Did Modeling Overestimate the Transmission Potential of Pandemic (H1N1-2009)? Sample Size Estimation for Post-Epidemic Seroepidemiological Studies

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1 Georgia State University Georgia State University Public Health Faculty Publications School of Public Health 2011 Did Modeling Overestiate the Transission Potential of Pandeic (H1N1-2009)? Saple Size Estiation for Post-Epideic Seroepideiological Studies Carlos Castillo-Chavez Arizona State University, ccchavez@asu.edu Gerardo Chowell Georgia State University, gchowell@gsu.edu Hiroshi Nishiura The University of Tokyo, nishiurah@.u-tokyo.ac.jp Follow this and additional works at: Part of the Public Health Coons Recoended Citation Nishiura H, Chowell G, Castillo-Chavez C (2011) Did Modeling Overestiate the Transission Potential of Pandeic (H1N1-2009)? Saple Size Estiation for Post-Epideic Seroepideiological Studies. PLoS ONE 6(3): e doi: / journal.pone This Article is brought to you for free and open access by the School of Public Health at Georgia State University. It has been accepted for inclusion in Public Health Faculty Publications by an authorized adinistrator of Georgia State University. For ore inforation, please contact scholarworks@gsu.edu.

2 Did Modeling Overestiate the Transission Potential of Pandeic (H1N1-2009)? Saple Size Estiation for Post-Epideic Seroepideiological Studies Hiroshi Nishiura 1,2,3 *, Gerardo Chowell 4,5, Carlos Castillo-Chavez 4,6 1 PRESTO, Japan Science and Technology Agency, Saitaa, Japan, 2 Theoretical Epideiology, University of Utrecht, Utrecht, The Netherlands, 3 School of Public Health, The University of Hong Kong, Hong Kong, China, 4 Matheatical and Coputational Modeling Sciences Center, School of Huan Evolution and Social Change, Arizona State University, Tepe, Arizona, United States of Aerica, 5 Fogarty International Center, National Institutes of Health, Bethesda, Maryland, United States of Aerica, 6 Santa Fe Institute, Santa Fe, New Mexico, United States of Aerica Abstract Background: Seroepideiological studies before and after the epideic wave of H1N are useful for estiating population attack rates with a potential to validate early estiates of the reproduction nuber, R, in odeling studies. Methodology/Principal Findings: Since the final epideic size, the proportion of individuals in a population who becoe infected during an epideic, is not the result of a binoial sapling process because infection events are not independent of each other, we propose the use of an asyptotic distribution of the final size to copute approxiate 95% confidence intervals of the observed final size. This allows the coparison of the observed final sizes against predictions based on the odeling study (R = 1.15, 1.40 and 1.90), which also yields siple forulae for deterining saple sizes for future seroepideiological studies. We exaine a total of eleven published seroepideiological studies of H1N that took place after observing the peak incidence in a nuber of countries. Observed seropositive proportions in six studies appear to be saller than that predicted fro R = 1.40; four of the six studies sapled seru less than one onth after the reported peak incidence. The coparison of the observed final sizes against R = 1.15 and 1.90 reveals that all eleven studies appear not to be significantly deviating fro the prediction with R = 1.15, but final sizes in nine studies indicate overestiation if the value R = 1.90 is used. Conclusions: Saple sizes of published seroepideiological studies were too sall to assess the validity of odel predictions except when R = 1.90 was used. We recoend the use of the proposed approach in deterining the saple size of post-epideic seroepideiological studies, calculating the 95% confidence interval of observed final size, and conducting relevant hypothesis testing instead of the use of ethods that rely on a binoial proportion. Citation: Nishiura H, Chowell G, Castillo-Chavez C (2011) Did Modeling Overestiate the Transission Potential of Pandeic (H1N1-2009)? Saple Size Estiation for Post-Epideic Seroepideiological Studies. PLoS ONE 6(3): e doi: /journal.pone Editor: Alessandro Vespignani, Indiana University at Blooington, United States of Aerica Received Deceber 14, 2010; Accepted February 15, 2011; Published March 24, 2011 Copyright: ß 2011 Nishiura et al. This is an open-access article distributed under the ters of the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original author and source are credited. Funding: HN was supported by the Japan Science and Technology Agency PRESTO progra. GC received financial support fro the College of the Liberal Arts and Sciences of Arizona State University. National Science Foundation (NSF - Grant DMS ), U.S. Departent of Defense (NSA - Grant H ), the Alfred T. Sloan Foundation and the Office of the Provost of Arizona State University support CCC s research. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the anuscript. Copeting Interests: The authors have declared that no copeting interests exist. * E-ail: nishiura@hku.hk Introduction Influenza A (H1N1-2009) caused the first influenza pandeic of the twenty-first century [1]. A substantial fraction of the world population has probably been infected already with this virus, but a direct estiation of the infected fraction of the population is not feasible by relying only on available epideiological case data (e.g. surveillance data consisting of confired cases or influenzalike illness cases). In particular, influenza is known to involve asyptoatic infections [2], and disease severity tends to be selfliiting aong healthy individuals who often do not require edical attention. Moreover, due to the non-specific nature of syptos, influenza-like illness is insufficient to confir or exclude the diagnosis of influenza [3]. Therefore, seroepideiological studies before and after an epideic wave are crucial for estiating the population attack rate (i.e. infected fraction of a population) [4], here also referred to as the final size or the proportion of infected individuals in a population at the end of an epideic. In addition, population-wide seroepideiological surveys are useful for onitoring epideiological dynaics in realtie, assessing effectiveness of certain interventions [5], and deterining prioritization strategies of vaccination during the course of an epideic (e.g. identifying subpopulations that should be vaccinated at particular ties during an ongoing epideic) [6,7]. Both serological and epideiological odeling studies have increased our understanding of the transission dynaics of H1N fro the beginning of the pandeic [4,8]. In particular, the reproduction nuber, R, defined as the average nuber of secondary cases generated by a single priary case PLoS ONE 1 March 2011 Volue 6 Issue 3 e17908

3 Saple Size for Post-Epideic Serological Studies throughout its entire course of infection [9], was estiated using epideiological data during the early stages of the pandeic. One of the iportant features of R is its potential to provide early and crude predictions of the expected final epideic size [10]. For instance, the frequently cited initial estiate for H1N is R = 1.40 [8], and the final size equation of any hoogeneously ixing odel (with an initially fully susceptible population) predicts that 51.1% of the population would experience infection by the end of the epideic (see next section). Nevertheless, several seroepideiological studies have suggested that the infected fraction was likely to be saller than 51.1% [11], a result that has led researchers to speculate on additional (often unforeseen) echaniss or factors influencing the transission dynaics. Hence, seroepideiological studies play a key role in validating crude predictions based on R. Further, whenever the observed (saple) final size is saller than that based on R, the use of seroepideiological studies ay provide indirect evidence of the positive effect of particular public health interventions. A glance at the literature shows that various seroepideiological studies published so far have adopted a binoial sapling process to quantify the uncertainty of the proportion of infected individuals (e.g. [12,13]). Accordingly, the confidence intervals of the proportion have also been derived fro a binoial distribution using exact or approxiate ethods [6,14,15]. Perhaps one of the ain reasons for widespread use of the binoial proportion in this context can be attributed to a well-known and siple forula for the saple size deterination of the binoial proportion [16]. Nevertheless, it should be noted that H1N is transitted fro huan to huan, and the risk of infection in one individual depends on other individuals in the sae population unit. This highlights the need to account for the so-called dependent happening [17,18]. Moreover, an observed final size represents a single stochastic realization aong all possible saple paths of the epideic, indicating a need to explicitly account for deographic stochasticity. These issues call for a foral fraework for deterining the saple size of post-epideic seroepideiological studies. The purpose of the present study is to introduce an approxiate ethod for the coputation of the uncertainty bound of the final epideic size, which also perits us to discuss siple ethods for saple size calculations. We reanalyze published datasets of postpeak seroepideiological studies of H1N and explicitly test if early estiates of R for H1N indicated a biased estiate of the final epideic size. Materials and Methods Seroepideiological data As a way to otivate our study, we start by presenting suary results of the seroepideiological studies of H1N Table 1 suarizes a total of eleven seroepideiological studies that were conducted after observing peak incidence of H1N in various populations [6,7,11 15,19 22]. If the epideic curve revealed a ultiodal distribution with clearly distinct peaks, the post-peak datasets can either be after the first wave (e.g. England [14], but we restrict our interest to London and the West Midland, because other areas were far less affected) or after the second wave (e.g. USA [13]). The ajority of studies sapled seru fro hospital laboratory, registered patients at clinics or blood donors, except for a defined cohort population in Singapore [22] and a Table 1. Post-peak seroepideiological studies of pandeic influenza (H1N1-2009) aong a general population. Country Survey location Subjects{ Saple size{ Prop before (%){ Prop after (%){ Sapling period{ After peak1 Vac Lab ethod" Australia [19] New South Wales Clinical cheistry laboratories Canada [15] British Colubia Patient service center China (1) [11] Beijing Blood donors and Patients China (2) [6] Hong Kong Blood donors, pediatric cohort Aug Sep 09 Yes No HI$ * May 10 Yes Yes HI$40 & MN$ * Nov Dec 09 No Yes HI$ Nov Dec 09 Yes No MN$40 Gerany [7] Frankfurt Hospitalized adults 225 * Nov 09 No No HI$40 India [20] Pune School children & Sep Oct 09 No No HI$40 general population Japan [21] entire Japan Healthy individuals Jul Sep 10 Yes Yes HI$40 New Zealand [12] Auckland region Registered patients Nov 09 Mar 10 Yes Yes HI$40 Singapore [22] Singapore Adult cohort Oct 09 Yes No HI ($4 fold rise) UK [14] England Patients accessing Sep 09 No No HI$32 health care USA [13] Pittsburgh Clinical laboratories Nov 09 No Yes HI$40 { Subjects, saple size and sapling period refer to those after observing the peak incidence of H1N For several studies exaining pre-existing iunity, the sae or additional saples before the 2009 pandeic were investigated at different tie periods, but are not included in this Table. { Estiated proportions seropositive before and after observing an epideic peak. When age-standardized estiate was given in the original study, we used it as the population ean. *Three studies did not estiate the proportion seropositive before the 2009 pandeic, and we assue that 7.5% of the population was initially iune based on a crude average aong other studies. 1 After peak colun represents if the sapling took place longer than 1 onth after observing the highest incidence of cases. Vaccination colun represents if a population-wide vaccination capaign of H1N took place prior to the sapling. " Laboratory ethods to deterine seropositivity; HI, heagglutination inbibition assay and MN, icroneutralization assay. doi: /journal.pone t001 PLoS ONE 2 March 2011 Volue 6 Issue 3 e17908

4 Saple Size for Post-Epideic Serological Studies saple of study volunteers of the general Japanese population [21]. Only the Japanese study has not been published in English; the data are based on National Epideiological Surveillance of Vaccine-Preventable Diseases which are annually conducted to understand the epideiological dynaics of a nuber of infectious diseases, involving at least 5,400 non-randoly sapled individuals across all age-groups in each year and covering 24 prefectures (225 individuals per prefecture) aong a total of 49 prefectures across Japan. Other published serological surveys were not included in Table 1, because they were conducted before the observed epideic peak or because they focused on a confined population (e.g. healthcare workers or ilitary personnel) [5,23 27], but a few of the have been discussed elsewhere [4]. The saple size of the eleven seroepideiological studies, which recorded post-peak seroprevalence, ranged fro 225 to 6035 individuals. Eight studies exained seroprevalence before the first wave, estiating the proportion of the population with preexisting iunity (Table 1). Where indicated, the saple size estiation of those studies relied on a binoial proportion [12 14,19]. The post-peak sapling period varied substantially with, for exaple, six studies sapling the post-peak seru ore than 1 onth after the peak incidence. Five studies clearly stated that a population-wide vaccination capaign against H1N had taken place prior to sapling. The laboratory ethod eployed in these studies was based on heagglutination inhibition assays (HI) or icroneutralization assays (MN) with eight studies setting the seropositive threshold level at HI$40. It is practically very difficult to deterine the end of an epideic, and thus, we regard the observed increase in seroprevalence (i.e. seroprevalence after the peak inus that before the peak) as an estiate of the fraction of infected individuals during the epideic. We used the agestandardized final size estiate for an entire population when given in the original study instead of using crude estiates of the seropositive fraction. The 2009 pandeic involved public health interventions, heterogeneous transission (e.g. age and spatial heterogeneities) and seasonality, but, as the first step to stiulate a relevant discussion on this subject, the present study adopts a hoogeneously ixing assuption without tie-dependent dynaics. Specifically, we focus on the difference between the observed final sizes for an entire population and the predictions of final size yielded by the odeling approach. Thus, the data in Table 1 are analyzed here under the assuption of a well-ixed population. It should be noted that, in the absence of any tiedependent factors, the final size is known to depend only on the reproduction nuber R, under the hoogeneous ixing assuption [9,10]. Following the earliest studies in Mexico [8,28], the estiation of R was conducted using the early epideic growth data in different locations across the world (yielding published estiates in 2009 [29 38], soe reassessed [39]). The estiated R, in different epideic settings and subpopulations, ranged fro less than 1 [40] to greater than 2 [28,29,35]. The definition of R also varied fro study to study. One study, for exaple, incorporated the ipact of seasonal variations in the force of infection [33]. Aong these, the earliest estiate of R was derived fro the early phase of the pandeic during the Spring 2009 in Mexico using various odeling ethods [8]. Using a Bayesian ethod, the posterior edian of R (and the 95% credible intervals) was estiated at 1.40 (1.15, 1.90) [8]. Since the posterior edian crudely represents id-point of estiates in other published studies, and because the lower and upper bounds roughly correspond to the range of R in other studies (with R,2), we focus on an estiate of R derived fro an exponential growth of cases in an outbreak in La Gloria, Mexico. Thus, we not only assess the prediction based on R = 1.40, but also on the lower and upper bounds of R. Note that the lower bound (1.15) is saller than the posterior edian of R obtained using other ethods in the sae study including a coalescent population genetic analysis (R = 1.22). Given an estiate of R for an initially fully susceptible population, and assuing that the initial nuber of infectives is sufficiently saller than the total population size, the final epideic size r satisfies 1{r~exp({ ^Rr), which is referred to as the final size equation [10]. Both sides of equation (1) represent the probability that an individual escapes infection throughout the course of an epideic. Since the presence of pre-existing iunity has yet to be clarified at the beginning of the 2009 pandeic, we use equation (1) to calculate the predicted final epideic size. Iteratively solving (1) for R being 1.15, 1.40 and 1.90, the final size r is 24.9%, 51.1% and 76.7%, respectively. We test these forecasts against the observed final sizes given in Table 1. For this reason, it is essential to copute uncertainty bounds (e.g. 95% confidence interval) of the observed final sizes in seroepideiological studies. Uncertainty bound for a binoial proportion As a prelude to discussing the uncertainty bound of final size, we first consider the confidence interval of a binoial proportion, which has been widely used in published seroepideiological studies shown in Table 1. Let X be a binoial rando variable for saple size n, and let r = X/n be the saple proportion positive. The ost well-known, parsionious, confidence interval of the binoial proportion, eploys a noral approxiation to binoial distribution, which is also referred to as the Wald confidence interval. The 100(1-2a)% confidence interval for the saple proportion r is written as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^r(1{^r) ^r+z a, ð2þ n where z a denotes 1-a quantile of the standard noral distribution (e.g. z a <21.96 for a = 0.025). The rules of thub suggest that the noral approxiation works well as long as nr.5 and n(1- r).5, but the rules of thub do not always work out well [41]. The coputation of the Wilson score interval is a better alternative, which is not coputationally difficult and yields better coverage of associated uncertainty [42,43]. Here, we focus on the Wald confidence interval in the present study, because we extend its principle to the coputation of the 95% confidence interval of the final epideic size. The idea behind the Wald confidence interval coes fro inverting the Wald test for r. Suppose that the null hypothesis H 0 :r = r 0 is tested where one wishes to detect a relevant alternative H 1 :r?r 0, where r 0 is the proposed value of the proportion. In the case of the prediction with R = 1.40, r 0 ight be set at (assuing that the final size follows a binoial distribution). The Wald statistic to be copared to a noral distribution is given by ^r{r 0 s:e:(^r), ð3þ where s.e.(^r) is the standard error of r, approxiated by the square root ter in (2). The saple size estiation of a binoial proportion can also eploy (3). In fact, if we let denote the argin of error, a ð1þ PLoS ONE 3 March 2011 Volue 6 Issue 3 e17908

5 Saple Size for Post-Epideic Serological Studies suary of sapling error that quantifies uncertainty, which corresponds to half the width of a confidence interval for the proportion r, then a desired argin of error of no ore than eans z a s:e:(^r)ƒ: By squaring both sides and using the approxiate standard error, we have z 2 ^r(1{^r) a ƒ 2 : n Solving equation (5) for n gives n z a 2^r(1{^r), ð6þ a well-known forula for estiating the iniu saple size n for a binoial proportion. Since the eventual r is unknown before the actual survey, one ay set r = or use a published seroprevalence estiate. It should be noted that equation (6) does not explicitly account for Type II error (i.e. power of the test) [44]. Hence, to incorporate the power in calculating the saple size, one can alternatively eploy the following forula ([45]): ð4þ ð5þ n z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 a ^r(1{^r) zz b (^rz)(1{^r{) : ð7þ Coparing (6) and (7), it is seen that the saple size n based on (6) corresponds to the case for a power of 50% in (7) (i.e. z b = z 0.5 =0). Uncertainty bound for a final epideic size An explicit derivation of final size distribution, which eploys a recursive equation, has been carried out through the so-called Sellke construction in a series of stochastic epideic odeling studies [46,47]. In addition, a nuber of stochastic odeling studies in the context of large populations have exained the asyptotic distribution of the final epideic size via the central liit theore [48,49]. Within a stochastic odeling fraework, it is known that an outbreak declines to extinction without causing a large epideic with a probability of extinction p (sall outbreaks are referred to as inor epideic). A ajor epideic occurs with probability 1-p. An approxiate standard error of the final size of the ajor epideic based on the asyptotic convergence result of the final size distribution is ([50,51]): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r(1{r)zr 2 s 2 u r(1{r) 2 s:e:(r)~ t N½1{R(1{r) Š 2, ð8þ where r now represents the observed final size and possibly the unique positive solution to (1) in case of an initially fully susceptible population. R is the reproduction nuber while and s denote the ean and standard deviation of the generation tie (and thus, s/ is the coefficient of variation (CV)), and N is the population size. This approxiation has been evaluated elsewhere [50,51]. If a proportion q of the population is initially iune, the reproduction nuber R estiated fro an exponential growth of cases in that population satisfies ([10]): { ln 1{ r R~ : ð9þ r The estiated R (e.g. in the range of 1.15 to 1.90 in Mexico) is not the basic reproduction nuber R 0 in a fully susceptible population, but satisfies R 0 = R/(1-q) [9]. Using the estiator of R in (9), the standard error in (8) can be rewritten as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 3 (1{r)z s 2 r(1{r) 2 ln 2 1{ r s:e:(r)~ u N rz(1{r)ln 1{ r 2 : ð10þ t Given that q and the CV of the generation tie are now known for H1N1-2009, the Wald confidence interval can eploy (10) for coputing the corresponding 95% confidence interval, for hypothesis testing and for estiating the iniu saple size required for post-epideic seroepideiological studies. One should bear in ind that the error estiate is nevertheless conservative (i.e. likely to be underestiated), because (i) the ethod is based on noral approxiation, (ii) we ignore tie-dependent dynaics including public health interventions, and (iii) we ignore heterogeneous transission (see Discussion for (ii) and (iii)). N is the population size in the above expressions. If we wish to replace N by saple size n, the binoial sapling error of n has to be accounted in the calculation of the variance. In the case of siple rando sapling, the resulting standard error is given by the su of the respective variance of two independent processes, i.e. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n(n{n) N 3 zs:e: 2 (r; n) ð11þ where n(n-n)/n 3 is an approxiate variance of the binoial sapling error, and s.e.(r;n) is the standard error of final size when the sapling error linked to n is ignored(i.e. what we replace N by n in equation (10)). The introduction of sapling error also applies to the standard error of the binoial proportion in (2), but this ter is usually ignored for very large N (because n(n-n)/n 3 is then negligibly sall) under an assuption that the randoly selected individuals sufficiently represent the entire population. Thus, we use only s.e.(r;n) in the following analyses. If n involves non-negligible fraction of N (e.g..5%), one ay use the above expression (11) or introduce the so-called finite population correction factor (FPC) for the calculation of the error [52]. Given an observed final size r, the 100(1-2a)% confidence interval for r is calculated as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^r 3 (1{^r)z s 2^r(1{^r) 2 ln 2 1{ ^r ^r+z a u n ^rz(1{^r)ln 1{ ^r 2 : ð12þ t Suppose that we have an unbiased estiate of q and a known CV of the generation tie (e.g. fro separate datasets). To copare the observed final size r against the prediction based on R = 1.40, r 0 would be 0.511, with the Wald statistic copared to a noral distribution given by PLoS ONE 4 March 2011 Volue 6 Issue 3 e17908

6 Saple Size for Post-Epideic Serological Studies Results ^r{r v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ^r 3 (1{^r)z s 2^r(1{^r) 2 ln 2 1{ ^r : ð13þ u n ^rz(1{^r)ln 1{ ^r 2 t p ffiffiffi Let n(r)~s:e:(r) n. The iniu saple size which explicitly accounts for only Type I error is calculated fro n z a 2n(^r) 2 : ð14þ If we account for both Type I and II errors, we have n z an(^r)zz b n(^rz) 2 : ð15þ It should be noted that the ethod used to account for the power (equation (15)) can only exaine the range of r,1-q- because the approxiate standard error of final size includes the logarithic function. Application and illustration To highlight the iportance of explicitly accounting for the variance of the final size distribution, the following two exercises are perfored. First, we exaine post-peak seroepideiological studies of H1N1-2009, coparing the 95% confidence intervals generated by two ethods; binoial proportion and asyptotic final size distribution. For this reason, when calculating the uncertainty bounds, we regard the data as if they were generated fro a binoial process or the final epideic size of a hoogeneously ixing population. For siplicity, we assue that we have an unbiased estiate of the proportion of population with pre-existing iunity based on the observed seropositive proportion prior to the epideic wave in Table 1. We consider uncertainty of the observed final size, which corresponds to the difference in infected fraction before and after observing the peak incidence. Subsequently, we test the significance of the observed final size against odel predictions (i.e. 24.0%, 51.1% and 76.7% based on R = 1.15, 1.40 and 1.90, respectively). The ean and standard deviation of the generation tie are fixed at 2.7 and 1.1 days, respectively (and so, the CV is 0.41) based on contact tracing data in the Netherlands [40]. To address the uncertainty with respect to the shape and scale of the generation tie distribution, we also consider hypothesis testing of two other scenarios in which the CV is 0 (i.e. a constant generation tie) and 1 (i.e. exponentially distributed generation tie). Second, as sensitivity analysis of the selected epirical illustrations, we present the desired iniu saple size of final epideic size by eploying the approxiate standard error of the final size. Exaining various argins of error ranging fro 0% to 50% with R being 1.15, 1.40 and 1.90 and the CV of the generation tie ranging fro 0 to 1, the above entioned forulae (14) and (15) are used with significance level at a = 0.05 and, for the latter forula, the power is set at 12b = Moreover, for this sensitivity analysis the proportion of the population with pre-existing iunity q is fixed at 7.5%, which corresponds to the ean based on eight published studies in Table 1. Subsequently, we also exaine the sensitivity of the iniu saple size required as a function of R and q. Confidence intervals Table 2 suarizes the epirical results of eleven seroepideiological studies of H1N The saple proportion infected ranged fro 4.5% to 38.5%. The sallest three final sizes resulted fro saples within 1 onth after observing peak incidence, and the largest three involved a population-wide vaccination capaign prior to the survey. Whereas the 95% confidence interval of the binoial proportion was narrow with the standard errors ranging fro 0.6% to 1.6%, the 95% confidence interval of final size was uch broader ranging fro 6.6% to 76.9%, which led to include 0% within the confidence liits of seropositive in nine studies, calling for ad-hoc truncation (or calling for an alternative ethod of coputation that ay include the F distribution). The broader uncertainty bound fro the odel-based final size than the binoial proportion can be analytically deonstrated as follows. First, the sallest standard error in (12) is seen when the CV of the generation tie is 0, i.e., vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^r ^r+z 3 (1{^r) a u n ^rz(1{^r)ln 1{ ^r 2 : ð16þ t Because 0#r#1 and 0#q#1, we have Therefore, it is proven that ^r ^rz(1{^r)ln 1{ ^r 1: ð17þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^r 3 (1{^r) ^r(1{^r) u n ^rz(1{^r)ln 1{ ^r t 2 : ð18þ n The equality holds when r =1. Hypothesis testing Assuing CV of the generation tie at 0.41, six serological studies appeared to have yielded significantly saller final sizes than that predicted by R = 1.40 (Table 2). Nevertheless, four of the six studies sapled seru within 1 onth after observing peak incidence, and four of the reaining five studies with insignificant result sapled seru longer than 1 onth after the peak (no significant association between the significant test result and sapling within 1 onth after the peak; p = 0.24, Fisher s exact test). Populations in four of the six studies with significantly saller final sizes were unvaccinated prior to sapling, and three of the five studies with insignificant results involved vaccination prior to the survey (p = 0.57, Fisher s). Taken together, five of the six studies with significantly saller final sizes sapled seru within 1 onth after peak incidence or exained unvaccinated population, while all the five reaining studies with insignificant test results conducted sapling longer than 1 onth after the peak or the population involved vaccination (p = 0.55, Fisher s). When coparing observed final sizes against R = 1.15, results of all studies were not found to be significantly different. Eight studies indicated that the observed final sizes were significantly saller than that predicted by R = Varying the CV of the generation tie fro 0 to 1 with R = 1.40, the significance levels with CV = 0 did not vary fro those of CV = 0.41, but the results with CV = 1 indicate PLoS ONE 5 March 2011 Volue 6 Issue 3 e17908

7 Saple Size for Post-Epideic Serological Studies Table 2. Uncertainty bounds and hypothesis testing of the post-peak seroepideiological studies of influenza (H1N1-2009). Country Saple size{ Prop infected (%){ 95% CI of binoial prop (%){ 95% CI of final size (%){ After peak1 Vac P-values$ R CV Australia [19] , , 50.2 Yes No * *,0.01 * Canada [15] , , 60.6 Yes Yes *, China (1) [11] , 8.1 0, 46.8 No Yes * *,0.01 * China (2) [6] , , 67.8 Yes No * Gerany [7] , 7.3 0, 56.0 No No * *,0.01 * India [20] , , 30.1 No No *, *,0.01 *,0.01 *,0.01 Japan [21] , , 45.6 Yes Yes *, *,0.01 *,0.01 *0.02 New Zealand [12] , , 81.4 Yes Yes * Singapore [22] , , 94.4 Yes No UK [14] , , 35.3 No No *, *,0.01 *,0.01 *0.01 USA [13] , , No Yes { Saple size refers to the nuber of enrolled subjects to easure the seroprevalence after observing an epideic peak. Proportion infected is given by the proportion after observing peak inus the proportion before the peak in Table 1. { 95% confidence intervals (CI) show lower and upper confidence intervals of the proportion. The 95% CI of binoial proportion is derived fro a noral approxiation to binoial distribution, while the 95% CI of final size is siilarly derived fro the Wald ethod eploying asyptotic convergence result of final size distribution. 1 After peak colun represents if the sapling took place longer than 1 onth after observing the highest incidence of cases. Vaccination colun represents if a population-wide vaccination capaign of H1N took place prior to the sapling. $ p-values are based on two-sided Wald test eploying the approxiate standard error of final epideic size. R, the estiated reproduction nuber in Mexico against which we would like to test our hypothesis; CV, the coefficient of variation of the generation tie. Significant difference is indicated by * ark followed by p-value. doi: /journal.pone t002 that only three observed final sizes were significantly saller than that predicted by R =1.40. Saple size estiation Figure 1 shows the iniu saple sizes required for postepideic seroepideiological studies to test the final size against R = 1.15, 1.40 and 1.90 with CV being 0, 0.41 and 1. Whereas edian (and lower and upper quartiles) saple size of epirical studies in Table 1 was 1127 (710, 2913), such saple sizes can only explicitly prove a difference fro the prediction of R = 1.90 at a argin of error 5%. To argue the significant difference fro prediction based on R = 1.40 with the identical argin of error and with varying CV of the generation tie 0.41 (range: 0, 1), we ideally need 8665 (range: 7215, 15947) individuals at the power of 50% and (13423, 29680) individuals at the power of 80%. At the argin of error 10%, these nubers are reduced to 2167 (1804, 3987) and 3715 (3093, 6841), respectively. As R gets closer to the lower uncertainty bound, and as the variance of the generation tie becoes larger relative to the ean, the iniu saple size required increases. Figure 2A exaines the sensitivity of the iniu saple size to the reproduction nuber R. Ignoring pre-existing iunity (q = 0), R = 2 with the CV of the generation tie 0.41 (0, 1) requires at least 201 (177, 320) individuals at power of 50% and 317 (281, 500) individuals at power of 80%. As R is reduced and approaches the critical level, uch greater saple sizes are required. For instance, the iniu saple size for R = 1.2 is ore than 2-fold higher than that required for R = 1.4. Figure 2B illustrates the relationship between iniu saple size and the proportion of the population with pre-existing iunity q (with fixed R = 1.40). Interestingly, the iniu saple size hits the largest value around q = For exaple, q = yielded the largest saple size with CV = 0. This can be inspected by taking first and second derivatives of (16) with respect to q (with the CV = 0), leading to: ^r q ax ~1{, ð19þ ^r 1{exp ^r{1 which is the ost difficult situation in which the hypothesis testing against the predicted final size requires us to collect an unrealistically large nuber of blood saples. q ax leads the denoinator of the approxiate standard error in (16) to be 0. Discussion We have introduced a fraework to copute the uncertainty bounds of the final epideic size that eploys the Wald approxiation, an approach otivated by the absence of a readily available ethodology to estiate the saple size of postepideic seroepideiological studies. Published seroepideiological studies of H1N so far have coputed the confidence interval of the observed final size as if it were a binoial proportion. However, the data generating process behind the dynaics of infectious diseases involves dependence between infected individuals [17], which does not lead to a binoial proportion. Moreover, the observed final size represents a single stochastic realization aong all possible saple paths (i.e. all PLoS ONE 6 March 2011 Volue 6 Issue 3 e17908

8 Saple Size for Post-Epideic Serological Studies Figure 1. Miniu saple sizes required for post-epideic seroepideiological studies of final size as a function of the argin error, the reproduction nuber, and the coefficient of variation of the generation tie. (A & B) Saple size with three different reproduction nubers as a function of the argin of error. (A) eploys an estiation forula based Type I error alone (at a = 0.05), while (B) accounts for both Type I and II errors (at a = 0.05 and 12b = 0.80). The argin of error represents rando sapling error, around which the reported percentage would include the true percentage. Since (A) is a special case of (B) (with b = 0.50), R = 1.40 in (A) is also shown as dotted line in (B). The coefficient of variation (CV) of the generation tie and the proportion of population with pre-existing iunity are fixed at 40.7% and 7.5%, respectively. (C & D) Saple size with three different coefficients of variation as a function of the argin of error. (C) accounts for Type I error alone (a = 0.05), while (D) accounts for both Type I and II errors (a = 0.05 and 12b = 0.80). The reproduction nuber and the proportion of population with pre-existing iunity are fixed at 1.40 and 7.5%, respectively. CV = 0 corresponds to a constant generation tie, whereas CV = 1 represents an exponentially distributed generation tie. In (B) and (D), several lines are truncated, due to ipossibility to account for larger argins of error in the estiation forula. doi: /journal.pone g001 possible probabilistic trajectories of the epideic), requiring us to consider stochastic variations in the data. To account for these issues, we eployed the approxiate standard error of the final size given as a convergence result of a hoogeneously ixing stochastic epideic odel. The calculation of the standard error was shown to be siple to copute (spreadsheet progras are sufficient). By applying the proposed uncertainty bound of final size to influenza (H1N1-2009), we have also shown that all the seroepideiological studies published to date did not necessarily indicate an overestiation of prediction based on R = 1.40, and oreover, all the observed final sizes did not reveal significant deviation fro prediction with the lower liit R = Published seroepideiological studies agree that the upper bound R = 1.90 (and thus, other published estiates of R.2 [29,30]) was likely an overestiation [39]. One ay still speculate that R = 1.40 ay well be an overestiation (because all of the observed final sizes were PLoS ONE 7 March 2011 Volue 6 Issue 3 e17908

9 Saple Size for Post-Epideic Serological Studies Figure 2. Sensitivity of iniu saple size for post-epideic seroepideiological studies to the reproduction nuber and the proportion of population with pre-existing iunity. (A). The iniu saple size with three different coefficients of variation (CVs) as a function of the reproduction nuber. (B). The iniu saple size with three CVs as a function of the proportion of population with pre-existing iunity. In (A), the proportion of population with pre-existing iunity is fixed at 0, and the estiates correspond to the argin of error of 10% and Type I and II errors at a = 0.05 and 12b = 0.50, respectively. In (B), the reproduction nuber is fixed at 1.40, and the estiates correspond to the argin of error of 10% and Type I and II errors at a = 0.05 and 12b = 0.50, respectively. doi: /journal.pone g002 saller than 51.1%), but the saple sizes of published seroepideiological studies turned out to be too sall to answer this question. Although forulae for variance of the final size distribution (i.e. the square root of which we regarded as an approxiate standard error) has been known aong stochastic odeling experts [50], the present study extended its use to the coputation of the 95% confidence interval of the observed final size by replacing the reproduction nuber by its estiator. This also led us to consider a parsionious Wald test and saple size estiation. What the present study suggests for post-epideic seroepideiological studies is to eploy the proposed forula (12) to calculate the 95% confidence interval and (14) or (15) to help deterine the saple size for seroepideiological surveys. For the latter, the following siplification of (14) ight be useful: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n(^r) 2 ^r 3 (1{^r)z s 2^r(1{^r) 2 ln 2 1{ ^r s:e:(^r) ~ s:e:(^r) ^rz(1{^r)ln 1{ ^r : ð20þ The standard error s.e.(^r) is calculated by using the specified confidence interval (i.e. twice the argin of error) and the confidence level (i.e. noinal coverage probability). For instance, if the argin of error is 5% and the confidence level is 95%, the standard error is 0.05/1.96 = Siilarly, the standard error is and at the confidence levels of 90% and 99%, respectively. It is worth stressing that the purpose of post-epideic seroepideiological studies is not necessarily to test the observed final size against a predicted value, but includes real-tie onitoring of an epideic and various considerations of public health interventions. As long as there is no better alternative ethod for coputing the uncertainty, the proposed approach should also be used for those other purposes to calculate conservative uncertainty bounds. The proposed ethod has a potential for explicitly discussing a posteriori effectiveness of interventions through the direct coparison of observed final sizes in different settings. Hence, we believe that the proposed calculation of the 95% confidence interval will greatly help progressing this area of research. It should also be noted that the use of the proposed uncertainty bounds plays an iportant role especially for influenza transission with R,2 (Figure 2A). Our illustration of the proposed ethod posed four technical challenges for the coputation of the uncertainty bound of final size; (i) the coefficient of variation of the generation tie has to be known, (ii) the proportion of pre-existing iunity before an epideic critically influences the bounds, (iii) sapling of several seroepideiological studies took place shortly after an epideic peak and (iv) vaccination and other public health interventions during the course of an epideic can odify the observed final size. As for (i), the present study deonstrates a critical need to estiate the variance of the generation tie in addition to the ean. That is, the distribution of the generation tie plays a key role not only in estiating R [53,54] but also in characterizing the variance of final epideic size. With respect to (ii), although we did not include seroepideiological studies prior to the 2009 pandeic [24,25,27], we have shown that such a survey of q is a key to deterine the saple size after the epideic [55]. In addition to the estiation of q itself, it should be noted that our ethod adopted an assuption that the pre-existing iunity offered a coplete protection fro infection (i.e. all-or-nothing protection). If the pre-existing iunity is iperfect and described by the so-called leaky protection (e.g. partial reductions in susceptibility per contact and in infectiousness upon infection), those quantifications will be required in addition to the estiation of the proportion of the initially iune population. Issues (iii) and (iv) pose further technical challenges to precisely estiate uncertainty bounds of seroprevalence in epirical studies. Given PLoS ONE 8 March 2011 Volue 6 Issue 3 e17908

10 Saple Size for Post-Epideic Serological Studies that the observation of incidence is given in every discrete tie unit, a possible way forward ay be to eploy a parsionious discrete tie stochastic odel (e.g. branching process or chain binoial odel) [56], which ay well enable us to draw the 95% confidence interval in a given reporting interval by conditioning the distribution to previous reporting intervals. Proposing siple ethods to address these issues is part of our future studies. Our ethod relied on the hoogeneous ixing assuption and ignored tie dependent factors that include seasonality and public health interventions. In this sense, the proposed uncertainty is regarded as an underestiate, because the tie-dependent variations in the transission potential can increase the variance of the final size distribution, and also because heterogeneous transission (e.g. age-dependent ixing) can also increase variance (e.g. an epideic with extreely high assortativity could generate ultiodal final size distribution for an entire population [57]). If an intervention is focused only on a portion of cases or if disease-induced deaths occur in non-negligible order, not only the variance but also the forulae for the final size relation (our equation (1)) have to be reassessed [58 60]. Moreover, in the presence of strong seasonality, a deterinistic odeling study has deonstrated a very liited predictive perforance of R alone in anticipating the final epideic size [61,62]. Given that seroepideiological studies tend to stratify population by age-group (to capture the age-dependency of the risk of infection), and considering that the final size of age-structured odels can be different fro that of hoogeneous population [63], further work could at least incorporate heterogeneous ixing by eploying the existing siilar convergence result of the final size distribution References 1. Neuann G, Noda T, Kawaoka Y (2009) Eergence and pandeic potential of swine-origin H1N1 influenza virus. Nature 459: Carrat F, Vergu E, Ferguson NM, Leaitre M, Caucheez S, et al. (2008) Tie lines of infection and disease in huan influenza: a review of volunteer challenge studies. A J Epideiol 167: Call SA, Vollenweider MA, Hornung CA, Siel DL, McKinney WP (2005) Does this patient have influenza? JAMA 293: Anonyous (2010) Seroepideiological studies of pandeic influenza A (H1N1) 2009 virus. Wkly Epideiol Rec 85: Lee VJ, Yap J, Cook AR, Chen MI, Tay JK, et al. (2010) Effectiveness of public health easures in itigating pandeic influenza spread: a prospective seroepideiological cohort study. J Infect Dis 202: Wu JT, Ma ES, Lee CK, Chu DK, Ho PL, et al. (2010) The infection attack rate and severity of 2009 pandeic H1N1 influenza in Hong Kong. Clin Infect Dis 51: Allwinn R, Geiler J, Berger A, Cinatl J, Doerr HW, et al. (2010) Deterination of seru antibodies against swine-origin influenza A virus H1N1/09 by iunofluorescence, haeagglutination inhibition, and by neutralization tests: how is the prevalence rate of protecting antibodies in huans? Med Microbiol Iunol 199: Fraser C, Donnelly CA, Caucheez S, Hanage WP, Van Kerkhove MD, et al. (2009) Pandeic Potential of a Strain of Influenza A (H1N1): Early Findings. Science 324: Diekann O, Heesterbeek JAP (2000) Matheatical Epideiology of Infectious Diseases: Model Building, Analysis and Interpretation. New York: Wiley. 10. Ma J, Earn DJ (2006) Generality of the final size forula for an epideic of a newly invading infectious disease. Bull Math Biol 68: Deng Y, Pang XH, Yang P, Shi WX, Tian LL, et al. (2010) Serological survey of 2009 H1N1 influenza in residents of Beijing, China. Epideiol Infect;in press (doi: /S ). 12. Bandaranayake D, Huang QS, Bissielo A, Wood T, Mackereth G, et al. (2010) Risk factors and iunity in a nationally representative population following the 2009 influenza A(H1N1) pandeic. PLoS One 5: e Zier SM, Crevar CJ, Carter DM, Stark JH, Giles BM, et al. (2010) Seroprevalence following the second wave of Pandeic 2009 H1N1 influenza in Pittsburgh, PA, USA. PLoS One 5: e Miller E, Hoschler K, Hardelid P, Stanford E, Andrews N, et al. (2010) Incidence of 2009 pandeic influenza A H1N1 infection in England: a crosssectional serological study. Lancet 375: Skowronski DM, Hottes TS, Janjua NZ, Purych D, Sabaiduc S, et al. (2010) Prevalence of seroprotection against the pandeic (H1N1) virus after the 2009 pandeic. CMAJ 182: using a ultitype epideic odel (e.g. age-structured odel). An elegant forula for the asyptotic final size distribution of ultitype epideic odels has been derived by Ball and Clancy [64], yielding a variance atrix (which is siilar to but a little ore coplicated than that discussed in the present study). Nevertheless, it should be noted that the eleents of the nextgeneration atrix (or the reproduction atrix) would be included as the solution of the final size equation for ultitype odels [64,65], and those cannot be siply replaced by the estiator of R using final size (i.e. as was done in the present study using hoogeneous odel), and thus, the coputation of 95% confidence interval ay well require full quantification of the next-generation atrix (in addition to observation of final sizes for each type). Each of the aboveentioned issues should be addressed in the future, ideally in the context of epirical applications. Until that tie, rather than relying on a binoial proportion, we recoend the use of the approach introduced in this study if the goal is to deterine the saple size of post-epideic seroepideiological studies, to calculate the 95% confidence interval of observed final size, or to conduct relevant hypothesis testing. Author Contributions Conceived and designed the experients: HN. Perfored the experients: HN. Analyzed the data: HN GC. Contributed reagents/aterials/analysis tools: HN. Wrote the paper: HN GC C-CC. 16. Desu MM (1988) Saple Size Methodology. New York: Acadeic Press. 17. Halloran ME, Struchiner CJ (1995) Causal inference in infectious diseases. Epideiology 6: Nishiura H, Kakehashi M, Inaba H (2009) Two critical issues in quantitative odeling of counicable diseases: Inference of unobservables and dependent happening. 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JAMA 303: Chan YJ, Lee CL, Hwang SJ, Fung CP, Wang FD, et al. (2010) Seroprevalence of antibodies to pandeic (H1N1) 2009 influenza virus aong hospital staff in a edical center in Taiwan. J Chin Med Assoc 73: Chen H, Wang Y, Liu W, Zhang J, Dong B, et al. (2009) Serologic survey of pandeic (H1N1) 2009 virus, Guangxi Province, China. Eerg Infect Dis 15: Hancock K, Veguilla V, Lu X, Zhong W, Butler EN, et al. (2009) Cross-reactive antibody responses to the 2009 pandeic H1N1 influenza virus. N Engl J Med 361: Aho M, Lyytikaïnen O, Nyhol JE, Kuitunen T, Rönkkö E, et al. (2010) Outbreak of 2009 pandeic influenza A(H1N1) in a Finnish garrison a serological survey. Euro Surveill 15: pii = Ikonen N, Strengell M, Kinnunen L, Osterlund P, Pirhonen J, et al. (2010) High frequency of cross-reacting antibodies against 2009 pandeic influenza A(H1N1) virus aong the elderly in Finland. Euro Surveill 15: pii = Boëlle PY, Bernillon P, Desenclos JC (2009) A preliinary estiation of the reproduction ratio for new influenza A(H1N1) fro the outbreak in Mexico, March April Euro Surveill 14: pii = PLoS ONE 9 March 2011 Volue 6 Issue 3 e17908