High-Fidelity Simulation of Pathogen Propagation, Transmission, and Mitigation

The current COVID-19 pandemic has stimulated a renewed interest in pathogen propagation, transmission, and mitigation [1–3, 9]. In particular, the relative impact of transmission via “large droplets” versus “small droplets” or aerosols—combined with possible changes to existing heating, ventilation, and air conditioning (HVAC) systems, building floorplans, pedestrian traffic management, and the installation of ultraviolet (UV) lights—has been the topic of thorough debate over the last year and a half. Natural and forced convection, the presence of moving pedestrians or objects, and accurate computation of droplets in airflow motion in the context of HVAC systems are all key requirements for the development of quantitative predictions of pathogen propagation, transmission, and mitigation in the built environment. Numerical techniques that satisfy these requirements have reached high degrees of sophistication and offer a quantum leap in accuracy when compared to simpler probabilistic models [6, 7].

Mitigation Techniques

When developing pathogen mitigation or elimination strategies, one must visualize the movement of the droplets that carry the pathogen:

• Larger \((1\: \rm{mm} \ge d \ge 0.1 \:\rm{mm})\) droplets follow a ballistic path, are not significantly slowed by the surrounding air, and drop and attach to the floor or any surface in a time of approximately \(O(1)\) seconds without considerable evaporation. Spitting saliva is one such example.
• Smaller \((d < O(0.01) \: \rm{mm})\) droplets are immediately slowed by the surrounding air, evaporate in a fraction of a second, and are transported by the air itself. One can conceptualize these aerosolized droplets as (invisible) cigarette smoke or sprays (e.g., hair spray or deodorants).

Figure 1 summarizes a list of common mitigation techniques and evaluates their effectiveness for a variety of transmission/infection mechanisms.

Physical Modeling of Aerosol Propagation

When solving the two-phase equations, one can best represent air (as a continuum) with the Navier-Stokes equations. These equations describe the conservation of momentum, mass, and energy for incompressible Newtonian flow and are given by

\[\rho \mathbf{v}_{,t} + \rho \mathbf{v} \cdot \nabla \mathbf{v} + \nabla p = \nabla \cdot \mu \nabla \mathbf{v} + \rho \mathbf{g} + \beta \rho \mathbf{g}(T - T_0) + \mathbf{s}_v,\tag1\]

\[\nabla \cdot \mathbf{v}= 0,\tag2\]

\[\rho c_p T_{,t} + \rho c_p \mathbf{v} \cdot \nabla T = \nabla \cdot k \nabla T + s_e.\tag3\]

Here, \(\rho\), \(\mathbf{v}\), \(p\), \(\mu\), \(\mathbf{g}\), \(\beta\), \(T\), \(T_0\), \(c_p\), and \(k\) respectively denote the density, velocity vector, pressure, viscosity, gravity vector, coefficient of thermal expansion, temperature, reference temperature, specific heat coefficient, and conductivity. In addition, \(\mathbf{s}_v \) and \(s_e\) are momentum and energy source terms (e.g., due to particles or external forces/heat sources). One can obtain both the viscosity and conductivity for turbulent flows either from additional equations or directly via a large eddy simulation (LES) assumption through monotonically-integrated LES. Adding the appropriate advection-diffusion equations with source terms allows researchers to attain the concentration of pathogens, age of air, and possible UV radiation.

We model the droplets/particles—which are relatively sparse in the flow field—with a Lagrangian description, which monitors and tracks individual particles (or groups of particles) in the flow. This allows for an exchange of mass, momentum, and energy between the particles and the air. The position \(\mathbf{x}_p\), velocity \(\mathbf{v}_p\), and temperature \(T_p\) of each particle are given by ordinary differential equations (ODEs) of the form

\[\frac{d\mathbf{x}_p}{dt}=\mathbf{v}_p,\tag4\]

\[\frac{d\mathbf{v}_p}{dt}=\mathbf{f}(\mathbf{v},\mathbf{v}_p,...),\tag5\]

\[\frac{dT_p}{dt}=s(\mathbf{v},\mathbf{v}_p, T, T_p, ...).\tag6\]

Empirical relationships that depend on both the particle and flow variables yield the terms on the right-hand side [4, 6, 7].

The Navier-Stokes equations are integrated using FEFLO via a so-called projection scheme [4] that is comprised of an “advective predictor” and a subsequent Poisson solver to re-establish incompressible flow. The Lagrangian particles are integrated with an explicit fourth-order Runge-Kutta scheme, and immersed body techniques handle the presence of moving, inhaling, and exhaling pedestrians [4, 6].

Modeling Pedestrian Motion

Modeling pedestrian motion has been the focus of research and development for more than two decades [5, 8]. An example simulation with moving pedestrians to assess pathogen propagation and/or mitigation options is presented here. This simulation utilizes the PEDFLOW model [5].

Necessary information from computational fluid dynamics (CFD) codes consists of pathogen distribution and the spatial distribution of temperature, smoke, and other toxic or movement-impairing substances in space. We interpolate this information to the computational crowd dynamics code at every timestep to properly calculate the visibility/reachability of exits, routing possibilities, smoke, toxic substance or pathogen inhalation, and any other flow field variable that the pedestrians require.

Illustrative Examples

The following examples are by no means exhaustive or unique; the simulation of aerosol transmission via high-fidelity CFD techniques has received considerable attention in recent years, and researchers worldwide have carried out such work with both commercial and open-source software. Further cases with videos and descriptions are available online. 

Sneezing in Subway Car

An obvious vector for pathogen contamination and spread is mass transport, as passengers are in extremely close proximity and airflow might result in considerable mixing. We therefore chose to analyze a sneeze in a subway car. The flow enters through two parallel slits in the ceiling and exits through the ceiling at both ends of the car. A detailed STL triangulation yielded the geometry [2]. An immersed technique was used for the passengers, who were located randomly throughout in the car; the mesh had approximately \(10^7\) elements.

As one might expect, the flow field is highly turbulent. Figure 2 illustrates the distribution of particles after a sneeze in the middle of the car by someone who is facing one end. The large red particles follow a ballistic path and fall to the ground. The air quickly stops the green particles of size \(d=0.1\: \rm{mm}\) which then sink slowly towards the floor in close proximity to the person who is sneezing. The blue particles, which are even smaller, rise with the cloud of warmer air that is exhaled by the sneezing individual and disperse much further at later times.

Sneezing in an Airplane Cabin

The air flow in airplane cabins has been a media focal point throughout the COVID-19 pandemic. Given that the air in planes is renewed much more often than in air-conditioned buildings—one exchange every two minutes versus one exchange every 12-15 minutes—people would likely assume that airplanes are much safer. Indeed, influenza studies have shown that the “radius of transmission” on a plane is limited to two-three rows; large droplets that move from surfaces to hands and then to noses/eyes/mouths are the most common route of infection. Nevertheless, regions of stagnant flow could be conducive to pathogen transmission. This prompted us to analyze flow and sneezing in a Boeing 737-500.

Unlike older models, the flow in this cabin enters through two parallel slits in the ceiling that are close to the windows, moves towards the center, and exits through holes in the floor below the windows. A detailed STL triangulation once again yielded the geometry [3], and the mesh had approximately \(0.89 \cdot 10^9\) elements. The run was carried out to one minute of physical time, which required approximately 24 hours on nearly 8,000 cores. Different “sneezing positions” were also considered, and only the smaller particles—which move much further—were accounted for. Figure 3 depicts the movement and extent of the “sneeze clouds” that originated at different positions. Each of these clouds is marked with a unique color, and the clouds remain bounded to two-three seats/rows in every direction. This result is qualitatively in line with studies of influenza and COVID-19 transmission on airplanes.

Corridor with Pedestrians

This example considers a corridor of size \(10.0 \: \rm{m} \times 2.0 \:\rm{m} \times 2.5\: \rm{m}\). Both entry and exit sides have two doors, each of size \(0.8\: \rm{m} \times 2.0\: \rm{m}\). For the purpose of climatisation, four entry vanes and one exit vane are placed in the ceiling. The vertical air velocity for the entry vanes is set to \(v_z=0.2\: \rm{m/sec}\), while the horizontal velocity is set as increasing proportionally to the distance of the vane’s center to a maximum of \(v_r=0.4 \: \rm{m/sec}\). The CFD mesh has approximately \(3.5\cdot 10^6\) elements of uniform size, and two streams of pedestrians enter and exit through the doors over time.

The case shown here considers two pedestrian streams in counterflow mode. Figure 4 indicates that the different velocities between walking pedestrians—and particularly counterflows—lead to large-scale turbulent mixing, therefore enhancing the spread of pathogens that emanate from infected victims. More cases/options are available in [6].

Outlook

As with any technology, further advances are clearly possible. The list is long, so we just mention the following areas for further development:

• Improved knowledge of the infectious dose that is required to trigger infection/illness
• Improved boundary conditions for HVAC exits
• Improved modeling of particle retention and movement through cloths (e.g., for masks).

The basic physical phenomena—and the partial differential equations and ODEs that describe them—have been known for more than a century, and solvers have advanced considerably over the last four decades. Nevertheless, we need a vigorous experimental program to complement and validate the numerical methods and establish firm “best practice” guidelines.

Acknowledgments: The authors acknowledge the help of Mika Gröndahl, a graphics editor at The New York Times, for the detailed STL models and thoroughly researched boundary conditions of the subway train car and airplane cabin. These runs would have been either impossible or meaningless without his efforts.

References

[1] Dietz, L. Horve, P.F., Coil, D.A., Fretz, M., Eisen, J.A., & Van Den Wymelenberg, L. (2020). 2019 novel coronavirus (COVID-19) pandemic: Built environment considerations to reduce transmission. mSystems, 5(2), e00245-20.
[2] Gröndahl, M., Goldbaum, C., & White, J. (2020, August 10). What happens to viral particles on the subway? The New York Times. Retrieved from https://www.nytimes.com/interactive/2020/08/10/nyregion/nyc-subway-coronavirus.html.
[3] Gröndahl, M., Mzezewa, T., & Fleisher, O. (2021, April 18). How safe are you from covid when you fly? The New York Times. Retrieved from https://www.nytimes.com/interactive/2021/04/17/travel/flying-plane-covid-19-safety.html.
[4] Löhner, R. (2008). Applied computational fluid dynamics techniques: An introduction based on finite element methods (2nd ed.). Hoboken, NJ: J. Wiley & Sons.
[5] Löhner, R. (2010). On the modeling of pedestrian motion. Appl. Math. Modelling, 34(2), 366-382.
[6] Löhner, R., & Antil, H. (2021). High fidelity modeling of aerosol pathogen propagation in built environments with moving pedestrians. Int. J. Num. Meth. Biomed. Engng., 37(3), e3428.
[7] Löhner, R., Antil, H., Idelsohn, S., & Oñate, E. (2020). Detailed simulation of viral propagation in the built environment. Comput. Mech., 66, 1093-1107.
[8] Thalmann, D., & Musse, S.R. (2007). Crowd simulation. London, England: Springer-Verlag.
[9] World Health Organization. (2020, July 9). Transmission of SARS-CoV-2: Implications for infection prevention precautions (Scientific brief). Retrieved from https://www.who.int/news-room/commentaries/detail/transmission-of-sars-cov-2-implications-for-infection-prevention-precautions.