A Physics Modeling Study of SARS-CoV-2 Transport in Air
AA Physics Modeling Study of SARS-CoV-2 Transport in Air
Luis A. Anchordoqui,
1, 2, ∗ James B. Dent, † and Thomas J. Weiler ‡ Department of Physics and Astronomy, Lehman College, City University of New York, NY 10468, USA Department of Physics, Graduate Center, City University of New York, NY 10016, USA Department of Physics, Sam Houston State University, Huntsville, TX 77341, USA Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA (Dated: August 2020)
General Idea:
Health threat from SARS-CoV-2 airborne infection has become a public emergency ofinternational concern. During the ongoing coronavirus pandemic, people have been advised by theCenters for Disease Control and Prevention to maintain social distancing of at least 2 m to limit therisk of exposure to the coronavirus. Experimental data, however, show that infected aerosols anddroplets trapped inside a turbulent pu ff cloud can travel 7 to 8 m. We carry out a physics modelingstudy for SARS-CoV-2 transport in air. Methodology:
We propose a nuclear physics analogy-based modeling of the complex gas cloud andits payload of pathogen-virions. The cloud is modeled as a spherical pu ff of hot moist air (withmucosalivary filaments), which remains coherent in a volume that varies from 0.00025 to 0.0025 m .The pu ff propagates scattering o ff the air molecules. We estimate the pu ff e ff ective stopping rangeadapting the high-energy physics model that describes the slow down of α -particles (in matter) viainteractions with the electron cloud. Research Findings:
We show that the stopping range is proportional to the product of the pu ff ’s diameterand its density. We use our pu ff model to determine the average density of the buoyant fluid in theturbulent cloud. A fit to the experimental data yields 1 . < ρ P /ρ air < .
0, where ρ P and ρ air are theaverage density of the pu ff and the air. We demonstrate that temperature variation could cause an O ( ± ff ect in the pu ff stopping range for extreme ambient cold or warmth. We also demonstratethat aerosols and droplets can remain suspended for hours in the air. Therefore, once the pu ff slowsdown su ffi ciently, and its coherence is lost, the eventual spreading of the infected aerosols becomesdependent on the ambient air currents and turbulence. Practical Implications:
Viral transmission pathways have profound implications for public safety. Ourstudy forewarns a health threat of SARS-CoV-2 airborne infection in indoor spaces. We argue in favorof implementing additional precautions to the recommended 2 m social distancing, e.g. wearing aface mask when we are out in public.
I. INTRODUCTION
The current outbreak of the respiratory disease identi-fied as COVID-19 is caused by the severe acute respira-tory syndrome coronavirus 2, shortened to SARS-CoV-2 [1–4]. The outbreak was first reported in December2019, and has become a worldwide pandemic with over10 million cases as of 1 July 2020. SARS-CoV-2 have beenconfirmed worldwide and so the outbreak has been de-clared a global pandemic by the World Health Organi-zation. The pandemic has spread around the globe toalmost every region, with only a handful of the WorldHealth Organization’s member states not yet reportingcases. Most of these states are small island nations in thePacific Ocean, including Vanuatu, Tuvalu, Samoa, andPalau.The coronavirus can spread from person-to-person inan e ffi cient and sustained way by coughing and sneez-ing. The virus can spread from seemingly healthy carri-ers or people who had not yet developed symptoms [5]. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
To understand and prevent the spread of the virus, it isimportant to estimate the probability of airborne trans-mission as aerosolization with particles potentially con-taining the virus. Before proceeding, we pause to notethat herein we follow the convention of the World HealthOrganization and refer to particles which are (cid:38) µ mdiameter as droplets and those (cid:46) µ m as aerosols ordroplet nuclei [6].There are various experimental measurements sug-gesting that SARS-CoV-2 may have the potential to betransmitted through aerosols; see e.g. [7–11]. Indeed,laboratory-generated aerosols with SARS-CoV-2 werefound to keep a replicable virus in cell culture through-out the 3 hours of aerosol testing [12]. Of course theselaboratory-generated aerosols may not be exactly analo-gous to human exhaled droplet nuclei, but they helpedin establishing that the survival times of SARS-CoV-2 de-pend on its environment, including survival times of: upto 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 [12]. On first glimpse this find-ing is surprising, as one would expect that the propertiesof air that degrade the SARS-CoV-2 exterior should abateat roughly half that time if it were adhered to a surface(i.e. at least half the solid angle is mostly exposed toair). However, the laboratory-generated aerosols have a r X i v : . [ phy s i c s . b i o - ph ] S e p shown that a precise description of SARS-CoV-2 maincharacteristics requires more complex systems in whichthe virus would be chemisorbed by some surfaces andrepelled by the others. More concretely, the survivalprobability of the virus is associated with the surface en-ergies of the various materials that can reduce the solidangle exposed to air molecule collisions. These proper-ties can lead to remarkable di ff erences , for example thatbetween copper and stainless steel. Despite the fact bothare metals, copper causes destruction of the virus muchmore rapidly than does stainless steel.The number of virions needed for infection is yet un-known. However, it is known that viral load di ff ersconsiderably between SARS-CoV and SARS-CoV-2 [13].A study of the variance of viral loads in patients of dif-ferent ages found no significant di ff erence between anypair of age categories including children [14].Beyond a shadow of a doubt, a major question of thispandemic has been how far would be far enough to eludedroplets and to di ff use droplet nuclei if a person nearbyis coughing or sneezing. The rule of thumb for this pan-demic has been a 2 m separation. Nevertheless, this hasnever been a magic number that guarantees completeprotection. Indeed, experiment shows that: (i) respi-ratory particles emitted during a sneeze or cough areinitially transported as a turbulent cloud that consists ofhot and moist exhaled air and mucosalivary filaments; (ii) aerosols and small droplets trapped in the turbulentpu ff cloud could propagate 7 to 8 m [15–18]. Moreover,once the cloud slows down su ffi ciently, and its coherenceis lost, the eventual spreading of the infected aerosolsbecomes dependent on the ambient air currents and tur-bulence [19]. In this paper we provide new guidance toaddress this question by introducing a physics model forSARS-CoV-2 transport in air.To develop some sense for the orders of magnitudeinvolved, we begin by reviewing the experimental data.A survey of 26 analyses reporting particle sizes gener-ated from breathing, coughing, sneezing and talking in-dicates that healthy individuals generate particles withsizes in the range 0 . (cid:46) D V /µ m (cid:46) . (cid:46) D V /µ m (cid:46) D V is the diameter ofa respiratory particle (droplet or droplet nucleus) con-taining the virus [20]. The majority of the particlescontaining the virus have outlet velocities in the range10 (cid:46) v V , / (m / s) (cid:46)
30 [18, 21, 22]. Up to 10 . particles areexpelled at an initial velocity of 30 m / s during a sneeze,and a cough can generate approximately 10 . particleswith outlet velocities of 20 m / s [23]. 97% of coughed par-ticles have sizes 0 . (cid:46) D V /µ m (cid:46)
12, and the primary sizedistribution is within the range 1 (cid:46) D V /µ m (cid:46) A ∼ π D V , while the parti-cle’s volume scales as V ∼ π D V /
6. Therefore, the ratioof area to volume is A / V ∝ / D V , and it is the smallestdroplets that will live the longest.The layout of the paper is as follows. In Sec. II we review the generalities of aerodynamic drag force andestimate the terminal speed of aerosols and droplets. InSec. III we model the elastic scattering of the turbulentcloud with the air molecules and estimate the pu ff stop-ping range assuming standard ambient temperature andpressure conditions. After that, we use our pu ff modelto determine the average density of the buoyant fluidin the turbulent cloud. The paper wraps up with someconclusions presented in Sec. IV. II. TERMINAL SPEED
When a particle propagates through the air, the sur-rounding air molecules have a tendency to resist its mo-tion. This resisting force is known as the aerodynamicdrag force. For a spherical particle, the aerodynamicdrag force is given by F d = π η air D V v V κ . (1)where η air (cid:39) . × − kg / (m · s) is the dynamic viscosityof air and v V is the virus velocity vector. Eq.(1) is the well-known Stokes’ law, with the Cunningham slip correctionfactor κ ; see Appendix for details. Stokes’ law assumesthat the relative velocity of a carrier gas at a particle’ssurface is zero; this assumption does not hold for smallparticles. The slip correction factor should be applied toStokes’ law for particles smaller than 10 µ m.The particle Reynolds number, R = D V v V ρ air η air , (2)is a dimensionless quantity which represents the ratio ofinertial forces to viscous forces, where ρ air (cid:39) . / m is the air density at a temperature of 20 ◦ C (293 K). For R <
1, the inertial forces can be neglected. The dragcalculated by Eq.(1) has an error of about 12% at
R ≈ R >
1. In the vertical direction,the upward component of the aerodynamic drag force F d , ⊥ is counterbalanced by the excess of the gravitationalattraction over the air buoyancy force F g = π D V ( ρ H O − ρ air ) g , (3)where ρ H O (cid:39)
997 kg / m and g (cid:39) . / s is the ac-celeration of gravity. Since ρ air (cid:28) ρ H O the air buoy-ancy force becomes negligible, and so F g ≈ M V g , with M V the aerosol mass. When the upward aerodynamicdrag force equals the gravitational attraction the dropletreaches mechanical equilibrium and starts falling with aterminal speed v V , f , ⊥ ≈ M V g κ π η D V . (4) TABLE I: Cunningham slip correction factor and terminalspeed. D V ( µ m) κ v V , f , ⊥ (m / s)0 .
001 215 . . × − .
010 22 .
05 6 . × − .
100 2 .
851 8 . × − .
500 1 .
327 1 . × − .
000 1 .
163 3 . × − .
500 1 .
109 7 . × − .
000 1 .
081 1 . × − .
000 1 .
054 2 . × − .
000 1 .
033 7 . × − .
000 1 .
023 1 . × − .
000 1 .
016 3 . × − The terminal speed is ∝ D V (due to the diameter depen-dence of the mass), and hence larger droplets would havelarger terminal velocities thereby reaching the groundfaster. The terminal speed for various particle sizes isgiven in Table I. The time t f it will take the virus to fall tothe ground is simply given by the distance to the grounddivided by v V , f , ⊥ . For an initial height, h ∼ D V = µ m, t f = hv V , f , ⊥ ∼ . (5)The time scale as a function of the droplet size and heigthis shown in Fig. 1.The aerodynamic drag force holds for rigid sphericalparticles moving at constant velocity relative to the gasflow. To determine the stopping range, in the next sec-tion we model the elastic scattering of the turbulent pu ff cloud with the air molecules. III. STOPPING RANGE
Respiratory particles of saliva and mucus are expelledtogether with a warm and humid air, which generatesa convective current. The aerosols and droplets are ini-tially transported as part of a coherent gas pu ff of buoy-ant fluid. The ejected pu ff of air remains coherent ina volume that varies from 0 . . [26].This corresponds to a pu ff size 0 . (cid:46) D P / m (cid:46) . ffi cient [27] of α = .
1. The pu ff is ejected with1 (cid:46) v V , , (cid:107) / (m / s) (cid:46)
10 [26]. The turbulent pu ff cloudconsists of an admixture of moist exhaled air and mu-cosalivary filaments. Next, in line with our stated plan,we use the experimental data to calculate the range ofthe average density of the buoyant fluid in the turbulentcloud.The mass ratio of the average air molecule comparedto the aerosol, m air / M V , is roughly 10 − (since the size of diameter [ μ m ] h e i gh t [ m ] FIG. 1: Contours of the time t f in minutes in the h − D V plane. the aerosol and the mass for its chief constituent, H O,compared to the air molecule are 10 and 10 ), thoughthere is an obvious variation with aerosol size at constantdensity. If we consider instead the mass inside the pu ff M P the ratio R ≡ m air / M P is even smaller. Due to theenormous mass ratio, the virions inside the pu ff will notundergo large angular deflections, so we will treat thevirions as having the same direction for its initial andfinal velocities (since we are looking at a stopping dis-tance, this is a reasonable assumption). Starting with thenon-relativistic one-dimensional equation for the virusvelocity β we have in the lowest nontrivial order (in R (cid:28)
1) and any frame (cid:32) β v air , f (cid:33) = M (cid:32) β v air , (cid:33) , (6)where the matrix M is derived by imposing conservationof energy and momentum, and is given by M = (cid:32) − R R − (cid:33) , (7)with β = v V , , (cid:107) , and v air , and v air , f the initial and finalvelocities of the air molecule, respectively. As the veloc-ity β falls with each interaction, the velocity loss remainsconstant; the target particle is a new air molecule at eachinteraction.Though individual air molecules are traveling at anaverage speed of a few hundred meters per second,throughout we assume the medium to be stationary. Inanalogy with the description of the slowing down ofalpha particles in matter (which assumes the electroniccloud is at rest), we can describe the scattering of thepu ff in the frame in which the air molecule is at rest, i.e., v air , = d β/ dx = ∆ β/λ V mfp = R β/λ V mfp , (8)with solution ln β = (2 R /λ V mfp ) (cid:82) dx . Finally, we have forthe stopping distance L = λ V mfp R ln (cid:32) β β f (cid:33) , (9)with β f ≡ v V , f , (cid:107) . Note that L /λ V mfp is not only the numberof mean free paths traversed by the fiducial virus, butis also the number of interactions of the virus with airmolecules; of course, there is a one-to-one correlationbetween the number of mean free paths traveled andinteractions.Since β is homogeneous and the mass ratio R is a con-stant for a given pu ff size D P , we have the above simpleequation. The mass ratio R is very small, and (2 R ) − is correspondingly very large. There are a tremendousnumber of mean free paths / interactions involved as thevirions bowling ball rolls over the air molecule.Finally, we must calculate λ V mfp = / ( n air σ ). The airmolecules 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 theBoltzmann constant. We assume a contact interactionequal to the cross-sectional hard-sphere size of the pu ff ,i.e. σ = π ( D P / . Substituting into Eq.(9) we obtain thefinal result for the stopping distance L = k B TP π ( D P / R ln (cid:32) β β f (cid:33) . (10)We take the sneeze or cough which causes the dropletsexpulsion to be at a standard ambient air pressure of P =
101 kPa and a temperature of T ∼
293 K. It is important tostress that temperature variation could cause an O ( (cid:46) ± e ff ect in L for extreme ambient cold or warmth . We nowproceed to fit the experimental data. For L ∼ v V , f , (cid:107) ∼ / s [16], we obtain 1 . < ρ P /ρ air < . . (cid:46) D P / m (cid:46) .
68, where ρ P is the average densityof the fluid in the pu ff .A point worth noting at this juncture is that our modelprovides an e ff ective description of the turbulent pu ff cloud. Note that independently of their size and theirinitial velocity all respiratory particles in the cloud ex-perience both gravitational settling and evaporation.Aerosols and droplets of all sizes are subject to continu-ous settling, but those with settling speed smaller thanthe fluctuating velocity of the surrounding pu ff wouldremain trapped longer within the pu ff . Actually, becauseof evaporation the water content of the respiratory parti-cles is monotonically decreasing. At the point of almostcomplete evaporation the settling velocity of the aerosolsis su ffi ciently small that they can remain trapped in the pu ff and get advected by ambient air currents and dis-persed by ambient turbulence. The size of the pu ff thencontinuously grows in time [26]. Our result can equiva-lently be interpreted in terms of the e ff ective coherencelength of the turbulent cloud assuming ρ P ∼ ρ air . Thee ff ective size of the pu ff and its e ff ective density are en-tangled in Eq. (10). Numerical simulations show thatduring propagation the pu ff edge grows ∝ t / [16]. Af-ter a 100 s the pu ff would grow by a factor of 3 (see Fig. 7in [26]), in agreement with our analytical estimates. Inclosing, we note that if we ignore the motion of the airpu ff carrying the aerosols, as in the analysis of [28], it isstraightforward to see substituting R by m air / M V ∼ − into Eq. (10) that the individual aerosols would not travelmore than a few cm away from the exhaler, even underconditions of fast ejections, such as in a sneeze. Thisemphasizes the relevance of incorporating the completemultiphase flow physics in the modeling of respiratoryemissions when ascertaining the risk of SARS-CoV-2 air-borne infection. IV. CONCLUSIONS
We have carried out a physics modeling study forSARS-CoV-2 transport in air. We have developed a nu-clear physics analogy-based modeling of the complexgas cloud and its payload of pathogen-virions. Using ourpu ff model we estimated the average density of the fluidin the turbulent cloud is in the range 1 . < ρ P /ρ air < . ff slows down su ffi ciently, and its coherence is lost,the eventual spreading of the infected aerosols becomesdependent on the ambient air currents and turbulence.De facto, as it was first pointed out in [19] and later de-veloped in [29, 30] airflow conditions strongly influencethe distribution of viral particles in indoor spaces, culti-vating a health threat from COVID-19 airborne infection.Altogether, it seems reasonable to adopt additionalinfection-control measures for airborne transmission inhigh-risk settings, such as the use of face masks whenin public. If the results of this study - t f of O (hr) foraerosols, for example - are borne out by experiment, thenthese findings should be taken into account in policydecisions going forward as we continue to grapple withthis pandemic. Appendix
There are important considerations in the develop-ment of Stokes’ law, including the hypothesis that thegas at particle surface has zero velocity relative to theparticle. This hypothesis holds well when the diameterof the particle is much larger than the mean free pathof gas molecules. The mean free path λ airmfp is the av-erage distance traveled by a gas molecule between twosuccessive collisions. In analyses of the interaction be-tween gas molecules and particles, it is convenient to usethe Knudsen number Kn = λ airmfp / D V , a dimensionlessnumber defined as the ratio of the mean free path to par-ticle radius. For Kn (cid:38)
1, the drag force is smaller thanpredicted by Stokes’ law. Conventionally this conditionis described as a result of slip on the particle surface. Theso-called slip correction is estimated to be [31] κ = + Kn (cid:2) . + . − . / Kn) (cid:3) . (11)In our calculations we take λ airmfp = η air ρ air (cid:18) π m air k B T (cid:19) / , (12)where k B is the Boltzmann constant, T is the temperaturein Kelvin, and the density of air is given by ρ air = PR g T , (13)with P =
101 kPa, and where R g = .
058 J / (kg · K)is the ideal gas constant. The molar massof air is m mol =
29 g / mol, which leads to m air = . × − kg / molecule. Funding / Support:
The research of L.A.A. is supportedby the U.S. National Science Foundation (NSF GrantPHY-1620661). J.B.D. acknowledges support from theNational Science Foundation under Grant No. NSFPHY182080. The work of T.J.W. was supported in partby the U.S. Department of Energy (DoE grant No. DE-SC0011981).
Role of the Funder / Sponsor:
The sponsors had no rolein the preparation, review or approval of the manuscriptand decision to submit the manuscript for publication.Any opinions, findings, and conclusions or recommen-dations expressed in this article are those of the authorsand do not necessarily reflect the views of the NSF orDOE.
Declaration of Competing Interest:
The authors declarethat they have no known competing financial interestsor personal relationships that could have appeared toinfluence the work reported in this paper.
Ethical Approval:
The manuscript does not contain ex-periments on animals and humans; hence ethical per-mission not required. [1] C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, L. Zhang,G. Fan, J. Xu, X. Gu, Z. Cheng, T. Yu, J. Xia, Y. Wei, W.Wu, X. Xie, W. Yin, H. Li, M. Liu, Y. Xiao, H. Gao, L.Guo, J. Xie, G. Wang, R. Jiang, Z. Gao, Q. Jin, J. Wang,and B. Cao, Clinical features of patients infected with 2019novel coronavirus in Wuhan, China, Lancet , 497 (2020)doi:10.1016 / S0140-6736(20)30183-5[2] P. Zhou, X. Yang, X. Wang, B. Hu, L. Zhang, W. Zhang, H.Si, Y. Zhu, B. Li, C. Huang, H. Chen, J. Chen, Y. Luo, H.Guo, R. Jiang, M. Liu, Y. Chen, X. Shen, X. Wang, X. Zheng,K. Zhao, Q. Chen, F. Deng, L. Liu, B. Yan, F. Zhan, Y. Wang,G. Xiao, and Z. Shi, A pneumonia outbreak associatedwith a new coronavirus of probable bat origin, Nature , 270 (2020) doi:10.1038 / s41586-020-2012-7[3] N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, X.Zhao, B. Huang, W. Shi, R. Lu, P. Niu, F. Zhan, X. Ma, D.Wang, W. Xu, G. Wu, G. F. Gao, and W. Tan, A novel coro-navirus from patients with pneumonia in China, 2019, N.Engl. J. Med. , 727 (2020). doi:10.1056 / NEJMoa2001017[4] X. Tang, C. Wu, X. Li, Y. Song, X. Yao, X. Wu, Y. Duan, H.Zhang, Y. Wang, Z. Qian, J. Cui, and J. Lu, On the originand continuing evolution of SARS-CoV-2, Natl. Sci. Rev. , 1012 (2020). doi:10.1093 / nsr / nwaa036[5] C. Rothe, M. Schunk, P. Sothmann, G. Bretzel, G.Froeschl, C. Wallrauch, T. Zimmer, V. Thiel, C. Janke,W. Guggemos, M. Seilmaier, C. Drosten, P. Vollmar, K.Zwirglmaier, S. Zange, R. W¨olfel, M. Hoelscher Trans-mission of 2019-nCoV Infection from an AsymptomaticContact in Germany, N. Engl. J. Med., Jan. 30 (2020).doi:10.1056 / NEJMc2001468[6] E. Y. C. Shiu, N. H. L. Leung, and B. J. Cowl-ing, Controversy aorund airborne versus droplet trans-mission of respiratory viruses: implications for infec- tion prevention, Curr. Opin. Infect. Dis. , 372 (2019)doi:10.1097 / QCO.0000000000000563[7] S. W. X. Ong, Y. K. Tan, P. Y. Chia, T. H. Lee, O. T. Ng, M.S. Y. Wong, and K. Marimuthu, Air, surface environmen-tal, and personal protective equipment contamination bysevere acute respiratory syndrome coronavirus 2 (SARS-CoV-2) from a symptomatic patient, JAMA March 4 (2020).doi:10.1001 / jama.2020.3227[8] J. L. Santarpia, D. N. Rivera, V. Herrera, M. J. Mor-witzer, H. Creager, G. W. Santarpia, K. K. Crown, D.M. Brett-Major, E. Schnaubelt, M. J. Broadhurst, J. V.Lawler, St. P. Reid, and J. J. Lowe, Transmission poten-tial of SARS-CoV-2 in viral shedding observed at the Uni-versity of Nebraska Medical Center, medRxiv preprintdoi:10.1101 / / s41586-020-2271-3[10] J. Cai, W. Sun, J. Huang, M. Gamber, J. Wu, and G. HeIndirect virus transmission in cluster of COVID-19 cases,Wenzhou, China, 2020, Emerg. Infect. Dis. , 1343 (2020)doi:10.3201 / eid2606.200412[11] Z.-D. Guo, Z.-Y. Wang, S.-F. Zhang, X. Li, L. Li, C. Li,Y. Cui, R.-B. Fu, Y.-Z. Dong, X.-Y. Chi, M.-Y. Zhang, K.Liu, C. Cao, B. Liu, K. Zhang, Y.-W. Gao, B. Lu, and W.Chen, Aerosol and surface distribution of severe acuterespiratory syndrome coronavirus 2 in hospital wards,Wuhan, China, 2020, Emerg. Infect. Dis. , 1583 (2020)doi:10.3201 / eid2607.200885[12] N. van Doremalen, T. Bushmaker, D. H. Morris, M. G.Holbrook, A. Gamble, B. N. Williamson, A. Tamin, J. L. Harcourt, N. J. Thornburg, S. I. Gerber, J. O. Lloyd-Smith,E. de Wit, and V. J. Munster, Aerosol and surface stabilityof SARS-CoV-2 as compared with SARS-CoV-1 N. Engl. J.Med., March 17 (2020). doi:10.1056 / NEJMc2004973[13] R. W¨olfel, V. M. Corman, W. Guggemos, M. Seilmaier, S.Zange, M. A. M ¨uller, D. Niemeyer, T. C. Jones, P. Voll-mar, C. Rothe, M. Hoelscher, T. Bleicker, S. Br ¨unink,J. Schneider, R. Ehmann, K. Zwirglmaier, C. Drosten,and C. Wendtner, Virological assessment of hospital-ized patients with COVID-2019, Nature , 465 (2020)doi:10.1038 / s41586-020-2196-x[14] T. C. Jones, B. M ¨uhlemann, T. Veith, G. Biele, M.Zuchowski, J. Hofmann, A. Stein, A. Edelmann,V. M. Corman, C. Drosten, An analysis of SARS-CoV-2 viral load by patient age medRxiv preprintdoi:10.1101 / , e15 (2016) doi:10.1056 / NEJMicm1501197[16] L. Bourouiba, E. Dehandschoewercker, and J. W. M. J.Bush, Violent expiratory events: on coughing and sneez-ing, Fluid Mech. , 537 (2014). doi:10.1017 / jfm.2014.88[17] B. E. Scharfman, A. H. Techet, J. W. M. Bush, and L.Bourouiba, Visualization of sneeze ejecta: steps of fluidfragmentation leading to respiratory droplets, L. Exp. Flu-ids , 24 (2016). doi:10.1007 / s00348-015-2078-4[18] L. Bourouiba, Turbulent gas clouds and respiratorypathogen emissions: Potential implications for reduc-ing transmission of COVID-19, JAMA March 26 (2020).doi:10.1001 / jama.2020.4756[19] L. A. Anchordoqui and E. M. Chudnovsky, A physi-cist view of COVID-19 airborne infection through con-vective airflow in indoor spaces, SciMedJ , 68 (2020)doi:10.28991 / SciMedJ-2020-02-SI-5 [arXiv:2003.13689].[20] J. Gralton, E. Tovey, M. L. McLaws, and W. D.Rawlinson, The role of particle size in aerosolisedpathogen transmission: A review, J. Infect. , 1 (2011)doi:10.1016 / j.jinf.2010.11.010[21] X. Xie, Y. Li, A. T. Y. Chwang, P. L. Ho, W. H. Seto, Howfar droplets can move in indoor environments – revisitingthe Wells evaporation – falling curve, Indoor Air , 211 (2007). doi:10.1111 / j.1600-0668.2006.00469.x[22] J. Wei and Y. Li, Airborne spread of infectious agents inthe indoor environment, Am. J. Infect. Control , 5102(2016). doi:10.1016 / j.ajic.2016.06.003[23] E. C. Cole and C. E. Cook, Characterization of infectiousaerosols in health care facilities: An aid to e ff ective engi-neering controls and preventive strategies, Am. J. Infect.Control , 453 (1998). doi:10.1016 / S0196-6553(98)70046-X[24] J. P. Duguid, The size and the duration of air-carriage ofrespiratory droplets and droplet-nuclei, J. Hyg. (Lond.) , 471 (1946). doi:10.1017 / s0022172400019288[25] S. Yang, G. W. M. Lee, C. M. Chen, C. C. Wu, and K. P.Yu, The size and concentration of droplets generated bycoughing in human subjects, J. Aerosol Med. , 484 (2007)doi:10.1089 / jam.2007.0610[26] S. Balachandar, S. Zaleski, A. Soldati, G. Ahmadi, and L.Bourouiba, Host-to-host airborne transmission as a mul-tiphase flow problem for science-based social distanceguidelines, arXiv:2008.06113.[27] B. R. Morton, G. I. Taylor, and J. S. Turner, Turbulent grav-itational convection from maintained and instantaneoussources, Proc. Roy. Soc. of Lond. Ser. A. Math. , 1 (1956).doi:10.1098 / rspa.1956.0011[28] W. F. Wells, On air-borne infection study II: droplets anddroplet nuclei, Am. J. Hyg. , 611 (1934).[29] B. L. Augenbraun, Z. D. Lasner, D. Mitra, S. Prabhu, S.Raval, H. Sawaoka, and J. M. Doyle, Assessment andmitigation of aerosol airborne SARS-CoV-2 transmissionin laboratory and o ffi ce environments J. Occup. Environ.Hyg., in press (2020).[30] M. J. Evans, Avoiding COVID-19: Aerosol guidelinesmedRxiv preprint doi:10.1101 / ff ects of particle mor-phology on lung delivery: Predictions of Stokes’ lawand the particular relevance to dry powder inhaler for-mulation and development, Pharm. Res. , 239 (2002)doi:10.1023 //