Jpn. J. Infect. Dis., 63, 2010 When a novel influenza virus emerges, it is crucial to gain an understanding of the virulence at the very early stage of a pandemic. If a high mortality is predicted, then corresponding countermeasures can be chosen and implemented, and the extent of the pandemic in the presence and absence of interventions is subsequently estimated. In the ongoing pandemic involving the influenza A (H1N1) virus 2009, such an early assessment was initially made based on confirmed

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... ases whose risk of death was estimated at up to 0.5% (1-3). Although it later appeared that the risk of death among all symptomatic cases with medical attendance (i.e., symptomatic case fatality ratio [sCFR]) ranged from approximately 0.02-0.05% (4,5), a ratio which is more useful for the assessment of virulence and prediction of the mortality impact, epidemiological interpretation of minimally available data during the very early stages of the pandemic in the present study involves confirmed cases alone. Yoshikura (6) recently reported a linear relationship between the logarithms of the cumulative numbers of confirmed cases and deaths. In general, a log-log plot enables the relationship between two items to be interpreted and the relevant parameters estimated, given that an underlying power law mechanism exists (such as that observed for the individual contact heterogeneity of infectious diseases (7) ). To facilitate epidemiological understanding of the underlying mechanisms behind the empirically observed linear relationship in a bottom-up fashion, a mechanistic statistical model is needed. This article aims to explicitly clarify the relationship between the cumulative numbers of confirmed cases and deaths, to apply this relationship to estimate the confirmed case fatality ratio of the 2009 H1N1 pandemic virus, and to compare the estimates between different countries. Let the cumulative numbers of confirmed cases and deaths by calendar time t be C t and D t , respectively. Based on a wellknown mechanistic relationship (1), the expectation of D t given C t is written as where p is the confirmed case fatality ratio (cCFR), defined as an unbiased risk of death among confirmed cases. u t is referred to as a factor of underestimation (1), which reflects the proportion of confirmed cases C t whose risk of death has been observed by calendar time t, i.e., [2] where F s is the conditional cumulative distribution function of the time from onset to death (given a fatal outcome). Usually, u t varies as a function of calendar time during the nonlinear epidemic phase, but is independent of time during the initial exponential growth phase, resulting in [3] where M(-r) is the moment-generating function of the time from onset to death, given an exponential growth rate of confirmed cases r. For instance, if F s follows a Weibull distribution with the shape and scale parameters and , respectively, we have [4] From equation [1], the log-log relationship between C t and D t is given by [5] It should be noted that during the exponential growth phase, equation [1] yields a linear relationship between C t and D t even without logarithmic transformation. In other words, the slope k described in Yoshikura (6) is always 1 as long as an epidemic follows an exponential growth phase, and the intercept L is the product of two strictly interpretable quantities, i.e., p and u, during that growth phase (it should be noted that L varies with calendar time during the non-linear phase). Figure 1 compares the early growth phases in five different countries that reported at least 10 deaths by 1 July 2009 (8). Consistent with equation [1], the relationship between C t and D t appears to be linear (Fig. 1A) . However, the slopes are different. Argentina yielded the largest estimate (0.034) followed by Mexico (0.014), while the slopes of the remaining three countries, the United States, Canada, and Chile, ranged from 0.002-0.003. Figure 1B shows the exponential growth phase of confirmed cases during the corresponding period of observation, and compares the observed and predicted exponential growth of C t . While the growth rate in Mexico alone was higher than the other growth rates (0.093 per day), the remaining four countries yielded similar rates to one another (0.05-0.06 per day), and thus experienced invasion of the H1N1 virus with similar transmission poten-