Letter to the editor on simple short range transport and dispersion (T&D) modeling of COVID-19 virus, indoors and outdoors

Editor’s note: the author advocates the use of simple short-range dispersion models for estimating the exposure to the COVID-19 virus.

Dear Dr. Rao,

The purpose of this letter is to comment on current approaches and outline some simple basic methods for modeling the transport and dispersion of COVID-19 from a single source (e.g., a person coughing) at small distances and small times, both indoors and outdoors. Many groups are doing laboratory experiments and applying models to assess aspects of this complex problem, which covers a broad variety of topics, ranging from the sneeze scenario to the reaction of a person’s body to the inhaled virus. A good comprehensive discussion, including explanations of physical, chemical, and biological processes, is given by Vuorinen et al. (2020).

I believe that there should be more use of basic science concepts regarding atmospheric transport and dispersion (T&D) processes in current comprehensive model systems. The T&D model treats what happens to the virus after it enters the atmosphere and as it forms a cloud of aerosols that move downwind, disperse, and may evaporate and/or settle out. Currently, there are some oversimplifications and misapplications of the science and unjustified recommended public actions. For example, the widely-mentioned unofficial guidance is that the COVID 19 aerosols from a sneeze would travel six feet, and so persons were urged to keep six feet away from others. This is a rough rule of thumb that may be good on average over all sneezes and scenarios, but basic T&D science says that the distance will vary significantly with volume of sneeze, aerosol sizes, and ambient conditions such as flow speed (outdoors or indoors). Another unjustified rule-of-thumb is that there is no need to worry if you are outside, because the aerosols just diffuse away.

Background on Transport and Dispersion (T&D) models

The T&D model can calculate concentrations of virus in the air as a function of space and time. The space domain of present interest is about 1 to 10 or 20 m and the time domain is 1 second to several minutes, for both indoor and outdoor releases. The preferred approaches to the problem depend a great deal on the time scale of the health effects. For example, if a few virus particles can cause health effects, then all it takes is a few seconds of exposure, and it is important to be able to predict the time variation of virus concentrations ever few seconds. This would involve modeling a single sneeze and the T&D of the virus puff. Current estimates of the numbers of viruses (virons) to initiate COVID-19 in a person range from 100 to 500, which could be inhaled in a few seconds.

The problem of short-range transport and dispersion of hazardous aerosols at small distances and time scales outdoors first rose to the level of a major scientific problem during World War I, when both sides used chemical and biological agents offensively. The top fluid dynamicists in the world worked on this problem, leading to fundamental (and straightforward) T&D models (analytical formulas) for application to distance scales from next to the source to 1 km, and time scales from a few seconds to a few minutes (e.g., Taylor 1921; Richardson 1923). The resulting formulas for the spread of a cloud in the atmosphere formed the basis for many of today’s operational T&D models. Fundamental atmospheric dispersion textbooks (e.g. Arya 1999; Hanna, Briggs, and Hosker 1982; Pasquill and Smith 1974) cover the basic science aspects of outdoor T&D and suggest several analytical formulas that can be used for the momentum jet containing aerosols from a sneeze, the subsequent transport and dispersion of the cloud, and the evaporation and settling of the aerosols.

There are also “basic physics” models for indoor scenarios. The CONTAM indoor air pollution model (Dols and Polidoro 2015) is a widely-used general purpose software tool. However, the core of CONTAM includes a simple well-mixed room model (the concentration, C, in mass per cubic meter, is assumed to be the same throughout the room). This model would be useful for averaging times of about 5 or 10 minutes or more, when the pollutant would become mixed around the room, but is not useful for short-distance and short-time problems such as the transport and dispersion within the room of the aerosols released in a sneeze over distances less than 10 m and with time resolution of a second. The indoor small plume problem has received little attention in indoor field studies.

For both indoor and outdoor scenarios, many researchers use Computational Fluid Dynamics (CFD) models, which can produce T&D results at high resolution in space and time. They provide precise details, but have been shown to be no more accurate (when compared with field observations of inert tracers) than much simpler models (e.g., see Hernández-Ceballos et al. 2019). The CFD models need a turbulence closure assumption, and, indoors, they perform better if they have a good input of the characteristics of forced ventilation. Turbulence has been extensively measured outdoors and can be estimated using boundary layer parameterizations in, for example, Arya (1999). Indoors, there are only a few comprehensive studies of mean flow and turbulence. As examples of potentially useful indoor turbulence studies, sonic anemometers (Wasiolek et al. 1999) and hot wires (Zhang et al. 1995) have been used to measure mean winds and turbulence at several locations in a few typical ventilated room configurations, finding relatively low mean flow speeds (about 5 to 20 cm/s) and relatively large turbulence intensities (σ_v/u equal to about 0.2 to 1.0).

Suggested simple basic science models

As described above, there is a wide range of categories of approaches to T&D modeling of COVID 19 viruses. At the ultra-simple end, there are the “six foot” rule-of-thumb, and the “well-mixed room” indoor model. At the complex end, there are CFD models. Here, I suggest a simple model, to handle cloud T&D in the initial stages before the material is influenced by the walls, floors and ceilings, and flow constraints. Because of the analytical formulation, it is relatively easy to understand the effects of variations in scenarios and inputs. In two recent papers, I have used similar simple basic science models (a Gaussian puff model for instantaneous tracer releases in an urban downtown region (Hanna, Chang, and Mazzola 2019), and the Britter and McQuaid Workbook nomograms for dense gas releases for the Jack Rabbit II chlorine field trials (Hanna 2020)), and shown that they perform as well as many of the more complex T&D models.

To better protect the public, a “worst case” calculation is of most interest. It is assumed that the health effect is primarily determined by virus particles breathed in a few seconds. It is also assumed that the person being affected by the virus is in a location such that the middle of the cloud passes him, with the highest concentration of viruses. The worst case also usually assumes that all of the aerosol mass, Q (g), remains airborne (i.e., does not settle out to the ground), and that all virons remain viable.

For most of the real-world T&D scenarios that I have studied, I start out by doing a simple screening analysis, to get an idea of what the magnitude of the concentrations might be. Surprisingly, I often find that this initial screening prediction turns out to be fairly close (within a factor of two) to the predictions of much more detailed models. I attribute this to the fact that the screening model assumptions are based on “fitting” parameters using others’ analyses of cloud growth observations in many field experiments.

The initial “cloud” formed due to the sneeze or cough or breathing has large random variability, so it is adequate to initialize the cloud by assuming a typical median shape and size. For example, assume that a person’s sneeze (containing a mass Q of aerosols) produces a cube-shaped cloud (say with a side dimension D_o of 0.5 m), centered at the height of the person’s mouth, and with a uniform concentration of emitted tiny aerosol mass in the cube. Alternatively, I could assume a spherical cloud with a Gaussian distribution, but the well-mixed cube is better for explaining the concept.

To calculate concentrations along the path of the sneeze cloud traveling with the air flow, I’m suggesting a simple screening model for T&D of tiny aerosols within about 10 to 20 m of the person who breathed them out, or for the first 10 to 60 seconds of cloud travel.

For turbulent dispersion in outdoor air during typical conditions, it is widely recognized that, on average in the near field, the cloud dimension D grows approximately linearly with distance, such that the cloud size is proportional to about 10% of the distance traveled (on average). The 10% is roughly equivalent to a turbulence intensity (ratio of turbulent speed to mean speed). It is assumed that the cube dimension, D, starts at D_o and increases linearly with distance, x:

(1) $\mathrm{D}={\mathrm{D}}_{\mathrm{o}}+0.1\mathrm{x}$(1)

The assumption of linear cloud growth at small times and distances is consistent with Taylor’s (1921) almost-century-old seminal theory of diffusion. The concentration, C (in g/m³), in the cloud equals the mass released divided by the volume of the cube:

(2) $\mathrm{C}=\mathrm{Q}/{\mathrm{D}}^3\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{s}$(2)

This is a good screening approximation for distances within about 10 or 20 m of the source (a person coughing or sneezing) both indoors and outdoors. Although the flow speed, u, is not in Equation (2)(2) $\mathrm{C}=\mathrm{Q}/{\mathrm{D}}^3\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{s}$(2) , it is assumed that the cloud/puff is moving downwind at the flow speed; consequently, x = ut, where t is time after puff emission.

Although there are few detailed observations of indoor turbulence in rooms, Wasiolek et al. (1999) report turbulence intensities, in two typical rooms with HVAC systems, that are usually larger than 10%. As an example of the application of Equations (1)(1) $\mathrm{D}={\mathrm{D}}_{\mathrm{o}}+0.1\mathrm{x}$(1) and (2(2) $\mathrm{C}=\mathrm{Q}/{\mathrm{D}}^3\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{s}$(2) ), assume that Q = 2 g (hypothetical, and not based on any observations) and D_o = 50 cm (somewhat in agreement with observations of sneeze cloud dimensions). The concentration, C_o, in the initial cloud is Q/D_o³, or 16 g/m³. Then, at a distance, x, of 5 m, C equals 2 g/m³, and at a distance of 10 m, C equals 0.59 g/m³. Thus, between the initial position and x = 5 m, the cloud is diluted by a factor of 16/2 = 8, and by x = 10 m, the cloud is diluted by a factor of 16/0.59 = 27.1. To determine the number of virons, multiply C by the number of virons per unit mass in the initial cough or sneeze. Note that, with this model, it is easy to see how concentration, C, varies with mass emitted, assumed initial cloud dimension and distance.

The above screening formula is for instantaneous clouds or puffs such as resulting from a cough or sneeze. For breathing or talking or singing, the release of aerosols can be assumed to be continuous for the time period (about 10 to 60 s) when the cloud is traveling the distance of 10 or 20 m. In that case, a slightly different screening formula can be used, valid for continuous releases. It is necessary to know the mean flow speed, u, along the cloud trajectory, which is better known outdoors than indoors. The source term, Q_c, now has units of mass per unit time. Assume (again, completely fictional) that 0.2 g are released in 20 seconds, giving Q_c = 0.01 g/s. In this case, we initialize the cloud as a certain cross-wind area (a square with dimensions D_o, where it is still assumed that D = D_o + 0.1x). The initial size, D_o, would be less (assume 20 cm for the purposes of this calculation) than for a cough or sneeze. The concentration is given by:

(3) $\mathrm{C}={\mathrm{Q}}_{\mathrm{c}}/\left(\mathrm{u}{\mathrm{D}}^2\right)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{s}$(3)

Assuming D_o = 0.2 m, Qc = 0.01 g/s and u = 0.5 m/s, this formula gives an initial concentration, C_o, of (0.01 g/s)/(0.5 m/s*0.04 m²) = 0.5 g/m³, a concentration C equal to 0.0408 g/m³ at x = 5 m and C = 0.0139 g/m³ at x = 10 m. Thus, between the initial position and x = 5 m, the cloud is diluted by a factor of 0.5/0.0408 = 12.25. Between the initial position and x = 10 m, the cloud is diluted by a factor of 0.5/0.0139 = 36.0. The dilution is slightly greater for the continuous example than for the instantaneous example which is the opposite of what would be intuitively expected from Equations (2)(2) $\mathrm{C}=\mathrm{Q}/{\mathrm{D}}^3\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{s}$(2) and (3(3) $\mathrm{C}={\mathrm{Q}}_{\mathrm{c}}/\left(\mathrm{u}{\mathrm{D}}^2\right)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{s}$(3) ), for all conditions the same. However, a major difference is that the initial cloud size, D_o, was assumed to equal 0.5 m for the instantaneous release (sneeze) and 0.2 m for the continuous release (talking). The smaller initial cloud will show more relative dilution.

From the form of this simple formula, some interesting results can be interpreted. For example, with D_o reduced to 0.2 m, by a distance of travel of 5 or 10 m, the second term (the dispersion component) in Equation (1)(1) $\mathrm{D}={\mathrm{D}}_{\mathrm{o}}+0.1\mathrm{x}$(1) is more than twice as large as the initial cloud size and therefore the initial cloud size has little effect on the concentration. Also, at larger distances, the concentration for the continuous source is decreasing with distance according to a square power law (C is proportional to 1/x²), instead of the cube power law found for puffs in Equtaion (2)(2) $\mathrm{C}=\mathrm{Q}/{\mathrm{D}}^3\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{s}$(2) . These are well-known results for continuous plumes and instantaneous puffs (Hanna, Briggs, and Hosker 1982). If the flow speed, u, is uncertain or approaches zero, just make a worst-case assumption (say, u = 0.2 m/s). It is always turbulent in the ambient air. Indoor flows (mean and turbulence) are well-known only when a special study takes place in that room; otherwise, there is much uncertainty.

Because of the simple form of the above screening equations, the recommended methods make it easy to account for differences in inputs and other physical and chemical processes. Effects of aerosol evaporation and settling and virus viability can be incorporated using straightforward basic science formulas (see Section 10–5 in Hanna, Briggs, and Hosker 1982). The major point is that, with the use of basic science principles, it is possible to quickly understand the resulting concentration predictions.

Many persons use T&D screening models such as those described above to determine whether the concentrations are high enough to cause problems. If the screening concentrations are three orders of magnitude lower than the trigger for health effects, then there is no point pursuing more detailed analysis. If the opposite occurs, then it may be useful to do a more detailed analysis.