A Physics Modeling Study of SARS-CoV-2 Transport in Air

Health threat from SARS-CoV-2 airborne infection has become a public emergency of international concern. During the ongoing coronavirus pandemic, people have been advised by the Centers for Disease Control and Prevention to maintain social distancing of at least 2 m to limit the risk of exposure to the coronavirus. We carry out a physics modeling study for SARS-CoV-2 transport in air. We show that if aerosols and droplets follow semi-ballistic emission trajectories, then their horizontal range is proportional to the particle's diameter. For standard ambient temperature and pressure conditions, the horizontal range of these aerosols remains safely below 2 m. We also show that aerosols and droplets can remain suspended for hours in the air, providing a health threat of airborne infection. The latter argues in favor of implementing additional precautions to the recommended 2~m social distancing, e.g. wearing a face mask when we are out in public.


I. INTRODUCTION
The current outbreak of the respiratory disease identified as COVID-19 is caused by the severe acute respiratory syndrome coronavirus 2, shortened to SARS-CoV-2 [1]. The outbreak was first reported in December 2019, and has become a worldwide pandemic with over 10 million cases as of 1 July 2020. SARS-CoV-2 have been confirmed worldwide and so the outbreak has been declared a global pandemic by the World Health Organization. The pandemic has spread around the globe to almost every region, with only a handful of the World Health Organization's member states not yet reporting cases. Most of these states are small island nations in the Pacific Ocean, including Vanuatu, Tuvalu, Samoa, and Palau.
The coronavirus can spread from person-to-person in an efficient and sustained way by coughing and sneezing. The virus can spread from seemingly healthy carriers or people who had not yet developed symptoms [2]. To understand and prevent the spread of the virus, it is important to estimate the probability of airborne transmission as aerosolization with particles potentially containing the virus. Before proceeding, we pause to note that herein we follow the convention of the World Health Organization and refer to particles which are 5 µm diameter as droplets and those 5 µm as aerosols or droplet nuclei [3].
There are various experimental measurements suggesting that SARS-CoV-2 may have the potential to be transmitted through aerosols; see e.g. [4][5][6][7][8]. Indeed, laboratory-generated aerosols with SARS-CoV-2 were found to keep a replicable virus in cell culture through-out the 3 hours of aerosol testing [9]. Of course these laboratory-generated aerosols may not be exactly analogous to human exhaled droplet nuclei, but they helped in establishing that the survival times of SARS-CoV-2 depend on its environment, including survival times of: up to 72 hours on plastics, up to 48 hours on stainless steel, up to 24 hours on cardboard, up to 4 hours on copper, and in air for 3 to 4 hours [9]. On first glimpse this finding is surprising, as one would expect that the properties of air that degrade the SARS-CoV-2 exterior should abate at roughly half that time if it were adhered to a surface (i.e. at least half the solid angle is mostly exposed to air). However, the laboratory-generated aerosols have shown that a precise description of SARS-CoV-2 main characteristics requires more complex systems in which the virus would be chemisorbed by some surfaces and repelled by the others. More concretely, the survival probability of the virus is associated with the surfaces' energies of the various materials that can reduce the solid angle exposed to air molecule collisions. These properties can lead to remarkable differences , for example that between copper and stainless steel. Despite the fact both are metals, copper causes destruction of the virus much more rapidly than does stainless steel.
The number of virions needed for infection is yet unknown. However, it is known that viral load differs considerably between SARS-CoV and SARS-CoV-2 [10]. A study of the variance of viral loads in patients of different ages found no significant difference between any pair of age categories including children [11].
Beyond a shadow of a doubt, a major question the pandemic has been how far would be far enough to elude droplets and to diffuse droplet nuclei if a person nearby is coughing or sneezing. The rule of thumb for this pandemic has been a 2 m separation. Nevertheless, this has never been a magic number that guarantees complete protection. Indeed, a gas cloud model suggests that a cough or sneeze could send respiratory particles as far as 8 m [12][13][14], and actually airflow patterns in a room might influence the distance even a droplet could travel [15,16]. In this paper we provide new guidance to address this question by introducing a physics model for SARS-CoV-2 transport in air.
To develop some sense for the orders of magnitude involved, we begin by reviewing the experimental data. A survey of 26 analyses reporting particle sizes generated from breathing, coughing, sneezing and talking indicates that healthy individuals generate particles with sizes in the range 0.01 D V /µm 500, whereas individuals with infections produce particles in the range 0.05 D V /µm 500, where D V is the diameter of a respiratory particle (droplet or droplet nucleus) containing the virus [17]. The majority of the particles containing the virus have outlet velocities in the range 10 v V,0 /(m/s) 100 [18,19]. Up to 10 4.6 particles are expelled at an initial velocity of 100 m/s during a sneeze, and a cough can generate approximately 10 3.5 particles with outlet velocities of 20 m/s [20]. 97% of coughed particles have sizes 0.5 D V /µm 12, and the primary size distribution is within the range 1 D V /µm 2 [21,22]. The evaporation rate depends on the exposed surface area of the droplet, A ∼ πD 2 V , while the droplet volume scales as V ∼ πD 3 V /6. Therefore, the ratio of area to volume is A/V ∝ 1/D V , and it is the smallest droplets that will live the longest.
The layout of the paper is as follows. In Sec. II we review the generalities of aerodynamic drag force and estimate the terminal speed of aerosols and droplets. In Sec. III we model the elastic scattering of aerosols with the air molecules and estimate the horizontal range assuming standard ambient temperature and pressure conditions. The paper wraps up with some conclusions presented in Sec. IV.

II. TERMINAL SPEED
When a particle propagates through the air, the surrounding air molecules have a tendency to resist its motion. This resisting force is known as the aerodynamic drag force. For a spherical particle, the aerodynamic drag force is given by where η air 1.8×10 −5 kg/(m · s) is the dynamic viscosity of air and v V is the virus velocity vector. Eq.(1) is the wellknown Stokes' law, with the Cunningham slip correction factor κ; see Appendix for details. Stokes' law assumes that the relative velocity of a carrier gas at a particle's surface is zero; this assumption does not hold for small particles. The slip correction factor should be applied to Stokes' law for particles smaller than 10 µm. The particle Reynolds number, is a dimensionless quantity which represents the ratio of inertial forces to viscous forces, where ρ air 1.2 kg/m 3 is the air density at a temperature of 20 • C (293 K). For R < 1, the inertial forces can be neglected. The drag calculated by Eq. (1) has an error of about 12% at R ≈ 1. The error decreases with decreasing particle Reynolds number.
For the case at hand, R > 1. In the vertical direction, the upward component of the aerodynamic drag force F d,⊥ is counterbalanced by the excess of the gravitational attraction over the air buoyancy force where ρ H 2 O 997 kg/m 3 and g 9.8 m/s is the acceleration of gravity. Since ρ air ρ H 2 O the air buoyancy force becomes negligible, and so F g ≈ M V g. When the upward aerodynamic drag force equals the gravitational attraction the droplet reaches mechanical equilibrium and starts falling with a terminal speed The terminal speed is ∝ D 2 V (due to the diameter dependence of the mass), and hence larger droplets would have larger terminal velocities thereby reaching the ground faster. The terminal speed for various particle sizes is given in Table I. The time t f it will take the virus to fall to the ground is simply given by the distance to the ground divided by v V, f,⊥ . For an initial height, h ∼ 2 m, we find that for D V = 2 µm, The aerodynamic drag force holds for rigid spherical particles moving at constant velocity relative to the gas flow. To determine the horizontal range, in the next section we model the elastic scattering of aerosols with the air molecules.

III. HORIZONTAL RANGE
The mass ratio of the average air molecule compared to the aerosol, R ≡ m air /M V , is roughly given by R ∼ 10 −12 (since the size of the aerosol and the mass for its chief constituent, H 2 O, compared to the air molecule are 10 4 and  the virus velocity β we have in the lowest nontrivial order (in R 1) and any frame where the matrix M is derived by imposing conservation of energy and momentum, and is given by with β 0 = v V,0, , and v air,0 and v air, f the initial and final velocities of the air molecule, respectively. As the velocity β falls with each interaction, the velocity loss remains constant; the target particle is a new air molecule at each interaction.
Though individual air molecules are traveling at an average speed of a few hundred meters per second, throughout we assume the medium to be stationary. In analogy with the description of the slowing down of alpha particles in matter (which assumes the electronic cloud is at rest), we can describe the scattering of the aerosol in the frame in which the air molecule is at rest, i.e., v air,0 = 0 (in essence, adopting a stationary medium on average). The stopping power is given by the velocity-loss equation with solution ln β = (2R/λ V mfp ) dx. Finally, we have for the stopping distance with β f ≡ v V, f, . Note that L/λ V mfp is not only the number of mean free paths traversed by the fiducial virus, but is also the number of interactions of the virus with air molecules; of course, there is a one-to-one correlation between the number of mean free paths traveled and interactions.
Since β is homogeneous and the mass ratio R is a constant for a given D V , we have the above simple equation. The mass ratio R is very small, and (2R) −1 is correspondingly very large. There are a tremendous number of mean free paths/interactions involved as the virus bowling ball rolls over the air molecule.
Finally, we must calculate λ V mfp = 1/(n air σ). The air molecules act collectively as a fluid, so the volume V over the air density is given by the ideal gas law as k B T/P, where P is the pressure, T the temperature, and k B is the Boltzmann constant. We assume a contact interaction equal to the cross-sectional hard-sphere size of the aerosol, i.e. σ = π(D V /2) 2 . Substituting into Eq. (9) we obtain the final result for the stopping distance We take the sneeze or cough which causes the droplets expulsion to be at a standard ambient air pressure of P = 101 kPa and a temperature of T ∼ 293 K. It is important to stress that temperature variation could cause an O( ±8%) effect in L for extreme ambient cold or warmth.
Our results are encapsulated in Fig. 1, where we show contour plots for the stopping range in the D V − v V, f plane. We can see that the horizontal range of aerosols and droplets remains safely below the 2 m social distancing recommended by the Centers for Disease Control and Prevention. Our results are in sharp contrast with the findings of [18] for propagation of particles emitted in isolation. This is because the authors of [18] consider only the degradation of the horizontal velocity due to droplet evaporation that has a less significant impact on the stopping power than the elastic scattering on air molecules, and consequently this could lead to a horizontal range beyond 2 m. However, a description of SARS-COV-2 transport in air in terms of particles emitted in isolation could be an oversimplification. It is important to stress that if the SARS-CoV-2 respiratory infection is transmitted through a multiphase turbulent gas cloud containing clusters of droplets the horizontal range may reach 7 to 8 meters [14]. In addition, airflow patterns in a room could influence the distance isolated aerosols would travel, extending the horizontal range [15].

IV. CONCLUSIONS
We have carried out a physics modeling study for SARS-CoV-2 transport in air. We have shown that aerosols and droplets can remain suspended for hours in the air, providing a health threat of airborne infection. We have also shown that if aerosols and droplets follow semi-ballistic emission trajectories, then their horizontal range is proportional to the particles diameter. For standard ambient temperature and pressure conditions, the stopping distance of these semi-ballistic aerosols remains safely within the recommended 2 m social distancing by the Centers for Disease Control and Prevention. Whilst the conceptual framework investigated in this paper can be useful for generic situations, the dichotomy between large droplets and small airborne particles emitted in isolation could be an oversimplification. In this direction, it has been suggested that the SARS-CoV-2 respi-ratory infection is transmitted primarily through a multiphase turbulent gas (a puff) cloud containing clusters of droplets (with a continuum of droplet sizes) that can travel 7 to 8 meters [14]. In addition, airflow patterns in a room could influence the distance isolated aerosols would travel, extending the horizontal range [15]. Altogether, it seems reasonable to adopt additional infectioncontrol measures for airborne transmission in high-risk settings, such as the use of face masks when in public. If the results of this study -t f of O(hr) for aerosols, for example -are borne out by experiment, then these findings should be taken into account in policy decisions going forward as we continue to grapple with this pandemic.
There are important considerations in the development of Stokes' law, including the hypothesis that the gas at particle surface has zero velocity relative to the particle. This hypothesis holds well when the diameter of the particle is much larger than the mean free path of gas molecules. The mean free path λ air mfp is the average distance traveled by a gas molecule between two successive collisions. In analyses of the interaction between gas molecules and particles, it is convenient to use the Knudsen number Kn = 2λ air mfp /D V , a dimensionless number defined as the ratio of the mean free path to particle radius. For Kn 1, the drag force is smaller than predicted by Stokes' law. Conventionally this condition is described as a result of slip on the particle surface. The so-called slip correction is estimated to be [23] κ = 1 + Kn 1.257 + 0.4 exp(−1.1/Kn) .
In our calculations we take where k B is the Boltzmann constant, T is the temperature in Kelvin, and the density of air is given by