Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009
 Hiroshi Nishiura^{1, 2}Email author,
 Gerardo Chowell^{3, 4},
 Muntaser Safan^{5} and
 Carlos CastilloChavez^{3, 6}
DOI: 10.1186/1742468271
© Nishiura et al; licensee BioMed Central Ltd. 2010
Received: 15 December 2009
Accepted: 7 January 2010
Published: 7 January 2010
Abstract
Background
In many parts of the world, the exponential growth rate of infections during the initial epidemic phase has been used to make statistical inferences on the reproduction number, R, a summary measure of the transmission potential for the novel influenza A (H1N1) 2009. The growth rate at the initial stage of the epidemic in Japan led to estimates for R in the range 2.0 to 2.6, capturing the intensity of the initial outbreak among schoolage children in May 2009.
Methods
An updated estimate of R that takes into account the epidemic data from 29 May to 14 July is provided. An agestructured renewal process is employed to capture the agedependent transmission dynamics, jointly estimating the reproduction number, the agedependent susceptibility and the relative contribution of imported cases to secondary transmission. Pitfalls in estimating epidemic growth rates are identified and used for scrutinizing and reassessing the results of our earlier estimate of R.
Results
Maximum likelihood estimates of R using the data from 29 May to 14 July ranged from 1.21 to 1.35. The nextgeneration matrix, based on our agestructured model, predicts that only 17.5% of the population will experience infection by the end of the first pandemic wave. Our earlier estimate of R did not fully capture the populationwide epidemic in quantifying the nextgeneration matrix from the estimated growth rate during the initial stage of the pandemic in Japan.
Conclusions
In order to quantify R from the growth rate of cases, it is essential that the selected model captures the underlying transmission dynamics embedded in the data. Exploring additional epidemiological information will be useful for assessing the temporal dynamics. Although the simple concept of R is more easily grasped by the general public than that of the nextgeneration matrix, the matrix incorporating detailed information (e.g., agespecificity) is essential for reducing the levels of uncertainty in predictions and for assisting public health policymaking. Modelbased prediction and policymaking are best described by sharing fundamental notions of heterogeneous risks of infection and death with nonexperts to avoid potential confusion and/or possible misuse of modelling results.
Background
The reproduction number, R, the average number of secondary cases generated by a typical (or "average") single primary case, of influenza A (H1N1) 2009 is a summary measure of the transmission potential in the population of interest. It has been estimated using the early epidemic growth data in different locations across the world [1–12]. The estimations have been based primarily on models that include one or a limited number of aspects of heterogeneous transmission. The scientific community has been attracted to R because it provides a first aggregated measure of the overall transmissibility of an emerging infection [13]. Further, the estimate of R, based on homogeneously or nearly homogeneously mixing population models that by design ignore most individual differences, is not only used to assess the initial growth of an epidemic but also the extent to which the population will experience infection by the end of a first pandemic wave [14, 15]. Except for a unique study estimating R using a data set of international spread [7], the exponential growth rate, r, of cases during the initial epidemic phase has been investigated using a simple procedure that involves translating r into R through the use of the estimator R = 1/M(r) where M(r) is the momentgenerating function of the generation time distribution, given the growth rate r [16]. Naturally, the higher the growth rate r of the number of cases, the larger the estimate of R.
The majority of R estimates for this ongoing pandemic have ranged from 1.11.8 [17] while our estimate of R in Japan was in the range of 2.02.6 under the assumption of a mean generation time of 1.9 days through May 2009 [5]. The most plausible reason for this estimate, as noted in our earlier study [5], involved the role of initial conditions as the very early growth was driven by the high contact rates that are common to school settings [17, 18]. In addition to the phenomenological explanation, it is important to assess whether or not the methodology used to estimate R was adequate. We do this here using data that go beyond those used in our estimation of R for the earlier epidemic period of May 2009 in Japan. Here we provide an updated estimate of R for the novel influenza A (H1N1) 2009 in Japan, summarising the relevant methodological issues in estimating R from the growth rate of cases and initiating a dialogue on how estimates of the transmission potential should be shared with nonexperts, including the general public.
Discussion
The epidemic data in Japan
Since our original data indicate that the 97.5 percentile point of the reporting delay distribution (i.e., the time from illness onset to notification) is 8 days, we analyse a total of 3480 cases that developed the disease on or before 14 July. Figure 1A shows the temporal distribution stratified by agegroup of all identified cases. Of the 3480 cases, 67.0% were among individuals 19 years of age or less. The population of those aged from 2039 years accounted for 24.2% of the total, and the remaining (older adult) cases accounted for only 8.8%. The contributions from imported cases to the early epidemic growth in this island nation, in addition to the local (indigenous) transmissions, are also critical (Figure 1B). Of the 3840 confirmed cases, 694 (19.9%) had a history of overseas travel within 10 days preceding the onset of illness, and we refer to them as imported cases in the present study.
Growth rates of two different phases
We model the expected value of the incidence of illness onset at calendar time t as E(c(t)) = k exp(r(tτ)) where k is a constant, r is the growth rate of the corresponding period, and τ is the starting time point of exponential growth (assumed as 5 May and 29 May, respectively). Minimizing the sum of squares between the observed data and expectation, r is estimated as 0.37 and 0.08 per day, respectively, for the former and latter periods. The estimate for the former period is smaller than that reported in our earlier study in May (i.e., 0.47 per day) [5], because of our use of refined dates of onset and the use of a simpler statistical method in the present study. The estimates of the exponential growth rates differ by almost a factor of five (i.e., 0.37/0.08) in the two windows in time, indicating that the cases in the former period experienced a 1.3 times greater daily growth rate (i.e., exp(0.37)/exp(0.08)) than those in the latter period. A glance at the agespecific data show that the disease spread from an initial cluster that mix primarily in an assortative manner into the "general" Japanese population is the most likely key to this dramatic difference. Since the latter period reflects the early populationwide spread of H1N1 involving the entire Japanese community, R for this period is estimated using the following methodology.
Modelling methods
We employ an agestructured model to derive an estimate for R since the transmission of influenza A (H1N1) 2009 is known to differ greatly among age groups [1, 3, 5, 9]. Spatial heterogeneity, social heterogeneity (e.g. differing patterns of transmission between household, school and workplacesettings), or potential changes in behaviour are mostly ignored.
where G(s) is the cumulative distribution function of the generation time distribution, which we assume to be known and to follow a gamma distribution. In the early modelling studies, the mean generation time was estimated at 1.9 days [1], 2.63.2 days [3] and 2.5 days [4]. From contact tracing data in the Netherlands, the mean and standard deviation (SD) were estimated at 2.7 and 1.1 days, respectively [22]. We adopt 2.7 days as the mean and fix the coefficient of variation to 40.7% as calculated from the Dutch study. We partly address issues of uncertainty by measuring the sensitivity of R to differing mean generation times ranging from 2.1 to 3.3 days.
where R is the reproduction number to be estimated (i.e., scalar quantity), s_{i} measures the susceptibility of agegroup i given a contact, and m_{ij} is the frequency of contact made by an individual in agegroup j with that in i (which is assumed known and is extracted from a contact survey in the Netherlands [23]); let S and M be square matrices. S is the diagonal matrix in which the diagonal elements (i, i) are s_{i} and the entries outside the main diagonal are all zero. The (i, j) element of M is m_{ij} with which we adopt frequencydependent assumption, and we ignore more detailed contact including the "type" and "duration" [24]. We normalize the product SM (i.e., the dominant eigenvalue of SM is set to 1) so that R scales the nextgeneration matrix. We aggregate the population into six discrete age groups (05, 612, 1319, 2039, 4059 years and 60 and older) in order to be able to adhere to the precision of the contact survey [23]; consequently, the nextgeneration matrix has dimensions 6 × 6 (36 elements).
where E(c_{i, t}) is the conditional expectation (i.e., the righthand side of (1)) and c_{i, t} is the observed number of cases of agegroup i on day t. Maximum likelihood estimates of the parameters are obtained by minimizing the negative logarithm of (4) with the 95% confidence intervals (CI) derived from profile likelihood.
Modelling results
Those aged from 1319 years appear to be able to maintain the transmission by themselves (i.e., meeting the definition of maintenance host, R_{33} > 1 [26]). Nevertheless, age groups 1 and 2, children aged from 012 years, appear incapable of maintaining transmission (i.e., the dominant eigenvalue of the 2 × 2 matrix involving transmissions among and between those aged from 05 and 612 years is less than 1). The maximum likelihood estimate of R is 1.28 (95% CI: 1.23, 1.33). The relative contribution of imported cases to secondary transmission, α, is estimated at 0.15 (standard error = 0.14).
Limitation of the growth rate r
As expected from the greatly differing exponential growth rates between early May and from 29 May to 14 July (Figure 2), the reproduction number for the latter period, ranging from 1.21 to 1.35, is much smaller than our previously reported estimate for the former time period when the transmission was mainly confined to school settings [5]. The estimate in the latter period is consistent with the estimates of R in other countries [14, 16]. The situation is not straightforward, however, as the estimation was carried out using confirmed cases (which may be biased towards severe cases). Further, it should be noted that various interventions, including reactive school closure and contact tracing, were instituted during the whole period of observation, so the R value for the latter period and especially the entries R_{ij} involving school children might potentially lead to underestimates for R in the present study.
Since the small outbreak in the former time period was restricted to a limited number of schools and the contacts made by the students in Osaka and Kobe (and as Japan was unique in successfully "containing" the local schoolbased outbreak before actual pandemic overshoot), the depletion of susceptibles in May and undiagnosed cases are unlikely to have played a significant role in our estimates of a smaller R for the epidemic in the latter transmission period, which saw the pandemic takeover. Rather, as we discussed above, the local networks of interactions (i.e., transmission within networks that connect to other networks in time), and consequently the initial conditions (i.e., which network gets infected first), played a key role in our estimates for the initial outbreak growth. The earlier estimates of R captured the initial role in the generation of secondary cases from schools where the frequency of transmission among school children greatly exceeded those of the community (and the "type" of their contact is perhaps more dense (or close) than those in the community [24, 30]). Although the sensitivity of R to differing mean generation times was examined within a relatively narrow range, this aspect could not account for the high R estimates obtained in Japan [5].
One important conclusion is drawn from the present study. The lessons learnt from our estimation over the two windows in time has forced us to revisit the role of using summary statistics to characterize transmission potential from the data generated by heterogeneous contact patterns. As a network expands, the structure of networks involved in transmission changes, and consequently the summary statistics of cases, also change in time [13]. The initial summary statistics therefore depend in a rather critical way on the initial conditions (i.e., where and how the disease was introduced) [31], which is not always captured well by homogeneous mixing models. Since the very early stage of this epidemic alone involves primarily a few specific subgroups of the population, it is difficult to quantify the nextgeneration matrix fully and estimate a reproduction number that adequately captures the transmission potential for the entire population. Whereas the nextgeneration matrix includes representative levels of population heterogeneity, the infected individuals during the very initial epidemic stage were clearly not representative of the entire population of interest. The use of the nextgeneration matrix involves the introduction of a "typical" infectious individual into the population, but such an individual cannot be properly characterized if the matrix involves unavoidable approximations (due to limited availability of structured data) when an outbreak happens to be mostly confined to a single cluster whose average individual is "atypical" of the entire population.
The previously reported expected value, R = 2.3, for the May outbreak might well approximate the intensity of transmissions in schools (and indeed, is consistent with the estimate in school settings in the USA [3]). This is another example of what is often referred as core group effects in the epidemiological literature [32]. Naturally, the use of the empirical data from school clusters does not provide sufficient information to carry out precise estimation of the agedependent nextgeneration matrix, so the resulting dominant eigenvalue in the earlier study should not be regarded as R but rather as a measure of transmissibility conditioned on the initial conditions. The need for the collection of additional data may be critical when agespecific transmission is highly assortative and/or when agespecific susceptibility is highly heterogeneous (as is the case for influenza A (H1N1) 2009). Summary statistics based on highly aggregated populations are in general not helpful in identifying the pressure points of a heterogeneous network, which is essential in the identification and assessment of the most effective (e.g., agespecific) intervention policies. In other words, the finer details of epidemic data (i.e., epidemiological information at a local level, e.g., active surveillance of cases) need to be taken into account in the modelling. Not only school outbreaks, but also other social factors and settings (e.g. transportation, hospital settings and mass gatherings) can play enhancing or reducing transmission roles.
In addition to the challenges posed by our need to average over different levels of heterogeneous mixing, quantification of the growth rates involves the additional challenges that come from underreporting, notably ascertainment of cases and reporting bias. Further, imputation of onset dates for missing data is sometimes required, and moreover, the timevarying reporting frequency may even call for naïve adjustment of the growth rate of confirmed cases by the growth rate of hospitalized (or other severe portions of) cases [4]. The data set we examined in Japan involved contact tracing efforts at all local levels, so the growth rate of confirmed symptomatic cases is thought to have captured the actual increase in infection appropriately. Nevertheless, achieving precise estimation of incidence for this mild disease remains an open question, particularly if the proportion of asymptomatic infections among the total of infected individuals is high.
Is the epidemic growth rate useless?
and thus the estimator of R is given by 1/M(r) (see Background) [16]. Hence, and not surprisingly, as long as the intrinsic growth rate and generation time are the same among subpopulations, the estimator of R for the multitype epidemic model can be identical to that of singlehost epidemic model [34].
The incorporation of additional levels of detail into the basic model used to generate growth estimates depends on the model's ability to capture the underlying transmission dynamics in the data and the purposes of the research questions or public health policymaking goals. These issues are particularly relevant when a clustering of cases is observed [35, 36]; as we saw in Japan, clusters of cases caused a delay in accurately estimating the true population average of transmissibility. The epidemic growth rate remains a useful quantity for estimating the transmission potential at the population level in the absence of obvious clusters of cases and as long as the approximately modelled transmission sufficiently captures the actual heterogeneity. We start from the premise that the use of heterogeneous mixing models is essential in the assessment of critical theoretical claims and policymaking decisions. Hence, it is worth noticing that in this context, technical questions remain regarding the use and applicability of the exponential growth rate. They include (i) the development of methods for estimating the generation time distribution and (ii) the determination of an appropriate length of the exponential growth period [37–39].
How should we communicate the estimate?
Extracting the agespecific population estimate in Japan [41], z is estimated at 17.5%, not surprisingly much less than predicted by (10), a value that is indeed close to the actual range of the impact of first pandemic wave in Northern Hemisphere countries [17]. The "real" value of z may be even smaller if we account for additional levels of heterogeneity in transmission. The reproduction number, R, for the entire population may be useful for obtaining a rough estimate of how much vaccine we need (e.g., deciding the total number of vaccines to be manufactured), while R_{ij} is far more essential for structuring the most effective strategy of vaccination and planning the optimal prioritization schemes [42, 43]. Given that R can also be calculated from R_{ij}, communicating R_{ij} rather than R to the general public would be the most informative strategy of science communication for modelling results. When one explains the concept of R_{ij} to nonexperts, it's ideal to mention the limitation due to its nature of approximation because of limitations in structured data in any empirical observation.
The case fatality ratio (CFR), an epidemiological measurement of virulence, would also benefit from the use of detailed (e.g., agestructured) information. Whereas the confirmed CFR (cCFR) for the entire population conditioned on confirmed cases has been estimated at approximately 0.5% during the very early stage of the pandemic [1, 44, 45], the symptomatic CFR (sCFR), which is conditioned on symptomatic cases, later appeared to be 0.048% [46]. The CFR estimate for the entire population is regarded as a summary measure of virulence, so the reduced order of virulence (approximately by a factor of 10) provides not only more accurate information but a more useful measure for assessing the impact of this pandemic. Nevertheless, knowledge based on additional information would be even more useful. Ideally, we would like to compare epidemiologically the virulence between different settings (e.g., comparison between age groups or countries) and to use this knowledge in the development of relevant public health policy. Understanding the heterogeneous risks of death by age and riskgroups is critical in the effective design of intervention strategies. In other words, apart from discussing the changing assessment of the CFR as a whole, CFR_{i} for agegroup i is probably more informative than CFR for the entire population.
Conclusions
We analysed the temporal distribution of the influenza A (H1N1) 2009 epidemic in Japan from MayJuly 2009, estimating the reproduction number for the time period from 29 May to 14 July  a followup to our earlier estimates derived primarily from data on school outbreaks in May. The updated estimate of R, an average over a wider spectrum of the Japanese population, ranged from 1.21 to 1.35. Our higher estimate reported in [5] was based on a data set that was incapable of quantifying the nextgeneration matrix for the whole country from the growth rate of cases in a restricted school setting alone. The upshot of this analysis is tied to the fact that in regions where disease transmission is highly heterogeneous the initial conditions will have a strong effect on the variation in estimates of the rate of epidemic growth. To quantify R and R_{ij} appropriately from the growth rate of cases, it is essential that the model not only captures the underlying transmission dynamics behind the empirical data but also that we have sufficiently structured data for each subpopulation to estimate the role of variability in our estimates. In addition to extending the window in time for developing accurate population level measurements [39], it may be useful to examine additional epidemiological information (e.g., active surveillance of local transmission patterns and the severe portion of cases including hospital admissions).
Communicating R to nonexperts is simpler than R_{ij}, but R_{ij} contains the information that is most useful for the development of specific intervention efforts. Developing simple interpretations and effective ways of communicating R_{ij} (i.e., the average number of secondary cases in host i generated by a single primary case in host j) must be pursued. This retrospective study highlights the importance of incorporating the concept of heterogeneous risks of infection and death in the context of the assessment of pandemic potential.
List of abbreviations
 R :

the reproduction number
 RTPCR:

Reverse transcription polymerase chain reaction
 CI:

confidence intervals
 SD:

standard deviation
 CFR:

case fatality ratio.
Declarations
Acknowledgements
The work of H Nishiura was supported by the JST PRESTO program. The authors are grateful to the editor and the three reviewers for their useful comments, advices and expediting publication of this article.
Authors’ Affiliations
References
 Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, Hollingsworth TD, Griffin J, Baggaley RF, Jenkins HE, Lyons EJ, Jombart T, Hinsley WR, Grassly NC, Balloux F, Ghani AC, Ferguson NM, Rambaut A, Pybus OG, LopezGatell H, AlpucheAranda CM, Chapela IB, Zavala EP, Guevara DM, Checchi F, Garcia E, Hugonnet S, Roth C, WHO Rapid Pandemic Assessment Collaboration: Pandemic potential of a strain of influenza A (H1N1): early findings. Science. 2009, 324: 15571561. 10.1126/science.1176062.PubMed CentralView ArticlePubMed
 Boëlle PY, Bernillon P, Desenclos JC: A preliminary estimation of the reproduction ratio for new influenza A(H1N1) from the outbreak in Mexico, MarchApril 2009. Euro Surveill. 2009, 14 (19): pii: 19205
 Yang Y, Sugimoto JD, Halloran ME, Basta NE, Chao DL, Matrajt L, Potter G, Kenah E, Longini IM: The transmissibility and control of pandemic influenza A (H1N1) virus. Science. 2009, 326: 729733. 10.1126/science.1177373.PubMed CentralView ArticlePubMed
 White LF, Wallinga J, Finelli L, Reed C, Riley S, Lipsitch M, Pagano M: Estimation of the reproductive number and the serial interval in early phase of the 2009 influenza A/H1N1 pandemic in the USA. Influenza Other Respi Viruses. 2009, 3: 267276. 10.1111/j.17502659.2009.00106.x.PubMed CentralView Article
 Nishiura H, CastilloChavez C, Safan M, Chowell G: Transmission potential of the new influenza A(H1N1) virus and its agespecificity in Japan. Euro Surveill. 2009, 14 (19): pii: 19205
 Nishiura H, Wilson N, Baker MG: Estimating the reproduction number of the novel influenza A virus (H1N1) in a Southern Hemisphere setting: preliminary estimate in New Zealand. N Z Med J. 2009, 122: 7377.PubMed
 Balcan D, Hu H, Goncalves B, Bajardi P, Poletto C, Ramasco JJ, Paolotti D, Perra N, Tizzoni M, Broeck Van den W, Colizza V, Vespignani A: Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility. BMC Med. 2009, 7: 4510.1186/17417015745.PubMed CentralView ArticlePubMed
 Munayco CV, Gomez J, LagunaTorres VA, Arrasco J, Kochel TJ, Fiestas V, Garcia J, Perez J, Torres I, Condori F, Nishiura H, Chowell G: Epidemiological and transmissibility analysis of influenza A(H1N1)v in a southern hemisphere setting: Peru. Euro Surveill. 2009, 14 (32): pii: 19299
 McBryde E, Bergeri I, van Gemert C, Rotty J, Headley E, Simpson K, Lester R, Hellard M, Fielding J: Early transmission characteristics of influenza A(H1N1)v in Australia: Victorian state, 16 May  3 June 2009. Euro Surveill. 2009, 14: pii = 19363
 de Silva UC, Warachit J, Waicharoen S, Chittaganpitch M: A preliminary analysis of the epidemiology of influenza A(H1N1)v virus infection in Thailand from early outbreak data, JuneJuly 2009. Euro Surveill. 2009, 14: pii = 19292
 Pourbohloul B, Ahued A, Davoudi B, Meza R, Meyers LA, Skowronski DM, Villaseñor I, Galván F, Cravioto P, Earn DJ, Dushoff J, Fisman D, Edmunds WJ, Hupert N, Scarpino SV, Trujillo J, Lutzow M, Morales J, Contreras A, Chávez C, Patrick DM, Brunham RC: Initial human transmission dynamics of the pandemic (H1N1) 2009 virus in North America. Influenza Other Respi Viruses. 2009, 3: 215222. 10.1111/j.17502659.2009.00100.x.PubMed CentralView Article
 Tuite AR, Greer AL, Whelan M, Winter AL, Lee B, Yan P, Wu J, Moghadas S, Buckeridge D, Pourbohloul B, Fisman DN: Estimated epidemiologic parameters and morbidity associated with pandemic H1N1 influenza. CMAJ. 2009, 10.1503/cmaj.091807.
 Busenberg S, CastilloChavez C: A general solution of the problem of mixing of subpopulations and its application to risk and agestructured epidemic models for the spread of AIDS. IMA J Math Appl Med Biol. 1991, 8: 129. 10.1093/imammb/8.1.1.View ArticlePubMed
 World Health Organization: Mathematical modelling of the pandemic H1N1 2009. Wkly Epidemiol Rec. 2009, 84: 341348.
 Brauer F, Feng Z, CastilloChavez C: Discrete epidemic models. Math Biosci Eng. 2010, 7: 115. 10.3934/mbe.2010.7.1.View ArticlePubMed
 Wallinga J, Lipsitch M: How generation intervals shape the relationship between growth rates and reproductive numbers. Proc R Soc Lond Ser B. 2007, 274: 599604. 10.1098/rspb.2006.3754.View Article
 World Health Organization: Transmission dynamics and impact of pandemic influenza A (H1N1) 2009 virus. Wkly Epidemiol Rec. 2009, 84: 481484.
 Mikolajczyk RT, Akmatov MK, Rastin S, Kretzschmar M: Social contacts of school children and the transmission of respiratoryspread pathogens. Epidemiol Infect. 2008, 136: 813822. 10.1017/S0950268807009181.PubMed CentralView ArticlePubMed
 Ministry of Health, Labor and Welfare, Japan. Influenza A(H1N1). Tokyo: Ministry of Health, Labor and Welfare, (accessed on 10 December 2009), http://www.mhlw.go.jp/english/topics/influenza_a/index.htmlhttp://www.mhlw.go.jp/english/topics/influenza_a/index.html
 Diekmann O, Heesterbeek JAP: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. 2000, Chichester: John Wiley and Son
 Diekmann O, Heesterbeek JA, Metz JA: On the definition and the computation of the basic reproduction ratio R_{0} in models for infectious diseases in heterogeneous populations. J Math Biol. 1990, 28: 365382. 10.1007/BF00178324.View ArticlePubMed
 Hahné S, Donker T, Meijer A, Timen A, van Steenbergen J, Osterhaus A, van der Sande M, Koopmans M, Wallinga J, Coutinho R, Dutch New Influenza A(H1N1)v Investigation Team: Epidemiology and control of influenza A(H1N1)v in the Netherlands: the first 115 cases. Euro Surveill. 2009, 14 (27): pii: 19267
 Wallinga J, Teunis P, Kretzschmar M: Using data on social contacts to estimate agespecific transmission parameters for respiratoryspread infectious agents. Am J Epidemiol. 2006, 164: 936944. 10.1093/aje/kwj317.View ArticlePubMed
 Smieszek T: A mechanistic model of infection: why duration and intensity of contacts should be included in models of disease spread. Theor Biol Med Model. 2009, 6: 2510.1186/17424682625.PubMed CentralView ArticlePubMed
 Nishiura H: Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 191819. Theor Biol Med Model. 2007, 4: 2010.1186/17424682420.PubMed CentralView ArticlePubMed
 Nishiura H, Hoye B, Klaassen M, Bauer S, Heesterbeek H: How to find natural reservoir hosts from endemic prevalence in a multihost population: A case study of influenza in waterfowl. Epidemics. 2009, 1: 118128. 10.1016/j.epidem.2009.04.002.View ArticlePubMed
 Centers for Disease Control and Prevention (CDC): Serum crossreactive antibody response to a novel influenza A (H1N1) virus after vaccination with seasonal influenza vaccine. Morb Mortal Wkly Rep. 2009, 58: 521524.
 Itoh Y, Shinya K, Kiso M, Watanabe T, Sakoda Y, Hatta M, Muramoto Y, Tamura D, SakaiTagawa Y, Noda T, Sakabe S, Imai M, Hatta Y, Watanabe S, Li C, Yamada S, Fujii K, Murakami S, Imai H, Kakugawa S, Ito M, Takano R, IwatsukiHorimoto K, Shimojima M, Horimoto T, Goto H, Takahashi K, Makino A, Ishigaki H, Nakayama M, Okamatsu M, Takahashi K, Warshauer D, Shult PA, Saito R, Suzuki H, Furuta Y, Yamashita M, Mitamura K, Nakano K, Nakamura M, BrockmanSchneider R, Mitamura H, Yamazaki M, Sugaya N, Suresh M, Ozawa M, Neumann G, Gern J, Kida H, Ogasawara K, Kawaoka Y: In vitro and in vivo characterization of new swineorigin H1N1 influenza viruses. Nature. 2009, 460: 10211025.PubMed CentralPubMed
 Reichert T, Chowell G, Nishiura H, Christensen RA, McCullers JA: Does Glycosylation as a modifier of Original Antigenic Sin explain the case age distribution and unusual toxicity in pandemic novel H1N1 influenza?. BMC Infect Dis. 2010, 10: 510.1186/14712334105.PubMed CentralView ArticlePubMed
 Britton T, Nordvik MK, Liljeros F: Modelling sexually transmitted infections: the effect of partnership activity and number of partners on R0. Theor Popul Biol. 2007, 72: 389399. 10.1016/j.tpb.2007.06.006.View ArticlePubMed
 Newman MEJ: The structure and function of complex networks. SIAM Review. 2003, 45: 167256. 10.1137/S003614450342480.View Article
 Hethcote HW, Yorke JA: Gonorrhea Transmission Dynamics and Control (Lecture Notes in Biomathematics, 56). 1980, Berlin:SpringerVerlag
 Goldstein E, Apolloni A, Lewis B, Miller JC, Macauley M, Eubank S, Lipsitch M, Wallinga J: Distribution of vaccine/antivirals and the 'least spread line' in a stratified population. J R Soc Interface. 2009, 10.1098/?rsif.2009.0393.
 Nishiura H, Chowell G, Heesterbeek H, Wallinga J: The ideal reporting interval for an epidemic to objectively interpret the epidemiological time course. J R Soc Interface. 2010, 7: 297307. 10.1098/rsif.2009.0153.PubMed CentralView ArticlePubMed
 Keeling M: The implications of network structure for epidemic dynamics. Theor Popul Biol. 2005, 67: 18. 10.1016/j.tpb.2004.08.002.View ArticlePubMed
 Smieszek T, Fiebig L, Scholz RW: Models of epidemics: when contact repetition and clustering should be included. Theor Biol Med Model. 2009, 6: 1110.1186/17424682611.PubMed CentralView ArticlePubMed
 Gurevich KG: Formal kinetics of H1N1 epidemic. Theor Biol Med Model. 2009, 6: 2310.1186/17424682623.PubMed CentralView ArticlePubMed
 Chowell G, CastilloChavez C: WorstCase scenarios and epidemics. Bioterrorism: Mathematical and Modeling Applications in Homeland Security, (Frontiers in Applied Mathematics 28). Edited by: Banks T, CastilloChavez C. 2003, Philadelphia, PA: Society for Industrial Mathematics
 Bettencourt LMA, Ribeiro RM, Chowell G, Lant T, CastilloChavez C: Towards Real Time Epidemiology: Data Assimilation, Modeling and Anomaly Detection of Health Surveillance Data Streams. Intelligence and Security Informatics: Biosurveillance (Lecture Notes in Computer Science, volume 4506). 2009, Berlin: Springer
 Ball F, Clancy D: The final size and severity of a generalised stochastic multitype epidemic model. Adv Appl Prob. 1993, 25: 721736. 10.2307/1427788.View Article
 Statistics Bureau, Ministry of Internal Affairs and Communications. Population estimates. Tokyo: Statistics Bureau (in Japanese). (accessed on 10 December 2009), http://www.stat.go.jp/data/nihon/02.htmhttp://www.stat.go.jp/data/nihon/02.htm
 Longini IM, Ackerman E, Elveback LR: An optimization model for influenza A epidemics. Math Biosci. 1978, 38: 141157. 10.1016/00255564(78)900238.View Article
 Britton T: On critical vaccination coverage in multitype epidemics. J Appl Prob. 1998, 35: 10031006. 10.1239/jap/1032438395.View Article
 Garske T, Legrand J, Donnelly CA, Ward H, Cauchemez S, Fraser C, Ferguson NM, Ghani AC: Assessing the severity of the novel influenza A/H1N1 pandemic. BMJ. 2009, 339: b284010.1136/bmj.b2840.View ArticlePubMed
 Nishiura H, Klinkenberg D, Roberts M, Heesterbeek JA: Early epidemiological assessment of the virulence of emerging infectious diseases: a case study of an influenza pandemic. PLoS ONE. 2009, 4: e685210.1371/journal.pone.0006852.PubMed CentralView ArticlePubMed
 Presanis AM, De Angelis D, New York City Swine Flu Investigation Team, Hagy A, Reed C, Riley S, Cooper BS, Finelli L, Biedrzycki P, Lipsitch M: The severity of pandemic H1N1 influenza in the United States, from April to July 2009: a Bayesian analysis. PLoS Med. 2009, 6: e100020710.1371/journal.pmed.1000207.PubMed CentralView ArticlePubMed
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