Aerosol Particles Generated by Coughing and Sneezing of a SARS-CoV-2 (COVID-19) Host Travel over 30 m Distance

Received: July 30, 2020
Revised: November 26, 2020
Accepted: December 5, 2020

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Cite this article:

Gorbunov, B. (2021). Aerosol Particles Generated by Coughing and Sneezing of a SARS-CoV-2 (COVID-19) Host Travel over 30 m Distance. Aerosol Air Qual. Res. 21, 200468. https://doi.org/10.4209/aaqr.200468

HIGHLIGHTS

• The 2 m Safe Distance concept was examined by modelling of the aerosol dispersion. 
• It was found that aerosol particles generated by coughing can travel over 30 m. 
• Deposition doses up to 200 virus copies in 2 min were predicted by modelling. 

Keywords: COVID-19 transmission, Viral load, Size distribution, Modelling

1 INTRODUCTION

Understanding transmission of COVID-19 infection is one of the key issues the scientific community faces. Investigating the main virus transmission routes is vital for policy development to curb the spread of the current pandemic. The US Centres for Disease Control and Prevention recommends a 6-foot (1.83 m) separation distance. This exclusion distance is based on estimates of the range that have not considered convection flow and atmospheric diffusivity that affect dispersion of exhaled aerosol clouds considerably. Recent works on COVID-19 have shown that sneezes and coughs not only consist of large mucosalivary droplets that quickly follow short-range deposition trajectories downwards but, importantly, are primarily made of a multiphase turbulent gas (a puff) cloud that entrains ambient air and carries aerosol particles (Bourouiba, 2020).

The initial stage of airborne viral disease transmission is caused by virus laden droplets that are generated mainly by coughing, sneezing (Jayaweeraa et al., 2020; Mittal et al., 2020). This creates a population of airborne particles in the vicinity of a host. There is an ambiguity about terminology used for these particles: in some publications they are described as droplets, but in others as aerosol particles (Yang et al., 2007; Vuorinen et al., 2020). Here to avoid an ambiguity, airborne particles (diameter < 100 µm) generated by infected persons are described as aerosol particles assuming that these include other terms such as droplets, microdroplets and nuclei with and without viruses (Pruppacher and Klett, 1997; Baron and Willeke, 2001).

There are two major routes whereby infection can be transmitted from a host to another person: first, viruses are directly transmitted through the air by aerosol particles that are inhaled into the oronasopharynx and second, an indirect transmission when viruses are transferred by contact with contaminated objects (Pica and Bouvier, 2012; Mittal et al., 2020). Respiratory infectious diseases can be spread by airborne transmission (Hui, 2010; Rosa et al., 2013; Jin et al., 2020; Morawska and Cao, 2020). Airborne routes seem to be important for spreading COVID-19 for both presymptomatic and asymptomatic individuals (Domingo et al., 2020; Eissenberg and Shihadeh, 2020; Howard et al., 2020; Setti et al., 2020a). The presence of COVID-19 (SARS-CoV-2) virus on particulate matter have been confirmed in Northern Italy (Setti et al., 2020b) and in China (Liu et al., 2020).

van Doremalen et al. (2020) evaluated the stability of SARS-CoV-2 and SARS-CoV-1 in aerosols and on various surfaces and estimated their decay rates. SARS-CoV-2 remained viable in aerosols throughout the duration of the experiment (3 hours). Their results indicate that aerosol transmission of SARS-CoV-2 (COVID-19) is plausible, since the virus can remain viable and infectious in aerosols for hours and on surfaces up to days. On the contrary, Xiao et al. (2020) stated that evidence from some trials did not support a substantial effect on transmission of laboratory-confirmed influenza via the aerosol route. The complexity of experimental data on COVID-19 in field and laboratory fuels interests to modelling aerosol particles transmission in both open spaces and confined environments.

Recently a number of works on modelling aerosol transport in relation to SARS-CoV-2 transmission have been published. For example, a computational fluid-particle dynamics (CFPD) model was employed to simulate the transport, condensation/evaporation, and deposition of SARS-CoV-2 laden droplets emitted by coughs, with different wind velocities and humidity in a small 3 m cube (Feng et al., 2020). Numerical results indicate that the convection will enhance the complexity of the secondary flows with recirculation between the two virtual humans in the cube. With the complex environmental wind and relative humidity (RH) conditions, the 6-feet social distancing policy may not be sufficient to protect the direct person-to-person aerosol transmission.

Four different 3D computational fluid dynamics (CFD) simulation software were employed to investigate the effect of air ventilation on aerosol evolution in indoor environment (Vuorinen et al., 2020). They show that the exposure time to inhale > 100 aerosol particles could range from seconds to hours depending on the situation.

The COVID-19 pandemic has exposed significant scientific gaps in our understanding of critical issues, ranging from the transmission pathways of such respiratory diseases, to the strategies used for mitigating these transmissions (Feng et al., 2020; Jayaweeraa et al., 2020; Mittal et al., 2020; Morawska and Cao, 2020). Discrepancies in collected data suggest that more research is necessary to determine with increased certainty the role that environmental factors play in the transmission of viral pathogens exhaled by infected individuals.

In this paper, methods of atmospheric physics well developed for air quality modelling are employed to evaluate importance of the aerosol route for COVID-19 transmission. Here we are considering aerosol dispersion in complex geometries by employing compressible Navier-Stokes equations, with an integrated turbulent diffusion, aerosol-gas mass exchange, heat transfer with gravitational forces and buoyancy. The aerosol exposure-dose relationship will be investigated using the International Commission on Radiological Protection (ICRP) lung deposition model (ICRP, 1994). The main aim of this work is to: a) reduce uncertainties associated with aerosol transmission of COVID-19 in public places and evaluate the travel distances of larger particles (inertial size range) under typical conditions for different geometries; b) quantify effects of heat transfer on aerosol dispersion; c) investigate exposure-dose relationships for aerosol particles of relevant sizes and d) consider possible metrics for quantification of the risk mitigation associated with exposure to virus laden aerosols.

 
2 MODELLING OF COVID-19 SPREAD VIA THE AIR

The aerosol transformation processes relevant to dispersion of SARS-CoV-2 viruses were studied employing validated Comsol v5.5 finite element software (see ref. Comsol Multiphysics and Steer et al., 2014). The software comprises the Fluid Dynamics module with Navier-Stokes compressible flow (Ma < 0.3) equations, Heat Transfer module and Transport of Diluted Species with atmospheric turbulent diffusion module. These modules were solved together using Comsol Multiphysics software. It should be noticed that both Comsol versions of Navier-Stokes equations (compressible and incompressible) can be employed here because they produce identical results for the problems solved here.

The Navier-Stokes equations especially combined with heat and mass transfer are notoriously difficult for numerical modelling. The modelling procedure involves a set of software runs with different mesh sizes starting from coarse towards finer meshes until the mesh refinement does not influence the result. Both global and local mesh refinement metric were used to ensure efficiency of convergence and accuracy of the solution. A local metric is defined at a specific location in the model but the global - over the entire model space. The Mesh Builder module automatically generates non-uniform meshes with smaller sizes near sharp corners or surfaces with small curvature radius.

The PARDISO direct solver and iterative GMRES solver have been employed for stationary and time dependent velocity fields (u,v,w,t), temperature field T_X,Y,Z,t and the number concentration of aerosol particles C_X,Y,Z,t. An additional mass transfer of water vapor between aerosol particles and the gas phase was simplified and employed as a parameterization of a direct modelling of the mass exchange (Pruppacher and Klett, 1997; Baron and Willeke, 2001; Vuorinen et al., 2020). The lung deposition model was based on ICRP (1994) with lung deposition efficiency from Gorbunov et al. (2004); Ruzer et al. (2005); Karl et al. (2016).

In modelling, the dispersion of the aerosol cloud was calculated from a simplified initial aerosol cloud of an elliptical shape with a given aerosol number concentration and cloud boundaries. The subsequent evolution of the aerosol particles concentration was recorded as a concentration C_X,Y,Z for a steady-state modelling or as C_X,Y,Z,t for time dependent solutions.

In this work we developed a simple human head model where we used a cylinder for neck and ellipsoids for the head and nose. This model is a useful simplification. We tested how variations in the size of the model and features of the head affect the aerosol concentration field. It was found that variation in size of parts (by 10%) has less than 1% influence on the aerosol/droplet concentration at distances more than 2 m from the source. Therefore, the simplified human model seems to be enabled to obtain sufficiently accurate results.

2.1 Steady-state Aerosol Source X–Z Plane and Boundary Conditions

First, an open space aerosol dispersion 2D steady-state model was developed and employed to evaluate the influence of the wind velocity on the possibility of transferring aerosol particles from the source to an uninfected person. In the model, aerosol particles of sizes close to those observed in coughing, talking and sneezing experiments were considered. The concentration field of aerosol particles (C_X,Z) have been calculated in X–Z space (–3 m < X < 17 m; –2.5 m < Z < 2.5 m). The mouth of the host head was at X = 0, Z = 0.8m or 1.7m from the ground.

The velocity field boundary conditions for the ground and the human surface were u = 0, without slip (No slip). The right boundary was at X = 17 m was pressure p = 1atm. The top boundary condition at Y = 2.5 m was u·n = 0 with slip. The left boundary at X = –3 m was u = U₀·n with U₀ - wind speed.

Different boundary conditions for the convection flow inlet were tested, for example a more realistic linear condition U₀ = 2·(Z + 2.5 m)·U₀, but the influence on the concentration field profile was negligible (less than 10% in number concentration) if the total mass flow was constant. It could be explained by the position of the aerosol source quite far above the ground (1.7 m). Therefore, the effect of the ground boundary was small at 1.7 m. A randomization of U₀ along the Z co-ordinate also gave rather negligible variations in the number concentration field.

The aerosol particles boundary conditions for the ground and the human surface were C_X,Z = 0. The right boundary condition at X = 17 m was the outflow. The top boundary condition at Y = 2.5 m was “No Flux” (Comsol Multiphysics). The left boundary at X = –3 m was C_X,Z = 0.

The initial aerosol cloud shape was an ellipsoid with the diameter along the X co-ordinate - 1 m and the vertical diameter - 0.5 m. The cloud was at X = 1 m, Z = 0.8 m or 1.7 m from the ground. The initial aerosol particle concentration inside the cloud was C_X,Z = 5 cm^–3. In preliminary test runs, it was established that variations in the shape and the volume of the cloud did not influence the results considerably if the larger diameter of the cloud was less than 1.5 m.

Here, simulations were carried out for relatively simple geometries of open spaces. Various wind speeds and turbulent diffusion coefficients were considered. An example of confined environments with and without ventilation facilities was studied in the model too.

2.2 Atmospheric Diffusivity and Aerosol Evolution

Dispersion of aerosol particles in the atmosphere is mainly influenced by the convection, heat transfer and turbulent diffusion. The atmosphere especially near the boundary layer at the size of the human is a non-steady-state non-equilibrium system with extremely complicated structure of boundary conditions. Atmospheric turbulence is intermittent in space and time and comprises eddies ranging from microscale size to meters even near the ground.

The diffusivity of such a patchy turbulence as atmospheric is mainly related to statistical parameters describing the morphology of turbulent events: filling factor, lifetime and height of the patches, etc. (Wilson, 2004). A statistical description of the turbulent characteristics is often employed in order to evaluate the impact of small-scale turbulence on the transport of aerosols. In-situ measurements of atmospheric diffusivity coefficients range from 0.2 to 0.8 m² s^–1 (Alisse and Sidi, 2000). There are many uncertainties with the atmospheric turbulence, for example greater values and variations in the atmospheric diffusivity from circa 1 to 100 m² s^–1 have been reported (Kennedy and Shapiro, 1980). In the boundary layer the atmospheric diffusion coefficient is in the range from 0.1 to 160 m² s^–1 depending on the stability of the atmosphere, the urban geometry and traffic (Hanna et al., 1982). In street canyons the atmospheric diffusion coefficient is likely to be in the range from 0.1 to 10 m² s^–1, ibid.

To investigate aerosol dispersion over long distance simplifications are required (e.g., see Feng et al., 2020; Vuorinen et al., 2020). Here we employed software where compressible (Ma < 0.3) Navier-Stokes equations are solved to generate the larger relatively stable eddies, but the smaller short-lived eddies are modelled with integrated atmospheric diffusion approach using the diffusion of diluted species module (COMSOL Multiphysics). The diffusion coefficients are taken from experimental works. This simplification allows modelling of aerosol dispersion on larger scales and it is widely used in studying atmospheric pollution studies from the global scale down to human size scales. This COMSOL Multiphysics approach is similar but not identical to Reynolds averaged Navier-Stokes approach (RANS) in which turbulent motion is not explicitly resolved and its effects on the velocity flow field are modelled using approximate turbulence models. The advantage of the current approach is the computational speed (~10 min for 3D models) and ability to incorporate the heat transfer, turbulent diffusion and external forces modules. This has been widely used in air quality control and many other applications (COMSOL Multiphysics).

 
2.3 Size and Concentration of Droplets Generated by Coughing and Sneezing

Sneezing and coughing generate clouds of largely liquid aerosol particles in the air mainly in the size range from 1 to 100 µm (Zhao et al., 2005; Han et al., 2013).

Particles of different sizes have significantly different dynamic characteristics. Relatively large droplets are deposited within 1 metre due to gravitational settling (Leder and Newman, 2005). Smaller particles can travel longer distances. Gustavo (2012) has detected high concentrations of submicron particles in cough aerosols. The total number concentration of aerosol particles (integrated over all sizes) in some papers is from 10³ cm^–3 to 2·10³ cm^–3 (Yang et al., 2007). Average size range of the particles was 1–10 µm, and 50% of particles have diameter > 10 mm. However, some of small particles may or may not be directly related to coughing but to the background aerosol. The estimated droplet concentrations for coughing ranged from 2.4 to 5.2 cm^–3 per cough (Chao et al., 2009). There are many inconsistencies in the literature on particle number and size distribution of aerosol particles generated by speaking, singing, coughing and sneezing. In this paper, the initial concentration of aerosol particles generated by a host considered as an unknown parameter in modelling that does not affect the dispersion of aerosols. If the initial concentration is known, then the modelling results can be readily converted into aerosol particle number concentrations.

The majority of data suggests that size range of particles generated by coughing and sneezing by infected humans is from 1 µm to 100 µm. The particles in the size range from 1 µm to 10 µm are part of PM₁₀ characteristic and well researched, for example it is known that they stay airborne for some time (minutes to hours) and travel long distances (more than 10 s of meters). Dispersion of larger particles 10 µm to 100 µm is less understood. The larger particles have a greater viral load than the small particles (on average 1000 times greater), therefore, a contribution of larger particles to the COVID-19 aerosol transmission route should be better understood. Here we investigate dispersion of aerosol particles of the size range from 10 µm to 100 µm.

2.4 Viral Load

There are many uncertainties with quantification of the viral load. The median viral load in posterior oropharyngeal saliva or other respiratory specimens is 5·2 log copies per ml and maximal virus loading is found circa 10⁸ copies mL^–1 (To et al., 2020). The viral loads in throat swab and sputum samples peaked at around 5–6 days after symptom onset, ranging from around 10³ to 7.11·10⁸ copies mL^–1 according to Health Information and Quality Authority (2020). The viral loads, as identified by values of RT-PCR assay, have been reported to be highest soon after the onset of symptoms reaching up to 10⁸ copies mL^–1 (Zou et al., 2020). There is no direct evidence on the viral load in aerosol particles generated by an infected person. The data obtained from swabs are used here as a proxy to evaluate the number of virus copies in aerosol particles. In the absence of direct measurements, the assumption of equal viral loads in the posterior oropharyngeal saliva and in aerosol particles is accepted here.

Here we model atmospheric transfer at various conditions including atmospheric diffusion coefficient (0.1 m² s^–1–10 m² s^–1), size of aerosol particles/droplets (10 µm–100 µm) and concentration is modelled in dimensionless units that can be easily adjusted to any starting value. The range of parameters is essential to limit uncertainties with assumptions on parameters to ensure that using variations in parameters would not change the conclusions dramatically.

 
3 RESULTS AND DISCUSSIONS

Two types of aerosol source have been modelled: a steady-state source and time-dependent source. The first one is a simplification of a constant aerosol generation process such as during talking or singing. The latter is related to a single coughing or sneezing act, generating a cloud that evolves and moves with time.

 
3.1 Steady-state Aerosol Source X–Z Plane

It was found that aerosol particles were carried out with the convection flow and spread from the source along the Z-coordinate, Fig. 1. The aerosol particle concentration field forms a narrow plume from the source directed slightly to the ground. If we assume that the initial concentration of particles in the cloud is C₀ = 5 cm^–3 then the concentration shown with a red contour line (C,_XY = 0.3·C₀) reaches almost 6 m distant from the source for wind speed 1 m s^–1, Fig. 1(A). The concentration shown with the yellow contour line (C_X,Y = 0.2·C₀) covers over 10-meter distance from the source of the aerosol particles. Interestingly the green contour line (C_X,Y = 0.1·C₀) is spread beyond X = 12 meters. The concentration field is asymmetric due to influence of the ground boundary conditions. The upwind area is completely clean from aerosol particles emitted from the source. Modelling demonstrates a strong influence of convection on the spread of aerosol particles along the X-axis, but diffusivity is responsible for dispersion of particles along the Y-axis and Z-axis.

Fig. 1. Number concentration of aerosol particles field in cm^–3 modelled for an aerosol open space geometry with a single aerosol source (brown figure looking to the right). Diameter of aerosol particles 100 µm; RH = 90%. T = 20°C. (A) wind speed - 1 m s^–1 and the turbulent diffusion coefficient Dif = 0.01 m² s^–1; (B) wind speed - 3 m s^–1 and the turbulent diffusion coefficient Dif = 0.01 m² s^–1; (C) wind speed - 1 m s^–1 and the turbulent diffusion coefficient Dif = 0.1 m² s^–1. Magenta streamlines in Fig. 1(C) show an unrealistic case of aerosol particle trajectories without the atmospheric diffusion (Dif = 0).

On a quiet day, the average air wind speed at the height of 1.5 to 2 meters above the ground is from 0.3 m s^–1 to 1 m s^–1. On a moderate day, with a higher wind speed, for example 3 m s^–1, red and yellow contour lines are shrunk, but the green line extends beyond the model border, X = 17 m, Fig. 1(B). This shows that the faster movement of the aerosol decreases the concentration of particles due to a greater dilution of the aerosol. This effect is well known in air pollution research when higher wind speed episodes are associated with lower concentrations of aerosol particles. This decreases the area of the plume and therefore, chances that the general public will be exposed to the virus laden aerosol particles. Therefore, in a windy day the viral infection transfer rate should be lower than in a quiet day.

The atmospheric diffusivity also influences the aerosol dispersion. An increase in the diffusion coefficient from Dif = 0.01 m² s^–1 (Fig. 1(A)) to Dif = 0.1 m² s^–1 (Fig. 1(C)) increases the vertical spread of the contour plots but shortens them in the X-axis. A spectacular manifestation of the effects of diffusion is shown in Fig. 1(C) with magenta streamlines obtained with Dif = 0. This is the case of an unrealistic steady-state laminar flow without turbulent diffusion when an aerosol cloud travels the distance determined by the gravitational force and the wind speed.

The model built enables complicated geometries to be studied. If a car is placed in an open space, for example a parking area near a supermarket then the air flow can be disturbed by the car and change the aerosol concentration field. The flow of aerosol particles is carried over a car near the source and extended zone of high concentration, Fig. 2.

Fig. 2. Number concentration of aerosol particles field in cm^–3 modelled for an aerosol open space geometry with a single aerosol source (brown figure looking to the right). Diameter of aerosol particles 100 µm; RH = 90%. T = 20°C. Wind speed - 1 m s^–1 and the turbulent diffusion coefficient Dif = 0.01 m² s^–1. A car is placed at X = 6 m. The concentration color legend is the same as in Fig. 1.

In this case a person at X = 10 m is in the middle of the green contour line and exposed to aerosol concentration in the range 0.1·C₀ < C_X,,Z < 0.2·C₀. Without car a person in the same place would be exposed the aerosol concentration C_X,,Z < 0.1·C₀, Fig. 1(C). There is also a large stagnation zone behind the car where several large eddies are formed between X = 7 m and 17 m, Fig. 3. It should be noticed that the largest eddy at ~12 m causes the backwind trajectories of the magenta streamlines. Similar effects were observed in models with other obstacles: trees, shrubs, people, etc. The considered cases were modelled in a humid air Rh = 90% with aerosol particles of 100 µm diameter. It is a worst-case scenario when evaporation is negligible, and the terminal velocity of particles is largest. Therefore, in all other cases travel distances of aerosol particles should be greater. Thus, in open spaces various obstacles to the wind may increase the spread of viral infections.

Fig. 3. A magnified air velocity field for of aerosol particles of 100 µm; RH = 90%. T = 20°C. Wind speed - 1 m s^–1 and the turbulent diffusion coefficient Dif = 0.01 m² s^–1. A car is placed at X = 6 m (not in the frame). Symbols of humans have been removed to make eddies clear visible. The velocity color legend is in m s^–1.

The influence of heat transfer on the air dynamic and the aerosol particle concentration was investigated in cases when a car shown in Fig. 2 has an elevated boundary temperature as after been driven. In the model runs the surface of the car boundary vas varied from 30°C to 40°C. the influence of the elevated temperature was found to be negligible. Thus, this type of heat transfer was not essential for aerosol cloud dispersion. However, for other geometries and conditions effects of heat transfer can be greater.

 
3.2 Steady-state Aerosol Source X–Y Plane

The aerosol cloud dispersion in the horizontal X–Y plane was calculated at Z = 1.7 m in the same way as it was done for modelling aerosol travel in the vertical plane (X–Z). In the model, a concentration field of smaller 10 µm diameter particles have been calculated in X–Y space (–3 m < X < 17 m; –5m < Y < 5 m). The aerosol source (the mouth of the host head) was at X = 0, Y = 0 at 1.7 m from the ground.

The velocity field boundary conditions for boundaries at Y = 5 m and at Y = –5 m were allowing slip. The rest of boundary conditions were the same as in the X–Z modelling. 

It was found that aerosol particles were carried out with the convection flow and spread from the source in all directions. An open space geometry modelled with wind speed 0.3 m s^–1 (0.66 mph) shows that small particles travel considerable distances, greater than 15 meters, Fig. 4.

Fig. 4. Contour plot for the concentration field of aerosol particles (10 µm diameter) from a single source in X–Y plane 20 m × 10 m. Wind speed - 0.3 m s^–1 (0.66 mph), atmospheric diffusivity – 0.05 m² s^–1. Contour lines show the concentration of aerosol particles: (C,_XY = 0.3·C₀) – red; (C,_XY = 0.2·C₀) – yellow; (C,_XY = 0.1·C₀) – green. RH = 50%. T = 20°C. The initial concentration of particles in the cloud is C₀ = 5 cm^–3.

An aerosol particle concentration field forms a plume from the source directed downwind. The concentration field is highly asymmetric. The upwind area is completely clean from aerosol particles emitted from the source. Modelling demonstrates a strong influence of convection on the spread of aerosol particles along the X-axis and diffusivity - along the Y-axis.

An increase in the wind speed from 0.3 m s^–1 to 1 m s^–1 leads to decrease of the length of each contour plot. For example, the yellow contour line covers less than 9-meter length, Fig. 5 where in Fig. 4 it is 15 meters. This is explained by a greater dilution of the aerosol at higher speed of wind. This supports findings in the modelling in X–Z plane that an increase in the wind speed reduces exposure to aerosol particles.

Fig. 5. Contour plot for the concentration field of aerosol particles from a single source in the area of 40 m × 10 m. Velocity of air - 1 m s^–1 (2.2 mph), atmospheric diffusivity - 0.05 m² s^–1. Contour lines show the concentration of aerosol particles: (C,_XY = 0.3·C₀) – red; (C,_XY = 0.2·C₀) – yellow; (C,_XY = 0.1·C₀) – green. With RH = 50% and T = 20°C. The initial concentration of particles in the cloud is C₀ = 5 cm^–3. An extended version of the geometry is shown in Fig. 4.

 
3.3 Time-dependent Modelling of Evolution of an Aerosol Cloud in an Open Space

A supermarket car park (10 m × 20 m) with a moderate wind speed in a dry weather is considered. A single infected person generates an aerosol cloud (by a single cough) containing small (1 µm < Dp < 10 µm) and larger (100 µm > Dp > 10 µm) aerosol particles; where Dp is the aerodynamic diameter of a particle, Baron and Willeke (2001). In typical air (RH = 50%, T = 20°C) the larger aerosol particles quickly release the water vapor that reduces the terminal velocity considerably (Vuorinen et al., 2020). The convection is also influenced by buoyancy, movement of people, machines and animals. In this modelling, all the driving forces of convection except for the wind velocity were integrated into an atmospheric diffusion coefficient.

The initial size of the aerosol cloud generated by a single cough was assumed to be 0.5 m × 1 m ellipse with concentration of particles C₀ = 5 cm^–3. After the first 6 seconds the cloud has grown up to 6 meters, see aerosol particles concentration contour line C_XY = 0.1·C₀ (green line) in Fig. 6. Then the cloud moved with the wind further along the X-axis and become wider in the X-Y plane, see a supplementary video. At the same time the particle concentration in the cloud is decreasing due to dispersion by atmospheric diffusion. The black contour line disappeared first and then the red line disappeared also. Therefore, non-steady-state modelling confirms that aerosol particles generated by a cough can travel considerable distances from the source in excess of 10 meters.

Fig. 6. An XY cross-section of number concentration field of aerosol particles at the beginning of the aerosol cloud evolution, time - 6 s. Wind speed 0.3 m s^–1; T = 20°C; RH = 50%; diffusivity Df = 0.03 m² s^–1. The contour lines are: (C,_XY = 0.3·C₀) – red; (C,_XY = 0.2·C₀) – yellow; (C,_XY = 0.1·C₀) – green. Other conditions as in previous steady-state model.

3.4 3D Modelling in Confined Spaces

The modelling in open spaces was adapted to confined spaces. For this the wind speed was replaced by convection driven by buoyancy (e.g., caused by heating radiators and electronic devices) and ventilation. These create a complex non-uniform 3D velocity field. In modelling, integrated diffusion coefficients were employed as in the modelling of the open spaces.

The rectangular geometry with a 4 m × 6 m footprint and 3 m height with an inlet and an outlet ventilation and with 3 beds was modelled, Fig. 7. The ventilation flow rate was Q = 0.1 m³ s^–1 and the volume exchange is 5 times per hour (V = 72 m³). Boundary conditions were as described in the 2D sections.

Fig. 7. Geometry for 3D modelling of aerosol dispersion. The model includes three blue top rectangular beds at the ground level, a ventilation inlet at {1 m < X < 3 m; 0 < Y < 0.4 m; Z = 3.1 m} and a ventilation outlet at {1 m < X < 3 m; 5.6 m < Y < 6 m; Z = 3.1 m}.

It was found that from the inlet the ventilation flow was directed to the floor where it changed direction and moved almost horizontally, see the Y–Z plane at X = 0.5 m in Fig. 8. The effects of buoyancy (driven by differences in temperature) on the convection flow were found to be greater in the confined space than in open spaces. If a radiator in a confined space (not shown in Fig. 7) was at T = 50°C or 30°C higher than the wall temperature, the heat induced convection velocity flow was the same order of magnitude than the ventilation flow velocity. Therefore, in confined spaces heat transfer has to be taken into account.

Fig. 8. Velocity field for the ventilation velocity vairin = 0.1 m s^–1; Difcov = 0.1 m² s^–1; T = 20°C; RH = 50%; The color-coded legend shows velocity in m s^–1.

A possible scenario of contaminated ventilation with the initial aerosol concentration C₀ = 1 cm^–3 was modelled. Modelling shows that in the confined space, aerosol particles occupy a considerable part of the room volume. If a person is in the bed at Y = 1 m he/she is exposed to C,_XY > 0.2·C₀ that is greater than the exposure level for a person in the bed at Y = 5 m where C,_XY < 0.1·C₀., Fig. 9.

Fig. 9. Aerosol particle concentration field in a Y-Z plane at X = 2 m. The conditions are the same as in Fig. 8. Contour lines are: (C,_XY = 0.3·C₀) – red; (C,_XY = 0.2·C₀) – yellow; (C,_XY = 0.1·C₀) – green. C₀ = 1 cm^–3.

The difference between an open space and a confined space scenario is the volume where the aerosol is dispersed and the number of aerosol emission acts. 3D modelling does not contradict to a possibility that viral infections can be transformed via ventilation in confined spaces. However, there are very limited data available on direct measurements of viral presence in ventilation facilities. Airborne viral infection transmission via ventilation is an important issue for multistore buildings including hospitals and care homes and should be investigated along with other transmission routes.

 
3.5 Accumulated Dose of Viral Load and Exposure to Large Aerosol Particles/Droplets

This modelling confirms a potential possibility of the aerosol route for transmission of COVID-19 infection. In this section we are using the modelling results and the literature data to investigate exposure – dose relationships relevant to viral load that can be transferred from a host to a healthy individual. To evaluate transfer of a viral infection, first an aerosol particle concentration field C_X,Y,Z,t was calculated as described above and then the ICRP lung deposition model was applied to estimate the number of aerosol particles deposited in the respiratory tract (Hinds, 1999; Gorbunov et al., 2009), Fig. 10. 

Fig. 10. Aerosol particle deposition efficiency curves (E_Dp) vs. diameter of aerosol particles. The total deposition in the respiratory system shown with the dashed black curve and regional depositions are shown with colored lines: the upper respiratory tract (red line), tracheobronchial region (green line) and alveolar region (blue line).

The total number of particles deposited in the respiratory system of a person who is at a point {X,Y,Z,t} after a single cubic centimetre of the aerosol breathed in (N_pd) can be found from:

Here E_Dp is the deposition efficiency of aerosol particles in the respiratory system as function of the particle size, Fig. 10. The aerosol particle number size distribution has been included in expression (1) and indicated as a symbol Dp in C_X,Y,Z,t,Dp (Gorbunov et al., 2013). So expression (1) C_X,Y,Z,t,Dp is a 5-dimensional function of space, time and particle size.

According to the ICRP deposition efficiency curve, the particles relevant to COVID-19 transmission (1 µm < Dp < 100 µm) are captured mainly by the upper respiratory tract. Therefore, the face masks for COVID-19 should be designed to filter out particles in the range from 1µm to 100 µm preferably with maximal efficiency at the E_Dp maximum – 5 µm. The current face mask industry standard does not include testing in this size range (Lee et al., 2016). The initial size distribution of droplets was taken from Yang et al. (2007).

First, to calculate the viral load, one has to find the total volume of aerosol particles in 1 cm³ of air that deposited in the respiratory tract (V_pd).

The accumulated dose of the viral load (AD_pd) is the total number of virus copies deposited in the respiratory tract after number of breathing cycles n and the volume of a single breath (V₁). The viral load of the posterior oropharyngeal saliva is b. Evaporation of water from aerosol particles/droplets results in increasing the viral load (Baron and Willeke, 2001). This effect is accounted for by the humidity factor H_f that ranges from 1 for a humid air to 300 for the dry air (Vuorinen et al., 2020).

It should be noted that n may or may not be constant. It is influenced by the physical activity, fitness level, age and gender. The average breath rate is ~5 cycles per minute.

The number of breath cycles and the concentration of the aerosol particles define the deposition dose of a person that is exposed to an aerosol. The humidity in modelling was RH = 50% (low humidity) and RH = 90% that is close to humidity of the exhaled air Rh = 87%.

Let us consider a healthy person standing at point {12 m, 0, 1.7, t} as in Fig. 4 where he/she is exposed to aerosol particles Dp = 20 mm with C_X,Y,Z,t,Dp = 1 cm^–3, n = 10 and b = 10⁵ virus copies cm^–3. The viral dose received by the person was found to be 20 copies of viruses in a humid air and considerably more in dryer air (circa 100 copies depending on the RH).

The critical viral load dose for COVID-19 is not well studied and further research is required. However, there is a demand to quantify the risk of the infection transfer via the aerosol route. The expressions (2) and (3) can be used to quantify the dose and potentially to be employed as a measure of the risk associated with exposure to airborne viruses like SARS-CoV-2.

The modelling developed here is a simplified approach where all statistical variabilities of the aerosol evolution in real environments are accounted for by the integrated turbulent diffusion coefficient superimposed on the laminar flow solution. There is a question of uncertainties associated with this approach. However, it is unlikely that these uncertainties affect the main conclusions of the current work. Under the circumstances a fast software (1–10 min computation time with PC) widely used and verified in a different field can be rather useful in evaluation of the airborne virus transfer and evaluating the risk associated with exposure to airborne viruses.

 
3.6 Limitations and Assumptions in Modelling

The general set of Navier-Stokes equations governs the motion of compressible Newtonian fluids. Running simulations using the original NS equations is often beyond the computational power of most of today's computers and supercomputers (Comsol Multiphysics software). Instead, a Reynolds-Averaged Navier-Stokes formulation of the Navier-Stokes equations, which averages the velocity and pressure fields in time is used. This assumption has been validated and widely accepted in air pollution modelling.

It is assumed here that the aerosol particle – air boundary can be treated according to Cunningham by introducing a slip correction factor (Baron and Willeke, 2001). This limitation is common in the aerosol science and very well validated. In addition, the density of aerosol particles/droplets was assumed to be 1.0 g cc^–1. It is a limitation, but it is unlikely that a small density change may noticeably affect the length of particle dispersion in the air.

Another assumption is the isotropic diffusion of the atmosphere along the model vertical dimension (from the ground level up to 5 m). The main reason for this is lack of reliable data on anisotropic diffusivity near the ground in complex environments and large uncertainties with experimental results. To minimize effects of these uncertainties modelling were carried out with a range of diffusion coefficients.

3.7 Possible Experimental Verification of the 30 m Travel of Aerosol Particles (10 µm to 100 µm Diameter)

SARS-CoV-2 viruses have been observed in the air, see for example Liu et al. (2020). The modelling predicts measurable concentrations at 30 m distance from the source. However, concentration of particles/droplets is small. From theoretical point of view, the particles can be studied employing a cascade impactor, e.g., Gorbunov et al. (2009) to fractionate particles, optical microscopy to count them and PCR technique for identification of viruses. However, in practice it might be challenging to synchronize emission act, wind direction and sampling device.

Alternatively, verification of modelling predictions could be done with a deactivated virus placed in a matrix of relevant body fluid, for example the lung fluid or saliva. This formulation can be aerosolized using either spinning disc aerosol generator, vibration orifice generator, or similar (Baron and Willeke, 2001). The generator can be positioned stationary and run in preferable stable atmospheric conditions of moderate turbulence. In this case an issue of synchronization is addressed. The concentration of aerosol particles/droplets generated can also be considerably greater than concentration in a single act of sneezing. This experiment will be a steady-state experiment. The aerosol particles can be collected with either a cascade impactor or an aerosol filter.

 
4 CONCLUSIONS

• The results obtained confirmed that aerosol particles from a host can travel distances considerably longer than the currently recommended 2m safe distance. Aerosol particles of diameters from 10 µm to 100 µm that potentially can carry SARS-CoV-2 (COVID-19) viruses travel over 30 m at some atmospheric conditions.
  
  
• Modelling the evolution of aerosol clouds generated by coughing and sneezing reveals that in realistic weather scenarios up to 200 virus copies can be deposited in the upper respiratory tract of a healthy individual.
  
  
• A metric based on aerosol particle size distribution and ICRP lung deposition model is suggested.
  
  
• The modelling demonstrates how the aerosol particle dispersion is influenced by the weather and geometry of the space.
  
  

 
ACKNOWLEDGMENTS

The author wishes to thank Dr. J. Rowles in this study for help with preparation of the manuscript.

 
DISCLAIMER

Reference to any companies or specific commercial products does not constitute or imply any endorsement.