Simulating and forecasting the cumulative confirmed cases of SARS-CoV-2 in China by Boltzmann function-based regression analyses
Authors: Xinmiao Fu, Qi Ying, Tieyong Zeng, Tao Long, Yan Wang
Journal: Journal of Infection (2020)

•Cumulative confirmed cases in China were well fitted with Boltzmann function.•Potential total numbers of confirmed cases in different regions were estimated.•Key dates indicating minimal daily number of new confirmed cases were estimated.•Cumulative confirmed cases of 2003 SARS-CoV were well fitted to Boltzmann function.•The Boltzmann function was, for the first time, applied to epidemic analysis. As reported in this Journal1Tang J.W. Tambyah P.A. Hui D.S.C. Emergence of a novel coronavirus causing respiratory illness from Wuhan, China.J Infect. 2020; https://doi.org/10.1016/j.jinf.2020.01.014Abstract Full Text Full Text PDF Scopus (113) Google Scholar and elsewhere,2Wang C. et al.A novel coronavirus outbreak of global health concern.Lancet. 2020; https://doi.org/10.1016/S0140-6736(20)30185-9Abstract Full Text Full Text PDF Scopus (4961) Google Scholar an outbreak of atypical pneumonia caused by the zoonotic 2019 novel coronavirus (SARS-CoV-2) is on-going in China and has spread to the world. As of Feb 16, 2020 (24:00, GMT+8), there have been 70,548 confirmed patients and more than 1700 deaths from SARS-CoV-2 infection in China, and 58,182 confirmed patients and 1696 deaths in the most affected province, Hubei Province. Much research progress has been made in dissecting the evolution and origin of SARS-CoV-2 and characterizing its clinical features.3Yang Y., et al., Epidemiological and clinical features of the 2019 novel coronavirus outbreak in China. doi:10.1101/2020.02.10.20021675, 2020.Google Scholar, 4Wu F. et al.A new coronavirus associated with human respiratory disease in China.Nature. 2020; (2020)https://doi.org/10.1038/s41586-020-2008-3Crossref Scopus (7481) Google Scholar, 5Zhou P. et al.A pneumonia outbreak associated with a new coronavirus of probable bat origin.Nature. 2020; (2020)https://doi.org/10.1038/s41586-020-2012-7Crossref Scopus (14262) Google Scholar, 6Lu R. et al.Genomic characterisation and epidemiology of 2019 novel coronavirus: implications for virus origins and receptor binding.Lancet. 2020; https://doi.org/10.1016/s0140-6736(20)30251-8Abstract Full Text Full Text PDF Google Scholar, 7Guan W.J., et al., Clinical characteristics of 2019 novel coronavirus infection in China. doi:10.1101/2020.02.06.20020974, 2020.Google Scholar While the outbreak is on-going, people raise grave concerns about the future trajectory of the outbreak, especially given that the working and schooling time has been already dramatically postponed after the Chinese Lunar New Year holiday was over (scheduled on Jan 31). In particular, a precise estimation of the potential total number of infected cases and/or confirmed cases is highly demanding. Earlier studies based on susceptible-exposed-infectious-recovered metapopulation and susceptible-infected-recovered-dead models revealed the number of potentially infected cases and the basic reproductive number of SARS-CoV-2.3Yang Y., et al., Epidemiological and clinical features of the 2019 novel coronavirus outbreak in China. doi:10.1101/2020.02.10.20021675, 2020.Google Scholar,8Wu J.T. Leung K. Leung G.M. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in wuhan, china: a modelling study.Lancet. 2020; https://doi.org/10.1016/S0140-6736(20)30260-9Abstract Full Text Full Text PDF Scopus (2933) Google Scholar,9Anastassopoulou, C., et al., Data-based analysis, modelling and forecasting of the novel coronavirus (2019-NCOV) outbreak. doi:10.1101/2020.02.11.20022186, 2020.Google Scholar These traditional epidemiological models apparently require much detailed data for analysis.3Yang Y., et al., Epidemiological and clinical features of the 2019 novel coronavirus outbreak in China. doi:10.1101/2020.02.10.20021675, 2020.Google Scholar,8Wu J.T. Leung K. Leung G.M. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in wuhan, china: a modelling study.Lancet. 2020; https://doi.org/10.1016/S0140-6736(20)30260-9Abstract Full Text Full Text PDF Scopus (2933) Google Scholar Here we explored a simple data-driven, Boltzmann function-based approach for estimation only based on the daily cumulative number of confirmed cases of SARS-CoV-2 (Note: the rational for Boltzmann function-based regression analysis is presented in supporting information (SI) file). We decided to collect data (initially from Jan 21 to Feb 10, 2020) in several typical regions of China, including the center of the outbreak (i.e. Wuhan City and Hubei Province), other four most affected provinces (i.e., Guangdong, Zhejiang, Henan, Hunan) and top-4 major cities in China (i.e., Beijing, Shanghai, Guangzhou, Shenzhen). During data analysis on Feb 13, 2020, the number of new confirmed cases on Feb 12 in Hubei Province and Wuhan City suddenly increased by 14,840 and 13,436, respectively, of which 13,332 and 12,364 are those confirmed by clinical features (note: all the number of confirmed cases released by Feb 12 were counted according to the result of viral nucleic acid detection rather than by referring to clinical features). We thus arbitrarily distributed these suddenly added cases to the reported cumulative number of confirmed cases from Jan 21 to Feb 14 for Hubei Province by a fixed factor (refer to Table S1), assuming that they were linearly accumulative in those days. It is the same forth with the data for Wuhan City. Regression analyses indicate that all sets of data were well fitted with the Boltzmann function (all R2 values being close to 0.999; Figs. 1A, B, S1, and Table 1). The potential total number of confirmed cases for mainland China, Hubei Province, Wuhan City, and other provinces were estimated as 72,800±600, 59,300±600, 42,100±700 and 12,800±100; respectively; those for provinces Guangdong, Zhejiang, Henan and Hunan were 1300±10, 1170±10, 1260±10, 1050±10, 1020±10 and 940±10, respectively (Table 1); those for Beijing, Shanghai, Guangzhou and Shenzhen were 394±4, 328±3, 337±3 and 397±4, respectively. In addition, we estimated the key date, on which the number of daily new confirmed cases is lower than 0.1% of the potential total number as defined by us subjectively (refer to Table 1).Table 1Regression analysis results of confirmed cases of SARS-CoV-2 in China.Regionswithout uncertaintywith uncertaintyaThe reported cumulative number of confirmed cases may have uncertainty. Assuming the relative uncertainty follows a single-sided normal distribution with a mean of 1.0 and a standard deviation of 10%, the potential total number and key dates were estimated at 95% CI. For detail, refer to the Methods section and Figs. 1C, D, S2 and S3.potential total numberkey datebKey date is determined when the number of daily new confirmed cases is less than 0.1% of the potential total number.R2potential total number (mean, 95% CI)key date (95% CI)bKey date is determined when the number of daily new confirmed cases is less than 0.1% of the potential total number.China72,800±6002/280.99979,589 (71,576, 93,855)(2/28, 3/10)Hubei Province59,300±6002/270.99964,817 (58,223, 77,895)(2/27, 3/10)Wuhan City42,100±7002/270.99946,562 (40,812, 57,678)(2/28, 3/10)Other provinces12,800±1002/270.99913,956 (12,748, 16,092)(2/27, 3/13)Guangdong Province1300±102/220.9991415 (1324, 1550)(2/22, 3/01)Zhejiang Province1170±102/200.9971269 (1204, 1364)(2/21, 2/27)Henan Province1260±102/240.9991372 (1271, 1559)(2/26, 3/09)Hunan Province1050±102/260.9991140 (1050, 1279)(2/28, 3/11)Beijing City394±42/250.999429 (395, 486)(2/25, 3/11)Shanghai City328±32/220.999356 (334, 388)(2/22, 3/01)Guangzhou City337±32/200.998365 (346, 393)(2/20, 2/28)Shenzhen City397±42/180.998430 (407, 461)(2/17, 2/25)a The reported cumulative number of confirmed cases may have uncertainty. Assuming the relative uncertainty follows a single-sided normal distribution with a mean of 1.0 and a standard deviation of 10%, the potential total number and key dates were estimated at 95% CI. For detail, refer to the Methods section and Figs. 1C, D, S2 and S3.b Key date is determined when the number of daily new confirmed cases is less than 0.1% of the potential total number. Open table in a new tab The above analyses were performed assuming that the released data on the confirmed cases are precise. However, there is a tendency to miss-report some positive cases such that the reported numbers represent a lower limit. One typical example indicating this uncertainty is the sudden increase of more than 14 000 new confirmed cases in Hubei Province on Feb 12 after clinical features were officially accepted as a standard for infection confirmation. Another uncertainty might result from insufficient kits for viral nucleic acid detection at the early stage of the outbreak. We thus examined the effects of such uncertainty using a Monte Carlo method (for detail, refer to the Methods section in SI file). For simplicity, we assumed that the relative uncertainty of the reported data follows a single-sided normal distribution with a mean of 1.0 and a standard deviation of 10%. Under the above conditions, the potential total numbers of confirmed cases of SARS-CoV-2 for different regions were estimated (Figs. 1C, D, S2 and S3) and summarized in Table 1. The potential total numbers for China, Hubei Province, Wuhan City and other provinces were 79,589 (95% CI 71,576, 93,855), 64,817 (58,223, 77,895), 46,562 (40,812, 57,678) and 13,956 (12,748, 16,092), respectively, indicating that overall the outbreak may not be so bad as previously estimated.9Anastassopoulou, C., et al., Data-based analysis, modelling and forecasting of the novel coronavirus (2019-NCOV) outbreak. doi:10.1101/2020.02.11.20022186, 2020.Google Scholar Such uncertainty analysis also allowed us to estimate the key dates at 95% CI. As summarized in Table 1, the key dates for mainland China, Hubei Province, Wuhan City, and other provinces would fall in (2/28, 3/10), (2/27, 3/10), (2/28, 3/10) and (2/27, 3/13), respectively. Finally, the ongoing SARS-CoV-2 outbreak has undoubtedly caused us the memories of the SARS-CoV outbreak in 2003. We thus collected the data from the WHO officiate website for analysis, and found that the cumulative numbers of confirmed cases of 2003 SARS-CoV both in China and worldwide were fitted well with the Boltzmann function, with R2 being 0.999 and 0.998, respectively (Figs. 1E and F). In summary, we found that all data sets, including both the on-going outbreak of SARS-CoV-2 in China and the 2003 SARS-CoV epidemic in China and worldwide, were well fitted to the Boltzmann function (Fig. 1 and S1). These results strongly suggest that the Boltzmann function is suitable for analyzing the epidemics of coronaviruses like SARS-CoV and SARS-CoV-2. One advantage of this model is that it only needs the cumulative number of confirmed cases, somehow as simple as the recently proposed model.10Huang N.E. Qiao F. A data driven time-dependent transmission rate for tracking an epidemic: a case study of 2019-nCoV.Sci Bull. 2020; https://doi.org/10.1016/j.scib.2020.02.005Crossref Scopus (52) Google Scholar In addition, the estimated potential total numbers of confirmed cases and key dates may provide valuable guidance for Chinese central and local governments to deal with this emerging threat at current critical stage. None. We thank graduate students (Boyan Lv, Zhongyan Li, Zhongyu Chen, Yu Cheng, Mengmeng Bian, Shuang Zhang, Zuqin Zhang, and Wei Yao; all from Prof. Xinmiao Fu's research group at Fujian Normal University) for data collection. This work is support by the National Natural Science Foundation of China (No. 31972918 and 31770830 to XF). Download .pdf (.73 MB) Help with pdf files