humidities decreased to values just above 50% as the water vapor plume dispersed throughout the domain. All simulations showed a similar trend at 1 s, although beyond ~0.8 m, the coarser meshes yielded slightly higher relative humidity values than the finer meshes. The grid convergence index (GCI) was calculated by selecting the distance the expelled vapor plume propagated along an isopleth of RH=0.5 at 0.05 s for fine (~480K cells), medium (~98K cells), and coarse (~12K cells) meshes. The GCI for the fine and coarse meshes were 0.078 and 0.13, respectively, and the ratio of the GCI accounting for the grid refinement ratio and convergence order was 1.04, indicating that the results were in the asymptotic range [46] . The simulated pressures, temperatures, and velocities were also evaluated as part of the grid convergence study, and results converged to an asymptotic solution at ~100K -200K cells.

As a result of these grid convergence studies, the simulations used for the comparative cough scenarios used meshes with at least ~10 5 elements, which corresponded to element sizes of ~1 mm near the mouth, ~1 cm within ~0.5 m of the mouth, and element sizes of several centimeters or more further away from the mouth, depending on the size of the simulated domain. CFD simulations were also qualitatively compared to high-speed images of a sneeze ( Figure 5 ).

The experimental visualizations revealed that the expelled plume of droplets exhibited a counterrotating flow pattern near the leading edge as the plume propagated forward (smoke-ring effect), entraining air along the flow path of the plume. The effects of buoyancy were also observed at later times as the plume propagated upward following the initial jet-like pulse from the sneeze.

Simulations of the physical sneeze using the CFD tools and models described in this study were performed. Boundary conditions described in Bourouiba et al. [12] were used ( Table 1 .

For further validation, the velocity distribution measured by Kwon et al. [11] during a cough was simulated. Data from Kwon et al. [11] was chosen because they used a non-invasive laserimaging method to measure the velocities induced by a cough from human subjects. The velocities of atomized oil droplets were recorded using particle image velocimetry within a field of view of ~0.25 m (horizontal x-direction) by ~0.18 m (vertical direction) immediately in front of the mouth. Simulations used the same methods and boundary conditions described in Sections 2.1.1 and 2.1.2 except that the mouth angle was increased to 38° to correspond to the values measured by Kwon et al. [11] . Figure 7 shows the measured and simulated velocity distributions during the peak cough velocity within the measured field of view. Results show that the simulated velocity distribution overpredicts the velocities greater than 6 m/s and underpredicts the velocities in the range of 1 -6 m/s. However, the general trend in the measured velocity distribution is captured by the simulations and provides further verification of the simulation method. 

The risk of exposure to expelled aerosolized pathogens during various expiratory events is modeled by using the simulated spatial and temporal expelled vapor concentrations (recall that airborne pathogens are assumed in this study to be uniformly distributed in the aerosol plume, which is represented by the bulk motion of the expelled vapor plume). The normalized cumulative exposure (exposure risk) at a particular location is defined in this study as the timeintegrated concentration at that location divided by the time-integrated concentration at the source (mouth) as a function of time following an expiratory event. The cumulative exposure was determined at several distances from the source as shown in Figure 8 . For violent expiratory events such as coughing and sneezing, the normalized cumulative exposure is expected to increase rapidly for distances close to the source. At further distances, the exposure risk is expected to decrease due to dilution and dispersion of the expelled aerosol plume. In addition, the exposure risk increases more slowly at further distances since it takes time for the aerosol plume to propagate to longer distances. For talking or breathing, the impacts of time and distance are expected to be even more pronounced since the velocity and number of expelled pathogens are less. Section 3.1.2 provides simulated results of the normalized cumulative exposures following a cough for different external wind conditions.

The modeling methods and tools described in the previous sections were used to simulate spatial and temporal vapor concentrations following an expiratory event under varying environmental conditions. This enabled a calculation of the normalized cumulative exposure risks by taking the ratio of time-integrated concentrations at various locations to the time-integrated concentration at the source and plotting those values as a function of time and distance (Section 3.1.2). This section presents a model of infection probability that uses the CFD results together with viral transmission parameters (e.g., viral load, lung deposition fraction, aerosol viability, infectivity).

The infection model in this study assumes an exponential probability density function for infection as a function of dose. Watanabe et al. [47] found that an exponential model fit the dose-response relationship for infection of both mice to the SARS coronavirus (SARS-CoV) and

humans to the HCoV-229E coronavirus. The probability of infection, P(x,t), can be expressed as follows for an exponential distribution:

where r = infectivity rate or probability that a single pathogen will cause an infection (1) and (2) yield the following expression for the probability of infection:

The infectivity rate, r, can be expressed in terms of the median infectious dose, d = ID50, required to infect 50% of a population by setting Eq. (1) to 0.5 and solving for r: 50 50 

Although the median infectious dose, ID50, has not been reported for SARS-CoV-2 as of the writing of this report, a range of ID50 (0.6 -3 TCID50) has been reported in the literature for influenza A aerosols (≤ 3 m) [49, 50] and is used in this study to determine the infectivity rate, which ranges from 0.23 to 1 (infections per TCID50).

The lung deposition fraction, , is a function of particle size and has been reported in the literature for particle sizes range from 0.015 -5 microns using 67 human subjects ranging in age from 7 -70 years [51] . The deposition fraction ranged from ~0.3 -0.9 and exhibited a nonmonotonic behavior with the lowest deposition occurring ~0.2 -0.5 microns and increasing deposition for smaller and larger particle sizes.

The viral load, c, for SARS-CoV-2 has been reported in the literature and can vary by orders of magnitude depending the patient, stage of illness, and sample location (e.g., throat, nose). Pan et al. [52] report median values of SARS-CoV-2 viral load from throat and sputum samples of 7.99x10 4 and 7.52x10 5 RNA copies/mL, respectively, with individual sample loads ranging from 641 to 1.34x10 11 copies/mL. Zou et al. [53] report SARS-CoV-2 viral loads ranging from ~1.5x10 4 to 1.5x10 7 in nasal and throat swabs, and Wolfel et al. [54] reported average and maximum SARS-CoV-2 viral loads in sputum of 7x10 6 copies/mL and 2.35x10 9 copies/mL, respectively.

It should be noted that viral load can be expressed as viral copies per mL, PFU/mL, or TCID50/mL, where PFU and TCID50 are based on culturable (infectious) viruses. Pantelic et al. [50] and Murphy et al. [55] report the median viral loads for influenza A from seven patients as 5x10 5 TCID50/mL. The correlation between SARS-CoV-2 viral load and culturable virus is currently unknown [53] , but some studies have shown that ~1 TCID50/mL or ~1 PFU/mL correspond to ~1000 viral copies/mL since not all viral copies are culturable [19, 56] . Based on these studies, despite the enormous reported range of viral loads, we assume a modest range of infectious viral loads for SARS-CoV-2 between ~10 5 and 10 6 TCID50/mL in this study.

Assuming ~1000 viral copies/TCID50 [19, 56] , this corresponds to ~10 8 -10 9 copies/mL, which falls well within the reported ranges for SARS-CoV-2 [52] [53] [54] .

The density of respiratory fluid, mucus, is assumed to be similar to that of liquid water with variations due to dissolved solids (e.g., salts) and oils [13] .

The inhalation rate is calculated from reported average values of inhalation volume per breath and breaths per minute during tidal breathing.

The density (or concentration) of exhaled water vapor, vapor, is obtained from CFD simulations described in earlier sections. The concentrations can vary in space and time depending on the expiratory event and environmental conditions.

The fraction of pathogens that remain viable in aerosol form as a function of time has been measured for SARS-CoV-2 by van Doremalen et al. [3] . We normalized the viability data to the maximum TCID50 per liter of air and assumed that the maximum viability once aerosolized was 10% for comparison to past data reported in Pantelic et al. [50] for influenza A. Figure 9 shows that the decay of SARS-CoV-2 matches the decay of influenza A. Assuming an initial viability of 10% once aerosolized, the viability of SARS-CoV-2 drops to ~2% after ~2 -3 hours. The expression for the viability of aerosolized SARS-CoV-2 as a function of time, t (hours), that is used in this study is given as follows: 

The value of 0.46 is calculated as the propagated relative errors from vapor(x,t) and f(t) (i.e., root sum square) Table 1 . The length of the simulated domain is 1 m. downwind of potential sources of airborne pathogens. These findings are consistent with those of Feng et al. [17] , which found that downwind exposure to expelled respiratory droplets could occur beyond 6 ft (1.8 m).

The impact of face coverings (face masks and face shields) on the expelled vapor plume and resulting exposure risks during a cough were evaluated using the CFD tools described in Section micron will be filtered). Although particle and pathogen filtration resulting from electrostatics and other deposition processes (e.g., turbophoresis, thermophoresis, inertial effects, Brownian motion) were neglected in these simulations, the pressure drop as a function of flow rate (e.g., permeability) for a MERV 11 material was used to analyze the impact of face masks on bulk flow and transport of small aerosols. In addition, Leung et al. [58] measured the efficacy of surgical masks as a function of respiratory droplet size and showed that during 30 minutes of breathing, masks were effective at blocking larger droplets (>5 microns) for both seasonal coronavirus and influenza virus. However, results were mixed for smaller aerosol droplets (<5 microns). Leung et al. [58] observed a reduction in detected small aerosols of coronavirus for those who wore a mask (from 4 of 10 to 0 of 11), but mask wearing did not yield a significant reduction in detected small aerosols (<5 microns) of influenza virus. Thus, while masks may be effective at blocking larger droplets, smaller aerosolized droplets may pass through. Figure 16 shows the simulated velocities near the mask at 0.05 s, which is at the time of the peak velocity (25 m/s). Results show that the velocity exiting the mouth is ~25 m/s but reduces to ~13 m/s near the interior side of the mask and to less than ~4 m/s at the exterior side of the mask.

The reduced velocities and flow rate across the mask are due to the reduced permeability and 1 From Solidworks Flow Simulation. MERV 11 is similar in particle filtration to a cloth mask. Higher MERV-rated filters have smaller pore sizes and are more effective at trapping smaller particles. Chart of MERV ratings and particle filtration can be found at this website: https://www.epa.gov/indoor-air-quality-iaq/what-merv-rating-1.

Permeability of various filter types can be found here: https://oaspub.epa.gov/eims/eimscomm.getfile?p_download_id=498205 prescribed pressure drop across the simulated porous mask as a function of velocity as shown in Figure 16 . Konda et al. [59] and Davies et al. [60] provide measurements of pressure drop and particle filtration efficiency for masks made of different materials. al. [58] showed that masks may not be as effective at filtering small aerosols (< 5 microns), the current study shows that masks are effective at reducing the forward propagation and momentum of the expelled breath and vapor. These effects, together with an increased time to exposure, are potential benefits that the mask provides to reducing probability of transmission by allowing additional dilution of the expelled plume. A caveat is that these simulations reveal that the mask may yield temporarily increased aerosol concentrations in the vicinity of the source for several seconds before the plume has time to disperse. If a receptor were to move into that region immediately after the expiratory event, the receptor may be exposed to higher concentrations relative to a plume that was dispersed by a cough without a mask. These results complement the CFD simulations performed by Feng et al. [17] and Khosronejad et al. [35] , which showed that masks can significantly reduce expelled droplets in the air. Face Shield. Figure 20 shows As discussed in the case of a face mask, these simulations reveal that a face shield may yield temporarily increased aerosol concentrations in the vicinity of the source for several seconds before the plume has time to disperse. If a receptor were to move into that region immediately after the expiratory event, the receptor may be exposed to higher concentrations relative to a plume that was dispersed by a cough without a face shield or a mask. Table 2 .

The expelled vapor concentrations were taken from the CFD simulations described in Section presented in Figure 21 are intended to illustrate general trends in the probability of infection.

Uncertainties in the infection parameters can vary widely as seen in Table 2 . To include and determine the impact of these uncertainties, probabilistic models are developed in the next section. Table 2 assuming the receptor is stationary and directly exposed to and inhaling the pathogen over the entire duration.

Significant uncertainties exist in the input parameters used in the infection model. In this study 300 realizations were found to be sufficient to adequately represent the cumulative distribution function of the probability of infection for the uncertainty distributions provided in Table 2 . Latin Hypercube Sampling was used to sample 300 values from each of the input parameters in Table 2 . Eq. (3) was then solved for each of the 300 realizations. 95% at distances of 1 ft and 3 ft, respectively. These simulations assume that the receptor is directly and continuously exposed to the expelled pathogen plume over the entire duration and that the receptor is inhaling continuously at an average rate for tidal breathing (Table 2) .

At 10 s (Figure 23 ), the expelled plume had not yet reached 9 ft, but distances of 6 ft or less yielded probabilities of infection. The median probability of infection at 10 s is ~100%, 98%, and 65% at 1 ft, 3 ft, and 6 ft, respectively. The 90 th percentile for the probability of infection is nearly ~100% for distances of 1 ft and 3 ft, and ~97% for a distance of 6 ft.

At 100 s (Figure 24 ), the median probability of infection is ~100%, 99%, 95%, and 80% at distances of 1 ft, 3 ft, 6 ft, and 9 ft, respectively. The 90 th percentile for the probability of infection approaches ~100% for all distances. It should be emphasized that these simulations assume direct and continuous exposure by the receptor with continuous inhalation and no external airflow. Nevertheless, such high likelihoods and probabilities of infection at long distances may indicate that the range of assumed infectiousness or viral load in Table 2 may be too high. In particular, the range of median viral load was taken from measurements of SARS-CoV-2 in units of RNA copies/mL, but as referenced in Table 2 , there may be only ~1 infectious virus out of every ~1000 measured RNA copies. Table 2 assuming the receptor is stationary and directly exposed to and inhaling the pathogen over the entire duration. Table 2 assuming the receptor is stationary and directly exposed to and inhaling the pathogen over the entire duration. Table 2 assuming the receptor is stationary and directly exposed to and inhaling the pathogen over the entire duration.

The cumulative probabilities span a large range due to the large uncertainties in the input parameters. A stepwise linear (rank) regression is performed to quantify the relative impact of each input parameter on the magnitude and spread of the simulated probability of infection. The stepwise regression generates a sequence of regression models that successively adds the most important input parameters to the regression to improve the overall correlation. In the end, the regression identifies those parameters that are significantly correlated to the performance metric and omits those parameters that are not. Results can help guide and prioritize future research to better characterize the most important parameters and processes.

The sensitivity of the independent variables can be represented by ranking of the independent variables using either R 2 or B are typically the same.

As an example, Figure 25 and Figure 26 show plots of the R 2 and the standardized regression larger mucus density, mucus (g/mL), means that for a given viral load, c (pathogens/mL), fewer pathogens will exist per mass of respiratory fluid (c/mucus). So, for a simulated mass of expelled respiratory fluid (simulated as pathogen-laded water vapor in this study), the simulation with a larger assumed mucus density will yield a smaller number of pathogens and a lower probability of infection. 

This study developed a modeling approach that coupled CFD models with exposure and infection models to quantify impacts of ventilation and face coverings on airborne transmission risks. CFD models were developed to evaluate aerosol plume dynamics and the impacts of face coverings and ventilation. The simulations were used to quantify relative exposure risks as a function of time and location for a cough under various airflow conditions. Infection models were also developed using the results of the spatial and temporal pathogen concentrations derived from the CFD simulations assuming a passive scalar representation of the aerosol plume.

Probabilistic analyses were performed to quantify uncertainties and determine important parameters associated with the large uncertainties in the infection-model parameters. Key findings are summarized as follows:

• Results showed that CFD simulations were useful in performing comparative analyses of different transmission scenarios. The simulations used in the current study provided a good balance of physics-based modeling and expedient run times to evaluate many scenarios and configurations.

• The normalized cumulative exposure at prescribed location was defined as the time- • Wearing a face mask or face shield significantly reduced the forward plume propagation and normalized cumulative exposure by an order of magnitude. However, the vapor concentrations near the face were more persistent than without face coverings due to the reduced forward momentum.

• Uncertainty distributions were assigned to each model parameter, and Latin Hypercube

Sampling was used to perform probabilistic simulations. A stepwise rank regression analysis was performed to identify key parameters that impacted the probability of infection. Viral load, pathogen concentration in the air, and infectivity were among the most important parameters impacting the variability and magnitude of the probability of infection.

Limitations of the current study and recommended future work include the following:

 We assumed that the expelled pathogens consisted of aerosols (< 5 microns), which could be represented by a passive scalar (e.g., water vapor) [17, 30, 31, 36] . Droplet dynamics and evaporation were neglected.

o Additional studies comparing the use of a passive scalar vs. evaporating droplets should be performed to elucidate discrepancies in relative exposure risks for simulated scenarios. Motion of the source and receptor, thermal plume emanating from the receptor, and inhalation/exhalation by the receptor may also impact fluid and pathogen transport.

 Simulations of coughs with a mask neglected filtering effects of particles; only permeability effects on the pressure drop and fluid hydraulics were investigated.

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