Figure 1.
The population and sampling process for estimating the unbiased confirmed case fatality ratio during the course of an outbreak.
At time t we know the cumulative number of confirmed cases and deaths, Ct and Dt, and wish to estimate the unbiased case fatality ratio π, by way of the factor of underestimation ut. If we knew ut we could specify the size of the population no longer at risk (utCt, shaded), although we do not know which surviving individuals belong to this group. A proportion π of those in the group still at risk (size (1- ut)Ct, unshaded) is expected to die. Because each case no longer at risk had an independent probability of dying, π, the number of deaths, Dt, is a sample from a binomial distribution with n = utCt, and pt = π.
Figure 2.
Temporal distribution of the date of onset for an H1N1 influenza epidemic in the USA and Canada, 2009.
Epidemic curves of confirmed cases of human infection with swine-origin influenza A (H1N1) virus (S-OIV) with known date of onset in (A) the USA (n = 394) and (B) Canada (n = 2004). The vertical dashed line is the date on which the Centers for Disease Control and Prevention identified S-OIV. The World Health Organization increased the pandemic alert level from 3 to 4 on April 27 (black arrow) and then to 5 on April 29 (gray arrow). It should be noted that confirmed cases include substantial numbers of imported cases from abroad. In Canada, a few cases whose dates of onset were unable to be traced are also included according to their dates when a specimen was collected (the exact number of such cases is not known). Assuming that their impact on our estimation procedure is negligibly small, we regard all cases in B as representing the dates of onset.
Figure 3.
Early determination of the unbiased confirmed case fatality ratio of severe acute respiratory syndrome (SARS) in Hong Kong, 2003.
(A & B) Cumulative numbers of confirmed cases and deaths. The increase in death is delayed in observation because of the time delay from onset to death. (C) Observed biased confirmed case fatality ratio (cCFR) estimates as a function of time (thick line) calculated as the ratio of the cumulative number of confirmed cases to deaths at time t. The estimate at the end of an outbreak (i.e. 302/1755 = 17.2 %) is the realized cCFR by the end of the epidemic. The horizontal continuous line and dotted lines show the expected value and the 95% confidence intervals of the predicted unbiased cCFR estimate (based on our method) only by using the observed data until 27 Mar 2003 (estimated at 18.1 % (95% CI: 10.5, 28.1). The 95% confidence interval was derived from profile likelihood. (D) The comparisons between the realized cCFR (horizontal grey line), the unbiased cCFRs based on observations by calendar time t, and the biased cCFR estimates, bt, given by the ratio of deaths to cases. Each prediction was obtained by using the exponential growth rate r up to time t and the cumulative numbers of deaths and cases at time t, and the mean time from onset-to-death of 35.9 days [11] which is assumed to follow an exponential distribution. Overestimation is seen in the early stages of the epidemic, but the 95% confidence limits in the later stages include the realized cCFR.
Figure 4.
Sensitivity of the unbiased confirmed case fatality ratio of an influenza virus (H1N1) infection to different means and coefficients of variation of the time from onset to death in the USA and Canada, 2009.
The contours show the maximum likelihood estimate of the unbiased confirmed case fatality ratio as a function of the mean and coefficient of variation of the time from onset-to-death in (A) the USA and (B) Canada. The estimates are based on observation by May 1 and June 10, respectively, with 2 and 4 deaths among a total of 399 and 2978 confirmed cases, respectively. A gamma distribution is employed for the time from onset to death, f(s). Both the quantitative and qualitative patterns of the USA differ from those of Canada, because the epidemic curve in the USA include more cases who developed the disease recently than those in Canada. It should be noted that the contour gray scales are different in (A) and (B).
Figure 5.
Upper bound of the confirmed case fatality ratio when there is no report of death.
Upper bound of the cCFR (confirmed case fatality ratio) estimates in (A) the USA and (B) Canada, given no deaths by April 21 and April 24, 2009, respectively (based on 42 and 91 cases). The upper bounds are examined for significance levels at 95% and 99% to find at least 1 death. Gamma and exponential distributions were employed to model the distribution of time from onset to death.
Figure 6.
Time variations in the biased confirmed case fatality ratio of an H1N1 influenza epidemic in the USA and Canada, 2009.
Cumulative numbers of confirmed cases and deaths in (A) the USA and (B) Canada. Cases (bars) and deaths (thick lines) are comparatively shown. (C) The biased estimates of confirmed case fatality ratio (cCFR) given by the ratio of deaths per confirmed cases. The data were extracted from irregular situation updates of the World Health Organization [25], and the horizontal axis (time) corresponds to the date of reporting. Therefore, it should be noted that the estimate suffers reporting delay, and in this sense, the calculated biased cCFR is different from our bt (based on date of onset) in the main text. The most recent report was made on June 19. Since the interval of update has been irregular, the cumulative number of cases and deaths is kept the same as the latest report when there was no update on the corresponding date.