dynamics of infectious dse.pdf

7/28/2019 dynamics of infectious dse.pdf

http://slidepdf.com/reader/full/dynamics-of-infectious-dsepdf 1/12

Dynamics of infectious diseasetransmission by inhalable

respiratory dropletsNikolaos I. Stilianakis1,2,* and Yannis Drossinos1,3

1Joint Research Centre, European Commission, Ispra (VA) 21027, Italy 2Department of Biometry and Epidemiology, University of Erlangen-Nuremberg,

Erlangen, Germany 3School of Mechanical and Systems Engineering, Newcastle University,

Newcastle upon Tyne NE1 7RU, UK

Transmission of respiratory infectious diseases in humans, for instance influenza, occurs byseveral modes. Respiratory droplets provide a vector of transmission of an infectious pathogen

that may contribute to different transmission modes. An epidemiological model incorporatingthe dynamics of inhalable respiratory droplets is developed to assess their relevance in theinfectious process. Inhalable respiratory droplets are divided into respirable droplets, withdroplet diameter less than 10 mm, and inspirable droplets, with diameter in the range10–100 mm: both droplet classes may be inhaled or settle. Droplet dynamics is determinedby their physical properties (size), whereas population dynamics is determined by, amongother parameters, the pathogen infectivity and the host contact rates. Three model influenzaepidemic scenarios, mediated by different airborne or settled droplet classes, are analysed.The scenarios are distinguished by the characteristic times associated with breathing atcontact and with hand-to-face contact. The scenarios suggest that airborne transmission,mediated by respirable droplets, provides the dominant transmission mode in middle andlong-term epidemics, whereas inspirable droplets, be they airborne or settled, characterizeshort-term epidemics with high attack rates. The model neglects close-contact transmission

by droplet sprays (direct projection onto facial mucous membranes), retaining close-contacttransmission by inspirable droplets.

Keywords: respiratory droplets; aerosol; influenza; dynamics; epidemic

1. INTRODUCTION

An essential element in understanding the epidemiologyof infectious diseases is their mode of transmission. Forsome infectious diseases, the routes of transmission aremultiple and their relative importance and dynamicsare neither well described nor well understood. Theidentification of the dominant transmission modeis important because efficient and effective controlstrategies depend on it.

For respiratory infectious diseases, influenza prob-ably being the most prominent example, threedifferent, mutually non-exclusive, main modes of respir-atory pathogen transmission have been identified(Tellier 2006; Brankston et al. 2007). The classifica-tion used in the medical literature considers ‘contact’,‘droplet’ and ‘airborne’ transmission. Contact trans-mission (be it direct or indirect) arises from contactwith pathogen-containing droplets: direct contacttransmission refers to physical contact and transfer of pathogens from an infected person to a susceptible,whereas indirect contact transmission refers to contact

with fomites and subsequent transport of the pathogenvia, for example, hands to the upper region of therespiratory tract (mouth, nose). Droplet transmissionoccurs via large droplets that are generated by a closeexpiratory event (coughing, sneezing): they depositimmediately onto a susceptible’s mucous membranes.As large droplets gravitationally settle quickly droplet

transmission constitutes a transmission mode only forclose contact. Airborne transmission (also referred toaerosol transmission) occurs via inhalation of small res-piratory droplets (also referred to as ‘droplet nuclei’)that are small enough to remain airborne.

Not all modes of transmission are relevant for all res-piratory infectious diseases; each transmission modemay affect different locations in the respiratory tractsince deposition in the respiratory tract depends on dro-plet size. For example, the airborne mode is the mostrelevant for tuberculosis since the reception site is thelower respiratory tract, whereas the influenza receptionsites may be anywhere in the respiratory tract rendering

all three modes relevant.Respiratory droplets, all droplets generated by an

expiratory event—coughing, sneezing, laughing, talk-ing, breathing—have diameters d that cover a large* Author for correspondence ([email protected]).

J. R. Soc. Interface (2010) 7, 1355–1366

doi:10.1098/rsif.2010.0026

Published online 29 April 2010

Received 19 January 2010Accepted 28 January 2010 1355 This journal is q 2010 The Royal Society

mailto:[email protected]

mailto:[email protected]



size range from approximately 0.6 to more than1000 mm. Droplets responsible for transmission byinhalation have also been classified as respirable andinspirable (Nicas & Sun 2006). Respirable dropletsremain airborne sufficiently long to provide a mechan-ism for airborne transmission. Inspirable droplets arelarger: they are either inhaled at close contact or they

gravitationally settle very fast. They contribute toinfection transmission by inhalation as they may beinhaled immediately after generation (e.g. duringthe first breath) instead of direct deposition onto theupper respiratory tract. We shall refer to these two dro-plet classes as ‘inhalable’ droplets. Nicas & Sun (2006)consider ‘respiratory droplet spray’ an alternativedescription to droplet transmission. Furthermore, theyargue that respirable droplets have an aerodynamicdiameter (after evaporation) d , 10 mm, whereas thesize of inspirable droplets is in the range [10, 100] mm.Atkinson & Wein (2008) also accept the distinctionbetween respirable and inspirable droplets (retaining

the d ¼

10 mm demarcation). Furthermore, the contri-bution of droplet sprays (Nicas & Sun 2006) ordroplet transmission (Atkinson & Wein 2008) duringclose expiratory events was estimated separately fromcontact and airborne transmission.

Therelative importance of the three transmission modesis a hotly debated issue; see, for example, the debate oninfluenza transmission (Gardam & Lemieux 2007; Tang& Li 2007; Tellier 2007; Vaille Lee 2007). Furthermore,recent modelling work on the interaction of humans withpathogens in the environment, using influenza trans-mission as an example, underlines the importance of environmental pathways such as air, hand, fomites, etc.

in pathogen transmission (Li et al. 2009).In this work, we investigate the relevance and impor-tance of disease transmission by inhalable, be theyairborne or settled, droplets. We neglect droplet-spraytransmission at close contact since relevant data arelimited and its contribution to transmission requiresseparate treatment. We develop a model for thedynamics of inhalable droplets that allows the quantifi-cation of their role in the transmission process via a judicious decomposition of the basic reproductionnumber. Settled droplets are not generated indepen-dently, but they arise as inhalable dropletsgravitationally settle. The model is an extension of a

susceptible–infective–recovered (SIR) epidemic modelfor a closed, homogeneously mixed population by incor-porating inhalable droplets as a vector of transmission.It describes the physical processes that determine thefate of a droplet, and it connects droplet properties toessential biological and behavioural features of thetransmission process, for example pathogen infectivityand contact rate of a susceptible with infected persons.

We argue, via the analysis of three model influenzaepidemic scenarios, that airborne respirable and air-borne or settled inspirable droplets can influence thedynamics of epidemics of aerially transmissible patho-gens. The scenarios are based on reasonable input

parameters differing in the choice of the characteristictimes associated with breathing at contact or thehand-to-face contact time. Each droplet-size class mayhave a different effect on the dynamics of a model

epidemic by determining the dominant transmissionmode, and the duration and incidence of the epidemic.Results suggest that the duration of an epidemic isinversely related to the pathogen load of the dropletsconstituting the dominant transmission mode. Wenote that the model is general enough to be applicableto the transmission of any respiratory infection whose

infectious agent is transported by inhalable respiratorydroplets.

2. A MODEL OF TRANSMISSIONDYNAMICS

2.1. Extended homogeneous population mixing

assumption

An essential ingredient of most epidemiological models isthe ‘homogeneous mixing’ assumption. According to it,population hosts have similar epidemiological propertiesand mix randomly. In the presence of droplets we extend

this assumption by considering that an infected personcarries, is associated with, a ‘cloud’ of pathogen-carryingaerosol droplets. The personal-cloud droplets are con-tinuously generated (e.g. by sneezing) and removed(e.g. by settling, inactivation), and they remain associ-ated with the infected person during his/hermovements in space. Furthermore, we associate anaverage number of droplets with each infected person,and a characteristic personal-cloud volume. The exist-ence of a personal cloud implies that settled dropletsmay also be associated with an infected individual.

If we consider that each infected person carries acloud of inhalable droplets and hosts of the SIR sub-

populations are randomly mixed, then droplets alsobecome homogeneously mixed with the two sub-populations S and I . Note that the spatial homogeneityof the average number of droplets does not arise fromdroplet diffusion (a very slow process), but from theirassociation with the homogeneously mixed infectedsubpopulation.

The existence of a cloud surrounding a person hasbeen stipulated in studies of personal exposure to ambi-ent-air particulate matter (Wallace 1996, 2000). Itscontribution to increased exposure to droplets or par-ticles, beyond a time-weighted average of indoor andoutdoor concentrations, has been referred to as the ‘per-

sonal cloud’ effect. Furthermore, Eichenwald et al.(1960), earlier and in a different context, suggestedthe existence of a personal cloud to justify superspread-ing effects of respiratory infections. They argued thatnewborns whose noses were colonized by Staphylococcus aureus disperse large quantities of airborne bacteria,becoming highly infectious. These babies were referredto as ‘cloud babies’, and the suggested mechanism of airborne dispersal of bacteria, the ‘cloud’ phenomenon.The same mechanism was investigated for adults, muchlater, by Sherertz et al. (1996), who referred to them as‘cloud adults’.

2.2. Model description

Emitted respiratory droplets are of varying sizes charac-terized by their number distribution, namely the

1356 Infectious disease transmission N. I. Stilianakis and Y. Drossinos

J. R. Soc. Interface (2010)



number of droplets per unit air volume as a function of diameter. Size, specifically diameter, is the most impor-tant physical property of a droplet. The dropletdiameter determines whether a droplet will settleunder gravity (it determines its airborne residencetime, and hence whether it is likely to be inhaled) orwhether it will evaporate before settling (evaporation

depends, in addition, on the chemical composition of the droplet), thereby converting a large to a smallerdroplet. Hence, the characteristic diameter of a dropletrelevant to transmission via inhalable droplets is deter-mined from its settling and evaporation times. Weneglect hygroscopic growth in the respiratory tract,a process inverse to ambient-air evaporation.

2.2.1. Monodisperse droplet number distribution. Con-sider a closed population of S (t ) susceptibles, I (t )infected, and R(t ) recovered persons with N ¼ S þ I þR the total (constant) population. Initially, let infectedpersons shed droplets of a single diameter d , taken to bethe droplet post-evaporation diameter. Moreover, let c be the (average) number of contacts a susceptible hasper unit time with any individual. Since susceptiblesand infected persons are homogeneously mixed a frac-tion I /N of these contacts will be with infectedindividuals. The expected number of contacts a suscep-tible has with infected individuals in a short time d t isc (I /N )d t . However, the infectious agent is not theinfected individual, but the pathogen carried by thedroplet. According to the extended homogeneousmixing assumption each infected is surrounded by apersonal cloud containing an average number of n d¼D /

I droplets within a characteristic personal-cloudvolume V cl, where D (t ) is the total number of airborne

droplets: the average droplet concentration per infectedbecomes n d/V cl. The contact of a susceptible with adroplet occurs by inhaling: if B is the average breathingrate (m3 s21) the average number of droplets perinfected person encountered by a susceptible per unit of time is B (n d/V cl). Finally, let t d be a characteristic timeof breathing during contact rendering B (n d/V cl)t dthe average number of droplets inhaled by a suscep-tible during the encounter. As each droplet isindependent and the droplet population is randomlymixed, the number of contacts a susceptible has with

a pathogen-carrying droplet becomes

c I

N B

n d

V clt d d t ¼ c

B

V clt d

D

N d t ; ~c d

D

N d t ; ð2:1Þ

where ~c d ¼ cB t d=V cl is the contact rate of a susceptiblewith a droplet. Given the number of contacts between asusceptible and a droplet and the independence of thenumber of contacts the transmission term in the dyna-mical equations of the SIR model may be easily derived.For completeness we reproduce the derivation, aspresented, for example, in Keeling & Rohani (2008),in appendix A. Accordingly, the rate of transmissionto all susceptibles becomes

dS

dt ¼ À

1

N ~b d D S ; ð2:2Þ

where ~b d ¼ ~c d ~pd is the transmission rate per droplet

with p d the probability that a contact with an inhaleddroplet results in successful transmission. Disease trans-mission depends indirectly on droplet size: airbornedroplets transmit the pathogen from ambient air tothe respiratory tract through inhalation and deposition.Thus, the probability of disease transmission dependson the pathogen load of the droplet and on its depo-

sition location in the respiratory tract. Theprobability of transmission may, thus, be expressed as

~pd ¼ pd q dðd Þ N pðd Þ; ð2:3Þ

where pd is the probability of transmission per inhaledpathogen, N p(d ) is the number of pathogens in a dro-plet of diameter d , and q d(d ) is the probability of deposition in the human respiratory tract (a functionof droplet size). The number of pathogens in a dropletis N p(d ) ¼V dr p, where V d is the (spherical) dropletvolume and r p the pathogen concentration in the lungfluid. Therefore, equation (2.2) becomes

dS

dt ¼ ÀN pðd Þb d q dðd Þ D

S

N ; ð2:4Þ

where the transmission rate per inhaled pathogen isb d ¼ ~c dpd. Henceforth, the subscript d will denoteairborne droplets.

Disease transmission via settled (deposited) dropletsC (t ) (of diameter d ) is governed by an equation similarto equation (2.4). Specifically, we associate an averagenumber n c ¼ C /I settled droplets with every infectedindividual. Let the settled-droplet removal rate byphysical transport of a settled droplet to the upper res-

piratory tract be h , a rate corresponding to the dropletremoval rate due to inhalation B /V cl. The characteristiccontact time t c is the time associated with hand-to-facecontact. Henceforth, the subscript c will refer to settleddroplets. Then, the number of contacts a susceptiblehas with a settled droplet in dt is c ht cC dt /N (comparewith equation (2.1)). If, as before, p c, the probabilitythat a contact with a settled droplet results in successfultransmission, is decomposed as p c ¼ pc q c(d )N p(d ), withq c(d ) the deposition probability in the facial mucoussurfaces for instantaneous contact and pc the prob-ability of transmission per settled pathogen, thetransmission rate per settled droplet becomes

~b c ¼ ~c c ~pc and the transmission rate per settled patho-gen b c ¼ ~c cpc with ~c c ¼ c ht c the contact rate of asusceptible with a settled droplet. The number of pathogens in a settled droplet transported to facialtissues, N p(d ), is calculated by assuming that thewhole droplet is transferred.

Therefore, the rate of change of the number of susceptibles due to pathogen-carrying droplets becomes

dS

dt ¼ ÀN pðd Þ b d q dðd Þ D þ b c q cðd Þ C ½

S

N : ð2:5Þ

Note that the transmission rates per pathogen b d and

b c do not depend on droplet size: they have beenexpressed as the product of a contact rate times theprobability of infection per contact with the infectiousagent, as in usual epidemiological SIR models.

Infectious disease transmission N. I. Stilianakis and Y. Drossinos 1357




If infected persons recover at a rate mI , with 1/mI theinfectivity period, their dynamics is determined by

dI

dt ¼ N pðd Þ b d q dðd Þ D þ b c q cðd Þ C ½

S

N À mI I

¼ ÀdS

dt À mI I : ð2:6Þ

The dynamics of airborne droplets D (t ) is determinedby the competition of generation and annihilation pro-cesses. Their number increases proportionally to thenumber of infected persons at the rate k pathogen-loaded droplets are shed. It decreases proportionallyto their gravitational settling rate u . They are removedby inhalation, and subsequent deposition, via two pro-cesses: either each infected person breathes her/hisown droplet cloud or any host encounters an infectedperson and she/he inhales the cloud droplets. Thefirst removal-rate term, independent of the contact

rate, is the product of the breathing rate B , times theaverage droplet concentration n d/V cl, the depositionprobability q d(d ), and the number of infected personsI : it evaluates to Bq d(d )D /V cl. The removal rate dueto an encounter of, say, a susceptible with an infectedindividual is proportional to the contact rate of a sus-ceptible with a droplet c ˜ d, the probability of encountering an infected I /N , the deposition prob-ability q d(d ), and the average droplet number perinfected n d: it evaluates to c ˜ dq d(d )D /N . Note that thedroplet-inhalation removal term ( per susceptible)differs from the corresponding infection termN pðd Þ~c dpdq dðd ÞD =N in equation (2.5). They differ

because the rate of decrease of D due to inhalationdoes not depend on the number of pathogens in the dro-plet: disease transmission depends on the pathogenload, but the droplet removal depends on its physicalproperties (e.g. droplet diameter). Furthermore, therate of decrease of D due to inhalation does notdepend on the probability of infection: regional depo-sition in the lung differs from infection. Thecorresponding total removal rate is multiplied by thenumber of susceptibles. The same term describesremoval by infected or recovered individual inhalationif we consider that the contact rate is the same forall hosts, and that I ) 1. Then, the removal term due

to encounters of all sub-populations becomes~c dq dðd ÞD ðS þ I þ RÞ=N ¼ ~c dq dðd ÞD , i.e., it reduces toa linear term for a constant population.

Droplets are effectively removed because pathogenslose infectivity. If md is the rate at which pathogenslose infectivity, i.e. dN pðd Þ=dt ¼ ÀmdN pðd Þ, then therate droplets are removed is mdD since the number of pathogens is proportional to the number of droplets(for a constant pathogen concentration in the lungfluid, and md independent of droplet size).

Therefore,

dD

dt ¼ k I ÀB

V cl þ ~c d

q dðd Þ þ md þ u !

D : ð2:7Þ

In a completely analogous manner mutatis mutandis the

number of settled droplets C (t ) changes according to

dC

dt ¼ u D À ½ðh þ ~c cÞq cðd Þ þ mcC ; ð2:8Þ

where droplets settled on surfaces lose infectivity at ratemc, and their number increases at the rate u airbornedroplets settle. As in the case of airborne-droplet

removal due to inhalation (and deposition), settled dro-plets may be removed by any infected, corresponding toa removal term h q c(d )C , or following contact of anyhost with an infected, leading to a removal termc ˜ cq c(d )C .

The rate of change of the number of persons whorecover is

dR

dt ¼ mI I : ð2:9Þ

The system of equations (2.5)– (2.9) is solved withappropriate initial conditions.

2.2.2. Polydisperse droplet number distribution. Theemitted polydisperse distribution is divided into l bins, each one characterized by an average dropletdiameter d i , an airborne droplet number D i (t ), anda corresponding number of settled droplets C i (t ).We consider that d j ! d i for j ! i . The dynamics of the infection for such a discretized distribution isdescribed by

dS

dt ¼ À

Xl

i ¼1

N pðd i Þ b di q d i

ðd i ÞD i þ b ci q c i

ðd i ÞC i

Â Ã S

N ;

ð2:10a Þ

dI

dt ¼ À

dS

dt À mI I ; ð2:10bÞ

dD i

dt ¼ k i I À

B

V clþ ~c di

q d i

ðd i Þ þ md þ u i

!D i

þX j .i

f ji D j À D i

X j ,i

f ij ; i ; j ¼ 1; . . . ; l ;

ð2:10c Þ

dC i

dt ¼ u i D i À ½ðh þ ~c c i

Þq c i ðd i Þ þ mc C i

þX j .i

~f ji C j À C i X j ,i

~f ij ; i ; j ¼ 1; . . . ; l ;

ð2:10d Þ

anddR

dt ¼ mI I ; ð2:10e Þ

with appropriate initial conditions. In equation (2.10c )the last two terms are evaporation terms with f ji theevaporation rate of droplet d j to become droplet d i ;the penultimate term models the increase of D i dropletsdue to evaporation of all larger droplets ( j . i ), and thelast term its decrease via evaporation to smaller dro-plets. Similarly, for settled droplets, equation (2.10d ),

the same evaporation terms for settled droplets aredenoted by f ˜

ij . We neglect nonlinear processes thatconvert smaller droplets into larger ones by coagulationand the inverse process of droplet break-up.





3. THE CASE OF INFLUENZA

3.1. Respiratory droplet number distribution

Experimental measurements of the respiratory aerosolnumber distribution shed during sneezing and coughingare relatively scarce. The size and number of dropletsgenerated can be very diverse (Knight 1980) and of sig-

nificant inter-subject variability (Edwards et al. 2004).Nicas et al. (2005) summarized and critically evaluatedthree detailed experimental studies (Duguid 1946;Loudon & Roberts 1967; Papineni & Rosenthal 1997).They suggested the use of the experimental data byLoudon & Roberts (1967). We, as well as Atkinson &Wein (2008), follow their recommendation: in particu-lar, we use the data reported in table I in Nicas &Sun (2006). It should be noted, however, that morerecent experimental measurements by Morawska et al.(2009) and Chao et al. (2009) found significant differ-ences from the Loudon & Roberts data; they alsoquestioned the data by Yang et al. (2007).

For simplicity and illustrative purposes, we approxi-mate the full droplet number distribution by a bimodaldistribution. Furthermore, we consider two bimodaldistributions: one to model respirable droplets, andhence airborne influenza transmission, and one tomodel inhalable droplets, and hence airborne trans-mission at close contact (neglecting, as mentionedearlier, droplet-spray transmission). Estimates of droplet sizes corresponding to these distributionsmay be obtained from their gravitational settling andevaporation rates. Evaporation, being a molecular pro-cess, is very fast (Nicas et al. 2005; Morawska 2006);for example, a 20 mm droplet evaporates to a 1 mm

diameter droplet within 0.24 s21

(at 50% ambientrelative humidity). Henceforth, we neglect dropletevaporation, and we follow Nicas et al. (2005) totake the post-evaporation diameter (approximately)half the pre-evaporation diameter.

The droplet residence time in air, and the impor-tance of a droplet size in airborne transmission, isdetermined from its gravitational settling velocity(Drossinos & Housiadas 2006); for example, a dropletof d ¼ 10 mm deposits (crosses a characteristic lengthof 1.5 m) within 8 min, whereas a d ¼ 4 mm dropletdeposits within 50 min. Consequently, and in accord-ance with Nicas et al. (2005), we consider that

droplets larger than 10 mm do not remain airborne suf-ficiently long to become respirable. The size of eachmode of the respirable droplet distribution was calcu-lated by mapping experimental data (Nicas et al.2005) to two sizes. Specifically, for a minimum pre-evaporation bin diameter d min and a maximum bindiameter d max the mean droplet diameter is obtainedfrom the mean droplet volume corresponding to thesebins, v ¼ p ðd 4max À d 4minÞ=ð24ðd max À d minÞÞ. We chosethe (pre-evaporation) pairs (d min, d max)) to be (2,11.6)and (11.6,20) to obtain post-evaporation diametersd 1¼ 3.89 $ 4 mm (small respirable droplets) and d 2¼8.08 $ 8 mm (large respirable droplets).

This bimodal representation is consistent withregional deposition of inhaled droplets in the respiratorytract (Hinds 1999; Drossinos & Housiadas 2006). Dro-plets with d . 8 mm deposit almost exclusively in the

extrathoracic region, whereas droplets with d , 4 mmmay reach the alveolar region.

Inspirable droplets are droplets that settle relativelyfast, but at close contact a fraction may be inhaled bya susceptible during his/her first breath if the infectedperson faces the susceptible (Atkinson & Wein 2008).According to Nicas & Sun (2006), their diameter

ranges from 10 to 100 mm. We consider that the inhal-able droplet distribution consists of a modecharacteristic of respirable droplets and a mode charac-teristic of inspirable droplets. We obtained theirdiameters as previously described; specifically, the res-pirable droplet diameter was calculated from themean droplet volume corresponding to d min¼ 2,d max ¼ 22.4 (table I, Nicas & Sun 2006), and the inspir-able droplet diameter to d min¼ 22.4, d max ¼ 228. Wefound the (post-evaporation) respirable-droplet diam-eter to be d 1 ¼ 7.3 mm and the inspirable-dropletdiameter d 2 ¼ 74 mm.

Droplet generation rates k are calculated by requir-

ing (pre-evaporation) volume (hence, pathogennumber) conservation. The number of emitted mean-volume-diameter droplets was calculated by ensuringthat the total emitted droplet volume (correspondingto the chosen mapping of bins) be preserved. For therespirable droplet distribution emitted during a coughwe found generation rates of pathogen-loaded dropletsk 1 ¼ 160 (d 1 ¼ 4 mm) and k 2 ¼ 7.5 (d 2¼ 8 mm),whereas for the inhalable droplet distribution k 1¼41.47 (d 1 ¼ 7.3 mm) and k 2¼ 138.48 (d 2¼ 74 mm).Nicas & Sun (2006), in addition, argued that only 50per cent of the emitted inspirable droplets enter thehead airways region; we, thus, take k 2¼ 69.24 inspir-

able droplets per cough. The respirable-dropletgeneration rates are consistent with the emission of 470 droplets of post-evaporation diameter greaterthan 1 mm per cough, half of which have a diameterless than 20 mm (Nicas et al. 2005). Note that thetotal emitted lung-fluid volume in a cough is 0.044 cm3,the volume in the respirable fraction 6.7 Â 1028 cm3

(approx. 1.5 Â 1024% of the total emitted volume), andthe volume in the inhalable fraction 2.38 Â 1024 cm3

(approx. 0.54%). The daily generation rates are obtainedby considering a 200-fold increase fora sneeze (Nicas et al.2005), and a total of 11 sneezes and 360 coughs per day(Atkinson & Wein 2008). They are reported in table 1.

3.2. Bimodal droplet number distribution

The model equations, equations (2.10a )–(2.10e ), for abimodal droplet number distribution, with d 2 . d 1and expressed in terms of droplet infectivities b ˜ , become

dS

dt ¼ Àð ~b d 1

D 1 þ ~b c 1C 1 þ ~b d 2

D 2 þ ~b c 2C 2Þ

S

N ; ð3:1a Þ

dI

dt ¼ À

dS

dt À mI I ; ð3:1bÞ

dD

1dt ¼ k 1I À nÀ1d 1 D 1; d

D 2

dt ¼ k 2I À nÀ1d 2 D 2; ð3:1c Þ

dC 1

dt ¼ u 1D 1 À nÀ1

c 1C 1;

dC 2

dt ¼ u 2D 2 À nÀ1

c 2C 2 ð3:1d Þ





Table 1. Parameter values for three model influenza epidemic scenarios. *All other parameters as in the epidemic scenariomediated by airborne inspirable droplets.

parameter value reference

r p pathogen concentration in the lung fluid 3.71 Â 106 pathogens ml21 Li et al. (2009)c contact rate between a susceptible and an

infected13 per day Mossong et al. (2008)

B breathing rate 24 m3

d21

Hinds (1999)V cl volume of the personal cloud of an infected

person8.0 m3 estimated

pd probability of infection by an inhaledpathogen

0.052 Li et al. (2009)

pc probability of infection by a settledpathogen

6.93 Â 1025 Li et al. (2009)

h removal rate of settled droplets by physicalcontact

72 per day Li et al. (2009)

q d(d ) inhaled-droplet deposition probability q d 1 ¼ 0:88; q d 2 ¼ 0:99 Drossinos & Housiadas(2006)

q c(d ) hand-to-face deposition probability q c 1 ¼ q c 2 ¼ 0:35 Li et al. (2009)mI infection recovery rate 0.20 per day Carrat et al. (2008)md inactivation rate of airborne pathogens 8.64 per day Hemmes et al. (1960)

mc inactivation rate of settled pathogens 2.88 per day Bean et al. (1982)epidemic mediated by airborne respirable droplets

d diameter ( post-evaporation) d 1 ¼ 4, d 2 ¼ 8 mm Nicas et al. (2005)N p(d ) number of pathogens per droplet 9.15 Â 1024 (d 1), 8.20 Â1023 (d 2) calculatedt d 1 ¼ t d 2 characteristic breathing time 20 min Mossong et al. (2008)t c 1 ¼ t c 2 characteristic hand-to-face

contact time15 s estimated

b d transmission rate per inhaled pathogen 0.027 per day calculated~b d transmission rate per inhaled droplet 2.2 Â 1025 (d 1), 2.2 Â 1024 (d 2) per

daycalculated

b c transmission rate per settled pathogen 1.13 Â 1025 per day calculatedb ˜ c transmission rate per settled droplet 3.6 Â 1029 (d 1), 3.24Â1028 (d 2) per

daycalculated

k respirable-droplet production rate 4.10 Â 105 (d 1), 1.92 Â 104 (d 2) per

day

Nicas et al. (2005)

u gravitational settling rate 28.80 (d 1), 113.2 (d 2) per day Hinds (1999)

epidemic mediated by airborne inspirable droplets

d diameter ( post-evaporation) d 1 ¼ 7.3, d 2 ¼ 74 mm Nicas & Sun (2006)N p(d ) number of pathogens per droplet 6.0 Â 1023 (d 1), 6.4 (d 2) calculatedt d 1 characteristic breathing time (respirable

droplet)20 min Mossong et al. (2008)

t d 2 characteristic breathing time (inspirabledroplet)

1.0 min estimated

t c 1 ¼ t c 2 characteristic hand-to-face contacttime

15 s estimated

b d 1transmission rate per inhaled pathogen

(respirable)0.027 per day calculated

b d 2 transmission rate per inhaled pathogen(inspirable) 1.4 Â 10

23

per day calculated~b d 1

transmission rate per inhaled respirabledroplet

1.4 Â 1024 per day calculated

~b d 2transmission rate per inhaled inspirable

droplet8.9 Â 1023 per day calculated

b c transmission rate per settled pathogen 1.13 Â 1025 per day calculatedb ˜ c transmission rate per settled droplet 2.37 Â 1028 (d 1), 2.53 Â 1025 (d 2) per

daycalculated

k droplet production rate 1.06 Â 105 (d1), 1.77 Â 105 (d 2) perday

Nicas et al. (2005)

u gravitational settling rate 94.40 (d 1), 9.523 Â 103 (d 2) per day Hinds (1999)

epidemic mediated by settled inspirable droplets *t c 1 ¼ t c 2 characteristic hand-to-face contact time 20 s estimated

b c transmission rate per settled pathogen 1.49 Â 102

5 per day calculatedb ˜ c transmission rate per settled droplet 3.13 Â 1028 (d 1), 3.33 Â 1025 (d 2 )

per daycalculated





and

dR

dt ¼ mI I ; ð3:1e Þ

where we introduced the droplet decay times

nÀ1d 1

¼ ð1 þ c t d 1 ÞB

V clq d 1 ðd 1Þ þ md þ u 1; ð3:2a Þ

nÀ1c 1

¼ ð1 þ c t c 1 Þh q c 1 ðd 1Þ þ mc; ð3:2bÞ

nÀ1d 2

¼ ð1 þ c t d 2 ÞB

V clq d 2 ðd 2Þ þ md þ u 2; ð3:2c Þ

and nÀ1c 2

¼ ð1 þ c t c 2 Þh q c 2 ðd 2Þ þ mc: ð3:2d Þ

In our simulations, we chose initial conditions S (0)¼103, I (0)¼ 1, D 1(0)¼D 2(0)¼C 1(0)¼C 2(0)¼R(0)¼ 0.As mentioned, evaporation terms have been neglected;note, however, that droplet dynamics is determined bythe post-evaporation diameter (e.g. settling rate), whereasthe pathogen load depends on the pre-evaporation

diameter.The basic reproduction number corresponding to

equations (3.1), calculated as suggested by Diekmannet al. (1990), evaluates to (S 0 $ N )

R0 ¼ ~b d 1

k 1

mI

nd 1 þ ~b c 1

k 1u 1

mI

nd 1nc 1 þ ~b d 2

k 2

mI

nd 2

þ ~b c 2

k 2u 2

mI

nd 2nc 2 ; Rd 10 þ Rc 1

0 þ Rd 20 þ Rc 2

0 : ð3:3Þ

The terms k i /mI and k i u i /mI (i ¼ 1,2) are ratios of dro-plet generation rates (settled droplets are onlygenerated by settling airborne droplets in this model)

to the infection recovery rate; they represent thenumber of droplets generated by an infectious individualduring his/her infectivity period. The last equality inequation (3.3) decomposes R0 into the sum of fourterms each one corresponding to a droplet class.For each class the basic reproduction number is theproduct of droplet infectivity, number of droplets gener-ated during the infectious period, and average dropletdecay time. A similar decomposition was proposedby Li et al. (2009) using a different model. This decompo-sition of R0 allows the quantification of the contributionof each droplet class to the spreading of an epidemic;we shall use it to identify dominant transmission

modes in §4.

3.3. Model parameters

In our model the transmission rates per droplet~b i ; ~c i ~pi ¼ b i q i ðd ÞN pðd Þði ¼ d ; c Þ that depend on thetransmission rates per pathogen b d ; ~c dpd ¼ cB t dpd=V cl

or b c ; ~c cpc ¼ c ht cpc are derived quantities. The par-ameters required for their determination are presentedin table 1.

The average contact rate c is taken to be 13 contactsper day (Mossong et al. 2008). We take the probabilityof infection per inhaled pathogen to be pd ¼ 0.052 and

per pathogen originating from settled droplets pc ¼

6.93 Â 1025, as suggested by Li et al. (2009). Fromthe same source, we take the pathogen concentrationin the lung fluid to be r p¼ 3.71 Â 106 pathogens

cm23 to obtain the number of pathogens per droplet-size class N p(d i ). The average breathing rate perperson B is approximately 1 m3 h21 (Hinds 1999).The volume of the personal cloud of an infectedperson V cl is estimated to be 2 m Â 2 m Â 2 m ¼ 8 m3.The removal rate of settled droplets (per person) h isrelated to the rate a susceptible touches the face, e.g.

nose picking, eye rubbing. It may be estimated fromthe number of hand contacts with a surface and self-inoculation, approximately 15 per hour (Nicas & Best2008), assuming that these contacts occur for16 h d21, and that the probability a surface-to-handcontact leads to the face of a susceptible is 0.3 (Liet al. 2009); this evaluates to h ¼ 72 per day.

The average deposition probability in the respiratorytract is taken to be q d 1 ðd 1Þ ¼ 0:88 for small respirabledroplets and q d 2 ðd 2Þ ¼ 0:99 for large respirable or inhal-able droplets (Hinds 1999; Drossinos & Housiadas2006). The deposition probability of settled dropletsonto facial surfaces is taken to be q c 1 ðd 1Þ ¼

q c 2 ðd 2Þ ¼ 0:35, an estimate based on the efficiency of pathogen transfer from hand to face (Li et al. 2009).The transmission rates depend crucially on the

characteristic time scales of breathing at contact t d orthe hand-to-face contact time t c. We consider threepossible epidemic scenarios distinguished by differentcharacteristic time scales. These scenarios are discussedin detail in §4. We first consider an epidemic mediatedby respirable droplets. The characteristic times of breathing and hand-to-face contact are taken to beindependent of size, the breathing time during contactestimated to be t d 1 ¼ t d 2 ¼ 20 min, in agreement withMossong et al. (2008), who examined number and dur-

ation of susceptible–infected person contact rates. Thecharacteristic hand-to-face contact time is estimated byrequiring that it is smaller than t d and that it reflectsthe high inactivation rate of influenza virus on hands(a process not explicitly included in the model). Sincevirus survival on hands is of the order of a minute oreven less (Bean et al. 1982; Schurmann & Eggers1983) we take t c 1 ¼ t c 2 ¼ 15 s. As a second scenariowe consider an influenza epidemic mediated by inhal-able droplets. The breathing time associated withinspirable droplets is much smaller since inspirable dro-plets settle very fast, remaining close to the infected;they are inhaled in the first breath after generation.

We, thus, take t d 1 ¼ 20 min; t d 2 ¼ 1 min retaining thehand-to-face contact time scales t c 1 ¼ t c 2 ¼ 15s. Thethird scenario is an epidemic mediated by settled inspir-able droplets: we retain the breathing time scales of thesecond scenario, t d 1 ¼ 20 min and t d 2 ¼ 1 min, but weincrease the hand-to-face contact time, a quantity diffi-cult to quantify, to t c 1 ¼ t c 2 ¼ 20s. Table 1 summarizesthe pathogen and droplet transmission rate for all threescenarios.

The infectivity period varies, depending on the sever-ity of the infection. Although it differs for subclinicallyinfected persons and persons with clinical symptoms, aperiod of 2–6 days is considered common in the natural

history of influenza (Carrat et al. 2008). We chose anaverage of 5 days (mI ¼ 0.20 per day).

The droplet settling rates u were taken from Hinds(1999) and Drossinos & Housiadas (2006).





The inactivation rates md and mc, possibly droplet-

size dependent, are taken to be independent of size.Early aerosol studies on influenza survival showedthat the inactivation rate of influenza virus is on aver-age approximately 8.64 per day (Hemmes et al. 1960).Virus survival studies on surfaces estimate influenzasurvival between 30 min and 8 h depending on thetype of the surface (Bean et al. 1982). We chose aninactivation rate md¼ 8.64 per day of virus in airbornedroplets, and an inactivation rate mc ¼ 2.88 per day of virus in settled droplets.

4. RESULTS

We explored the dynamics of three model influenzaepidemic scenarios identified by their dominanttransmission mode.

4.1. An influenza epidemic mediated by airborne respirable droplets

Figure 1 describes the model dynamics of an epidemicwave for an influenza outbreak lasting about 150 daysusing the parameters presented in table 1. The modelreproduces the characteristic transmission dynamics of influenza by showing a typical epidemic wave of infected persons and the decrease of the susceptible

population. The basic reproduction number R0¼ 1.28is close to values empirically estimated for influenza(e.g. Wallinga et al. 2006; Yang et al. 2009). The epi-demic starts right after initial contacts between

susceptibles and virus-containing droplets shed by

infected (source) persons. The number of infectedpersons increases, reaching a maximum after about78 days, decreasing afterwards. The number of suscepti-bles decreases monotonically, reaching a steady stateafter about 150 days. Approximately 44 per cent of the population gets infected. The number of dropletsexhibits a dynamical behaviour similar to the dynamicsof infected persons. All classes peak at day 78, but theirnumbers differ significantly. Small settled and air-borne droplets are much more numerous than largeairborne and settled droplets.

The relative importance of each mode is also quanti-fied in figure 1d , where the cumulative number of

infected persons by each mode ( proportional to the cor-responding force of infection) is presented. Smallairborne respirable droplets are the dominant trans-mission mode, followed by large airborne, and lastlyby settled respirable droplets whether small or large.Even though small respirable droplets have low infectiv-ity per droplet ( proportional to the droplet volume)they have high impact in viral transmission becausethey are numerous and they remain airborne for longperiods. Large respirable droplets are generated at alower rate but their infectivity is high; however, theysettle faster, becoming of secondary importance as atransmission mode. The infectivity of settled droplets,

irrespective of whether they originate from small orlarge respirable droplets, depends strongly on theduration of contact between a pathogen and a suscep-tible, and therefore on the pathogen inactivation rate

0

200

400

600

800

1000(a) (b)

(c) (d )

n u m b e r o f S ,

I ,

R

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

n u m b e r o f a i r b o r n e d r o p l e t s

( × 1 0 5 )

0 50 100 150

0.5

1.0

1.5

2.0

2.5

3.0

3.5

time (days)

n u m b e r o f s e t t l e d d r o p l e t s

( × 1 0 5 )

0 50 100 150

100

200

300

400

time (days)

c u

m u l a t i v e i n c i d e n c e

Figure 1. Model dynamics of an influenza outbreak mediated by airborne respirable droplets. ( a ) Number of susceptible (solidline), infected (dotted) and recovered persons (dashed). (b) Number of airborne small (solid) and airborne large (dashed) respir-able droplets. (c ) Number of settled small (solid) and settled large (dotted) respirable droplets. ( d ) Cumulative number of infections attributable to a transmission mode: airborne small (solid) and airborne large (dashed) droplets; number of infectionsattributable to settled respirable droplets (total number) not visible.





on the hands. Experimental studies suggest that therate is high, rendering contact a transmission mode of negligible importance (Bean et al. 1982; Schurmann &Eggers 1983). It should be noted though that estimatesof the pathogen inactivation rate on the hands are rare,partially inconsistent and differ by several orders of magnitude (e.g. Boone & Gerba 2007; Weber &Stilianakis 2008), indicating the need for consistentexperimental work.

The relative importance of transmission modes mayalso be estimated by comparing the relative contri-bution of each droplet class to R0. They evaluate toRd 1

0 ¼ 1:11; Rc 10 $ 10À4; Rd 2

0 ¼ 0:17 and Rc 20 $ 10À4,

confirming that the dominant transmission mode issmall airborne droplets (Rd 10 ) followed by large airborne

droplets(Rd 20 ). The contribution of settled droplets is

almost non-existent. This scenario suggests that aninfluenza epidemic lasting a few months may bedominated by airborne transmission of respirabledroplets.

4.2. An influenza epidemic mediated by airborne inspirable droplets

The possible contribution of inspirable droplets is inves-tigated in this and the following model scenarios. Even

though inspirable droplets remain airborne for a veryshort time, settling rapidly, they may be importantfor close-contact transmission due to their high patho-gen load (large volume). In addition, settled inspirable

droplets may be transferred to facial tissues to contrib-ute significantly to increased transmission by contact,again due to their pathogen load.

The difference between this scenario and the pre-vious one is, apart from the droplet sizes, thebreathing time associated with inspirable droplets(t d 2 ¼ 1 min). As argued, we take the inspirable-dropletbreathing time to be shorter than the respirable-dropletbreathing time at contact (t d 2 , t d 1 ).

The simulation results show an epidemic character-ized by faster dynamics with a peak on day 27 andR0¼ 2.31. This type of dynamics is common in closedpopulations such as schools, nursing homes, etc.

(Stilianakis et al. 1998; Nishiura et al. 2009). The hightransmissibility of inspirable droplets results in astrong epidemic wave with a higher attack rate (87%).Furthermore, the relative importance of transmissionmodes changes. The cumulative incidence, figure 2d ,shows that the dominant transmission modes areattributable to inspirable droplets, first to airbornedroplets (Rd 2

0 ¼ 0:82) and then to settled ones(Rc 2

0 ¼ 0:79). Large respirable droplets also contributeto the spreading of the epidemic (Rd 1

0 ¼ 0:70), but notsettled respirable droplets (Rc 1

0 $ 10À4).

4.3. An influenza epidemic mediated by settled inspirable droplets

Settled droplets provide an important transmissionmode if the hand-to-face characteristic contact time is

0

200

400

600

800

1000

n u m b e r o f S ,

I ,

R

0

0.5

1.0

1.5

2.0

2.5

n u m b e r

o f a i r b o r n e d r o p l e t s

( × 1 0 5 )

0 20 40 60 80 100

24

6

8

10

12

14

time (days)

n u m b

e r o f s e t t l e d d r o p l e t s

( × 1 0 5 )

0 20 40 60 80 100

100

200

300

400

time (days)

c u m u l a t i v e i n c i d e n c e

(a) (b)

(c) (d )

Figure 2. Model dynamics of an influenza outbreak mediated by airborne inspirable droplets. (a ) As in figure 1. (b) Numberof airborne respirable (solid line) and airborne inspirable (dashed line) droplets. (c ) Number of settled respirable (dotted line)and settled inspirable (solid line) droplets. (d ) Cumulative number of infections attributable to airborne respirable (solidline), airborne inspirable (dashed line) and settled inspirable (dotted line) droplets; number of infections attributable to settledrespirable droplets not shown.





increased from t c¼ 15 to 20 s. The epidemic wave unfoldsvery fast with a peak at day 23 and lasts less than twomonths. The basic reproduction number R0¼ 2.57 ishigh but typical of outbreaks in closed populations. The

dominant contribution to transmission modes arisesfrom settled inspirable droplets (Rc 20 ¼ 1:05), followed

by airborne inspirable droplets (Rd 20 ¼ 0:82) and by air-

borne respirable droplets (Rd 10 ¼ 0:70). As in all three

scenarios considered herein settled respirable dropletsdo not constitute a transmission mode (Rc 1

0 $ 10À3).These numerical results indicate that contact throughsettled droplets as a dominant mode of transmission isa dynamically possible scenario too. Comparison of thelast two scenarios suggests that a very fast epidemic ischaracterized by the presence of three modes of trans-mission (except for settled respirable droplets) of varying relative importance.

5. DISCUSSION

The aim of this work was to develop an epidemiologicalmodel of infection by inhalable respiratory droplets able(i) to describe the dynamics of transmission by explicitconsideration of the vector, taken to be respirable (dro-plet diameter d 10 mm), inspirable (10 mm , d ,100 mm), or the corresponding settled droplets; (ii) totreat aerosol physical processes, such as release and per-sistence in the air, gravitational settling, evaporation,and, indirectly, regional deposition in the respiratorytract, and to couple the associated droplet dynamics

to biological processes at the population level; and(iii) to provide a theoretical framework for the investi-gation of the relative importance of transmissionmodes of respiratory infections, such as influenza, and

the associated control strategies. A decomposition of the basic reproduction number into the contributionof each droplet-size class allowed a quantitative assess-ment of the relative contribution of different modes to

overall transmission. The model does not incorporateinfectious disease transmission by droplet transmission(also referred to as droplet-spray transmission) associ-ated with close expiratory events. It does, however,include transmission at close contact by inhalation of inspirable droplets.

The following three model influenza epidemic scen-arios, differing in the characteristic times associatedwith breathing at contact and the hand-to-face contacttime, were considered: an epidemic mediated only byairborne respirable droplets (long characteristic timescales), one mediated by airborne inspirable droplets(shorter breathing contact time), and one by settled

inspirable droplets (longer hand-to-face contact time).Epidemics mediated predominantly by inspirabledroplets, either airborne or settled, are characterizedby fast dynamics (short-term epidemic) as observed inclosed populations; close contacts are common andtransmission modes such as inhalation of airborneinspirable droplets or contact with settled inspirabledroplets provide a selective advantage. Long-term epi-demics with low attack rates may be attributed torespirable droplets. Hence, numerical results suggestthat epidemic duration is inversely proportional to thepathogen load of the droplet-size class associated withthe dominant transmission mode.

Model results, and in particular the relative impor-tance of transmission modes, depend on parametersthat are, in general, difficult to estimate and for whichexperimental evidence is conflicting. Parameters such

0

200

400

600

800

1000

n u

m b e r o f S ,

I ,

R

0

0.5

1.0

1.5

2.0

2.5

3.0

n u m b e r

o f a i r b o r n e d r o p l e t s

( × 1 0 5 )

0 20 40 60 80 100

5

10

15

time (days)

n u m

b e r o f s e t t l e d d r o p l e t s

( × 1 0 5 )

0 20 40 60 80 100

100

200

300

400

time (days)

c u

m u l a t i v e i n c i d e n c e

(a) (b)

(c) (d )

Figure 3. Model dynamics of an influenza outbreak mediated by settled inspirable droplets. Captions are as in figure 2.





as number distribution of expelled respiratory dropletsand their pathogen load, pathogen inactivation rateson hands and surfaces, and pathogen removal ratesfrom surfaces determine the speed and extent of an epidemic, modify the relative importance of trans-mission modes, and determine the severity of theepidemic. As such, they may have dramatic impli-

cations for the assessment of the importance of transmission modes, and far-reaching implications forcontrol strategies.

Nevertheless, we note that a proper characterizationof all physical and microbiological parameters is notnecessarily needed. As model results provide infor-mation on population dynamics, such as observedduration of outbreaks, incidence, etc., the model canalso be used to help quantify model parameters. Inthis way model-predicted information on populationdynamics could complement some of the more difficultmicro-scale quantifications, although it is essential thatconsistency with plausible ranges of the unknown

micro-scale parameters is required.The model can be used to provide an initialassessment of the impact of control strategies thatblock a specific mode of transmission. It may be usedto evaluate control strategies that are based either onthe use of masks with several levels of protection fordifferent droplet sizes or on interventions with combi-nations of measures, such as the use of masks andantivirals.

We thank Balint Alfoldy, Lorenzo Isella, Thomas P. Weberand Dieter Schenzle for useful discussions.

APPENDIX A

Let p d be the probability that a contact with a dropletresults in successful disease transmission: the prob-ability that transmission does not occur is 1 2 p d. Forhomogeneously mixed populations, and by associatinga personal cloud with each infected individual, thenumber of contacts of a susceptible with a pathogen-carrying inhalable droplet is c ˜ dD d t /N with c d¼ cB t d/V cl. Since contacts are independent the probabilitythat a susceptible escapes infection by any of these con-tacts with infected droplets during d t is (12 p d)c ˜ dD d t /N .Therefore, the probability d q that transmissionoccurs is

d q ¼ 1 À ð1 À ~pdÞ~c dD d t =N :

For an infinitesimal time interval, d t 0, the prob-ability becomes d q ¼ 2c dD ln(1 2 p d)d t /N , and for asmall probability of transmission ( p d 0) the overalltransmission probability becomes

d q ¼ ~c d ~pd

D

N d t for d t 0; ~p 0;

or, equivalently, the transmission rate per susceptibleis

dq

dt ¼ ~c d ~pd

D

N :

Hence, the total rate of transmission to all

susceptibles is

dS

dt ¼ À

1

N ~b dDS ; ðA 1Þ

with ~b d ¼ ~c d ~pd the transmission rate. As expectedequation (A 1) shows that airborne-disease trans-mission is frequency dependent, i.e. it is independent

of population size. Such a description is appropriatefor transmission where contacts are determined bysocial constraints, as, for example, for influenza trans-mission (Keeling & Rohani 2008).

REFERENCES

Atkinson, M. P. & Wein, L. M. 2008 Quantifying the routes of transmission for pandemic influenza. Bull. Math. Biol. 70,820–867. (doi:10.1007/s11538-007-9281-2)

Bean, B., Moore, B. M., Sterner, B., Peterson, L. R., Gerding,D. N. & Balfour, H. H. 1982 Survival of influenza virus onenvironmental surfaces. J. Inf. Dis. 146, 47–51.

Boone, S. A. & Gerba, C. P. 2007 Significance of fomites inthe spread of respiratory and enteric viral disease. Appl.

Environ. Microb. 73, 1687–1696. (doi:10.1128/AEM.02051-06)

Brankston, G., Gitternman, L., Hirji, Z., Lemieux, C. &Gardam, M. 2007 Transmission of influenza A in humanbeings. Lancet Infect. Dis. 7, 257–265. (doi:10.1016/S1473-3099(07)70029-4)

Carrat, F., Vergu, E., Ferguson, N. M., Lemaitre, M.,Cauchemez, S., Leach, S. & Valleron, J. 2008 Time linesof infection and disease in human influenza: a review of volunteer challenge studies. Am. J. Epidemiol. 167,775–785. (doi:10.1093/aje/kwm375)

Chao, C. Y. et al . 2009 Characterisation of expiration air

jets and droplet size distributions immediately at themouth opening. J. Aerosol Sci. 40, 122–133. (doi:10.1016/ j.jaerosci.2008.10.003)

Diekmann, O., Heesterbeek, J. A. P. & Metz, J. A. J. 1990On the definition and the computation of the basic repro-duction ratio R0 in models for infectious diseases inheterogeneous populations. J. Math. Biol. 28, 365–382.(doi:10.1007/BF00178324)

Drossinos, Y. & Housiadas, C. 2006 Aerosol flows. In Multi-

phase flow handbook (ed. C. T. Crowe), pp. 6-1– 6-58.Boca Raton, FL: CRC Press.

Duguid, J. P. 1946 The size and duration of air-carriage of respiratory droplets and droplet-nuclei. J. Hyg. 4, 471–480. (doi:10.1017/S0022172400019288)

Edwards, D. A., Man, J. C., Brand, P., Katstra, J. P.,Sommerer, K., Stone, H. A., Nardell, E. & Scheuch, G.2004 Inhaling to mitigate exhaled bioaerosols. Proc. Natl.

Acad. Sci. USA 101, 17383–17 388. (doi:10.1073/pnas.0408159101)

Eichenwald, H. F., Kotsevalov, O. & Fasso, L. A. 1960 Thecloud baby: an example of bacterial-viral interaction.Am. J. Dis. Child. 100, 161–173.

Gardam, M. & Lemieux, C. 2007 Transmission of influenza Ain human beings, Authors’ reply. Lancet Infect. Dis. 7,761–763. (doi:10.1016/S1473-3099(07)70271-2)

Hemmes, J., Winkler, K. & Kool, S. 1960 Virus survivalas a seasonal factor in influenza and poliomyelitis. Nature 188, 430–431. (doi:10.1038/188430a0)

Hinds, C. W. 1999 Aerosol technology , 2nd edn. New York,

NY: Wiley.Keeling, M. J. & Rohani, P. 2008 Modeling infectious diseases

in humans and animals . Princeton, NJ: PrincetonUniversity Press.



http://dx.doi.org/doi:10.1007/s11538-007-9281-2



http://dx.doi.org/doi:10.1128/AEM.02051-06




http://dx.doi.org/doi:10.1016/S1473-3099(07)70029-4



http://dx.doi.org/doi:10.1093/aje/kwm375





http://dx.doi.org/doi:10.1016/j.jaerosci.2008.10.003




http://dx.doi.org/doi:10.1007/BF00178324



http://dx.doi.org/doi:10.1017/S0022172400019288



http://dx.doi.org/doi:10.1073/pnas.0408159101







http://dx.doi.org/doi:10.1038/188430a0



















Knight, V. 1980 Viruses as agents of airborne contagion. Ann.

N.Y. Acad. Sci. 353, 147–156. (doi:10.1111/ j.1749-6632.1980.tb18917.x)

Li, S., Eisenberg, J. N. S., Spicknall, I. H. & Koopmann, J. S.2009 Dynamics and control of infections transmitted fromperson to person through the environment. Am. J.Epidemiol . 170, 257–265. (doi:10.1093/aje/kwp116)

Loudon, R. G. & Roberts, R. M. 1967 Relation between the

airborne diameters of respiratory droplets and the diam-eter of the stains left after recovery. Nature 213, 95–96.(doi:10.1038/213095a0)

Morawska, L. 2006 Droplet fate in indoor environments, orcan we prevent the spread of infection? Indoor Air 16,335–347. (doi:10.1111/ j.1600-0668.2006.00432.x)

Morawska, L., Johnson, G. R., Ristovski, Z. D., Hargreaves,M., Mengersen, K., Corbett, S., Chao, C. Y. H., Li, Y. &Katoshevski, D. 2009 Size distribution and sites of originof droplets expelled from the human respiratory tractduring expiratory activities. J. Aerosol Sci. 40, 256– 269.(doi:10.1016/ j.jaerosci.2008.11.002)

Mossong, J. et al . 2008 Social contacts and mixingpatterns relevant to the pread of infectious diseases.

PLoS Med. 5, 381–391. (doi:10.1371/ journal.pmed.0050074)Nicas, M. & Best, D. 2008 A study quantifying hand-to-face

contact rate and its potential application to predictingrespiratory tract infection. J. Occup. Environ. Hyg. 5,347–352. (doi:10.1080/15459620802003896)

Nicas, M. & Sun, G. 2006 An integrated model of infectionrisk in a health-care environment. Risk Anal. 26, 1085–1095. (doi:10.1111/ j.1539-6924.2006.00802.x)

Nicas, M., Nazaroff, W. W. & Hubbard, A. 2005 Towardsunderstanding the risk of secondary airborne infection:emission of respirable pathogens. J. Occup. Environ. Hyg.2, 143–154. (doi:10.1080/15459620590918466)

Nishiura, H., Castillo-Chavez, C., Safan, M. & Chowell, G.

2009 Transmission potential of the new influenzaA(H1N1) virus and its age-specificity in Japan. Euro

Surveill. 14, pii: 19227.Papineni, R. S. & Rosenthal, F. S. 1997 The size distribution

of droplets in the exhaled breath of healthy humansubjects. J. Aerosol Med. 10, 105–116. (doi:10.1089/ jam.1997.10.105)

Schurmann, W. & Eggers, H. J. 1983 Antiviral activity of analcoholic hand disinfectant. Comparison of the in vitro

suspension test with in vivo experiments on hands, and

on individual fingertips. Antiviral Res. 3, 25–41. (doi:10.1016/0166-3542(83)90012-8)

Sherertz, R. J., Reagan, D. R., Hampton, K. D., Robertson,K. L., Streed, S. A., Hoen, H. M., Thomas, R. & Jack Jr,G. M. 1996 A cloud adult: the Staphylococcus aureus -virus interaction revisited. Ann. Intern. Med. 124,539–547.

Stilianakis, N. I., Perelson, A. S. & Hayden, F. G. 1998 Emer-

gence of drug resistance during an influenza epidemic:insights from a mathematical model. J. Infect. Dis. 177,863–873.

Tang, W. J. & Li, Y. 2007 Transmission of influenza A inhuman beings. Lancet Infect. Dis. 7, 758. (doi:10.1016/S1473-3099(07)70268-2)

Tellier, R. 2006 Review of aerosol transmission of influenza Avirus. Emerg. Infect. Dis. 12, 1657–1662.

Tellier, R. 2007 Transmission of influenza A in human beings.Lancet Infect. Dis. 7, 759–760. (doi:10.1016/S1473-3099(07)70269-4)

Vaille Lee, R. 2007 Transmission of influenza A in humanbeings. Lancet Infect. Dis. 7, 760–761. (doi:10.1016/S1473-3099(07)70270-0)

Wallace, L. 1996 Indoor particles: a review. Air and Waste Manage. Assoc. 46, 98– 126.Wallace, L. 2000 Correlations of personal exposure to particles

with outdoor air measurements: a review of recent studies.Aerosol Sci. Technol. 32, 15 –25. (doi:10.1080/027868200303894)

Wallinga, J., Teunis, P. & Kretzschmar, M. 2006 Using dataon social contacts to estimate age-specific transmissionparameters for respiratory-spread infectious agents.Am. J. Epidemiol. 164, 936–944. (doi:10.1093/aje/kwj317)

Weber, T. P. & Stilianakis, N. 2008 Inactivation of influenzaA viruses in the environment and modes of transmission:a critical review. J. Infect. 67, 361–373. (doi:10.1016/ j.

jinf.2008.08.013)Yang, S., Grace, W., Lee, M., Chen, C. M., Wu, C. C. & Yu,K. P. 2007 The size and concentration of droplets gener-ated by coughing in human subjects. J. Aerosol Med. 20,484–494. (doi:10.1089/ jam.2007.0610)

Yang, Y., Sugimoto, J. D., Halloran, M. E., Basta, N. E.,Chao, D. L., Matrajt, L., Potter, G., Kenah, E. & LonginiJr, I. M. 2009 The transmissibility and control of pandemicinfluenza A (H1N1) virus. Science 326, 729–733. (doi:10.1126/science.1177373)



http://dx.doi.org/doi:10.1111/j.1749-6632.1980.tb18917.x




http://dx.doi.org/doi:10.1093/aje/kwp116








http://dx.doi.org/doi:10.1111/j.1600-0668.2006.00432.x






http://dx.doi.org/doi:10.1371/journal.pmed.0050074




http://dx.doi.org/doi:10.1080/15459620802003896









http://dx.doi.org/doi:10.1089/jam.1997.10.105





http://dx.doi.org/doi:10.1016/0166-3542(83)90012-8

















http://dx.doi.org/doi:10.1093/aje/kwj317





http://dx.doi.org/doi:10.1016/j.jinf.2008.08.013




http://dx.doi.org/doi:10.1089/jam.2007.0610



http://dx.doi.org/doi:10.1126/science.1177373


































dynamics of infectious dse.pdf

Documents