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An opinion on the multi-scale nature of Covid-19 type disease spread

Swetaprovo Chaudhuri, Abhishek Saha, Saptarshi Basu

PII: S1359-0294(21)00046-7

DOI: https://doi.org/10.1016/j.cocis.2021.101462

Reference: COCIS 101462

To appear in: Current Opinion in Colloid & Interface Science

Received Date: 9 March 2021

Revised Date: 15 April 2021

Accepted Date: 23 April 2021

Please cite this article as: Chaudhuri S, Saha A, Basu S, An opinion on the multi-scale nature ofCovid-19 type disease spread, Current Opinion in Colloid & Interface Science, https://doi.org/10.1016/j.cocis.2021.101462.

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An opinion on the multi-scale nature of Covid-19 type disease spread

Swetaprovo Chaudhuria,∗, Abhishek Sahab, Saptarshi Basuc,∗

aInstitute for Aerospace Studies, University of Toronto, Toronto, ON M3H 5T6, CanadabDepartment of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093, USA

cDepartment of Mechanical Engineering, Indian Institute of Science, Bengaluru, KA 560012, India

Abstract

Recognizing the multi-scale, inter-disciplinary nature of the Covid-19 transmission dynamics, we discuss some recentdevelopments concerning an attempt to construct a disease spread model from the flow physics of infectious dropletsand aerosols, and the frequency of contact between susceptible individuals with infectious aerosol cloud. Such anapproach begins with the exhalation event specific, respiratory droplet size distribution (both airborne/aerosolized andballistic droplets), followed by tracking its evolution in the exhaled air to estimate the probability of infection and therate constants of the disease spread model. The basic formulations and structure of sub-models, experiments involvedto validate those sub-models are discussed. Finally, in the context of preventive measures, respiratory droplet-facemask interactions are described.

Keywords: Respiratory droplets, aerosols, Covid-19, disease spread model, facemasks

1. Introduction1

At the time of writing this paper, the cumulative num-2

ber of Covid-19 positive cases in the world is close3

to 135 million, with nearly 3 million deaths. How-4

ever, it has been recognized that the actual number of5

cases could be far higher than the reported number. Li6

et al. [1] estimated that in China “86% of all infec-7

tions were undocumented [95% credible interval (CI):8

82–90%] before the 23 January 2020 travel restrictions.”9

A major impediment in the direct estimation of the total10

number of actual positive cases in the Covid-19 pan-11

demic has been the vast number of asymptomatic cases12

[2–5]. Subramanian et al. [5] noted, “Using a model13

that incorporates daily testing information fit to the14

case and serology data from New York City, we show15

that the proportion of symptomatic cases is low, rang-16

ing from 13 to 18%, and that the reproductive number17

may be larger than often assumed.” However, in many18

cases, for predictions, epidemiological models utilize19

the reported/available data to obtain the model (Ordi-20

nary Differential Equations) parameters towards estima-21

tion of the basic reproduction number. The role and22

power of present epidemiological models in determin-23

ing policy and implementation of non-pharmaceutical24

∗Corresponding authors: S. Basu (sbasu@iisc.ac.in) and S.Chaudhuri (schaudhuri@utias.utoronto.ca)

interventions like social distancing, lockdown cannot25

be overemphasized [6–8]. At the same time, the lack26

of availability of actual infection numbers, possible de-27

pendency and sensitivity of the model parameters on28

seasonality [9–11] and mechanism of disease transfer,29

behooves a physics based approach where the model30

parameters are derived from first principle calculations.31

Indeed, such an approach, if possible, is not without32

limitations and uncertainties, and must be reconciled33

with data for practical implementations. More impor-34

tantly, such an approach calls for a detailed mechanis-35

tic understanding of each of the several sub-components36

that spans across scientific disciplines and integration37

of ostensibly decoupled fields like virus kinetics inside38

aerosols, droplet evaporation, turbulent diffusion, hu-39

man mobility, human behavior, their modeling and the40

uncertainties thereof. These sub-components are not41

uniformly at the same level of understanding. Yet, given42

the extremely high stakes - the catastrophe that a res-43

piratory disease pandemic like Covid-19 inflicts (and44

probably will inflict in future) upon billions of human45

beings, such a scientific endeavour is probably required46

more than ever before. It is to be recognized that this47

opinion article is not a comprehensive review of the48

Covid-19 disease spread; it is inherently biased towards49

a physical science based viewpoint of the disease trans-50

mission problem with a focus on the recent experiences51

Preprint submitted to Current Opinion in Colloid and Interface Science April 27, 2021

and works of the authors. We hope the readers will par-52

don such a bias in view of the interdisciplinary connec-53

tions that the approach attempts to achieve: connecting54

an epidemiological model with the science of colloidal55

suspension of virus particles in respiratory liquids along56

with the spatio-temporal interfacial processes of respira-57

tory droplet evaporation and their conversion to droplet58

nuclei. For a comprehensive review of the state of the59

art, the readers can refer to the review article by Pohlker60

et al. [12].61

2. Physics based disease spread model62

Ironically, the first mathematical model for infectious63

disease spread was formulated by mathematician/fluid64

dynamicist Daniel Bernoulli in 1760 [13, 14] leading to65

the Kermack–McKendrick model in 1927 [15]. Since66

then epidemiological models have evolved in great so-67

phistication and complexity [16]. Mean-field compart-68

mental models can lack the heterogeneity information69

in terms of population mobility and density - this has70

led to development of network models [17] and agent71

based models [18] which attempts to model human mo-72

bility using different techniques - complex networks and73

mobile agents, respectively. In parallel to the popula-74

tion dynamics viewpoint, epidemiological model like75

the SEIR model could be viewed as a chemical reaction76

mechanism. Here, the reaction is the infection transmit-77

ting from infected I individuals to susceptible individu-78

als S who are converted to exposed E. Subsequently, E79

can become recovered R or deceasedD.80

As shown in [23]

S + Ik1,αβγ−−−−→ E + I [R1αβ]

Ek2−−−→ I [R2]

Ik3−−−→ 0.97 R + 0.03D [R3]

Here, the kis are the rate constants of the correspond-ing “reactions”. α corresponds to the particular expira-tory event like breathing, coughing, singing, sneezingor talking, whereas β corresponds to the specific vec-tor of transmission like respiratory droplet (all possiblesizes, thereby including both large and small droplets -the latter often referred as aerosols in transmission liter-ature), droplet-nuclei (typically aerosols) or fomites. γcorresponds to the specific location where the infectionis occurring. The reaction [R3] assumes 97% recoveryrate and 3% mortality rate, respectively. Probability dis-tributions of k1,αβγ could be obtained, from which anappropriate statistic representing mean-field k1 serving

as critical parameter for the corresponding mean-fielddisease spread Eqn. 1, for example over a city couldbe derived. Alternatively, a network model could beused where the following Ordinary Differential Equa-tion (ODE) could be solved at different nodes of a net-work, in a network model. The mean-field ODE withthe representative mean-field k1 is shown below:

d[I]dt

= k2[E] − k3[I]

d[E]dt

= k1[I][S ] − k2[E]

d[R]dt

= 0.97k3[I]

d[D]dt

= 0.03k3[I]

[S ] + [E] + [I] + [R] + [D] = 1

(1)

The [] represents the respective fraction of population.81

The skeletal structure of SEIR model presented above82

is well known. So, what has been achieved? What we83

get by casting the SEIR model in the form of a chemical84

reaction mechanism is that now the rate constants: k1,85

can have a sound physical interpretation and thus, paves86

the way for a first principle model. The rate constants k287

and k3 are controlled by virus-human body(virology and88

physiology) interaction and can be assumed constants89

to a first approximation; whereas for a Covid-19 type90

infectious respiratory disease, the k1 is primarily con-91

trolled by the flow physics of human exhalations, virus92

kinetics and statistics of human interactions. In chemi-93

cal reaction mechanisms involving gas-phase reactions,94

the Arrhenius rate constants have been historically de-95

rived using collision theory and later by more sophisti-96

cated transition state theory [24]. Quantum mechanical97

modeling utilizing Density Functional Theory (DFT)98

[25] has proved pivotal in estimating the structure of the99

transition states and the resulting potential energy land-100

scapes. If quantum mechanics can be used to calculate101

rate constants of chemical reaction mechanisms, could102

the fluid mechanics of aerosols that fundamentally un-103

derpins the disease transmission of Covid-19 be used to104

obtain the rate constants of the SEIR model? Indeed,105

the challenges in obtaining such a rate constant from106

first principles are many, the first being that infection107

does not occur in a fixed environment with fixed physi-108

cal dimensions. Human behavior even collectively may109

not be predictable. In such a scenario, a possible ap-110

proach could be the following. First, estimate the local111

rate constant k1,γ from k1,αβγ for an idealized, yet suffi-112

ciently generalized and well defined condition. Once113

that is obtained, a distribution of such rate constants114

2

120nm 0.5µm – 1mm 2m 20m 16,600 km

SARS-CoV-2

Respiratory droplet

Respiratory sprays: airborne and ballistic droplets

Infectious aerosol dispersion in crowds, people inhale aerosols: Covid-19 is airborne

World-wide spread10s:

Settling time for 100 µm droplet

2 min – 30 hr:Half-life of SARS-CoV-2 virus in aerosols

300 hr+:Settling time for

1µmdroplet

3 months

120 hr: incubation period

Figure 1: Multi-length scale, multi-time scale, multidisciplinary nature of the Covid-19 disease spread. Sub-images adapted from [19–22]

could be obtained for the different situations involved.115

The mean and/or other relevant statistics could be ob-116

tained from the corresponding moments of the resulting117

probability density functions to yield the mean-field k1.118

Covid-19 has been recognized as an airborne disease[26, 27], though the ballistic droplet and fomite routecould provide non-negligible transmissions. There areseveral studies which analyzed large scale transmis-sion events, that would could not have occurred unlessthe virus was airborne. In particular the Skagit Val-ley Chorale superspreading event [28], outbreak in aGuangzhou restaurant [29] and the Diamond PrincessCruiseship [30] are some prominent examples wherethe infected cases were at a sufficient distance from theindex case to be explained by the ballistic droplet orfomite route of transmission. As such, Lednicky et al.[31] could isolate SARS-CoV-2 virus from aerosols col-lected 2 to 4.8 m from Covid-19 positive patients. Oneof the most influential theoretical contributions in esti-mating the probability of indoor infection by airbornedisease is by Riley et al. [32] and is given by the Wells-Riley equation.

P = 1 − e(−I pqt/Q) (2)

Here P is the probability of infection, I is the numberof infected individuals, p being the rate of breathing perperson, q being the quantum (of infection) generationrate by I, t is the exposure time and Q is the volume flow

rate of incoming air. Rudnick and Milton [33] extendedthis further to define

P = 1 − e−µt (3)

Here µ = f Iq/n and f is the average fraction of in-door air that is exhaled as breadth, n being the numberof people present in the space. From this they derivedthe “reproductive number for an infectious disease in abuilding environment” as

R0A = (n − 1)e( f Iqt/n) (4)

This can be readily converted into k1 of Eqn. 1. As119

such this formulation has been extensively used as in120

[34] to estimate the number of new infections and is121

an incredibly powerful tool to estimate infection risks.122

However, inherent in the simplicity, such a formula-123

tion have the following limitations embedded through124

its assumptions. (i) Spatially homogeneous aerosol con-125

centration. While this assumption eliminates the need126

for flow physics considerations, it is a rather restric-127

tive assumption since aerosols are exhaled from specific128

sources (of very small dimensions compared to room129

size) and mostly disperse by progressively weakening130

turbulent diffusion. For continuously periodic activity131

like breathing, even in steady state, it is unlikely that132

the aerosol concentration will be homogeneous in an in-133

door space at any time. Infectious aerosols will be much134

more concentrated close to the infected individual than135

3

in a remote corner of a room. If it were, any necessity of136

social distancing would be meaningless. As such Chen137

et al. [35] emphasized the importance of short range air-138

borne route dominated by small droplets in close contact139

(< 2m) disease transmission (ii) Absence of considera-140

tion of size distribution of aerosols. This also has an im-141

portant bearing since larger sized droplets continuously142

settle, smaller ones remain airborne for longer times,143

the initial size distribution and the local thermal-fluid144

conditions determine the fraction of the originally ex-145

haled aerosol population that will remain airborne after146

a particular time from the exhalation event. (iii) Ab-147

sence of droplet physics and virus kinetics. The SARS-148

CoV-2 virus has half-life that can range from minutes149

to hours, in aerosols, depending on the local thermo-150

dynamic conditions and level of UV irradiation. The151

virus half-life could well be dependent on the chemical152

and/or thermodynamic state of the respiratory droplet153

or the droplet nuclei. (ii) and (iii) could be however ad-154

dressed in modified forms of the Wells-Riley model as155

has been recently done by [36]. An alternative analy-156

sis approach could be to estimate the basic reproduc-157

tion number directly from the flow physics of exhala-158

tions starting from the exhaled respiratory droplet size159

distribution, drawing inspiration from the rate constants160

of chemical reactions. Indeed such an approach is more161

complex, often involves idealizing assumptions and in-162

volves sub-disciplines ranging from turbulent disper-163

sion to virus kinetics in evaporation respiratory droplets.164

More importantly, instead of assuming a dominant path-165

way (aerosols), could an analysis be performed, such166

that the dominant pathway emerges from the analysis167

itself?168

2.1. Analyzing airborne pathways169

Involving several idealizing assumptions and simpli-170

fications, the rate constants required for solving Eq. 1171

and a basic reproduction number estimate starting from172

Duguid’s [37] cough droplet size distribution were ob-173

tained Chaudhuri et al. [23]. Distributions of differ-174

ent parameters were not considered in this work though175

the sensitivity of viral load on the rate constants were176

shown. Here, only the basic ideas are described. Draw-177

ing inspiration from molecular collision theory, the rate178

constants k1,αβ for each exhalation event α (breath,179

cough, sing, sneeze or talk) and transmission vector β180

(droplet, droplet nuclei or fomite as shown in Table 1)181

appearing in Eqns. [R1αβ] could be derived as a func-182

tion of the collision volume, probability of infection and183

concentration of I and S individuals. The following184

paragraph shows how.185

k1,αβ droplet nucleus fomitebreath k1,bd k1,bn k1,b f

cough k1,cd k1,cn k1,c f

sing k1,gd k1,gn k1,g f

sneeze k1,sd k1,sn k1,s f

talk k1,td k1,tn k1,t f

Table 1: Infection rate constants for different expiratory events andmodes of transmission. Adapted from Chaudhuri et al. [23]

Let’s assume that in a unit volume, the number ofinfected and susceptible individuals are nI and nS , re-spectively. Therefore, the number of collisions thatwould occur between the exhaled, infectious gas cloudD (ejected by I) and the susceptible individuals S isVcnInS . Vc is the collision volume, the volume swept bya cylinder of diameter σDS = (σD + σS )/2 in unit time.Here, σS is the diameter of the hemispherical, inhala-tion volumeVb = (1/12)πσ3

S , whereas σD is the diam-eter of the aerosol cloud which grows in time. When theaerosol cloud volume is less than the total volume underconsideration and when the relative velocity between Dand S are non-zero and given by VDS , the frequency ofcollision Z is given by πσ2

DS VDS nInS . However, all col-lisions will not result in infection since the probabilityof infection Pαβ (for expiratory event α and transmis-sion vector β) upon collision between S and D variesbetween 0 and 1. Including that Chaudhuri et al. [23]defined the generalized rate constant by Eqn. 5

k1,αβγ =πntotal,γ

tc

∫ τ

0σ2

DS (t)VDS (t)Pαβ(t)dt (5)

where ntotal,γ is the number density (number/volume) of186

the inhalation volumes at a location γ. Given the varia-187

tion in the height of the susceptible individuals, we as-188

sume the inhalation hemispheres (of volume Vb) are189

uniformly distributed within a given vertical distance190

H . If so, the population density ndensity,γ (number of191

people/area) could be converted to ntotal,γ = ndensity,γ/H .192

This also explains why a collision volume and not col-193

lision area has been considered. tc is the average time194

period between two expiratory events.195

Indeed the rate constant k1,αβγ is location, expiratory196

event specific while the ones required for solving Eqn.197

1 arises from a mean-field approximation. This can be198

addressed by i) considering actual distributions of ntotal,γ199

and other relevant parameters (like viral load) in Eqn. 5200

or ii) by constructing a network model where k1,αβγ is201

calculated at individual locations, after summing it over202

α, β.203

4

Rate constants

and local R0

from collision theory

SEIRD model

Jet/puff aerodynamics

Respiratory droplet/aerosol

model

Virus lifetime, viral

load & dose response

Probability of infection

1 10 100 1000

pdf

Ds,0(µm)

Cough droplet size pdf

Figure 2: The overall model structure as an assembly of several interacting sub-models.

In Eqn. 5, probability of infection Pαβ upon cross-ing the droplet cloud D is a key term. Its practical im-portance also cannot be overstated. Most works uti-lized the “quanta” of infectious emissions to estimatethe probability of infection. Approximating Duguid’scough droplet size distribution with a lognormal dis-tribution fα(D) = 1

√2πσD

exp[−(ln(D) − µ)2/2σ2] , thenumber of virions within droplet sizes D1 and D2 isgiven by Nvα =

πρvNtα6

∫ D2

D1D3 fα(D)dD. Incorporat-

ing the virus kinetics for the transmission vector β byψβ(t) = (1/2)

t/tβ 1

2 , Chaudhuri et al. [23] calculated thenumber of virions that would be inhaled from dropletsupon crossing the aerosol cloud D by

Nαd(t) =πρvNtαNbσ

3S (σS + σD(t))ψd(t)

12VDS (t)σ3D(t)

∫ D2(t)

D1(t)D3 fα(D)dD

(6)and the number of virions that would be inhaled fromthe dried droplet nuclei by

Nαn(t) =πρvNtαNbσ

3S (σS + σD(t))ψn(t)

12VDS (t)σ3D(t)

∫ Dn(t)

0D3 fα(D)dD

(7)ρv is the viral load, Ntα is the number of respiratorydroplets ejected. ψβ(t) is the fraction of infectious viri-ons available at time t in the vector β; tβ 1

2being the

corresponding half life of the virus. The limits of theintegrals in Eqns. 6, 7 are the initial droplet diametersthat evaporate and settle at time t, respectively and canbe estimated from a modified Wells’ plot where dropletevaporation and settling times are calculated account-ing from the accompanying warm and moist air jet thatis exhaled. The settling time is obtained from Stokesterminal velocity accounting for droplet size reductiondue to evaporation. The droplet size reduction due to

evaporation and the heat transfer to/from ambient canbe modeled using an energy balance equation. Here,the droplet was assumed to be homogeneous. It hasbeen shown that the effect of solute (nonvolatile compo-nents: mostly salts in respiratory fluids) concentrationin the respiratory droplets are critical in predicting theaccurate droplet lifetime [38]. This was incorporated inthe model by tracking the overall solute concentration inthe droplet. Its effect on evaporation rate was includedby evaluating the suppression of vapor pressure of theevaporating species (water, here) as prescribed by theRaoults law. The translation of the droplet in gas phasewas modeled using 1D drag equation. These approachleads to a set of linear differential equations, which canbe simultaneously solved to estimate the temporal evo-lution of location, size and temperature of the respira-tory droplet. Such approach readily incorporates effectsof ambient condition (temperature, humidity) and ini-tial droplet size on the evaporation rate and final size ofthe nuclei. A detailed version of this model has beendescribed by Chaudhuri et al. [23, 39]. From these thetotal probability of infection for the expiratory event αcould be defined utilizing the dose response model:

Pα(t) = 1 − e−rv∑βNαβ(t) (8)

The above dose response model is inspired by the semi-204

nal work of Haas [40]. The most important highlight of205

the combined system of equations 6-8 is that, here prob-206

ability of infection is directly from the exhaled respira-207

tory droplet size distribution without invoking “quanta”,208

as in previous studies. Of course the individual proba-209

bility of infection by the droplet route d and the dried-210

droplet nuclei route n could be calculated. For a non-211

ventilated large indoor space at typical air-conditioned212

state, the results for a single cough are shown in Fig. 3.213

5

100 101 102 10310-4

10-2

100

(a)

100 10110-4

10-2

100

(b)

Figure 3: Probability of infection Pcβ for droplet route d and drieddroplet nuclei route n, as well as total probability Pβ for (a) as a func-tion time from the instant of the beginning of a single cough (b) asa function of distance from location of the single cough along thecenter of the jet/puff trajectory. Note both d and n routes are es-sentially inhalation routes since large droplets settle in a very shorttime and are unlikely to contribute much in the disease transmission.T∞ = 21.44 oC, RH∞ = 50 %. The bold lines represent ρv = 7 × 106

copies/ml with td 12

= tn 12

= t 12

, where t 12

= 15.25 minutes. The greyshaded region denotes the lower limit tn 1

2= 0.01td 1

2and upper limit

tn 12

= 100td 12

, respectively, with td 12

= t 12

.Adapted from Chaudhuriet al. [23]

Such an analysis can answer three fundamental ques-214

tions, quantitatively with corresponding uncertainty:215

1. Why and how does the probability of infection re-216

duce with time and distance from the source? The217

reason why Pαβ reduces with time and distance218

are due to i) dilution of the aerosol cloud D by219

continuous entrainment of ambient air as captured220

by increasing σD(t) with time ii) droplet settling221

and evaporation if considering droplet route - re-222

ducing range within D1 and D2. But then droplet223

evaporates to produce droplet nuclei which can re-224

main infectious iii) finite lifetime of the virus in225

droplets/droplet nuclei. This is accounted in tβ 12

226

which is a function of temperature, relative humid-227

ity and UV index [41, 42]. Though ventilation is228

not considered, a practical method to reduce infec-229

tion probability is by increasing ventilation which230

would dilute the aerosol cloud even further along-231

side direct removal of the infectious aerosol parti-232

cles.233

2. Which is the most dominant route of transmission?234

Ostensibly from Fig. 3, the airborne droplet route235

has higher yet decaying probability. However,236

what is embedded in the log-log plot is that though237

the droplet nuclei has a much lower probability238

to infect, in absence of ventilation, it has much239

longer persistence. The time scales of its persis-240

tence is long enough and can match well with the241

lifetime of the virus in the aerosols. Summarizing,242

the d-route offers high infection probability over243

shorter time and distance whereas, n-route causes244

lower probability of infection but over longer time245

and distance. These two factors make the airborne246

dried droplet nulcei a very potent agent (possi-247

ble more than the airborne droplets) of transmis-248

sion. The role of droplet nuclei in infectious respi-249

ratory disease transmission was first identified by250

Wells. As shown in [23] the contribution of the251

dried droplet nuclei to the infection rate constant252

and corresponding basic reproduction number is253

much larger than the corresponding droplets. Of254

course this is under the assumption that the virus255

lifetime is same in both the droplet or the dried256

droplet nuclei. This uncertainty is also addressed257

in Fig. 3.258

3. What makes Covid-19 an airborne disease? Of259

course there is a very important biological aspect260

to this question which is beyond the scope of the261

article. In the framework of the analysis presented262

here, the factors that render the disease airborne263

are the following: i) high viral load ρv, this al-264

lows significant number of virions to be present265

within very small droplets/droplet nuclei that can266

float in air for a significant time and can be directly267

inhaled deep into the lungs. The effect of the dif-268

ferent viral loads on the corresponding basic repro-269

duction number R0,c was also shown [23]. ii) rela-270

tively small minimum infectious dose captured by271

rv. For Covid-19 the infectious dose is believed to272

be around 100−1000 copies though an exact num-273

6

ber remains eluding at this time. Based on the dose274

response analysis by Haas [43] in terms of plaque275

forming units, Schijven et al. [44] suggested 1440276

RdRP (RNA-dependent RNA polymerase) copies277

of SARS-CoV are required for infection iii) tβ 12

278

substantially large and of similar order as tsettle for279

small droplets and aerosols. Long half-life of the280

virus inside aerosol particles under most common281

indoor conditions allow the virus to be airborne.282

This is one of the most, if not the most, important283

parameter that qualifies the SARS-CoV-2 virus to284

be transmitted by the airborne route.285

3. Colloid and interfacial science of disease spread286

The previous discussion, has made it amply clear that287

long-term fate of the pathogen in respiratory fluid is crit-288

ical in understanding how long the virus remains po-289

tent airborne and thus how fast the disease spreads. For290

Covid-19 type pandemics, naturally, we ask what hap-291

pens to the pathogen during its transportation through292

respiratory droplets. While a detailed answer involves293

chemical, biological and virological aspects of the294

pathogen, many critical insights can be obtained from295

the interfacial dynamics between the pathogens and car-296

rier droplets. In fact, for a mechanistic understanding of297

the migration of these pathogens, one can assume them298

to be particles with fixed dimensions [45, 46]. The di-299

mension of most pathogens varies between hundreds of300

nm to few µm and their Stokes numbers in water-based301

solutions are small. Thus, their motion is primarily dic-302

tated by the flow patterns in the carrier droplet, the in-303

terfacial interactions, and their own motility, if any.304

As identified before, the respiratory droplets and305

droplet nuclei (including both large droplets and306

aerosols) can adopt three modes of infection transmis-307

sion: airborne, direct impact and fomite. The hydro-308

dynamics, evaporation, and flow structure inside a ses-309

sile droplet (fomite) are very different from an airborne310

moving droplet due to the solid substrate present in the311

former. Such dichotomy in hydrodynamics also creates312

a disparity in the migration of pathogens and their posi-313

tion in desiccated nuclei for these two modes. For ses-314

sile droplets, Deegan et al. [47, 48] explained that the315

very well-known coffee-ring structure formed by par-316

ticles in evaporating sessile droplets, arises from the317

strong evaporation at the pinned contact line, which318

is supported by strong radially outward flow in the319

droplet close to the substrate. However, such evapo-320

rating droplet also houses a couple of other effects in-321

cluding Marangoni and capillary flows, that can inhibit322

the formation of coffee-ring [49, 50]. An evaporating323

sessile droplet with unpinned contact line, on the other324

hand, can produce in general a more homogeneous dis-325

tribution of particles on the substrate [51]. In a recent326

effort, Zhao and Yong [52] used a Lattice Boltzmann327

Method with Brownian dynamics to access the effects of328

nanoparticles at the liquid-vapor interface of an evapo-329

rating sessile droplet. While the number density of par-330

ticles at the interface was found to influence the evap-331

oration rate, the wettability played a strong role in ac-332

cumulating the particles on the substrates. For weakly333

wettable surfaces, the particles were observed to be ag-334

gregated near the apex of the droplet, while for highly335

wettable substrates they were accumulated near the con-336

tact line. A more homogeneous distribution was ob-337

tained for neutral-wetting substrate.338

While the underlined basic mechanism in particle339

deposition in water-based solutions explained in these340

studies can be extended to pathogens, the dynamics341

in respiratory droplets are more involved due to the342

physico-chemical complexities and the resulting varia-343

tion in thermo-physical properties. Vejerano and Marr344

[53] showed that the surrogate respiratory (sessile)345

droplets containing dissolved salts undergo an efflores-346

cence process depending on the ambient relative hu-347

midity, which affects the crystallization process. Us-348

ing fluorescently tagged lipid, they could also isolate the349

virus locations in the drying fomites at various ambient350

conditions. The salt (NaCl) in the respiratory droplets351

crystallize on the substrate while the other components352

(Mucin, DPPC) can form a shell-like structure, encap-353

sulating the virus particles. Fig. 4a shows the possi-354

ble distribution of virus in a drying surrogate respira-355

tory droplet with φ6 virus exposed in an ambient of356

RH=29%. In particular, they highlighted that some357

preferential chemical affinity in available active organic358

and inorganic groups in the solution of water, DPPC and359

NaCl, might expedite the process of shell/layer forma-360

tion. Nevertheless as the evaporation continues, even-361

tually the nucleus of NaCl forms leading to crystal-362

lization. In a more recent effort, Rasheed et al. [54]363

highlighted the effects of substrate characteristics on364

fomites’ crystallization dynamics. They showed that365

dendritic or cruciform-shaped crystals were formed ma-366

jorly for most substrates, but regular cubical crystals367

were observed for machined steel substrates.368

The dynamics of pathogens in an airborne droplet are369

challenging to study experimentally, due to the involved370

complexity in isolating a single droplet in a controlled371

environment. An acoustic levitator, which can suspend372

a droplet by trapping it around a pressure anti-node in373

a standing wave, paved the way for studying isolated374

droplets in container-less environments [56–61]. More375

7

VLPs

(a)

(b) (c)

Figure 4: Drying of surrogate respiratory droplets. (a) Composite flu-orescent image of surrogate respiratory sessile droplet containing φ6virus exposed to 29% relative humidity. The bright green dots approx-imately 1µm in size may indicate the location of the virus. (adaptedfrom [53]). (b and c) Fluorescent VLPs inside crystal formed in dry-ing of surrogate levitated droplet with a (b) two component surro-gate respiratory fluid Compositions: 1% (w/w) aqueous NaCl solu-tion (adapted from [55]) and (c) four component surrogate respiratoryliquid: water, NaCl, DCCP and Mucin.

recently, Basu et al. [55] used such a setup in study-376

ing the surrogate respiratory droplet containing virus-377

like particles. They used 1% (w/w) NaCl aqueous so-378

lution as surrogate respiratory fluid and 100nm fluo-379

rescent particles as a surrogate virus, whose concen-380

tration was modulated from 0.005 to 0.1%. Although381

the chemical and biological signatures cannot be repli-382

cated, these VLPs can mimic the hydrodynamics of the383

SARS-CoV2 virus. The study confirmed that, irrespec-384

tive of the initial droplet diameter, the concentration of385

VLPs, the respiratory droplets evaporate and desiccate386

to form nuclei which are 20-30% of the initial droplet387

diameter in size. Importantly, they identified that at388

50% RH, the desiccated nucleus consists of a single389

NaCl crystal containing the VLPs. The detailed con-390

focal scanning of the crystals located a large portion of391

VLPs is in the crystal’s inner core, while the rest of them392

were distributed in the outer edges and on the surfaces393

on the crystal. The VLPs and NaCl’s relative diffu-394

sivities were compared with the droplet regression rate395

to demonstrate that, while homogeneity in fast-moving396

NaCl concentration was expected in the levitated respi-397

ratory droplet, VLPs were expected to be preferentially398

accumulated. Furthermore, the vortical motion created399

inside either levitated [61–63] or airborne droplets [64],400

induces the preferential distribution of VLPs mentioned401

above. Figures 4b and c show the final precipitates and402

VLP distribution in levitated respiratory droplets at RH403

=50% using two different surrogate droplet recipes. The404

study was then extended to actual human saliva, col-405

lected from various uninfected healthy subjects. While406

the variation of saliva content, arising from the subjects’407

eating, drinking, and physiological habits, resulted in a408

large scatter in the data, the average behavior closely409

followed the trend predicted with surrogate respiratory410

liquid. A similar study for a broader range of ambient411

conditions was reported recently by Lieber et al. [38],412

who also observed similar evaporation dynamics.413

Direct assessment of SARS-CoV-2 virus kinetics (in-414

fectivity) in aerosols and in sessile droplets has been re-415

ported by [41, 65] and in [8], respectively. All stud-416

ies found that SARS-CoV-2 virus stability monotoni-417

cally decreases with increase in temperature. Morris418

et al. [66] reported a U-shaped virus half-life behav-419

ior with ambient relative humidity. They attributed such420

response to the solute concentration in the respiratory421

droplets. Apparently, consistent with previous findings422

by Marr et al. [67], the enveloped viruses like SARS-423

CoV-2 survives well in droplets far from their dried424

state, as well as in the desiccated residue where the virus425

remains in a frozen state. In intermediate relative hu-426

midity levels (40-60%), in partially dried equilibrated427

droplets with highest solute concentration, the virus sur-428

vives for the least time, all other conditions being held429

equal. Indeed, virus survivability within desiccated nu-430

clei enables the virus to be airborne.431

4. Disease prevention: face masks432

Analysis presented in Section 2.1 can be used to un-433

derstand the effect of blocking droplets of different sizes434

on the probability of infection. Starting with Duguid’s435

cough droplet size distribution it has been shown that436

preventing ejection of all droplets above initial size of437

10µm can substantially reduce the probability of infec-438

tion [23]. Furthermore, it was also shown that pre-439

venting ejection of droplets above 5µm could result in440

a two order of magnitude reduction in the R0,c to a441

8

R0,c < 1 even for a very high viral load of ρV = 2.35 ×442

109copies/ml. Large R0,c are characteristics of super-443

spreader events, therefore blocking respiratory droplets444

upto a very small sizes like 5µm could ideally mitigate445

such events. Clearly such physical blockage of respi-446

ratory droplets at the point of source could be accom-447

plished with high quality, well-fitted face masks. Face448

masks provide a physical obstruction to the respiratory449

droplets, and their effectiveness in restricting droplet450

penetration depends on the effective pore size. Among451

the several local and medical-grade options available in452

the market, the N95 masks are proven to be most ef-453

fective. However, for community usage, their scarcity454

and high cost during pandemic have diverted the policy-455

makers to look for other economic and easily accessible456

alternatives like surgical-mask, cotton-mask, or home-457

made face masks. Historically, the usage of the face458

mask has been shown effective in restricting disease459

transmission during pulmonary tuberculosis (aerosol-460

based), Influenza (droplet-based) based illnesses, and461

restricting community transmission in Asian countries462

during the SARS-CoV-1 epidemic [68]. Face masks are463

therefore useful in obstructing the respiratory droplets464

during exhalation and inhalation and must be used ubiq-465

uitously to restrict the transmission of COVID-19.466

Although all face coverings provide some protection,467

their relative efficacy depends on the type of mask used.468

Several studies [69–73] have been carried out in re-469

cent times, focusing on investigating mask efficacy. Fis-470

cher et al. [69] used a laser-based optical measurement471

technique to determine the effectiveness of different lo-472

cally available face coverings (compared to no face-473

covering) in restricting ejected droplets during human474

speech. Home-made substitutes (like cotton masks) had475

equivalent effectiveness as that of surgical face masks,476

and compared to no mask were found to restrict ∼80%477

of the droplets from penetration. At the same time,478

some other substitutes like bandanas or neck gaiters479

have shown low or minimal protection. Hui et al. [74]480

used a high-fidelity human patient simulator (HPS) ly-481

ing at 45o on a hospital bed for investigating the dis-482

tance travelled by cough puffs when HPS was covered483

with no-mask, surgical mask, and N95 face mask. The484

velocities of ejected coughs were varied by controlling485

the flow rate of ejected fluid from 220-650 l/min, which486

resulted in maximum cough velocities of around 8 m/s.487

The turbulent cough flow without a mask was shown488

to traverse up to 70 cm, and the distance travelled re-489

duces by a factor of 2.3 and 4.5 with the usage of sur-490

gical and N95 face masks, respectively. N95 mask ef-491

fectively reduced the translational displacement of the492

ejected cough, but a significant transverse leakage was493

observed. Recently, Khosronejad et al. [75] numerically494

studied the effectiveness of face masks in indoor envi-495

ronments and showed that saliva droplets ejected dur-496

ing coughs in a stagnant environment without a mask497

could reach a distance of 2.62 m, and the same dis-498

tance reduces to 0.48 m and 0.78 m for medical and499

non-medical grade face masks. They also emphasized500

that under the influence of mild unidirectional breeze in501

the outdoor environment, saliva particulates can travel502

long distances in the flow direction in a short period,503

and face mask usage in such situations is necessary for504

preventing the inhalation of these droplets.505

Earlier works on face masks mostly focused on deter-506

mining filtration efficiency, comparing different types of507

medical grade and makeshift masks, and investigating508

the distance traveled by the ejected cough-puff and the509

capability of the face mask in restricting it. These inves-510

tigations were mainly carried out on small-sized cough511

droplets (∼0.1 to 100 µm) that can easily transmit or512

show minimal atomization during their passage through513

the mask layers. It must be noted that droplets (sub-514

micron to millimeters sizes) across a wide size range are515

generated during human coughs. Among these droplets,516

the larger-sized droplets (> 250µm) are lesser in num-517

ber but contribute to 90% of the ejected volume [73].518

Since the viral loading inside the droplet is volumetric,519

considering the fate of such large-sized droplets during520

their impact on the mask surface becomes essential. The521

large-sized droplets usually settle on the ground within522

seconds under the influence of gravity. However, on523

impacting a mask surface above certain thresholds ve-524

locities, these droplets undergo atomization into numer-525

ous tiny daughter droplets, most of which fall under the526

critical regime of aerosolization (< 100µm) [73]. This527

mechanism showed that the risk of infection through528

this route might be higher than what is predicted by529

considering mask filtration efficiencies alone. An ex-530

perimental study in this direction was recently carried531

by Sharma et al. [73], which uncovers the evolution of532

a large-sized cough droplet during its impingement on533

single- or multi-layer face masks. The aspects of droplet534

penetration, atomization mechanism, and final size dis-535

tribution were elucidated using side view shadowgraphy536

imaging, and the experimental results of droplet pene-537

tration and atomization were also validated using exist-538

ing theoretical framework.539

During this study, the primary experiments were con-540

ducted using surrogate cough droplets of size (Di) ∼541

620µm ejected at ∼ 10m/s (typical for a cough droplet)542

and impacting on a single-, double- or triple-layer sur-543

gical face mask. Figure 5a schematically shows the at-544

omization of a virus like particle laden surrogate cough-545

9

Figure 5: Secondary atomization of cough droplets through a face mask. (a) Schematic diagram representing atomization of impacting droplet intonumerous daughter droplets after penetration through the face mask. (b) Experimental images showing the extent of penetration through single-,double- and triple-layer face masks (from top to bottom in sequence). (c) Probability distribution for the diameter of atomized droplets showingmost daughter droplets lies in the critical regime. (figure adapted from Sharma et al. [73])

droplet while impacting the mask surface. Due to high546

impact velocity and Weber number (We = ρwv2i Di/σ ∼547

50 at penetration threshold), the inertia of the droplet548

dominates over surface tension forces, and thus the sur-549

face tension effects on the phenomenon were negligi-550

ble. The impacting droplet penetrates through the mask551

layers if the kinetic energy of the impacting droplet552

overcomes the energy loss due to viscous dissipation553

during liquid flow through the porous network of the554

mask. This penetration criterion was established else-555

where [76] and was validated experimentally for differ-556

ent droplet impact velocities (vi ∼ 2–10m/s) and droplet557

diameters (Di = 250 − 1200µm) [73]. It was shown that558

penetration of droplet through the face mask is indepen-559

dent of the impacting droplet diameter provided the ef-560

fective pore size (ε) of the mask is much smaller than561

the droplet size (Di), i.e., ε << Di, which was indeed the562

case during experiments. The penetration of the droplet563

through the different layered masks is shown in Fig. 5b.564

For a single-layer mask, the impacting droplet extrudes565

through the mask layer in the form of multiple cylin-566

drical ligaments that grow in length over time, because567

of which instabilities in the form of capillary waves are568

formed on the surface of the ligaments. The growth of569

these instabilities leads to the atomization into numer-570

ous tiny droplets. The estimation of droplet size and571

breakup time based on Rayleigh-Plateau Instability cri-572

teria was done and shown to be in good match with the573

experimental data (refer to [73]). Results showed that574

thicker ligaments lead to bigger droplet sizes and longer575

breakup times, and vice versa. During droplet penetra-576

tion through double- or triple-layer mask, the ligament577

formation was not observed due to the obstruction pro-578

vided by the multiple layers of the mask. The extent of579

penetration was lesser through a double-layer mask, and580

no or single droplet penetration was observed through a581

triple-layer mask (see Fig. 5b).582

The size distribution of atomized droplets is shown583

in Fig.1C, and the blue region in this curve indicates the584

droplet sizes < 100µ m, which have a higher aerosoliza-585

tion tendency and is referred to as a critical regime. The586

majority of atomized droplets lies in a critical regime587

for both single (∼ 60%) and double layer (∼ 80%)588

face mask. Each experimental run resulted in penetra-589

tion of ∼ 110 and ∼ 20 droplets (which corresponds to590

∼ 65% and ∼ 5% of the initial volume) for single and591

double-layer masks. Even though the number counts592

of penetrated droplets differ for single and double-layer593

masks, the probability distribution of atomized droplets594

remains similar for both masks. To consider the dif-595

ference in fluid properties of DI water droplet and ac-596

tual cough droplet (containing dissolved salt and pro-597

teins), experiments were also done with surrogate respi-598

ratory fluid. Similar results of droplet size distribution599

and penetration volume percentage were observed for600

this case (refer to [73]). This indicates that the pene-601

tration and atomization of high impact velocity droplets602

through the face mask is independent of the fluid proper-603

ties. Additional experiments result with variable impact604

velocities (3-10m/s) and impact angles (45o and 60o)605

were also demonstrated to bolster further this study’s606

relevance with droplet atomization scenarios during hu-607

man coughs. It was shown that at higher impact veloc-608

ities (7-10m/s) and different impact angles, no signifi-609

10

cant difference in the daughter droplet size distribution610

and penetrated volumes were observed (refer to [73]).611

In conclusion, a single layer mask that restricts 30%612

of the droplet volume was shown least effective in the613

study. A double-layer mask was more effective and ob-614

struct ∼ 91% of droplet volume, but 27.7% of trans-615

mitted volume fall in the critical regime, and at least a616

triple-layer mask is recommended for which only single617

or no droplet penetration was observed. It must also be618

noted that any face covering, even a single layer face619

mask provides some resistance to the exhaled respira-620

tory droplets during human coughs and, as such, should621

be used whenever required and mandated by the policy-622

makers.623

5. Conclusions624

In this article, we have provided an opinion on the625

need and the possible approach to develop a multi-scale626

physics based disease spread model from first princi-627

ples. This is an ambitious goal and an ongoing en-628

deavor. Such an effort is inherently multidisciplinary629

in that one has to combine expertise of aerosol scien-630

tists, fluid dynamicists, virologists, interfacial scientists631

under one common umbrella to develop such models632

that can be used for any respiratory disease transmis-633

sion for any viral outbreak. We have shown in details634

that chemical kinetics based pandemic model can be635

coupled with droplet level physics inclusive of evapo-636

ration, dispersion and precipitation to devise a hitherto637

new methodology to predict the infection spread in the638

context of COVID-19. In particular we also showed639

how the model can be improved by incorporating viral640

load distribution and kinetics by using novel experimen-641

tal techniques. Furthermore, the model can consummate642

new physics derived from social prevention measures643

like masks and social distancing. On a related topic,644

unique atomization pathways of cough droplets in sin-645

gle or double layer masks were investigated. We lastly646

harp the importance of a first principle model well vali-647

dated by experiments as a cornerstone of any pandemic648

macroscale model in the near future.649

6. Acknowledgements650

The authors thank Dr Prasenjit Kabi and Mr. Shub-651

ham Sharma for help with figures and some writeups.652

Comments and suggestions by Prof. Prasad Kasibhatla653

from Duke University is gratefully acknowledged. S.C.654

gratefully acknowledges the Heuckroth Distinguished655

Faculty Award in Aerospace Engineering from UTIAS.656

S.B. gratefully acknowledges the funding received from657

the DST-Swarnajayanti Fellowship and DRDO Chair658

Professorship awards. A.S. gratefully acknowledges659

funding received through UCSD’s internal grants.660

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13

Authors declare No Conflicts of Interest

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