Superspreading and the impact of individual variation on disease emergence

Transcription

1 Superspreading and the impat of individual variation on disease emergene Supplementary Information J.O. Lloyd-Smith,2, S.J. Shreiber 3, P.E. Kopp 4, W.M. Getz Department of Environmental Siene, Poliy & Management, 4 Mulford Hall, University of California, Berkeley CA Biophysis Graduate Group, University of California, Berkeley CA Department of Mathematis, The College of William & Mary, Williamsburg VA Centre for Mathematis, University of Hull, Hull HU6 7RX, United Kingdom

2 Table of Contents Table of Contents... 2 Additional Supplementary Materials Disussion Fators ontributing to variation in infetiousness Methods Candidate models for the offspring distribution Data analysis Parameter estimation and model seletion from full datasets Parameter estimation from mean and proportion of zeros Testing for deviation from Poisson homogeneity Confidene intervals for k Expeted proportions of transmission Superspreading events (SSEs) Dynami modelling Branhing proess model and analysis Branhing proess simulations Analysis of disease ontrol Control poliies theoretial framework Relative effiay of ontrol poliies Control poliies simulations Data Notes on outbreak and surveillane datasets SARS, Singapore SARS, Beijing Measles, US Measles, Canada Smallpox (Variola major), Europe Smallpox (Variola major), Benin Smallpox (Variola major), West Pakistan Smallpox (Variola major), Kuwait Smallpox (Variola minor), England Monkeypox, Zaire Pneumoni plague (Yersinia pestis), 6 outbreaks Hantavirus (Andes virus), Argentina Ebola Hemorrhagi Fever, Uganda Rubella, Hawaii Survey of superspreading events (SSEs) Superspreading events in the published literature Referenes

3 Additional Supplementary Materials The following materials are available as separate files from the Nature website: Supplementary Table A summary of results from our statistial analysis of unontrolled outbreaks, orresponding to the results shown in Figure a- of the main artile. Supplementary Table 2 Detailed results from our statistial analysis of unontrolled outbreaks (elaborating on the summary shown in Supplementary Table ), and from the analysis of data from four outbreaks before and after ontrol measures were applied. Supplementary Figures Supplementary Figure. Predition of SSE frequeny. Supplementary Figure 2. Branhing proess results for Z~NegB(R,k). Supplementary Figure 3. Impat of ontrol measures. Supplementary Figure 4. Estimation of the negative binomial dispersion parameter k from full datasets and from mean and proportion of zeroes. 3

4 . Supplementary Disussion. Fators ontributing to variation in infetiousness Here we summarize some of the known fators that ontribute to differenes in infetiousness among individuals, gathered from primary reports (inluding the SSE reports olleted in Setion 3.2., below) and from insightful disussions in the literature -8. This is a broad and omplex topi and we do not intend this setion as a omplete review we intend simply to delineate important issues and spur further researh, whih will be required to make pratial use of the findings presented in the main text, partiularly with regard to targeting moreinfetious individuals for ontrol. Variation in individual reprodutive number arises due to a ombination of host, pathogen and environmental effets. At the host level, distributions of ontat rates are often skewed 9-3 and index ases in SSEs are often noted to have high numbers of oupational or soial ontats 7,,4. Inreased transmission is orrelated with host ativities that failitate pathogen dispersion, suh as food handling 5 and singing 6,7. Transmission rates an exhibit strong agedependene,8, and previously vainated hosts often are less infetious 9,2. A reent experimental study doumented substantial variation among human hosts in the amount of exhaled bioaerosols (small droplets of airway-lining fluid) generated during normal breathing, suggesting a mehanism for variation in infetiousness for droplet- or aerosol-transmitted pathogens 2. (This study also demonstrated a potential means to redue infetiousness by altering airway surfae properties using inhaled saline solution.) Other relevant host fators may inlude hygiene habits, immunoompetene, norms regarding bodily ontat, and tendeny to seek treatment or omply with ontrol measures. Host-pathogen interations affet transmission rates via variation in pathogen load or shedding 5,2 and in symptom severity (whih may inrease transmission via greater shedding or derease transmission due to redued ontat rate,5,9,2 ). Severe oughing, due either to pulmonary involvement of the disease in question 22,23 or to oinfetions with other respiratory pathogens 2,24, is often linked to SSEs with suspeted airborne transmission. A series of observational and experimental studies has doumented the potential for upper respiratory trat infetions (with a respiratory virus, e.g. rhinovirus or adenovirus) to onvert nasal arriers of Staphyloous aureus into highly infetious loud patients, so-alled beause they are surrounded by louds of aerosolized bateria This mehanism has been proposed to underlie some SARS SSEs 29 a proposal that is untested, although generation of viral aerosols by a patient with SARS has been demonstrated so the potential for airborne spread exists 3,3. At the pathogen level, evolution of highly-transmissible pathogen strains is possible, but should lead to observable orrelations in Z within transmission hains if enough generations of uninterrupted transmission are traed losely (rarely the ase in any non-experimental system). An open question is the extent to whih pathogen biology influenes the different degrees of heterogeneity observed here. Environmental fators have a strong influene on transmission. Crowded or onfined settings suh as shools 32,33, nightlubs 7, markets 34, and airplanes 23 often lead to multiple infetions, as an funerals 35,36 and hospitals,37,38 for virulent diseases. Other important environmental fators are the suseptibility of an individual s ontats, due to age, illness, or lak of (suessful) vaination 9,2,39, and the state of medial knowledge, partiularly for a novel disease suh as SARS for whih misguided proedures and missed diagnoses are inevitable 4. The delay before an infetious patient is isolated is an important determinant of individual infetiousness 4, and is influened by auray of diagnosti riteria, publi health resoures, 4

5 severity of symptoms, and omorbid onditions,38,4. Imperfet disease ontrol measures an inrease variation in, if transmission is onentrated in a few missed ases or pokets of unvainated individuals,2,34,37,42. We emphasize that all of these host, pathogen and environmental fators join to omprise a ase s infetious history, whih in turn ditates the individual reprodutive number. Note that is a property of a given individual s infetious history, rather than a fixed property of the individual, beause an individual s infetiousness may hange with time due to differing irumstanes. 2. Methods 2. Candidate models for the offspring distribution The offspring distribution is the probability distribution for the number of seondary ases Z aused by eah infetious individual. We modelled the offspring distribution using a Poisson proess to represent the demographi stohastiity inherent in the transmission proess 4, with intensity that ould vary to reflet individual variation in infetiousness. The value of for a given individual s infetious history is thus the expeted number of seondary ases they will ause, i.e. their individual reprodutive number. Note that is an expetation and an take any positive real value, while Z is neessarily a non-negative integer (,,2,3, ). Owing to the influene of irumstane on disease transmission, is not neessarily a fixed attribute of eah individual host, but rather is a property of a partiular infetious history for a given host (i.e. the irumstanes throughout that host s infetious period). The offspring distribution is therefore a Poisson mixture 43-47, with mixing distribution given by the population distribution of, i.e. Z~Poisson(). We onsider three distint treatments of the individual reprodutive number, yielding three andidate models for the offspring distribution. To aid disussion of epidemiologial matters, we denote the sale parameter of all offspring distributions by R ; the relation to onventional notation is stated below. (Note that throughout this study, we use the basi reprodutive number R for unontrolled transmission in ompletely suseptible populations, and the effetive reprodutive number R when population immunity or ontrol measures are present. When either measure ould apply, we use R for notational larity.) The three andidate models for the offspring distribution are:. If individual variation is negleted and the individual reprodutive number for all ases is assumed to equal the population mean (=R for all ases), then the offspring distribution is Z~Poisson(R ). 2. In models with onstant per apita rates of leaving the infetious state (by reovery or death), the infetious period is exponentially distributed. If the transmission rate is assumed to be idential for all individuals, then the individual reprodutive number is exponentially distributed (~exponential(/r )). Using this expetation in the Poisson proess representing transmission yields a geometri offspring distribution, Z~geometri(R ) (Note: onventional notation is Z~geometri(p) where p=/(+r ).) 3. To inorporate variation in individual infetious histories (from a range of soures), we introdue a more general formulation in whih follows a gamma distribution with dispersion parameter k and mean R. As shown in Fig. 2a, this inludes =R and 5

6 ~exponential(/r ) as speial ases, and also allows enormous flexibility to fit realworld omplexities (at the expense of an added parameter). A Poisson proess with this gamma-distributed intensity yields a negative binomial offspring distribution with dispersion parameter k and mean R, Z~NegB(R,k) (Note: onventional notation is Z~NegB(p,k) where p = ( + R ) k.) When k= the NegB(R,k) distribution redues to Z~geometri(R ), and when k it redues to Z~Poisson(R ). In all three andidate models, the population mean of the offspring distribution is R. The variane-to-mean ratio differs signifiantly, however, equalling for the Poisson distribution, +R for the geometri distribution, and +R /k for the negative binomial distribution. 2.2 Data analysis The major purpose of our statistial analysis is to assess the empirial evidene for eah of the three andidate models desribed above, for a number of disease datasets. We approah this task using two parallel tehniques. In one approah, we apply maximum likelihood methods to estimate model parameters, then use information-theoreti model seletion to determine whih model is preferred. In a seond approah, we ondut a test for extra-poisson variability (using the Potthoff-Whittinghill statisti 48, related to the variane-to-mean ratio); if the Poisson model is deemed unlikely then we estimate the negative binomial dispersion parameter k for the dataset. Beause the Poisson and geometri models orrespond to speial values of k, then by estimating onfidene intervals on our estimate of k we gain insight into the likelihood that the Poisson or geometri model is supported by the data. Summarized results are given in Supplementary Table, and full results are shown in Supplementary Table 2. Two types of disease datasets were analysed: those with full distributions of Z and those where only the mean value of Z and the proportion of zeros (Z= values) are known. Desriptions of all outbreaks and issues speifi to eah dataset are outlined in Setion 3., below. When full ontat traing information was available, the dataset onsisted of a list of Z values for all infeted individuals prior to the imposition of ontrol measures. Some datasets are omposed of data from several outbreaks merged together, or ombined surveillane data for the first generation of transmission for many disease introdutions. In several surveillane datasets only limited information was available. When the mean number of ases aused per index ase and proportion of index ases that aused no further infetions are known, then the negative binomial parameters an be estimated as desribed below. In some instanes the total number of ases in subsequent generations of an outbreak was also reported, but this information was not used beause we ould not attribute these ases to speifi soures of infetion Parameter estimation and model seletion from full datasets When full datasets were available, model parameters were estimated by the method of maximum likelihood (ML). For the Poisson, geometri and negative binomial models, the ML estimate of the mean of the offspring distribution (i.e. the reprodutive number, R or R) is simply the sample mean 49,5. For the negative binomial distribution, the dispersion parameter k is asymptotially orthogonal to the mean and so is estimated independently after substituting the ML estimate of the mean into the likelihood expression 5,5. 6

7 Estimation of k from finite samples is a hallenging problem and has been the subjet of onsiderable researh This body of work shows that it is better to estimate k indiretly via its reiproal α=/k, as this avoids disontinuities for homogeneous datasets (i.e. inreasing homogeneity yields αø instead of kø ) 5,52,54,55. Furthermore the sampling distribution for α tends to be nearly symmetri 52, allowing a more rapid approah to asymptoti normality (see Fig. SI-). Many studies have employed simulation methods to assess the bias and effiieny of various statistial estimators for the dispersion parameter for finite sample sizes, though regrettably most studies investigating ML estimates have foused on k instead of the parameter range of greatest interest here (k<). Early work onluded that ML estimation has preferable small-sample bias and effiieny properties, and is generally superior (save for its omputational expense, whih is no longer a onern) ompared to the method of moments and other methods of estimating k 5,53. Reent work shows that ML estimates of k have only minor bias (-3%) for sample sizes N 2 and k<2 (values of k< were not tested but the bias appears quite stable for dereasing values; see Fig. b of Saha & Paul 5 ). In all ases where ML estimates of k have been tested by simulation, the bias of small-sample estimates has been to overestimate the true value of k 5,53,55. Gregory & Woolhouse 56 onduted an extensive simulation study of estimating k by the method of moments, inluding appliable parameter ranges (k<), and found a onsistent, larger positive bias in k estimates for small sample size. As they noted, the positive bias in k (i.e. underestimation of heterogeneity) arises beause smaller samples are less likely to inlude the rare extreme values through whih the negative binomial distribution manifests its heterogeneity. We therefore estimated k by applying ML to α=/k, and final values were onverted bak into dispersion parameters k beause this quantity is more familiar to epidemiologists and eologists. ML estimates based on the full distribution of Z are denoted here by k ˆ mle. The termination tolerane on numerial maximization was set suffiiently small that negligible auray was lost in inverting the estimates, and diret ML estimates of k mathed k=/α to beyond the fourth deimal plae exept as k rose toward infinity (and hene needed to be approximated by a large finite value in the diret estimation). We performed goodness-of-fit tests for the negative binomial model (i.e. the global model ) using maximum-likelihood parameter estimates for eah dataset, and in no ase were quasi-likelihood adjustments for overdispersed data required 57. Bootstrap sampling distribution Dispersion parameter, k Bootstrap sampling distribution Reiproal of dispersion parameter, α=/k Figure SI-. Bootstrap sampling distributions for the negative binomial dispersion parameter k and its reiproal α=/k. Distributions of maximum-likelihood estimates of k and α generated by, non-parametri resamples of the pneumoni plague dataset (N=74). 7

8 Having omputed the maximum likelihood sores for eah dataset, we ompared the Poisson, geometri and negative binomial models using Akaike s information riterion (AIC) 57 : AIC = -2 ln( L( ˆ θ data ) + 2K where ln( L( ˆ θ data ) is the log-likelihood maximized over the unknown parameters (θ), given the model and the data, and K is the number of parameters estimated in the model. Beause some of our datasets are small, we used the modified riterion AIC, whih redues to the onventional expression as sample size N beomes larger 57 : ( ( ˆ 2K ) ( K + ) AIC = -2 ln L θ data + 2K + N K We resaled the AIC by subtrating the minimum sore for eah dataset, and present the resulting values AIC. We then alulated Akaike weights w i for eah of the three andidate models: exp( 2 AIC, i ) w i = 3 exp AIC j= ( ). The Akaike weight w i an be interpreted as the approximate probability that model i is the best model of the set of andidate models onsidered, in the sense of ombining aurate representation of the information in the data with a parsimonious number of parameters Parameter estimation from mean and proportion of zeros When surveillane datasets did not inlude full information on the distribution of Z, but inluded the total number of disease introdutions and the number of these that led to no seondary ases, then ˆp, the proportion of primary ases for whom Z=, ould be estimated. If the total number of seond-generation ases is reported 58, then it was divided by the number of introdutions to estimate ˆR. In the studies on measles in the United States and Canada, data were not available to estimate ˆR ourselves so data-derived estimates of ˆR from the original reports were used 42,59. Given estimates of the mean ( ˆR ) and proportion of zeros ( ˆp ) of a negative binomial distribution, the dispersion parameter k an be estimated by solving the equation pˆ = ( + Rˆ k) k numerially 5. We denoted the resulting estimates k ˆ pz. This estimator is known to be less effiient and more biased than the ML estimator 5,53, but to asertain the auray of this method of estimation for our analyses, we ompared k ˆ pz and k ˆ mle for several outbreaks for whih we had full information on Z (Supplementary Fig. 4). The proportion of zeros estimate is quite aurate, partiularly for kˆ <, but is usually slightly higher than k ˆ mle and has a broader onfidene interval. Beause the estimates k ˆ pz were not obtained using ML methods, the AIC approah to model seletion was not appliable. Conlusions regarding these datasets were based entirely on onfidene intervals for k, desribed below. 2, j 8

9 2.2.3 Testing for deviation from Poisson homogeneity A great deal of researh has addressed the statistial question of assessing whether a ount dataset has signifiant deviations from a homogeneous Poisson distribution 44,47,48. After reviewing the performane of numerous possible test statistis 47, we seleted the Potthoff- Whittinghill index of dispersion test, whih is asymptotially loally most powerful against the negative binomial alternative 48. For a dataset X with N elements, this statisti is (N )*var(x)/mean(x) and its asymptotial distribution is hi-squared with N degrees of freedom. A p-value is obtained by determining the umulative density of the hi-squared(n ) distribution to the right of the test statisti, and represents the probability that the observed variane arose by hane from a Poisson distribution Confidene intervals for k Estimation of aurate onfidene intervals for the negative binomial dispersion parameter k estimated from finite samples is a diffiult hallenge. Many applied studies reporting values of k do not report onfidene intervals 6,6 ; those that do typially report a single measure, often the ML sampling variane 62. Beause of the reognized diffiulty of establishing aurate onfidene intervals for k, we adopted the onservative approah of applying multiple independent methods, from fully non-parametri to fully parametri, and evaluating their results in aggregate. Beause the intervals obtained using this suite of methods are very similar, we have onfidene in the reported intervals as approximate ranges of unertainty 63. We hose to report 9% onfidene intervals, sine the more extreme values (needed for, say, a 95% onfidene interval) are most diffiult to estimate aurately. We estimated 9% CIs for k using the following five methods. The first three approahes require a full dataset (i.e. the full observed distribution of Z), while the latter two require only the mean and proportion of zeros. All full datasets were analysed using all five methods, while redued datasets were analysed only using the latter two. See Supplementary Table 2 for these results. (i) Non-parametri bootstrap: Bootstrap datasets were generated by re-sampling with replaement from the original data. For eah bootstrap dataset, the ML estimates of ˆR and ˆ α = kˆ were determined as desribed above, generating a bootstrap sampling distribution. Confidene intervals were onstruted using the bias-orreted perentile method 64,65, beause both parameters are restrited to positive real values and tended to have skewed bootstrap distributions for whih the median of bootstrap estimates did not equal the parameter estimate from the original dataset. (Note that the sampling distribution of α is more symmetri than that of k, but bias-orretion was employed to remove any skew; see Fig. SI-). This method is seond order asymptotially aurate (i.e. the differene between real and desired overage is asymptotially O(/N) for sample size N) for even-tailed two-sided intervals 66, but bootstrap onfidene intervals of asymmetri distributions are still prone to errors in overage 65 so the displayed intervals are intended as approximate ranges of unertainty. We employed, resamples with replaement to generate our simulated bootstrap distributions. Datasets with very few non-zero values of Z generated signifiant proportions of bootstrapped datasets with all zeros. Suh all-zero datasets ontain insuffiient information to estimate kˆ, so when 5% or more of bootstrapped datasets ontained only zero values the bootstrap 9% onfidene interval was undefined. 9

10 (ii) Parametri bootstrap: Bootstrap datasets were generated using a negative binomial random number generator (nbinrnd in Matlab (v6. R3, MathWorks, Cambridge MA)) using the ML parameters estimated from the original data. This approah eliminates the influene of the partiular Z values in the original dataset, allowing for a more ontinuous distribution of Z in the bootstrap datasets, but makes a stronger assumption regarding the mehanism generating the data 64,66. Confidene intervals were generated exatly as for the non-parametri bootstrap datasets. (iii) Maximum-likelihood sampling variane: ML parameter estimates have large-sample variane given by the inverse of the Fisher information matrix, and thus asymptotially approah the Cramer-Rao bound for minimum-variane estimators 49. For the negative binomial dispersion parameter kˆ, or its reiproal αˆ, the asymptoti sampling variane annot be expressed in losed form but is easily alulated numerially 5,5 ; note the relationship 4 ( ˆ ) ( ˆ 52 Var α = k Var k). estimated the 9% onfidene interval for αˆ as [ αˆ z σ, α z ] 2 We alulated the large-sample variane for αˆ, denoted σ ˆα, and αˆ ˆ mle.95 mle +.95σ α ˆ, where z.95 is the 95 th perentile of the standard normal distribution 49. The onfidene interval for kˆ was then generated by inverting these two endpoints. (iv) Large-sample variane of k ˆ pz : The large-sample variane of k ˆ pz has been derived by Ansombe 5 using a general moment method. For all datasets (inluding the full datasets), this quantity was alulated and onfidene intervals generated using the approah outlined in method 3, above. (v) Binomial sampling variane in ˆp : In our final approah, informal inferene on k ˆ pz was performed based on the binomial sampling variability of ˆp, the proportion of infetious ases that ause no transmission. Exat 9% onfidene intervals on ˆp were obtained using the method of Clopper and Pearson 67 ; these intervals are the most onservative of many alternative binomial onfidene intervals, guaranteeing overage of at least 9% and often onsiderably more due to disreteness of the binomial distribution 68. Utilizing the fat that the asymptoti ovariane of ˆR and kˆ is zero 5, the estimate of ˆR (by other means) is taken as a given, and the onfidene interval for kˆ is determined by alulating k ˆ pz for eah endpoint of the onfidene interval for ˆp Expeted proportions of transmission The expeted proportion of transmission due to a given proportion of the population, plotted in Fig. b, was alulated as follows. First we estimated R and k, whih speify the pdf f (x) and df F (x) of the gamma-distribution desribing the individual reprodutive number for a given disease and population. We then alulated the umulative distribution funtion for transmission of the disease:

11 Ftrans ( x) = u f ( u) du R suh that F trans (x) is the expeted proportion of all transmission due to infetious individuals with < x. The expeted proportion of transmission due to individuals with > x is thus -F trans (x), while the proportion of individuals with > x is -F (x). These quantities were plotted parametrially as a funtion of x to make Fig. b. Similarly, the expeted proportion of transmission due to the most infetious 2% of ases, t 2, was alulated by finding x 2 suh that -F (x 2 )=.2, then t 2 =-F trans (x 2 ) (see Fig. ). 2.3 Superspreading events (SSEs) Fators ontributing to superspreading events are reviewed in Setion, above. Case reports orresponding to data in Fig. d are summarized in Setion 3.2. The perentile intervals in Fig. d were generated diretly from the Poisson distribution, with reprodutive numbers drawn from speifi studies of the relevant diseases where possible, or otherwise from ompiled estimates (see Setion 3.2). These latter estimates of R are intended to be indiative only, sine they do not neessarily desribe the same population setting or disease strain as the SSEs in question. Our proposed definition of superspreading events enables predition of the frequeny of SSEs, Ψ, for diseases with different degrees of individual variation (Supplementary Fig. ). One the threshold number of ases Z (99) has been defined for a 99 th -perentile SSE under effetive reprodutive number R, then for any k one an alulate from Z~NegB(R,k) the proportion of individuals Ψ R,k expeted to generate Z>Z (99). (Beause this requires estimates of R and k, real-time estimation of Ψ for an outbreak in progress is subjet to any biases in the available data. It is possible that SSEs will be over-represented in available datasets preisely beause of their important role in early survival of disease invasions when signifiant individual variation exists.) In a homogeneous population (k ), Ψ R,. by definition (where the lessthan arises beause the Poisson distribution is disrete; see below). When heterogeneity is aounted for, Ψ R,k >Ψ R, and varies strongly with both R and k, peaking between k=. and k= for the low R values of interest for emerging diseases. Beause the variane-to-mean ratio is fixed at for the Poisson distribution but inreases linearly with R for the NB model, for moderate k values Ψ R,k inreases strongly with R as the relative density of Z>Z (99) inreases. Note that the proportion of 99 th -perentile SSEs, Ψ Poisson, is often less than %, beause Poisson(R) is a disrete distribution and for arbitrary R there is unlikely to be an integer Z (99) suh that F Poisson(R) (Z (99) ) equals.99 exatly. As a result, the proportion of ases ausing SSEs under the negative binomial model, Ψ R,k, may approah some value less than. as kø. In plotting Supplementary Fig., we hose values of R suh that Ψ Poisson =Ψ R, =. and all plotted lines approahed the same asymptoti value. These values were omputed simply by examining Poisson df s for different R. Preise values of R in Supplementary Fig. are.48,.436,.279, 2.33, 3.57, 6.99,.345, and Note that this effet of the disreteness of the Poisson distribution, while a nuisane in making plots, has little pratial impat in this ontext beause most diseases have k<5 (Supplementary Table ). x

12 2.4 Dynami modelling 2.4. Branhing proess model and analysis We studied the properties of stohasti disease invasions using a single-type branhing proess model, whih allowed us to inorporate individual heterogeneity in infetiousness by varying the offspring distribution. This model of invasion assumes that the supply of suseptible individuals is not limiting for the outbreak, and that the numbers of seondary ases ( offspring ) aused by eah infetious individual are independent and identially distributed. Branhing proess models are summarized in depth elsewhere 69, as are their partiular appliations to modelling disease invasion 4. The heart of a branhing proess model is the offspring distribution, whih desribes the probability distribution of the number of new ases Z aused by eah infetious individual, i.e. it sets p k =Pr(Z=k) for k=,,2,3,. Analysis of branhing proess models enters on the probability generating funtion (pgf) of the offspring distribution, denoted g(s): k g s ( ) = p k s, s k = Two important properties of the epidemi proess follow diretly from g(s). The basi reprodutive number, R, is by definition the mean value of Z, and is equal to g (). The probability that an infetious individual will ause no seondary infetions, p =Pr(Z=), is g(). Thus a great deal an be learned about an outbreak from the y-interept of the pgf and its slope at s=. The n th iterate of the pgf, g n (s), is the pgf of Z n, the number of ases in the n th generation, and is defined as follows: g (s)=s, g (s)=g(s), and g n+ (s)=g(g n (s)) for n=,2,3, 69. The probability that the epidemi has gone extint by the n th generation is thus g n (). We denote the probability of extintion as nø by q, then q is a solution to the equation q=g(q) (from g n+ (s)=g(g n (s)) with nø ), whih from monotoniity and onvexity of g(s) has at most one solution on the interval (,) 69. When R, the only solution to q=g(q) is q= and disease extintion is ertain; when R >, there is a unique positive solution less than one 69. Finally, the pgf for the total number of individuals infeted in all generations of a minor outbreak (i.e. one that goes extint) is defined impliitly as G(s)=sg(G(s)) 69. The expeted size of a minor outbreak is then G (), and an be alulated numerially for a given g(s). For our treatment of the transmission proess, we assume that eah individual s infetious history has an assoiated individual reprodutive number, drawn from some distribution with pdf f v (u). Demographi stohastiity in transmission is then represented by a Poisson proess, as is standard in branhing proess treatments of epidemis 4. This yields the following pgf for a Poisson distribution with mean distributed as f v (u): If is a onstant, R, then the pgf is: g ( s ) u ( s) = e f ( u) du g( s) = e If is exponentially distributed with mean R, the resulting offspring distribution is geometri with mean R and pgf: ( ) g s = + R s R ( s ) ( ( )) 2

13 If is gamma distributed, with mean R and dispersion parameter k, the resulting offspring distribution is negative binomial, also with mean R and dispersion parameter k 43-45, with pgf: R g( s) = + ( s) k This expression was applied in all of the general branhing proess results shown above to derive our results. The expression q=g(q) was solved numerially to generate Fig. 2b and Supplementary Fig. 2b, showing the dependene of the extintion probability on R and k. The negative binomial pgf itself is plotted in Supplementary Fig. 2a, showing how the probability of infeting zero others (p ) inreases sharply with k for a given R. The expeted size of minor outbreaks (Supplementary Fig. 2) was plotted by solving G () numerially for a range of values of R and k. The probability of extintion in the n th generation (Supplementary Fig. 2d) was alulated using g n () g n (). These numerial solutions math the averaged output of many simulations preisely, for R above and below zero, and for kø and kø Branhing proess simulations To assess the growth rate of major outbreaks, a branhing proess epidemi was implemented by simulation, beginning with a single infetious individual (Fig. 2, Supplementary Figs. 2e,f). For eah infetious individual, the individual reprodutive number was drawn from a gamma distribution with hosen values of R and k, using the gamrnd funtion in Matlab (v6. R3, MathWorks, Cambridge MA) adapted to allow non-integer k. The number of seondary ases Z aused by that individual was then determined by drawing a Poisson random variable with mean, using the Matlab funtion poissrnd. Eah individual was infetious for only one generation, and the total number of infeted individuals in eah generation was summed. The first generation to reah ases was used as an arbitrary benhmark of epidemi growth rate. 2.5 Analysis of disease ontrol 2.5. Control poliies theoretial framework We onsider an epidemi that has a natural (i.e. unontrolled) offspring distribution Z~NegB(R,k), from whih we know the probability of infeting zero others is p =(+R /k) k. Under the population-wide ontrol poliy, every individual s infetiousness is redued by a pop fator so their expeted number of seondary ases is redued from to =(-) and the realized number is Z pop ~Poisson((-)). The reprodutive number under ontrol, R pop (denoted R in the main text, for simpliity), equals ( )R. If unontrolled individual reprodutive numbers are gamma-distributed, ~gamma(r,k), then only the sale parameter of the resulting negative binomial distribution is affeted by population-wide ontrol (the dispersion pop pop parameter k is unhanged) and Z ~NegB(( )R,k). The variane-to-mean ratio of Z is +( )R/k, and dereases monotonially as ontrol effort inreases. Under individual-speifi ontrol, eah infeted individual is ontrolled perfetly (suh that they ause zero seondary infetions) with probability. Imposition of individual-speifi ontrol influenes transmission only for the fration p of individuals whose natural Z value is greater than zero of these a fration have Z ind k =, while the remaining fration are 3

14 unaffeted and have Z ind =Z. Under an individual-speifi ontrol poliy, therefore, the ind proportion of ases ausing zero infetions is p = p + ( p ) and the population mean N N ind R = Zi Pr(ase i not ontrolled) = ( ) Zi = ( ) R. The exat distribution N N i= i= ind ind of Z is defined by Pr( Z =)= ind ind p and Pr( Z =j)=( )Pr(Z=j) for all j>, i.e. the ind distribution of Z has an expanded zero lass relative to Z, while for non-zero values its density is simply redued by a fator ( ) from Z~NegB(R,k). Hene, the offspring distribution under individual-speifi ontrol has pgf: k () ind ( ) R g s ( = + + s) k Applying a general result from the theory of branhing proesses 69, the variane-to-mean ratio of g + g g 2 g and shown to equal Z an be alulated from () () () ind ( ind ind ( ind ) ) ind () +R /k+r, whih inreases monotonially as inreases. For diret omparison with other offspring distributions in our analysis, this omposite distribution under individual-speifi ontrol an be approximated by a new negative binomial distribution, Z ind,nb ~NegB( R, k ind ind ) where R is given above and ind k is estimated using ind ind ind the proportion of zeros method as the solution to p p ( p) ( R k ) ind ind k = + = +. ind The approximated dispersion parameter k dereases monotonially as ontrol effort inreases (Fig. SI-2a). This approximation yields better than 95% overlap with the exat distribution for k, and better than 85% overlap for almost all of parameter spae (Fig. SI-2b). (The proportion ind ind,nb of overlap is alulated as Pr( Z = i) Pr( Z = i) 2, whih sales from to i= as the two distributions go from ompletely non-overlapping to idential.) The approximation approahes exatness for Ø and Ø, and is least aurate for large values of k beause it is unable to mimi the bimodal distribution of Z (Fig. SI-2). ind Relative effiay of ontrol poliies For population-wide ontrol, with all individuals transmission redued by a fator, the offspring distribution is Z ~NegB(( )R,k) and has pgf: pop () pop ( R ) ( g s = + s) k. For individual-speifi ontrol, with a random proportion of individuals ontrolled absolutely, the exat pgf (i.e. not the negative binomial approximation) is as given above: k () ind ( ) R g s ( = + + s) k. k 4

15 aapprox. shape parameter, kind.. R = R =3 R = b Proportion ontrolled, Probability density Exat distribution Approximate distribution From bottom to top, k =.,.5,, 3, Control effort,.. Dispersion parameter, k Z Figure SI-2. Negative binomial approximation for individual-speifi ontrol. (a) The approximated dispersion parameter k dereases monotonially as ontrol effort inreases. ind Curves depit unontrolled outbreaks with k= (blue), k= (green), and k=. (red), for R = (solid), R =3 (dotted), and R = (dashed). (b) Auray of the approximation whereby the offspring distribution under random individual-speifi ontrol is represented by a negative binomial distribution, Z ind,nb ~NegB( R, k ind ind ). Contours show the proportion of overlap between the exat and approximated offspring distributions, alulated as desribed in the text. () Exat and approximated negative binomial offspring distributions under individual-speifi ontrol for R =3. From bottom to top, five urves for both the exat and approximate distributions show k=.,.5,, 3, and. Claim: For all œ (,-/R ), the probability of extintion is always greater under individualspeifi ontrol than under population-wide ontrol. ( ) k ( k where X is a Bernoulli random variable with a probability of suess. Sine G is a onvex funtion, Jensen s inequality implies that g () s = G E X < E G X = g () s (*) R Proof of laim: Define G x) = + x ( s) pop ( ( )) ( ) ( ) whenever œ (,) and s œ [,). Furthermore, for the n th iterates of the pgf we have from (*) that gpop, n() < gind, n() so the probability of disease extintion by the n th generation is always greater under individualspeifi ontrol. Thus if œ (,-/R ), the probability of ultimate extintion under individualspeifi ontrol is greater than that under population-wide ontrol, i.e. q > q. If > ind pop pop ind ind pop -/R, then R = R < so that q = q =. That is, the threshold ontrol effort required to assure disease extintion is = -/R (provided individual-speifi ontrol is applied to randomly-hosen individuals). To onsider the effiay of ontrol poliies targeting the more infetious individuals in a population, we onsider a general branhing proess whose pgf is given by ind 5

16 g u( s ) () s = e f ( u) where f v (u) is the pdf of the individual reprodutive number for the outbreak in question. For a ontrol strategy C : [, ) [,] in whih the probability of absolutely ontrolling a ase with individual reprodutive number is C(), the pgf of the branhing proess beomes where g du C u( s ) () s = + e ( C( u) ) f ( u) = C du ( u) f ( u) is the fration of individuals ontrolled on average. For example, random individual-speifi ontrol orresponds to hoosing C()= for all. Maximally-targeted ontrol, in whih the top % of infetious individuals are ontrolled absolutely, orresponds to hoosing if < C( ) = if where satisfies f ( u) du. = Note that when is gamma-distributed with mean R and dispersion parameter k, the pgf under maximally-targeted ontrol is k R Γ () ( ) ( k, ( k R + s) ) g max s = + + s k Γ( k) k t where Γ( k, b) = t e dt and Γ ( k) = Γ( k,). b For any distribution of represented by f v (u), we an make the following laim: Claim: Let C and C 2 be two ontrol strategies that satisfy ( u) f ( u) du x du ( u) f ( u) du > C ( u) f ( u) 2 x C i = and C du (**) for all x >, so that C targets higher- individuals to a greater degree. Then the reprodutive C C2 C2 number under strategy ( R ) is less than that under strategy 2 ( R ). Moreover, if R >, then the probability of extintion is greater under strategy. 6

17 C C2 Proof of Claim: The laim R g C < g C Reall that if X and Y 2 are positive random variables suh that P(X > x) > P(Y > x) for all x >, then E(X)>E(Y) 3. Define X i to be the positive random variable with the pdf ( Ci ( u) ) f ( u) for i=,2. By (**) we have P( X 2 > x) = ( C2 ( u) ) f ( u) du > ( C( u) ) f ( u) du = P( X > x) x R < is equivalent to () (). for all x >. Hene g C ( ) = ( ) E( X ) ( ) ( ) (). 2 2 > E X = g C The seond assertion of the laim is equivalent to the statement that gc ( s) > g ( s) C for all s œ 2 [,). To prove this, define Y i =exp(-x i (-s)). Sine exp(-x(-s)) is a dereasing funtion of x for s œ [,) and P(X 2 >x)>p(x >x) for all x >, we have P(Y >x)>p(y 2 >x) for all x>. Hene, g s) = + E Y > + E Y g ( and as argued above we have ( ) ( ) ( ) ( ) ), C ( 2 = C s 2 C, ( ), () n gc2 n g > for all generations n and therefore q C C > q 2. To see the utility of this laim, let us onsider two ontrol strategies C and C 2 that ontrol two portions of the population in different ways. Suppose strategy C i ontrols the less-infetious * portion of the population (i.e. < ) with probability a i and ontrols the more-infetious portion * of the population (i.e. ) with probability b i. In other words * ai if < C i ( ) = * bi if Moreover, let us assume that both strategies ontrol the same fration of individuals, i.e. ( u) f ( u) du C i = for i=,2. Suppose that strategy targets more-infetious individuals to a greater degree than strategy 2, i.e. b > b 2 and thus a <a 2. This is a generalized formulation of the * targeted ontrol senario disussed in the main text (Figs. 3,d), for whih strategy defines as the solution to ( u) du =. 8 * x f and takes b 2 =4 a 2, whereas strategy 2 is non-targeted * individual-speifi ontrol with a 2 =b 2 =. For : C ( u) f ( u) du = b f ( u) du * and for < : > b 2 f ( u) du = C ( u) f ( u) 2 du 7

18 C ( u) f ( u) du = a f ( u) > a 2 f du ( u) du = C ( u) f ( u) du. C C2 C Condition (**) is fulfilled, so R < R and C q > q 2, orroborating the simulation results for targeted ontrol (Figs. 3,d; Supplementary Figs. 3,d). In general, the more a ontrol poliy targets the more-infetious individuals, the higher the probability of disease extintion and the slower the growth rate of an outbreak in the event of non-extintion. For any individual-speifi ontrol program that targets more-infetious individuals more than random (denoted by subsript tar ), then for a given ontrol effort œ (,) we have g () s > g () s > g () s tar ind pop for all s œ [,), so targeted individual-speifi ontrol is always more effetive than random individual-speifi ontrol, whih in turn is always better than population-wide ontrol Control poliies simulations To simulate the effet of different ontrol poliies (Figs. 3,d, Supplementary Figs. 3,d), the branhing proess simulation from Fig. 2 (desribed above) was adapted. For populationwide ontrol, every infeted ase s individual reprodutive number was redued to (-) before a Poisson random variate was drawn to determine the number of infetions aused. For random individual-speifi ontrol, every infeted ase had probability of having redued to zero before the Poisson random variate was drawn. For targeted individual-speifi ontrol, the total proportion of the population subjet to ontrol was, but the probability of ontrol for a top-2% individual was four times greater than that for a bottom-8% individual, e.g. Pr(ontrol, top- 2%)=/4 and Pr(ontrol, bottom-8%)=/6, yielding Pr(ontrol, overall)=/. Under this four-fold targeting, equal effort (in terms of total numbers ontrolled) is expended on top-2% and bottom-8% individuals. Targeted ontrol was simulated as follows. For eah ombination of R and k, the utoff value of dividing top-2% from bottom-8% infetiousness was established from the df of. During the simulation, after a value of was drawn from the gamma(r,k) distribution for eah infeted individual, they were assigned to the top-2% or bottom-8% ategories. For individuals in either ategory, a uniform random variate on [, ] was drawn, and if it was less than the probability of ontrol for that ategory then that individual s value of was reset to zero. The realized number of seondary infetions Z was then generated by drawing a Poisson random variate with mean. For the simulations shown in Fig. 3, ontrol was initiated in the seond generation (i.e. the index ase was not subjet to ontrol), representing a delay in reognition of the outbreak. Containment of an outbreak was defined as preventing it from growing to the point of a generation with ases. Sine a branhing proess that esapes ontrol will grow without bound, results were not sensitive to this arbitrary threshold. The relative effet of targeted ontrol (Fig. 3d) was omputed as follows. The unontrolled probability of a major outbreak for the 2 8

19 given R and k was omputed as Pr(ontainment % ontrol). The ontribution of ontrol efforts to ontainment was then alulated as: Contrib(ontrol poliy) = Pr(ontainment ontrol poliy) Pr(ontainment % ontrol). The relative effet of targeted ontrol, plotted in Fig. 3d, was then: Relative effet = Contrib(targeted indiv. ontrol)/contrib(random indiv. ontrol). This quantity equals for kø, sine targeting has no effet on a homogeneous population, but is greater than for all finite values of k. 3. Data 3. Notes on outbreak and surveillane datasets 3.. SARS, Singapore This dataset desribes the progression of SARS in Singapore, beginning with the index ase who imported the infetion from Hong Kong. The first ase had onset of symptoms on Feb 25, 23. The government was notified of an unusual luster of pneumonia ases on Marh 6, and again on Marh 4 for a luster of six persons, inluding two healthare workers (HCWs), with atypial pneumonia. A ase in the third generation had onset of symptoms on Marh 2, ten days before full ontrol measures were instituted. In the week of Marh, the serial interval (time from symptom onset of soure ase to symptom onset of seondary ase) for SARS in Singapore had a median of 6 days (interquartile range, 4-9 days) 4. Centralized ontrol measures were imposed on Marh 22, and tightened suessively on Marh 24 and April 9, so for our analysis we ombined the first three generations of transmission into one dataset representing spread prior to ontrol (N=57). Transmission data from the fourth through seventh generations were pooled to reate the dataset under ontrol measures (N=4). Control measures imposed during this period inluded use of isolation and full ontat preautions with all identified SARS patients, twie-daily sreening of HCWs for fever, limitation of hospital visitors, and later the shutdown of a vegetable market where a SSE that ourred after ontrol had been initiated 34. In the total Singapore dataset inluding seven generations of transmission and 2 probable SARS ases, 22 ases were not linked to the transmission hain due to transloation from other SARSaffeted regions or poorly-defined ontat history. Note that our maximum-likelihood estimate of R for the first three generations of SARS spread in Singapore (.63; 9% CI (.54,2.65)) is somewhat lower than other estimates for SARS in Singapore (3.; 95% CI (2.3,4.)) 7, though onfidene intervals overlap. This may be beause our dataset exludes unlinked ases, or beause we inlude the period between the WHO s global alert on SARS (Marh 2) and the imposition of entralized ontrol measures (Marh 22), during whih time transmission may have been redued by informal hanges of behavior or isolation of speifi patients. Analysis of a dataset inluding only the first two generations of transmission in Singapore (N=22) yields R ˆ,mle =2.55 (9% CI (.5,4.5)) and k ˆ mle =.2 (9% CI (.5, )) SARS, Beijing 23 This dataset desribes a hospital outbreak of SARS in the period before SARS was reognized in Beijing. The index ase was an elderly woman hospitalized for diabetes, who 9

20 aught SARS while a patient in the hospital, and diretly infeted 33 others. These seondgeneration ases inluded patients and visitors, and transmission by the seond generation ourred in the hospital (to patients and visitors), in homes, and in a workplae. The hospital had not implemented isolation or quarantine proedures during the seond generation s infetious period. Later in the outbreak administrative ontrols redued ontat rates, but infetion ontrol measures (masks, gloves, et.) and respiratory isolation were never in plae. We regard the first and seond generations of spread as a natural experiment in SARS nosoomial transmission. To diminish onern of seletion bias (i.e. that this outbreak ourred, and was traed and reported, beause it began with a superspreading event), we have removed the index ase (Z=33) from our main analysis, and used only the Z values from the seond generation ases (N=33) to alulate the values in Supplementary Table. Analysis inluding the index ase yields a higher estimate of R and more highly overdispersed distribution for (ˆ R,mle =.88, 9%CI (.4,3.32); k ˆ mle =.2, 9%CI (.78,.42); see Supplementary Table 2), as expeted given the addition of an extreme SSE. The dataset under ontrol was omprised of data from the third and fourth generations of ases (N=43), after the hospital s imposition of limits on visitors and soial ontats Measles, US In this summary of measles elimination efforts in the United States, 65 separate hains of measles transmission were identified (of whih 7 were lassified as importations). 22 outbreaks onsisted of a single ase with no seondary transmission (yielding an estimate of p =22/65). Insuffiient data were reported to estimate the effetive reprodutive number R diretly, but estimation of R was a major goal of the soure paper so we used their estimate and 95% onfidene interval. These estimates of R were derived from three approahes, all based on the assumption that Z~Poisson(R). Our analysis shows that the negative binomial offspring distribution is strongly favoured by AIC model seletion, but it is not lear what impat this would have on estimation of R using the methods desribed. We used the broadest onfidene interval reported to aount for this unertainty. Vaination levels in the US are reported to be above 9% in shool-aged hildren 7, but are possibly lower in other populations Measles, Canada As for the US measles dataset, this is routine surveillane data traking progress on elimination of measles from Canada. 49 outbreaks were reported, of whih 35 had only one ase. Again we were unable to estimate R diretly, and took estimates and onfidene intervals (based on Z~Poisson(R)) from the soure paper. The vaination level in the general population is reported to be 95-%. The authors raise the interesting point that long hains of transmission have ourred exlusively in religious ommunities that atively resist immunization, suggesting that an important determinant of the individual reprodutive number in this ontext is the suseptibility of one s ontats. 2, p Smallpox (Variola major), Europe This dataset is a summary of smallpox importations into Europe from , and thus ombines data olleted over a long time period in many ountries, probably with varying degrees of smallpox vaination. Two outbreaks were exluded from the analysis, beause one of them had three primary ases and the other had no primary ase (infetion was apparently transmitted 2

21 on a arpet). The remaining outbreaks eah had a single index ase, and the number of infetions in the first indigenous generation (i.e. ases within Europe) was taken as the Z value for eah index ase. Information on later generations is tabulated in the soure material, but was exluded from this analysis beause it was unlear if and when ontrol was imposed in eah outbreak, and there is no way to divide the total number of ases in the seond indigenous generation among the possible soure ases in the first indigenous generation Smallpox (Variola major), Benin A village-based outbreak ourred in Benin (formerly Dahomey) in 967. The existene of the outbreak was onealed from authorities for three months, after whih a vaination team arrived but is suspeted not to have affeted the natural die-out of the outbreak. Contat traing was by reolletion of the villagers and some links are unertain. Vaination sar rates were <2% among hildren, and >7% among adults. Transmission was predominantly by intimate ontats within households, rather than via frequent asual ontats among villagers. Limited ontrol measures were imposed by the villagers, but were judged by the authors of the report to have had little effet on transmission so we have not divided the dataset Smallpox (Variola major), West Pakistan This is surveillane data from 47 outbreaks in rural West Pakistan, fousing on transmission within ompounds inhabited by extended families. Of 47 outbreaks, 26 led to seondary transmission, with a total of 7 seond-generation ases. Sine all ompound residents were in reasonably lose ontat, generations of ases were assigned based on the interval between exposure to the index ase and onset of illness; for seond generation ases this interval was 9-2 days. The population is reported to be relatively homogeneous. There was no isolation of ontats from ases, and vaination is reported to have played a minor role, though it was also observed that previously-vainated index ases tended to be less infetious. Severe illness was assoiated with higher infetiousness in this study. A similar study in East Pakistan in 967 reported 3 smallpox outbreaks, with R~2.2 (stated verbally in the paper) and p =3/3, yielding an estimate of k ˆ pz = Smallpox (Variola major), Kuwait In this outbreak, smallpox was suspeted relatively quikly and ontrol measures were imposed rapidly in the affeted hospital. One unreognized ase had been transferred to another hospital, however, and initiated further spread there before the disease was reognized and ontrol was imposed. The outbreak was stopped by this expanded ontrol effort. The bakground level of vaination is not reported, but Kuwait had been free of endemi smallpox for a deade at the time of the outbreak. Control measures inluded intensive surveillane of hospitals suspeted to be infeted, with vaination of all patients. Household ontats of infeted individuals were vainated and plaed under surveillane, and a mass vaination ampaign was initiated that overed 8% of the total population of Kuwait by the midway point of the outbreak (i.e. the date by whih symptoms had appeared for roughly half of all ases) Smallpox (Variola minor), England This outbreak of Variola minor, the less ommon and less severe form of smallpox, was initiated by a laboratory release in Birmingham, England. Beause smallpox had been eliminated 2

22 from England for deades, the outbreak went unsuspeted until a ase in the fourth generation of transmission was diagnosed and ontrol efforts were initiated. Thorough investigations were onduted by British and US experts, but the results seem to have been published only as an appendix to a parliamentary inquiry into a 978 release of smallpox from the same laboratory in Birmingham 74. The ontat traing dataset is quite omplete, though there were several ases for whom a soure of infetion was not established. We have exluded the latter from our analysis. Vaination levels in the general population were roughly 6% 2, p Monkeypox, Zaire ,76 From , intensive surveillane and epidemiologi investigations were arried out in Zaire to monitor the risk of monkeypox emergene into the nihe left empty by the reent eradiation of smallpox. 47 monkeypox ases were judged to be primary ases infeted by an animal soure. These data are tabulated in several publiations, with the greatest detail shown in Jezek et al 75, who break down eah outbreak by number of seondary ases per index ase (Z) for eah generation. In our analysis, we used the data for the first generation of human-to-human transmission only, to minimize the influene of ontrol measures. Sars from smallpox vaination (whih is ross-protetive for monkeypox) were seen on 68% of investigated ontats, p. 99, but onern was expressed that vaine protetion may have been waning. Oasional instanes of sublinial infetion were reported, raising the possibility that these transmission figures are an underestimate. 3.. Pneumoni plague (Yersinia pestis), 6 outbreaks Datasets from six outbreaks of pneumoni plague (Yersinia pestis) were ompiled by Gani & Leah for their exellent reent analysis of the transmission and ontrol of plague outbreaks. They employ an approah similar to ours, omparing Poisson and geometri models for the offspring distribution with aggregated data on Z (for all six datasets, before ontrol measures), and onlude that the geometri distribution provides a superior fit. (Note that our analysis, while inluding the more flexible negative binomial distribution as a andidate model, also seleted the geometri model as the best ombination of auray and parsimony in fitting the aggregated data (Supplementary Table ).) Beause several of the soure reports were published in inaessible or foreign-language publiations, we ontated Dr. Raymond Gani diretly and he kindly provided the raw data from their analyses. We based our analysis of pneumoni plague on these data, with further referene to the soure report for the Mukden outbreak whih we analysed more losely in our work on ontrol measures 78. Mukden is a ity in Manhuria, China, whih experiened a pneumoni plague outbreak in 946 with 2 ases before ontrol measures and 27 ases after the advent of ontrol. Control measures inluded isolation and quarantine (in a suburban area) of all patients and ontats, disinfetion and loking of infeted houses, and wearing of masks required for all ontats and advised for the general population Hantavirus (Andes virus), Argentina This outbreak is the first reported instane of human-to-human transmission of a hantavirus, and is perhaps representative of a zoonoti pathogen beginning to adapt to a human host. It is definitely an anomalous pattern for hantavirus, as human-to-human transmission has not been reported elsewhere despite intensive surveillane. Contat traing for this outbreak was impreise, in part beause several of the infeted individuals had ontat with more than one earlier ase. The dataset of Z values analysed was drawn from a diagrammed transmission hain 22

23 and text desriptions in the outbreak report. In instanes where the soure of transmission was vague (i.e. transmission lines to two soure ases in the published transmission hain), we adopted the onservative poliy of dividing the seondary ases evenly between the possible soures in making our estimates of R ˆ,mle and k ˆ mle. The onfidene intervals reported in Supplementary Table inlude the upper and lower bounds of 9% onfidene intervals omputed for all alternative assumptions regarding these vaguely attributed ases. There is no mention of ontrol measures in the outbreak report, possibly beause human-to-human transmission was not thought to be a threat Ebola Hemorrhagi Fever, Uganda 2 8 These data ome from a traed portion of a large outbreak (425 presumptive ases) from Aug 2 to Jan 2. The study methodology was retrospetive ontat traing, with the stated goal of determining the original primary ases of the outbreak (i.e. those who had aquired infetion diretly from the zoonoti reservoir). Cases (or their next of kin) were asked to identify persons from whom they had probably aquired the disease, who were in turn asked to identify who had infeted them. Primary ases were defined as those whose soures of infetion ould not be identified. Prospetive ontat traing was onduted to the extent that lists of ontats of identified ases (information that was routinely olleted ) were mathed with a list of reported ases. This data olletion tehnique may bias the dataset toward surviving hains of transmission, sine these are the ones that led to the later-generation ases from whih ontat traing began. The effort at prospetive ontat traing would have mitigated this to some extent, but the level of traing effort was ertainly lower than for the retrospetive work. The resulting dataset is onspiuously low in Z= entries, just as we would expet for a methodology that is biased against deteting hains that have died out. Aordingly, we believe the results in Supplementary Table should be interpreted with aution, and have marked them as suh Rubella, Hawaii In this outbreak, an army reruit returned to Hawaii from the US mainland for the Christmas holidays. He imported rubella, and proeeded to infet every identified suseptible ontat he had during the 72-hour period of his prodromal illness 24. His extreme infetiousness may have been linked to a persistent nonprodutive ough linked to an earlier (separate) respiratory illness. The great majority of seondary ases did not ause further transmission; there was only one other infetion event reported in the outbreak. Several ases were not epidemiologially linked to any soure of transmission, and were omitted from the analysis. This outbreak is almost ertainly exeptional in the extreme infetiousness of the index ase, and the small number of transmitting individuals (i.e. only two ases had Z > ) prevented reliable estimation of model parameters. As a onsequene, we do not inlude results from this dataset in the main text or Supplementary Table, but show them in Supplementary Table 2 beause of the interesting disussion surrounding this outbreak. The authors of the original report onlude that highly heterogeneous infetiousness is neessary to explain observed patterns of rubella epidemiology in Hawaii. In partiular, they posit that During an unompliated rubella infetion the average individual may have minimal ontagious potential, while Other persons may have a substantially greater potential for spread. Proposed fators influening the potential for spread by individuals were age, sex, and oexisting or previous respiratory infetions (the latter fator supported by unpublished evidene from military amps). Spreader to spreader ontat is proposed to be neessary for sustained 23

24 rubella transmission in a population, explaining why extended rubella outbreaks are most often observed in large, rowded population groups. The authors onlude that the proposed individual variation in infetiousness, ombined with the sparse population distribution of Hawaii in the 96s, ould explain why the highly suseptible population of Hawaii an enounter dozens and perhaps hundreds of rubella introdutions eah year without resulting in a full-sale epidemi. This qualitative hypothesis is highly similar to the model-based onlusions reahed in our study. 3.2 Survey of superspreading events (SSEs) To demonstrate the universality of the superspreading phenomenon, and to identify reurrent themes in field reports of superspreading events, we have ompiled a list of superspreading events, their index ases, and the irumstanes surrounding them. This list is not intended to be omprehensive, but rather is a survey of the epidemiologial literature on diretlytransmitted infetions. This list was the basis for Fig. d in the main text. Also required for Fig. d were estimates of reprodutive numbers for the diretly-transmitted diseases shown. These were drawn from detailed studies where available, or else estimated from published ranges of values. For some diseases, various levels of population immunity (due to previous natural spread or vaination) may have been present for the different SSEs depited; beause these levels varied among settings and often were unknown, we adopted the most onservative approah of using estimates of basi reprodutive numbers in Figure d. R estimates and soure referenes are as follows: monkeypox, R =.32 (Supplementary Table ); Ebola hemorrhagi fever, R =.83 8 ; SARS, R =3 82 ; smallpox, R = ; rubella, R ~9 ; influenza, R ~4 84 ; measles, R ~6. Note that estimates for rubella, influenza and measles were drawn from published ranges of values, and are intended to be illustrative only Superspreading events in the published literature Disease Z Setting Patient Cirumstanes Ref. Ebola HF 46 Community?M Ative soial life, inluding workplae ontats; possibility of spread by injetion (re-used needles). Ebola HF Hospital 29M Popular dotor, with many visitors during hospitalization before death. Ebola HF 2+ Funeral 45F Misdiagnosed, leading to traditional funeral with washing and handling of adaver. Influenza 38 Airplane 2F All infetions ourred aboard grounded airplane with ventilation system turned off for three hours; severe ough. Lassa fever 6 Hospital 25F Misdiagnosed; atypial presentation with severe ough. Possible airborne spread via air urrents from bed to rest of ward

25 Measles 69 High shool 6F Haking ough; high shool setting 33 Measles 84 High shool 6M Haking ough; high shool setting 33 Measles 25 Dane party?m First arrival of measles in Greenland true virgin population. Index ase attended rowded daning-lik party. 85 Myoplasma pneumonia 26 Fraternity banquet Unk. * Gross bahanal fraternity banquet: inebriation, igar smoke membrane irritation, vomiting, shouting; partiipants drenhed with food missiles, drinks and gastri ontents. Pneumoni plague 32 Funeral?W Funeral attendees and visitors of an unreognized ase. Rubella 8 Home and parties 2M Previous (ongoing) respiratory illness with ough. Rubella 37+ Disotheque?M Crowded disotheque; possible airborne spread via air flow from index ase to rowd. Singing thought to aid aerosolization. SARS 3 Hotel and hospital 64M Undiagnosed: SARS not yet reognized. SARS 2 Hospital 47M Undiagnosed: SARS not yet reognized. SARS 87+ Apartment blok 26M Amoy Gardens outbreak. Hypothesis: unsealed plumbing and bathroom fans led to aerosolized virus, infeting many in apartment omplex. SARS 2 Hospital 22? Undiagnosed: SARS not yet reognized. SARS 23 Hospital 27? Undiagnosed: SARS not yet reognized. Patient was HCW infeted nosoomially. SARS 23 Hospital 53? Patient infeted nosoomially, omorbidities. SARS 4+ Hospital 6? Misdiagnosed. Patient infeted nosoomially, o-morbidities. SARS 2 Vegetable market, hospital 64? Misdiagnosed, with o-morbidities. Patient transmitted with minimal ontat (e.g. twie to taxi drivers)

26 SARS 44?? Co-morbidities. SARS 37 Hospital worker 43M Co-morbidities; popular hospital laundry worker, ontinued work despite symptoms SARS 33 Hospital 62W Undiagnosed: SARS not yet reognized. Patient infeted nosoomially, with o-morbidities. High ontat rate (many visitors) and no preautions in hospital. SARS Hospital 7W Undiagnosed: SARS not yet reognized. Patient infeted nosoomially, no preautions in hospital. SARS 8 Hospital 69W Undiagnosed: SARS not yet reognized. Patient infeted nosoomially, no preautions in hospital. SARS 2 Constrution site 23M High number of ontats at home and worksite. SARS 9 Home, hospital?m Misdiagnosed due to unknown ontat history, o-morbidities. SARS 24/2 Home, emergeny room, ICU, hospital?m Unproteted exposure to index patient and wife of emergeny personnel in ambulane, and of patients and staff in emergeny room. Intubation proedure infeted HCWs despite protetive equipment. Smallpox 9?? No details available , p.77 Smallpox Soial ontats 38M Undiagnosed: smallpox not suspeted. Visited with family and friends following travel abroad. Smallpox 38 Hospital spread to HCWs and patients 3M Undiagnosed: smallpox not suspeted. Noted as interesting ase and shown to students and staff in hospital. Smallpox 6? Undiagnosed: mild ambulant ase, not reognized as smallpox. 2, p.92 2, p.92 2, p.98 26

27 Smallpox 7 Hospital? Airborne spread despite rigorous isolation ; aided by severe bronhitis, low humidity, and strong air urrents Streptoous group A (type 46) Streptoous group A (type ) Army barrak?m Asymptomati ase, with strongly positive nose and throat ultures. + Hospital afeteria?m Food handler with strongly positive nose ulture and very high hand ultures; diretly handled eah piee of apple pie (popular item in afeteria). Tuberulosis 4/2 Rok onert? 2 index ases in rok band, infeted hundreds, if not thousands of fans, at least 4 ative ases. Airborne spread aided by singing. Tuberulosis 56 9M Undiagnosed ase, hildren not usually infetious with TB 2, p Notes Frational entries in Z olumn denote more than one possible index ase. Patient olumn shows age and sex of index ase, when known. * index ase not identified. HCW: healthare worker 4. Referenes. Anderson, R. M. & May, R. M. Infetious Diseases of Humans: Dynamis and Control (Oxford University Press, 99). 2. Beker, N. G. & Britton, T. Statistial studies of infetious disease inidene. J. R. Stat. So. B 6, (999). 3. Wallinga, J., Edmunds, W. J. & Kretzshmar, M. Perspetive: Human ontat patterns and the spread of airborne infetious diseases. Trends Mirobiol. 7, (999). 4. Diekmann, O. & Heesterbeek, J. A. P. Mathematial Epidemiology of Infetious Diseases: Model Building, Analysis, and Interpretation (John Wiley & Sons, Chihester, 2). 5. Anderson, R. M. et al. Epidemiology, transmission dynamis and ontrol of SARS: the epidemi. Phil. Trans. R.. So. Lond. B 359, 9-5 (24). 6. Koopman, J. Modeling infetion transmission. Annu. Rev. Publi Health 25, (24). 7. MDonald, L. C. et al. SARS in healthare failities, Toronto and Taiwan. Emerg. Infet. Dis., (24). 27

28 8. Grenfell, B. T., Wilson, K., Isham, V. S., Boyd, H. E. G. & Dietz, K. Modelling patterns of parasite aggregation in natural populations: Trihostrongylid nematode-ruminant interations as a ase study. Parasitology, S35-S5 (995). 9. Eubank, S. et al. Modelling disease outbreaks in realisti urban soial networks. Nature 429, 8-84 (24).. Shen, Z. et al. Superspreading SARS events, Beijing, 23. Emerg. Infet. Dis., (24).. Jezek, Z. & Fenner, F. Human Monkeypox (ed. Melnik, J. L.) (Karger, Basel, 988). 2. Woolhouse, M. E. J. et al. Heterogeneities in the transmission of infetious agents: Impliations for the design of ontrol programs. Pro. Natl. Aad. Si. U. S. A. 94, (997). 3. Meyers, L. A., Pourbohloul, B., Newman, M. E. J., Skowronski, D. M. & Brunham, R. C. Network theory and SARS: prediting outbreak diversity. J. Theor. Biol. 232, 7-8 (25). 4. Smith, D. H., Franis, D., Simpson, D. I. H. & Highton, R. B. The Nzara outbreak of viral haemorrhagi fever. in Ebola Virus Haemorrhagi Fever (ed. Pattyn, S. B.) 37-4 (Elsevier, 978). 5. Hamburger, M., Green, M. J. & Hamburger, V. G. The problem of the dangerous arrier of hemolyti Streptooi.2. Spread of infetion by individuals with strongly positive nose ultures who expelled large numbers of hemolyti Streptooi. J. Infet. Dis. 77, 96-8 (945). 6. Houk, V. N. Spread of tuberulosis via reirulated air in a naval vessel: the Byrd study. Ann. N. Y. Aad. Si. 353, -24 (98). 7. Marks, J. S. et al. Saturday night fever - a ommon-soure outbreak of rubella among adults in Hawaii. Am. J. Epidemiol. 4, (98). 8. Grenfell, B. T. & Anderson, R. M. The estimation of age-related rates of infetion from ase notifiations and serologial data. J. Hyg. (Lond). 95, (985). 9. Rao, A. R., Jaob, E. S., Kamalaks.S, Appaswam.S & Bradbury. Epidemiologial studies in smallpox. A study of intrafamilial transmission in a series of 254 infeted families. Ind. J. Med. Res. 56, 826-& (968). 2. Fenner, F., Henderson, D. A., Arita, I., Jezek, Z. & Ladnyi, I. D. Smallpox and Its Eradiation (World Health Organization, Geneva, 988). 2. Edwards, D. A. et al. Inhaling to mitigate exhaled bioaerosols. Pro. Natl. Aad. Si. U. S. A., (24). 22. Carey, D. E. et al. Lassa fever - epidemiologial aspets of 97 epidemi, Jos, Nigeria. Trans. R. So. Trop. Med. Hyg. 66, (972). 23. Moser, M. R. et al. Outbreak of influenza aboard a ommerial airliner. Am. J. Epidemiol., -6 (979). 24. Hattis, R. P., Halstead, S. B., Herrmann, K. L. & Witte, J. J. Rubella in an immunized island population. J. Am. Med. Asso. 223, 9-2 (973). 25. Sherertz, R. J. et al. A loud adult: The Staphyloous aureus - virus interation revisited. Ann. Intern. Med. 24, 539-& (996). 26. Bassetti, S. et al. "Cloud Adults" exist: Airborne dispersal of Staphyloous aureus (SA) assoiated with a rhinovirus infetion. Clin. Infet. Dis. 3, (2). 27. Sherertz, R. J., Bassetti, S. & Bassetti-Wyss, B. "Cloud" health-are workers. Emerg. Infet. Dis. 7, (2). 28. Bassetti, S. et al. Dispersal of Staphyloous aureus into the air assoiated with a rhinovirus infetion. Infet. Control Hosp. Epidemiol. 26, (25). 28

29 29. Bassetti, S., Bishoff, W. E. & Sherertz, R. J. Are SARS superspreaders loud adults? Emerg. Infet. Dis., (25). 3. Booth, T. F. et al. Detetion of airborne severe aute respiratory syndrome (SARS) oronavirus and environmental ontamination in SARS outbreak units. J. Infet. Dis. 9, (25). 3. Tong, T. R. Airborne severe aute respiratory syndrome oronavirus and its impliations. J. Infet. Dis. 9, 4-42 (25). 32. Riley, E. C., Murphy, G. & Riley, R. L. Airborne spread of measles in a suburban elementary-shool. Am. J. Epidemiol. 7, (978). 33. Chen, R. T., Goldbaum, G. M., Wassilak, S. G. F., Markowitz, L. E. & Orenstein, W. A. An explosive point-soure measles outbreak in a highly vainated population - Modes of transmission and risk-fators for disease. Am. J. Epidemiol. 29, (989). 34. Leo, Y. S. et al. Severe aute respiratory syndrome - Singapore, 23. Morbid. Mortal. Wkly. Rep. 52, 45-4 (23). 35. Hopkins, D. R., Lane, J. M., Cummings, E. C. & Millar, J. D. 2 funeral-assoiated smallpox outbreaks in Sierra-Leone. Am. J. Epidemiol. 94, 34-& (97). 36. Khan, A. S. et al. The reemergene of Ebola hemorrhagi fever, Demorati Republi of the Congo, 995. J. Infet. Dis. 79, S76-S86 (999). 37. Arita, I., Shafa, E. & Kader, M. A. Role of hospital in smallpox outbreak in Kuwait. Am. J. Publi Health 6, (97). 38. Varia, M. et al. Investigation of a nosoomial outbreak of severe aute respiratory syndrome (SARS) in Toronto, Canada. Can. Med. Asso. J. 69, (23). 39. Nkowane, B. M., Bart, S. W., Orenstein, W. A. & Baltier, M. Measles outbreak in a vainated shool population - Epidemiology, hains of transmission and the role of vaine failures. Am. J. Publi Health 77, (987). 4. Kamps, B. S. & Hoffmann, C. (eds.) SARS Referene, 3 rd ed. (Flying Publisher, 24) Aessed online at August 4, Lipsith, M. et al. Transmission dynamis and ontrol of severe aute respiratory syndrome. Siene 3, (23). 42. King, A., Varughese, P., De Serres, G., Tipples, G. A. & Waters, J. Measles elimination in Canada. J. Infet. Dis. 89, S236-S242 (24). 43. Patil, G. P. (ed.) Random Counts in Models and Strutures (Pennsylvania State University Press, University Park PA, 97). 44. Pielou, E. C. Mathematial Eology (Wiley, New York, 977). 45. Taylor, H. M. & Karlin, S. An Introdution to Stohasti Modeling (Aademi Press, San Diego, 998). 46. Douglas, J. B. Analysis with Standard Contagious Distributions (ed. Patil, G. P.) (International Cooperative, Fairland MD, 98). 47. Karlis, D. & Xekalaki, E. A simulation omparison of several proedures for testing the Poisson assumption. J. R. Stat. So. D 49, (2). 48. Potthoff, R. F. & Whitting.M. Testing for homogeneity.2. Poisson distribution. Biometrika 53, 83-& (966). 49. Rie, J. A. Mathematial Statistis and Data Analysis (Duxbury Press, Belmont CA, 995). 5. Ansombe, F. J. Sampling theory of the negative binomial and logarithmi series distributions. Biometrika 37, (95). 5. Saha, K. & Paul, S. Bias-orreted maximum likelihood estimator of the negative binomial dispersion parameter. Biometris 6, (25). 29

30 52. Ross, G. J. S. & Preee, D. A. The negative binomial distribution. Statistiian 34, (985). 53. Pieters, E. P., Gates, C. E., Matis, J. H. & Sterling, W. L. Small sample omparison of different estimators of negative binomial parameters. Biometris 33, (977). 54. Clark, S. J. & Perry, J. N. Estimation of the negative binomial parameter Kappa by maximum quasi-likelihood. Biometris 45, (989). 55. Piegorsh, W. W. Maximum-likelihood estimation for the negative binomial dispersion parameter. Biometris 46, (99). 56. Gregory, R. D. & Woolhouse, M. E. J. Quantifiation of parasite aggregation - a simulation study. Ata Trop. 54, 3-39 (993). 57. Burnham, K. P. & Anderson, D. R. Model Seletion and Multimodel Inferene: A Pratial Information-Theoreti Approah (Springer, New York, 22). 58. Heiner, G. G., Fatima, N. & MCrumb, F. R. Study of intrafamilial transmission of smallpox. Am. J. Epidemiol. 94, (97). 59. Gay, N. J., De Serres, G., Farrington, C. P., Redd, S. B. & Papania, M. J. Assessment of the status of measles elimination from reported outbreaks: United States, J. Infet. Dis. 89, S36-S42 (24). 6. Shaw, D. J., Grenfell, B. T. & Dobson, A. P. Patterns of maroparasite aggregation in wildlife host populations. Parasitology 7, (998). 6. Boye, M. S., MaKenzie, D. I., Manly, B. F. J., Haroldson, M. A. & Moody, D. Negative binomial models for abundane estimation of multiple losed populations. J. Wildl. Mgmt. 65, (2). 62. White, G. C. & Bennetts, R. E. Analysis of frequeny ount data using the negative binomial distribution. Eology 77, (996). 63. Kupper, L. L. Estimation, Interval. in Enylopedia of Biostatistis (eds. Armitage, P. & Colton, T.) (Wiley, Chihester, 998). 64. Efron, B. & Tibshirani, R. J. An Introdution to the Bootstrap (Chapman & Hall, London, 993). 65. Manly, B. F. J. Randomization, Bootstrap and Monte Carlo Methods in Biology (Chapman & Hall, London, 998). 66. Shao, J. & Tu, D. The Jakknife and Bootstrap (Springer, New York, 995). 67. Clopper, C. J. & Pearson, E. S. The use of onfidene or fiduial limits illustrated in the ase of the binomial. Biometrika 26, (934). 68. Newombe, R. G. Two-sided onfidene intervals for the single proportion: Comparison of seven methods. Stat. Med. 7, (998). 69. Harris, T. E. The Theory of Branhing Proesses (Dover, New York, 989). 7. Wallinga, J. & Teunis, P. Different epidemi urves for severe aute respiratory syndrome reveal similar impats of ontrol measures. Am. J. Epidemiol. 6, (24). 7. Papania, M. J. & Wharton, M. in VPD Surveillane Manual (Centers for Disease Control, 22). 72. Henderson, R. & Yekpe, M. Smallpox transmission in southern Dahomey - a study of a village outbreak. Am. J. Epidemiol. 9, (969). 73. Thomas, D. B. et al. Endemi smallpox in rural East Pakistan II. Intravillage transmission and infetiousness. Am. J. Epidemiol. 93, (97). 74. Shooter, R. A. Report of the investigation into the ause of the 978 Birmingham smallpox ourrene (H.M. Stationery Offie, London, 98). 3

31 75. Jezek, Z., Grab, B. & Dixon, H. Stohasti model for interhuman spread of monkeypox. Am. J. Epidemiol. 26, (987). 76. Fine, P. E. M., Jezek, Z., Grab, B. & Dixon, H. The transmission potential of monkeypox virus in human populations. Int. J. Epidemiol. 7, (988). 77. Gani, R. & Leah, S. Epidemiologi determinants for modeling pneumoni plague outbreaks. Emerg. Infet. Dis., (24). 78. Tieh, T. H., Landauer, E., Miyagawa, F., Kobayashi, G. & Okayasu, G. Primary pneumoni plague in Mukden, 946, and report of 39 ases with 3 reoveries. J. Infet. Dis. 82, (948). 79. Wells, R. M. et al. An unusual hantavirus outbreak in southern Argentina: Person-toperson transmission? Emerg. Infet. Dis. 3, 7-74 (997). 8. Franesoni, P. et al. Ebola hemorrhagi fever transmission and risk fators of ontats, Uganda. Emerg. Infet. Dis. 9, (23). 8. Chowell, G., Hengartner, N. W., Castillo-Chavez, C., Fenimore, P. W. & Hyman, J. M. The basi reprodutive number of Ebola and the effets of publi health measures: the ases of Congo and Uganda. J. Theor. Biol. 229, 9-26 (24). 82. Bauh, C. T., Lloyd-Smith, J. O., Coffee, M. & Galvani, A. P. Dynamially modeling SARS and other newly-emerging respiratory illnesses: past, present, future. Epidemiology (in press). 83. Gani, R. & Leah, S. Transmission potential of smallpox in ontemporary populations. Nature 44, (2). 84. Fraser, C., Riley, S., Anderson, R. M. & Ferguson, N. M. Fators that make an infetious disease outbreak ontrollable. Pro. Natl. Aad. Si. U. S. A., (24). 85. Christensen, P. E. et al. An epidemi of measles in southern Greenland, 95 - measles in virgin soil.2. The epidemi proper. Ata Med. Sand. 44, (953). 86. Evatt, B. L., Dowdle, W. R., Johnson, M. & Heath, C. W. Epidemi Myoplasma pneumonia. New Engl. J. Med. 285, 374-& (97). 87. Tsang, T. et al. Update: Outbreak of severe aute respiratory syndrome --- Worldwide, 23. Morbid. Mortal. Wkly. Rep. 52, (23). 88. Yu, I. T. S. et al. Evidene of airborne transmission of the severe aute respiratory syndrome virus. New Engl. J. Med. 35, (24). 89. Curtis, A. B. et al. Extensive transmission of Myobaterium tuberulosis from a hild. New Engl. J. Med. 34, (999). 3

32 Supplementary Figures From: Superspreading and the impat of individual variation on disease emergene J.O. Lloyd-Smith, S.J. Shreiber, P.E. Kopp, W.M. Getz Proportion of ases ausing SSEs,Ψ R,k R =.5.. Dispersion parameter, k Supplementary Fig. Predition of SSE frequeny. The expeted proportion of infetious ases ausing 99 th -perentile SSEs (Ψ R,k ) for outbreaks with Z~NegB(R,k), plotted versus k. Eah urve shows the relationship for a partiular value of the effetive reprodutive number, R. The values of R plotted were seleted suh that Pr(Z Z (99) Z~Poisson(R))=.. See Supplementary Notes for details of the alulation.

33 a g(s).5 g(s)= s k =. k =. k =.5 k = k = 4 k = inf b.5 s Pr(extintion), q.5 R =.5 R = 3 R = R = Dispersion parameter, k Supplementary Fig 2. Branhing proess results for Z~NegB(R,k). (a) The probability generating funtion of the negative binomial distribution, plotted for R =3 and different dispersion parameters k. The y-interept of the pgf equals p, the probability that an infeted individual will infet nobody, and is a major fator in the rising probability of extintion as k dereases. The extintion probability q is determined by the point of intersetion of the pgf with a line of slope (dashed) through the origin. (b) The probability of stohasti extintion given introdution of a single infeted individual, q, rises to as kø for any value of R. 2

34 Expeted size of minor outbreak k =. k =. k =.5 k = k = 4 k = inf d Pr(extintion) by n th generation, q n R k =. k =. k = k =4 k =inf Generation, n Supplementary Fig 2. Branhing proess results for Z~NegB(R,k) (ont). () Expeted size of a minor outbreak (i.e. an outbreak that dies out spontaneously) versus R. Curves for all k values are idential for R <. (d) The probability of stohasti extintion by the n th generation of transmission, q n, for R =3 and a range of k. Interestingly, for the third and subsequent generations, the k= ase has the highest ontinuing probability of extintion. 3

35 e R =. Dispersion parameter, k %.6% 6% 9% 4% inf 7% f First generation with ases R =3. Dispersion parameter, k % 6% 5% 67% 86% inf 5 5 First generation with ases 94% Supplementary Fig 2. Branhing proess results for Z~NegB(R,k) (ont). (e) Growth rate of simulated outbreaks with R =. and one initial ase, onditional on nonextintion. Boxes show interquartile range (IQR) and median (in grey) of the first disease generation with ases; whiskers show most extreme values within.5µiqr of the boxes, and rosses show outliers. Perentages show the proportion of, simulated outbreaks that reahed the -ase threshold (i.e. roughly -q). (f) Growth rate of simulated outbreaks with R =3. Both (e) and (f) are exatly analogous to Fig 2 exept for different values of R. 4

36 Supplementary Figure 3. Impat of ontrol measures. (a) Probability of stohasti extintion for diseases with different degrees of individual variation, k, under population-wide ontrol poliies where the infetiousness of all individuals is redued by a fator. (b) Probability of stohasti extintion under individual-speifi ontrol poliies where a randomly-seleted proportion of infetious individuals have their infetiousness redued to zero. In (a) and (b), outbreaks had R =3 and began with a single infetious ase, and ontrol was assumed to be present from the outset. The differene between (b) and (a) is shown in Fig. 3a in the main text. () Effet of random versus targeted ontrol measures. The plot is exatly analogous to Fig. 3 in the main text, exept that in the targeted ontrol senario individuals in the top 2% of infetiousness are ten-fold more likely to be ontrolled than those in the bottom 8% (rather than four-fold more likely as in Fig. 3), so 7% of ontrol effort is foused in the top 2% of ases (rather than 5% in Fig. 3). The probability of outbreak ontainment (i.e. never reahing ases) is shown for four diseases with R =3 and k=. (blue), k=.5 (green), k= (blak), or k (purple). Control poliies are population-wide (solid lines), random individual-speifi (dotted lines), or targeted individual-speifi (dashed lines). (d) The fator by whih targeting inreased the impat of ontrol on preventing a major outbreak relative to random individual-speifi ontrol, for the simulations shown in (). For k, targeting has no effet so this fator is, and dotted and dashed lines overlay one another in (). Results in () and (d) are the mean of, simulations, with ontrol beginning in the seond generation of ases. 5