A mathematical framework for estimating risk of airborne transmission of COVID-19 with application to face mask use and social distancing

Abstract

A 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 it can also be understandable by a broad audience outside science and academia. The caveats and limitations of the model are discussed in detail. The model is used to assess the protection from transmission afforded by face coverings made from a variety of fabrics. The reduction in the transmission risk associated with increased physical distance between the host and susceptible is also quantified by coupling the model with available and new large eddy simulation data on scalar dispersion in canonical flows. Finally, the effect of the level of physical activity (or exercise intensity) of the host and the susceptible in enhancing the transmission risk is also assessed.

FIG. 1.

Schematic depicting the key stages in the airborne transmission of a respiratory infection such as COVID-19.

FIG. 2.

The contagion airborne transmission (CAT) inequality that evaluates the conditions for the airborne transmission of a respiratory infection such as COVID-19. The left-hand side of the inequality represents the total inhaled viral dose, and the right-hand side is the minimum aerosol dose required to initiate an infection in the susceptible. The inequality is satisfied (and the transmission is successful) when the susceptible inhales a viral dose that exceeds the minimum infectious dose. The variables in the model can be segregated in different ways, as shown in the graphic.

FIG. 3.

Schematic depicting the inhalation volume of the susceptible that can be combined with the local concentration of the respiratory aerosol to estimate f_at.

FIG. 4.

Estimation of protection from aerosol transmission afforded by the donning of face coverings based on the published filtration efficiency data of Zangmeister et al. for 34 different fabrics/samples. The protection factor (PF) is a quantity that is normalized by the risk of transmission associated with the situation when neither the host nor the susceptible wears a mask.

FIG. 5.

Schematic showing various scenarios for which the effect of physical distancing on the transmission risk is assessed. The aerosol plume from the host consists of a near field, which can be highly variable, and a far field, where the plume exhibits more self-similar or universal characteristics within various classes of flow. The analysis in this section focuses on the far-field domain.

FIG. 6.

Results from the wall-modeled large eddy simulation (LES) of a breath generated aerosol plume (at x = 0) in a turbulent boundary layer. (a) Isosurfaces of instantaneous concentration of scalar C/C_o = 0.01, colored by the local streamwise velocity showing the breath aerosol puffs being transported in the turbulent flow, (b) contours of the mean concentration for a plume that is warmer than the ambient flow, and (c) contours of the mean concentration for a plume that is colder than the ambient flow. (d) Mean concentration with the streamwise distance (in meters) at a height of 1.5 m from the ground along with best fit power laws beyond the near-field region.

FIG. 7.

Estimate of the protection factor as a function of physical distance (in arbitrary units) between the host and susceptible for the five scenarios examined here. The protection factor is normalized for each case by the condition where both the host and susceptible are at a unit distance. The protection factor is inversely proportional to the decay of the concentration with distance between the host and the susceptible D_hs.

FIG. 8.

Estimate of the transmission risk increase due to physical activity/exercise induced increased ventilation rates for hosts and susceptibles. The assumed ventilation rate for each of the five levels is included in the legend. The increase in the transmission risk is normalized by the condition where both the host and susceptible are sedentary (sleeping or sitting).

FIG. 9.

Depiction of methodology for estimating the lower and upper bound of the average FE over the 50 nm–5000 nm range from the data of Zangmeister et al. This particular example is for the 1× Cotton 13, 2× Synthetic Blend 4 sample. The lower bound is obtained by assuming that the FE for PMDs greater than 825 nm is equal to the FE measured at 825 nm. The upper bound is obtained by employing a regression fit to the right side of the measured curve (beyond the PMD for minimum FE) and extending it to the FE = 100% line. Beyond this, the FE for the upper bound is assumed to equal 100%.