Results

Cumulative number of detected and diagnosed cases
The COVID-19 epidemic was initiated in Wuhan, the Provincial Capital of Hubei Province with a total population of 14.2 million, including 5.1 million mobile population. The mortality rate was 5.5/1000 for Wuhan residents with most available data in 2018. Assuming all diagnosed cases in China were infected in Wuhan (an exaggerated scenarios for illustration purpose), the two-month incidence rate of COVID-19 was 2.6/1000 among Wuhan residents. Based on reported case mortality of 2.3%, the population-based mortality of COVID-19 was 0.6/1000, or 1/9th of the mortality of Wuhan residents.
Figure 1 presents the cumulative diagnosed cases F(x) and major events during the study period from December 8, 2019 to February 8, 2020. During the period, a total of 37,198 cases were diagnosed and reported. The daily cases varied from 0 to 3886 with the median cases of 199 (January 8, 2020), and inter-quarter range (IQR) of 24 (December 23), and 830 (January 23, 2020).
Fig. 1 Cumulative number of diagnosed COVID-19(2019-nCoV) infection F(x) and key events before, during and after declaration of the outbreak in the first 2 months of the Epidemic in China

Dynamics of the epidemic and response to massive interventions
The dynamic changes based on the observed F(x) in Fig. 1 were presented in Fig. 2 using the first derivatives F ′ (x) (top panel of the figure) and the second derivative F ′  ′ (x) (bottom panel of the figure), respectively. Before the declaration of outbreak, information provided by the two dynamic measured was similar: not much variations were revealed relative to the changes after the outbreak. These findings suggest the nonlinear and chaotic character of the COVID-19 outbreak.
Fig. 2 The first F′(x) and second derivative F″(x) of diagnosed COVID-19 (formally 2019-nCoV infection) F(x) before, during and after declaring the outbreak in first 2 months of the Epidemic
After declaring the outbreak on January 20, information revealed by F ′  ′ (x) differed much from F ′ (x). Based on information from F ′ (x), the newly diagnosed F ′ (x) cases increased progressively with some fluctuation, then peaked on February 4, 2020, and followed by a decline. The increases in the diagnosed cases could be either due to the natural growth of the epidemic in itself, or due to the interventions to detect the infected or both. Furthermore, F ′ (x) provided no sign of epidemic decline until February 4, 2020. In other words, we have to wait for at least 14 days after the massive anti-COVID-19 epidemic without using information derived from F ′  ′ (x).
Quite different from F ′ (x), F ′′ (x) removed the time trend of F ′ (x) to show the acceleration/deceleration of diagnosed COVID-19. Consequently, F ′  ′ (x) was much more sensitive than F ′ (x) to gauge the intrinsic dynamics of the epidemic in response to the massive anti-COVID-19 action. Since January 21, 2020 after the massive anti-COVID actions, the F ′  ′ (x) suddenly became very active, as indicated by the alternative accelerations and decelerations. F ′  ′ (x) reached the peak on January 27 after the distribution of large number of test kits on January 26, which was an action based on the decision at the central government level in a meeting held by Chinese President Xi Jinping on January 24 and 25, the Chinese New Year’s Eve and New Year’s Day.
In addition, the estimated F ′′ (x) captured three significant decelerations on January 28, February 5 and 6 (two days in a row), and 8, 2020 respectively; corresponding to the intensified massive actions in locating and treating the infected, locking down more communities, plus mask use and massive pathogen sterilization in neighborhood environment. In addition to informing whether the epidemic was responsive to the massive interventions, information from F ′′ (x) signaled an overall downturn of the epidemic since the beginning of the massive anti-COVID-19 action on January 21, 2020. This was further pronounced by the band region between the two dotted lines in Fig. 2. Despite zigzags, an overall downward trend in F ′′ (x) was clearly revealed by the downward and progressively narrowing down band region. This trend strongly indicates that the epidemic could be brought under control soon with the current interventions in place.

Exponential growth and detection rate
The observed F(x) fit the exponential model of Eq. 4 well with R2 = 0.9778. The estimated α =1.1070, representing the first person who was infected and ignited the epidemic. The estimated β =0.1716, representing the growth rate. Using this estimated growth rate, it takes only 4 days for the diagnosed COVID-19 to double.
Figure 3 presents the daily detection rates, estimated with the fitted exponential model from day one of the epidemic to the last day of the study period. Based on findings in this figure and data from Figs. 1 and 2, we divided the COVID-19 epidemic during the first two months of the epidemic into five phases.
Fig. 3 Estimated daily detection rate Pi of COVID-19 (2019-nCoV) infection before, during and after declaration of the outbreak, the first 2 months of the Epidemic in China
Phase 1 was from December 8 to 25, 2019. During this period, the detection rate Pi was high overall, with fluctuations around and above 100%. This was corresponding to the early period after the first suspected case was identified and diagnosed.
Phase 2 was from December 26, 2019 to January 8, 2020, covering the New Year’s Day. The detection rate Pi fluctuated at around 50% with the lowest of 17% on December 31, 2019 and the highest of 108% on January 8.
Phase 3 was from January 8 to 20, 2020, and it was featured with a progressive decline in the estimated Pi from 105% on January 8, 2020 to 1% on January 20, 2020. This progressive declining period was the time for the Chinese to prepare for the traditional Chinese New Year’s with the longest and highest level of celebration. Unfortunately, the COVID-19 as an outbreak was silently stepping in during this period.
Phase 4 was from January 20 to 27, 2020 with the estimated Pi increased from 1% on January 20, 2020 to surpass 100%, and reached the peak of 170% on January 27, 2020. This period was corresponding to the initiation and progressive intensifying of the massive intervention organized and coordinated by the Central Government of China.
Phase 5 started from January 27, 2020 to the end of the study period, corresponding to the sustained massive national efforts, plus frequent emphases. Different from the previous four phases, reductions in the estimated Pi during this phase were not an indication of under-detection but an indication of declines in the epidemic reflected by the detected and confirmed cases of COVID-19. This is because the model predicted Pi did not consider any interventions but natural growth of the epidemic.
Based on Figs. 2 and 3 (Phase 4 and 5), three pieces of information can be derived: (1) The epidemic was highly sensitive to external interventions, supporting the nonlinear and chaotic characters revealed by the long latent period in the first three phases; (2) the massive national efforts were highly effective in detecting the detectable COVID-19; (3) signal for the COVID-19 in China to decline appeared on January 21 in 2020, 14 days before the start of eventual declines on February 4, as indicated by F ′  ′ (x) and F ′ (x) in Fig. 2 and Pi in Fig. 3.