Thus far, research on the basic reproduction number (R0) of COVID-19 has reported an estimated R0 in the range of 2.2 to 3.6 [1] [2] [3] [4] , which means that, on average, each infected person spreads the infection to more than two persons. Therefore, the majority of the population is at risk of infection if no intervention measures were undertaken. The true size of the COVID-19 epidemic remains unknown, as a significant proportion of infected individuals only exhibit mild symptoms or are even asymptomatic.
The intensive care needed to treat COVID-19 patients is adding pressure to the already stressed healthcare system worldwide. A recent report from WHO found that the case fatality rate was 5.8% in Wuhan, compared with 0.7% in the rest of the country [5] . The striking difference is mainly due to the sudden surge of severely ill people overwhelming the healthcare system. Hence a timely assessment of the evolving epidemic size is crucial for resource allocation and triage decisions.
In this article, we apply the back-calculation procedure developed by Brookmeyer and Gail to obtain a lower bound estimate of the number of infected individuals [6, 7] . The estimation procedure projects the observed numbers of confirmed cases to numbers previously infected, where the number of confirmed cases in each time interval follows a multinomial distribution with cell probabilities that can be expressed as a convolution of the density of the infection time and the distribution of time to diagnosis. As a result, the problem is reduced to estimating the size of a multinomial population. This approach has been applied to study infectious diseases such as acquired immunodeficiency syndrome (AIDS) and bovine spongiform encephalopathy (BSE) epidemic. (7, 8) Although the method can not predict future new infections, it provides a lower bound estimate for the number of confirmed cases in the near future to guide decision making on medical resource allocation.
Moreover, the estimation of infection density provides an assessment of the infection risk during and after the drastic measure undertaken by the Chinese authorities to lockdown the city of Wuhan. 
we set τ 0 ≡ −∞, so that no infection occurred prior to τ 0 , and τ K is the date of the last available report. Assume that the distribution of U given U ≤ τ K has a density function φ(u; α), u ≤ τ K , where α is a p α -dimensional vector of parameters. Moreover, we assume that, given U = u, T is a nonnegative, continuous random variable with the distribution function F u (t; β), where β is a p β -dimensional vector of parameters. To implement the maximum likelihood estimation, we assume that F u (t; β) ≡ F (t; β), that is, the time from infection to diagnosis is independent of the date when the individual was infected. This assumption may not be valid for the confirmed case in Wuhan or other cities in the Hubei province, as the medical resource in the early stage of the epidemic is extremely scarce and thus it may take a long time for infected individuals to be diagnosed. In fact, the surge in the number of confirmed cases in China on February 13 was due to a new diagnosis classification rule for cases in the Hubei province; the RT-PCR test for COVID-19 was not available for many of the previously infected individuals despite having symptoms of pneumonia. On the other hand, this assumption seems reasonable for areas outside of Hubei province as the healthcare system has not been overly stressed.
It follows from the assumption that T independent of U given U ≤ τ K that the probability of being diagnosed in the time interval [τ j−1 , τ j ), conditioning on being infected before 3 for use under a CC0 license. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted April 28, 2020. . the last examination time τ K , is given by
(1) Here π 1 , . . . , π K are the probabilities of being diagnosed in the K time intervals among those infected before time τ K .
Denote by N the (unknown) number of individuals infected before τ K and let X j , j = 1, . . . , K be the number of cases diagnosed in the time interval [τ j−1 , τ j ). Define n = K j=1 X j , so that a total of N − n individuals were infected but not diagnosed before 4 for use under a CC0 license.
This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted April 28, 2020. . Figure 1 depicts the structure of observed data and model parameters. Write X = (X 1 , . . . , X K , N − n) and π(θ) = (π 1 (θ), . . . , π K (θ), π K+1 (θ)) , so X follows the multinomial distribution with trial size N and cell-probabilities π(θ). Given the observed data, X 1 , . . . , X K , the likelihood function of (N, θ) is
The corresponding log-likelihood, up to a constant, is
where Γ(x+1) = x! is the Gamma function. We propose to estimate (N, θ) by the maximizer of likelihood function
Then the expected number of confirmed cases in the time interval [τ j−1 , τ j ) can be estimated
It is worthwhile to point out that the back-calculation algorithm described above estimates the two sets of parameters (α, β) simultaneously; this is different than the original algorithm where the parameters β in the Weibull distribution were replaced by values derived from other studies which provided information about the incubation time [6, 7] . In our analysis, T stands for the duration between infection and laboratory confirmation, whose information plays an important role in guiding medical resource allocation. Since no active surveillance testing and contact tracking were conducted in China before March 15, the length of time to confirmation is expected to be longer than the incubation time, that is, the duration between infection and symptom onset.
We have performed the analysis by incorporating the estimated incubation time distribution reported in [9] , but the model did not fit well. As a result, we decided to estimate the two sets of parameters simultaneously. However, one major consequence of such a strategy is that the distribution of U and T are estimated subject to a location shift factor. To see this, we note that the infection time, U , and the time from infection to diagnosis, T , are 5 for use under a CC0 license.
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The copyright holder for this preprint this version posted April 28, 2020. . not directly observed in the data, and we only get to observe U + T , that is, the time of confirmation. Hence a different set of random variables T * = T −δ and U * = U +δ yield the same distribution as T + U . On the other hand, although the location shift factor δ can not be estimated directly from the observed data, we can assess the magnitude of the location shift by comparing the estimated time to diagnosis distribution to what reported in the existing literature. Note that, under location shift, the relative difference in time between two landmark time points, such as the peak and lowest point in the infection density or last week compared to this week, can be estimated from the data.
The We model the time from infection to diagnosis using the Weibull distribution, which has been shown a reasonable approximation for the incubation time in infectious disease research. As argued before, the stationarity assumption for the time to diagnosis, that is, the distribution of duration between infection to diagnosis does not depend on the time of infection, should hold approximately in cases diagnosed outside the Hubei Province.
Moreover, for modeling the intensity function φ(t, α), we assume a step function with jump discontinuities every 7 days starting from January 20 (day 0), so that the risk of infection is constant within each week. Since the first two cases diagnosed outside of Hubei were reported to have visited Wuhan on January 7 and 9 and developed symptoms on January 13 and 14, respectively, we set τ 0 to January 1, 2020 to account for the possible infection period. Specifically, the incidence density function among cases infected before March 15 is of the form
where a 0 = −19, a j = 7×(j −1), j = 1, . . . , 9. Note that we require 9 j=1 α j (a j −a j−1 ) = 1 to ensure that φ(u, α) is a proper density function. 6 for use under a CC0 license. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted April 28, 2020. . q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 2B ). Moreover, the incidence increases 46% during the week of Wuhan lockdown from the preceding week, but dropped 76% in the next week, suggesting that the travel ban that started from Wuhan and its neighboring cities had initially increased the disease spread in other provinces, but was able to effectively slow down the spread of the disease in just two weeks.
Finally, continuous assessment of potential new cases to be diagnosed in the near future can be implemented by applying the back-calculation algorithm using up-to-date data. Figure 2C shows the projected number of confirmed cases with data available up to that time point, assuming no new infections afterward. The difference between the expected and observed cumulative number of confirmed cases gives the near-term prediction of additional cases to be diagnosed. Since the prediction is relatively unstable for the first few weeks of the epidemic due to the limited amount of data, we only perform prediction after February 10, that is, the fourth week into the epidemic. As shown in Figure 2C , the prediction obtained after February 24 is very close to the total number of confirmed cases at the end 8 for use under a CC0 license.
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The copyright holder for this preprint this version posted April 28, 2020. . https://doi.org/10.1101/2020.04.23.20074708 doi: medRxiv preprint of the study period (March 15). Ignoring the spike occurred around February 20, which was most likely caused by the delayed report of cases in two prisons, the prediction algorithm performs reasonably well after February 17. This shows that the back-calculation algorithm can potentially provide a useful utility for the planning of health care allocation, especially for an epidemic that is still growing.
Different countries have taken different measures in response to the novel coronavirus, and there has been a continuing heated debate on whether aggressive COVID-19 control measures cost more than they are worth. Among all countries, China has imposed the most sweeping restrictions in response to COVID-19. The authorities locked down cities, restricted movements of millions and suspended business operations to prevent further outbreak of the disease. South Korea, another country on the front-line of the epidemic, adopted the "test and trace" strategy to aggressively test people for the disease and quarantine those who tested positive, so that the rest of the population can go about their daily lives. Both countries have observed a significant decline in the number of confirmed cases. However, to the best of our knowledge, no formal quantitative evaluation of the effect of these aggressive control measures on the incidence of infection has been conducted. This research aims to provide some preliminary evidence on the effectiveness of these measures by analyzing data from China outside of the Hubei province. We conclude that the extreme measures undertaken by the Chinese government have effectively slowed down the spread of the disease outside of the Hubei province in about two weeks. We also demonstrate that the back-calculation algorithm can be used to estimate the number of infected individuals to be diagnosed in the near future. This provides a useful utility to guide the planning of medical resource allocation in the middle of the epidemic.
for use under a CC0 license. This article is a US Government work. It is not subject to copyright under 17 USC 105 and is also made available (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted April 28, 2020. . https://doi.org/10.1101/2020.04.23.20074708 doi: medRxiv preprint