decreasing after a short expansion lasting about two weeks from Jan 21.
The most intuitive metric to quantify the control efficacy is the effective reproduction number t R , which is defined as the mean number of secondary cases infected by a case with symptom onset at day t . Here we consider a slightly different one, temporal reproduction number, to include the period-dependent Here we consider an even-worse condition about data accessibility, where not only the complete tracing records, but also the full epidemic curves are unknown.
That is a usual situation in the early stage of an epidemic, for example, the number of confirmed cases of COVID-19 for each province in mainland China is made public every day, while the symptom onset of each case is not reported by Chinese CDC. We develop a Monte-Carlo method to infer the epidemic curve from a small number of recorded symptom onsets collected from scattered news reports (in total we have collected 3650 records with precise symptom onset time).
Combining it with the methods proposed in (5) and (6), we can estimate temporal reproduction number and thus evaluate the efficacy of control measures.
Technical details are given in (Appendix).
In addition to assumptions in (6), our method depends on another assumption that the distribution of the time intervals between symptom onsets and confirmations for each province, () pt  , is close to the synthesized one by scattered records. Based on the six provinces with the most records of symptom onsets, we have checked that the individual distributions are close to each other and can be well resembled by the synthesized distribution, which follows a translational Weibull distribution (Appendix). The province-level results (Table) demonstrate the impressive achievement of control measures, namely t R for the majority of provinces decreased to <1 within one week from the starting date of control. Even for Hubei, the epidemic was under control ( 1 t R  ) just in two weeks.
In addition, the average temporal reproduction number . We introduce the translational parameter because some cases are confirmed immediately so (0) 0 p  , while the original Weibull distribution gives (0) 0 p = for any shape parameter and scale parameter.
Appendix Figure 2 . Comparison between the synthesized distribution of time intervals between symptom onsets and confirmations (red circles) and the fitting curve (blue curve) that obeys a translational Weibull distribution.
We have collected the information about daily number of confirmed cases for all provinces in mainland China from Jan 11, 2020 to Feb 22, 2020. The number of cumulated confirmed cases is 76,936. For a very small fraction (4.74%) of these confirmed cases, we have found their symptom onset times by hand via scattered new reports. The confirmed cases for Tibet and Qinghai are only 1 and 15, so we do not analyze these two provinces. Appendix Figure 3 reports the estimated effective production number for each province from Jan 10, 2020 to Feb 21, 2020 by using the present method.
Though the synthesized distribution () pt  can well resemble individual distributions, using inferred data may still bring bias because the distribution () pt  is not stable, usually with smaller and smaller mean and standard deviation in the progress of an epidemic (8) . Appendix Figure 4 compares the estimates of effective reproduction numbers by the true and inferred records of symptom onsets for the six provinces with most known records. At the very beginning, the estimates from inferred data are smaller than the ones from true records, but they are getting closer and closer later and show almost the same * t .
Indeed, we still overestimate the reproduction number in the early stage, because a large fraction of cases (expect Hubei) are importations (7, 8) . Fortunately, the present method shows accordance with the one accounting for importations in the later part, for example, t R of the three example provinces (Guangdong, Hunan and Shandong) approach zero and then continuously decrease at Jan 23, Jan 26 and Jan 30 by the method in (8) and at Jan 25, Jan 25, Jan 27 by the present method.