Results

We first verified that the cumulative numbers of confirmed cases with respect to each region were all well fitted to the Boltzmann function (ie, all R ² close to 0.999) (Fig. 1A), consistent with our earlier report using the data from January 21 to February 14, 2020.^{Reference Fu3} Assuming that the number of deaths is proportional to the number of confirmed cases for the outbreak under specific circumstances, we speculated that the cumulative number of COVID-19 deaths would also obey the Boltzmann function. In support of this speculation, the cumulative numbers of COVID-19 deaths in the aforementioned regions were all well fitted to the Boltzmann function (R ² all being close to 0.999) (Figs. 1B, 1C and Table 1). The potential total numbers of deaths were estimated to be 3,200±40 in China, 108±1 in Hubei, 3,100±40 outside Hubei, 2,500±40 in Wuhan City, and 604±6 outside Wuhan (Table 1). These results, in conjunction with our earlier observation that the cumulative numbers of confirmed cases of 2003 SARS in mainland China and worldwide were well fitted to the Boltzmann function, prompted us to analyze the cumulative numbers of 2003 SARS deaths in the same way. The cumulative numbers of 2003 SARS deaths in mainland China, Hong Kong, and worldwide were all well fitted to the Boltzmann function (Fig. 1D), strongly suggesting that the Boltzmann function is suitable to simulate the course of deaths associated with coronavirus-caused diseases.

Fig. 1. Fitting the cumulative number of COVID-19 deaths to Boltzmann function. (A–C) Boltzmann function-based regressions analysis results on the cumulative numbers of confirmed COVID-19 cases (panel A) and deaths (panels B and C) in the indicated geographic regions. Parameters of the established functions for Wuhan City (panels A and B) and for other cities in Hubei (panel C) are shown in insets. Note: The reported cumulative number of confirmed cases of Hubei Province and Wuhan City were readjusted for data fitting due to the suddenly added cases determined using clinical features (for details, refer to Supplementary Table 1 online). (D) Boltzmann function-based analysis results on the cumulative numbers of 2003 SARS deaths in the indicated regions. Parameters of the established function for mainland China are shown in insets. (E). Regression analysis results for COVID-19 deaths in Wuhan City using the Boltzmann functions assuming that the relative uncertainty of the data follows a single-sided normal distribution with a mean of 1.0 and a standard deviation of 2.5%. Original data are shown as circles; simulated results are presented as colored lines as indicated. Inserts show key statistics. Results for other regions are presented in Supplementary Fig. 1 (online). (F). Prediction of COVID deaths in Wuhan City by Boltzmann function-based analyses. The real data from January 21 to different closing dates were arbitrarily analyzed (colored lines), and the potential total numbers of deaths under these analyses are shown in insets. Real data (○) from March 1 to 19 agree well with the predicted data (dotted red lines) that were derived from the real data (▪) from January 21 to February 29.

Table 1. Summary of the Estimated Total Numbers of COVID-19 Deaths in China

Note. SD, standard deviation; CI, confidence interval.

^a Boltzmann function–based regression analysis results assuming the reported cumulative number of deaths (from January 21 to February 29) are precise and have uncertainty (SD, 2.5%), respectively (see the Methods section in the Supplementary Information online and Fig. 1E and Supplementary Fig. 1 online). The same uncertainty was set for Richards function–based regression analysis (see the Methods section and Supplementary Fig. 2 online).

^b Officially reported cumulative numbers of COVID-19 deaths as of March 19, 2020.

One issue regarding our analyses is that some COVID-19 deaths might be misreported such that the reported numbers of deaths represent a lower limit. For instance, 134 new deaths were suddenly counted from >13,000 clinically diagnosed patients in Hubei Province on February 12, 2020 (as clearly indicated by a sudden increase in deaths) (Fig. 1B). Another uncertainty might result from those unidentified COVID-19 deaths that occurred in the early phase of the outbreak. We applied the Monte Carlo method (see the Methods section in the Supplementary Information online) to estimate such uncertainty assuming that the relative uncertainty of the reported numbers of deaths follows a single-sided normal distribution with a mean of 1.0 and a standard deviation of 2.5%. The potential total numbers of COVID-19 deaths were estimated to be 3,260 (95% confidence interval [CI], 3187–3394) in China; 110 (95% CI, 109–112) in Hubei Province; 3,174 (95% CI, 3095–3270) outside Hubei; 2,550 (95% CI, 2494–2621) in Wuhan City; and 617 (95% CI, 607–632) outside Wuhan (Fig. 1E and Supplementary Fig. 1 online). These numbers are slightly higher than those estimated without uncertainty (Table 1).

To verify our Boltzmann function–based estimates, we calculated the potential total numbers of deaths in the aforementioned regions by applying Richards function–based regression analyses, which were used to simulate the cumulative numbers of confirmed cases of 2003 SARS in different regions.^{Reference Hsieh, Lee and Chang4,Reference Wang, Wu and Yang5} The potential total numbers of COVID-19 deaths in mainland China, other provinces, Hubei Province, Wuhan City, and other cities were estimated to be 3,342 (95% CI, 3,214–3,527) in China; 111 (95% CI, 109–114) in Hubei Province; 3,245 (95% CI, 3,100–3,423) outside Hubei; 2,613 (95% CI, 2,498–2,767) in Wuhan City; and 627 (95% CI, 603–654) outside Wuhan (Supplementary Fig. 2 online). These simulated estimates are close to the numbers estimated by the Boltzmann function–based analyses (Table 1).

In further support of our estimates, we found that the established Boltzmann function was able to predict the death course in a short period such that the released cumulative deaths from March 1 to March 19 were close to the estimated numbers, for example, in Wuhan (Fig. 1F and Supplementary Table 1 online). When we analyzed the data from January 21 to different closing dates, we found that the courses of COVID-19 deaths could be largely be simulated based on the data through February 14 (Fig. 1F). These results indicate that the Boltzmann function-based analysis was able to predict the trend ahead by ~3 weeks.