We have introduced a framework to compute the uncertainty bounds of the final epidemic size that employs the Wald approximation, an approach motivated by the absence of a readily available methodology to estimate the sample size of post-epidemic seroepidemiological studies. Published seroepidemiological studies of H1N1-2009 so far have computed the confidence interval of the observed final size as if it were a binomial proportion. However, the data generating process behind the dynamics of infectious diseases involves dependence between infected individuals 
, which does not lead to a binomial proportion. Moreover, the observed final size represents a single stochastic realization among all possible sample paths (i.e. all possible probabilistic trajectories of the epidemic), requiring us to consider stochastic variations in the data. To account for these issues, we employed the approximate standard error of the final size given as a convergence result of a homogeneously mixing stochastic epidemic model. The calculation of the standard error was shown to be simple to compute (spreadsheet programs are sufficient). By applying the proposed uncertainty bound of final size to influenza (H1N1-2009), we have also shown that all the seroepidemiological studies published to date did not necessarily indicate an overestimation of prediction based on R
1.40, and moreover, all the observed final sizes did not reveal significant deviation from prediction with the lower limit R
1.15. Published seroepidemiological studies agree that the upper bound R
1.90 (and thus, other published estimates of R
) was likely an overestimation 
. One may still speculate that R
1.40 may well be an overestimation (because all of the observed final sizes were smaller than 51.1%), but the sample sizes of published seroepidemiological studies turned out to be too small to answer this question.
Although formulae for variance of the final size distribution (i.e. the square root of which we regarded as an approximate standard error) has been known among stochastic modeling experts 
, the present study extended its use to the computation of the 95% confidence interval of the observed final size by replacing the reproduction number by its estimator. This also led us to consider a parsimonious Wald test and sample size estimation. What the present study suggests for post-epidemic seroepidemiological studies is to employ the proposed formula (12) to calculate the 95% confidence interval and (14) or (15) to help determine the sample size for seroepidemiological surveys. For the latter, the following simplification of (14) might be useful:
The standard error s.e.
) is calculated by using the specified confidence interval (i.e. twice the margin of error) and the confidence level (i.e. nominal coverage probability). For instance, if the margin of error is 5% and the confidence level is 95%, the standard error is 0.05/1.96
0.025. Similarly, the standard error is 0.030 and 0.020 at the confidence levels of 90% and 99%, respectively. It is worth stressing that the purpose of post-epidemic seroepidemiological studies is not necessarily to test the observed final size against a predicted value, but includes real-time monitoring of an epidemic and various considerations of public health interventions. As long as there is no better alternative method for computing the uncertainty, the proposed approach should also be used for those other purposes to calculate conservative uncertainty bounds. The proposed method has a potential for explicitly discussing a posteriori effectiveness of interventions through the direct comparison 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 important role especially for influenza transmission with R
Our illustration of the proposed method posed four technical challenges for the computation of the uncertainty bound of final size; (i) the coefficient of variation of the generation time has to be known, (ii) the proportion of pre-existing immunity before an epidemic critically influences the bounds, (iii) sampling of several seroepidemiological studies took place shortly after an epidemic peak and (iv) vaccination and other public health interventions during the course of an epidemic can modify the observed final size. As for (i), the present study demonstrates a critical need to estimate the variance of the generation time in addition to the mean. That is, the distribution of the generation time plays a key role not only in estimating R 
but also in characterizing the variance of final epidemic size. With respect to (ii), although we did not include seroepidemiological studies prior to the 2009 pandemic 
, we have shown that such a survey of q
is a key to determine the sample size after the epidemic 
. In addition to the estimation of q
itself, it should be noted that our method adopted an assumption that the pre-existing immunity offered a complete protection from infection (i.e. all-or-nothing protection). If the pre-existing immunity is imperfect 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 estimation of the proportion of the initially immune population. Issues (iii) and (iv) pose further technical challenges to precisely estimate uncertainty bounds of seroprevalence in empirical studies. Given that the observation of incidence is given in every discrete time unit, a possible way forward may be to employ a parsimonious discrete time stochastic model (e.g. branching process or chain binomial model) 
, which may well enable us to draw the 95% confidence interval in a given reporting interval by conditioning the distribution to previous reporting intervals. Proposing simple methods to address these issues is part of our future studies.
Our method relied on the homogeneous mixing assumption and ignored time dependent factors that include seasonality and public health interventions. In this sense, the proposed uncertainty is regarded as an underestimate, because the time-dependent variations in the transmission potential can increase the variance of the final size distribution, and also because heterogeneous transmission (e.g. age-dependent mixing) can also increase variance (e.g. an epidemic with extremely high assortativity could generate multimodal final size distribution for an entire population 
). 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 formulae for the final size relation (our equation (1)) have to be reassessed 
. Moreover, in the presence of strong seasonality, a deterministic modeling study has demonstrated a very limited predictive performance of R
alone in anticipating the final epidemic size 
. Given that seroepidemiological 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 models can be different from that of homogeneous population 
, further work could at least incorporate heterogeneous mixing by employing the existing similar convergence result of the final size distribution using a multitype epidemic model (e.g. age-structured model). An elegant formula for the asymptotic final size distribution of multitype epidemic models has been derived by Ball and Clancy 
, yielding a variance matrix (which is similar to but a little more complicated than that discussed in the present study). Nevertheless, it should be noted that the elements of the next-generation matrix (or the reproduction matrix) would be included as the solution of the final size equation for multitype models 
, and those cannot be simply replaced by the estimator of R
using final size (i.e. as was done in the present study using homogeneous model), and thus, the computation of 95% confidence interval may well require full quantification of the next-generation matrix (in addition to observation of final sizes for each type).
Each of the abovementioned issues should be addressed in the future, ideally in the context of empirical applications. Until that time, rather than relying on a binomial proportion, we recommend the use of the approach introduced in this study if the goal is to determine the sample size of post-epidemic seroepidemiological studies, to calculate the 95% confidence interval of observed final size, or to conduct relevant hypothesis testing.