The ongoing pandemic of coronavirus disease 2019 , caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was first identified in Wuhan, Hubei, China in December 2019 (WHO, 2020; Huang et al., 2020; Guan et al., 2020; Li et al., 2020) , and then declared as a Public Health Emergency of International Concern (PHEIC) on Jan. 30 th , 2020 and a global pandemic on Mar. 11 st , 2020 (WHO, 2020 . To contain this rapid and wide spread of COVID-19 outbreak, it is imperative to quantify correctly the basic reproductive number (R0) defined as the average number of secondary cases generated from a primary case, a useful indicator of infectious disease transmission, locally and globally. We can use R0 larger or smaller than 1, being determined by the contact rate, transmission probability, and infectious period, to assess the possibility of potential spread or the control of COVID-19 in affected communities (Ferguson et al., 2005) .
At the beginning of COVID-19 epidemic, several studies have tried to quantify R0 by the models based on parameters derived from the SARS epidemic in 2003 (Wu et al., 2020; Imai et al., 2020a; Imai et al., 2020b) , which was estimated as 2-3, a value close to the R0 of SARS (Raily et al., 2003) . As the epidemic evolves over time, the difference between SARS and COVID-19 has been gradually noted in terms of transmission mode and infectious period in relation to clinical symptoms and signs (Huang et al., 2020; Guan et al., 2020; Li et al., 2020; Wang et al., 2020) . The property of asymptomatic infection and various clinical symptoms and signs of COVID-19 render the force of spread intractable to be estimated by the SARS models. The disparity of transmission between SARS and COVID-19 has been further speculated by the rampant spread in the . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . https: //doi.org/10.1101 //doi.org/10. /2020 Washington and the New York states of USA with a doubled increase from Mar. 26 th to Apr. 1 st and also in the beginning epidemics of European countries such as Italy and Spain (WHO, 2020) .
The COVID-19 outbreak on the Diamond Princess Cruise Ship (NIID, 2020a; NIID, 2020b) provides an unprecedented opportunity for us to understand its original transmissibility and force of spread by three unique epidemic-related characteristics.
First, susceptible, contact, transmission, and infectious period are clearly defined according to a fixed cohort of global passengers and crews living in the isolated setting of this cruise ship. Second, such a transmission mode through the isolated environment of the cruise ship provides strong human-to-human community-acquired infection in close setting distinct from the transmission mode through household or hospital and relevant healthcare institutions while assessing the transmissibility of this emerging infectious disease. The third is that the infected persons onboard consist of multi-nationals giving us a better sense of the transmissibility of COVID-19 across racial groups. Based on these three characteristics related to the transmission mode on the cruise ship, two thorny questions that have been not addressed in the previous studies are raised (Mizumoto et al., 2020; Sawano et al., 2020; Rocklöv et al., 2020) . Can we estimate and predict the epidemic curve to determine timely containment measures and evacuation? What is the effectiveness of containment measures on quarantine and isolation for the infected passengers before evacuation? Answering these two questions plays a crucial role in the guidance of how and when containment measures can be designed and offered for the coming cruise ships that are susceptible to the outbreak of COVID-19.
. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)
The copyright holder for this preprint this version posted June 23, 2020 . . https://doi.org/10.1101 Although the conventional deterministic approach by fitting the compartment models with the ODE system to observed number of reported cases can shed light on the dynamic of COVID-19 transmission and R0 at population level, such an approach may not feasible for the outbreak occurring on cruise ship with modest size of susceptibles. To cope with the uncertainty of outbreak in a confined environment, we thus developed a Bayesian compartment models for the COVID-19 outbreak on the Diamond Princess Cruise Ship.
The rest of the article is organized as follows. Section 2 briefs the empirical data on the outbreak of COVID-19 on the Diamond Princess Cruise Ship alone with the events took place in the quarantine measures on board. Section 3.1 gives the details on the specification of Bayesian ODE compartment model depicting the propagation of COVID-19 through the states of susceptibles, exposed, infectives, and removed. The methods for validating the fitted cases and evaluating the efficacy of containment measures are addressed in Section 3.2. Section 4 shows the estimated results on the dynamic of COVID-19 outbreak on the Diamond Princess Cruise Ship by fitting the Bayesian ODE compartment model to the empirical data. Section 5 gives the discussion of applications and implications on the proposed Bayesian ODE compartment model to assess the disease outbreak data.
. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)
The copyright holder for this preprint this version posted June 23, 2020 . . https://doi.org/10.1101 2 Empirical data
The data on the voyage of Diamond Princess cruise ship, the reported number of confirmed COVID-19 cases, and the quarantine process were retrieved from publicity available sources including the Princess Cruise website of the Carnival Cooperation (Princess Cruise Lines, 2020), the official website of Ministry of Health, Labor and Welfare, Japan (Ministry of Health, Labor and Welfare, Japan, 2020), and the National Institute of Infectious Disease, Japan (NIID, 2020a; NIID, 2020b) . The auxiliary information sources including the general media and scientific news and web-based summaries were also used for a cross-check on the cases numbers and the quarantine measures and the evacuation after the outbreak in Diamond Princess cruise ship (Princess Cruise Lines, 2020; Wikipedia, 2020a; Wikipedia, 2020b; Garcia-Navarro, 2020; Normile, 2020) . A prospective observational study design on the propagation of the COVID-19 outbreak was applied to evaluating the transmissibility of SARS-CoV-2.
Details on the derivation of information on COVID-19 outbreak on the Diamond Princess Cruise Ship are provided in the Supplementary material A.
The Diamond Princess Cruise Ship departed from Yokohama on Jan. 20 th , 2020.
There was a total of 3711 subjects including 2666 passengers from 11 countries (1285 from Japan, 470 from Hong Kong, 425 from USA, 215 from Canada, 40 from United Kingdom, 25 from Russia, 20 from Taiwan, 15 from Israel, 13 from New Zealand, and the rest of 158 from Australia and Germany) and 1045 crew members on the voyage of the Diamond Princess (NIID, 2020a; Ministry of Health, Labor and Welfare, Japan, 2020;
. CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136465 doi: medRxiv preprint Wikipedia, 2020a) . The outbreak started from a passenger from Hong Kong who had joined with part of the voyage and disembarked on Jan. 25 th , 2020 who was later confirmed as a COVID-19 case through laboratory testing (Ministry of Health, Labor and Welfare, Japan, 2020; Wikipedia, 2020a) . The Diamond Princess Cruise Ship was therefore quarantined at the Yokohama harbor since its dock on Feb. 5 th , 2020. Following the index case, 10 cases of COVID-19 were reported on Feb. 5 th , 2020, which heralded the outbreak yielding 762 cases in the following weeks (WHO, 2020; Ministry of Health, Labor and Welfare, Japan, 2020; Wikipedia, 2020a) . Up to Feb. 29 th , the COVID-19 outbreak on the Diamond Princess Cruis Ship have resulted in 6 deaths (WHO, 2020).
During the quarantine process, the personal protective equipement was provided for crew members. Passangers and crews were instructed to monitor body temperature by themselves. Subjecs with body temperature higher than 37.5 degree celsus or developed any illness were referred to the medical center on the ship where medical services were provided and the test for SARS-CoV-2 were referred. To maintain operations of the ship, some crew continued to perform essential but limited services while the ship remained in quarantine (NIID, 2020a) . This symptom-based approach started since the ship dock at Yokohama harbored on Feb. 5 th . With the gradual expansion of laboratory capacity from Feb. 11 st , the quarantine officers provided test systematically gauged by testing capacity and covered passengers without the manifestation of symptoms from Feb. 14 th (NIID, 2020a; NIID, 2020b) .
. CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136465 doi: medRxiv preprint Figure 1 shows both the epidemic curve and cumulative cases of all passengers on the Diamond Princess Cruise Ship during Jan. 20 th to Feb. 29 th , 2020. The cases were identified mainly based on the symptom of fever and the close contacts defined by sharing the cabin with confirmed cases during the quarantine process followed by laboratory test (NIID, 2020a; NIID, 2020b; Ministry of Health, Labor and Welfare, Japan, 2020). Passengers who were tested positive for SARS-CoV-2 were removed from the ship and transferred to health care facilities on land. Subjects with negative test result will be kept in the ship for the rest of quarantine period. The COVID-19 cases identified on the Diamond Princess Cruise Ship were reported on daily basis by the Ministry of Health, Labor and Welfare, Japan (2020). On Feb. 16 th , 2020, several countries started requesting the evacuation of their citizens from the cruise ship. The USA was the first country to pull out 380 Americans on Feb. 17 th , 2020. It should be noted that the quarantine measure has commenced from Feb. 5 th but isolation and decontamination were gradually expanded from Feb 14 th until the complete evacuation on Feb. 19 th , 2020 (NIID, 2020a; NIID, 2020b).
The data used for the following analysis consisted of the cases reported between Jan. 20 th to Feb. 19 th , 2020, the date of ending quarantine announced by Ministry of Health, Labor and Welfare, Japan. Detailed counts of the cases and the information sources are provided in Supplementary Material S- . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. .
To predict the propagation of COVID-19 outbreak on the cruise ship making use of the observed number of cases with the consideration of uncertainty, a Bayesian ODE SEIR model was developed. A four-compartment model (Figure 2 ) was applied to describe the daily flow of the population on the Diamond Princess cruise ship between the four status with respect to COVID-19 infection, including S (susceptible population), E (exposed persons to infectious persons), I (infectious persons with COVID-19), and R (recovered persons from infectious status or infectious persons being isolated or removed from the cruise ship, or dead) states.
Let s(t), e(t), i(t), and r(t) denote the numbers of susceptibles, exposed, infectives, and removed/isolated/recovered/dead on the cruise ship at time t. The four-compartment system can be depicted by using the following ordinary differential equations to describe the instantaneous change of the four types with time
With the solution of the four-state differential equations regarding the empirical data on the observed COVID-19 cases in Diamond Princess cruise ship, the parameters of transmission coefficient (β), progression rate (σ) to and recovery rate (α) from the infectious status can be estimated. The estimated results were further used to depict the . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. .
propagation on the number of four COVID-19 infective states by time. Note that because the population of the Diamond cruise ship is fixed, the following equation always holds.
( 2) where N is the initial size of this cohort on lockdown, namely the 3711 subjects being quarantined in Diamond Princess cruise ship.
The Bayesian Markov Chain Monte Carlo method for the ordinary differential equation system for the four-compartment SEIR model was used to estimate parameters of interests. The Bayesian approach facilitates us to incorporate the prior information of parameters. As COVID-19 is an emerging disease, little is known about the time interval between exposed and infected, as well as between infected and recovery. We borrowed the experience from the first report in Wu-Han that is the origin infectious resource for the outbreak of the Diamond Princess cruise ship. We then assigned Gamma(53, 278) and Gamma(24, 168) for and in order to fit the mean duration from exposed to infective and from infective to recovery for 5 (95% CI: 4-7) and 7 (95% CI: 5-12) days, respectively (Huang et al., 2020; Guan et al., 2020) .
Since only data on the number of reported confirmed cases per day were available, they contain those already infected, regardless of appearance of clinical symptoms and signs, and recovered cases, we therefore modeled two random variables, number of unknown infected status and the confirmed cases per day following two normal distributions of , 100
and , 100 ,
. CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136465 doi: medRxiv preprint respectively. Following the derivation of transmission coefficient, β, the basic reproduction number (R0) were estimated with function of transmission coefficient (β) and recovery rate (α) as follows,
The Bayesian Markov Chain Monte Carlo method for the ordinary differential equation system for the four-compartment SEIR model was used to estimate parameters of interests. A series of Bayesian ODE SEIR models were applied to quantify the force of COVID-19 spreading by using the data up to Feb. 19 th , 2020.
To assess the homogeneity of estimated results on transmission coefficient and . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. .
By using a Bayesian ODE SEIR model in conjunction with reported cases series, we estimated the transmission coefficient and R0 for the COVID-19 outbreak on the Diamond Princess Cruise Ship in Table Table 2 . Based on the transmission parameter estimated by the empirical data from Jan. 20 th to Feb. 19 th , the fitted cumulated cases on Feb. 19 th are 638 (95% CI: 558-715), which is comparable to the observed cumulative cases of 635 (χ 2 (1) = 0.02, P=0.90).
Based on the parameters of basic reproductive number trained by the data before CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136465 doi: medRxiv preprint only 16% passengers remain susceptible the outbreak would be expected to end on Mar. 7 th according to the theory of herd immunity when more than 82% [(1-1/5.71)100%] of total population have been infected. The evolution of COVID-19 outbreak can further be decomposed into the status of susceptible, exposed, infected, and removed (including recovery and death) among the 3711 passengers of the Diamond Princess Cruise Ship. Table 3 in the Supplementary Material). During the acceleration period of the outbreak (from Jan 10 th to Feb. 26 th ), the exposed subjects (Figure 4(b) , yellow line) were more than the infected (Figure 4(b) , blue line). The difference between two components was estimated as 234 on Feb. 19 th (S- represents the expected number of subjects who have been exposed but not confirmed as . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. .
infected cases yet would mix up with the susceptibles during evacuation. This accounted for the additional 126 confirmed cases requiring quarantine and isolation after the evacuation of those already exposed and still susceptible. The dynamic curve suggests Feb. 26 th , the crossed point between the exposed and the infected curve as shown in Figure 4(b) , may be a better evacuation time of the remaining uninfected passengers and crews than the real evacuation date of Feb. 19 th . . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136465 doi: medRxiv preprint
Due to a lack of preparedness on quarantine and isolation that have been often seen in the beginning stage of early epidemic on a new pathogen, the outbreak of COVID-19 in such a confined situation with the characteristics of fixed cohort and multi-national passengers provides a natural opportunity to estimate the transmissibility of this new SARS-CoV-2 and also insight into the deployment of containment measures for the outbreak of other cruise ships that are susceptible to COVID-19. It is very striking to see that the basic reproductive number in such a scenario was estimated up to 5.71 (95% CI: 4.08-7.55), which is unusually high compared with the transmissibility of SARS with the estimated R0 ranging from 2 to 3 (Riley et al., 2003), given its aerosols and fomite transmission mode because the persistence of SARS-CoV-2 on inanimate surfaces were reported. 22 More importantly, this unusually high value of R0 is reproducible in the beginning of several epidemics of COVID-19 such as Korea, Iran, and Italy and the subsequent initial epidemics waves in the Europe and in the USA. The high value of R0 also accounts for the soaring number of confirmed cases reported in a short period due to the accelerated transmission amplified by a series of clustered events due to public gathering. The recent outbreaks in the Washington and the New York state of USA triggered by a series of clustered events including long-term care facilities follow such a similar pattern (WHO, 2020). With such a high transmissibility, the virus can lead to an overwhelming number of COVID-19 cases and further bring extraordinary stress to health care system as well as social and economic infrastructure.
Based on the simulated projection of natural dynamic epidemic curve without containment measure, the entire infectious process took 47 days from the introduction of . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . index cases since Jan. 20 th until the end of epidemics on Mar. 7 th when 84% of passengers would be expected to be infected (3135 cases) to reach the phenomenon of herd immunity. Excluding two weeks of the incubation period from the introduction of index case, the epidemic cycle of COVID-19 on the Diamond Princess Cruise Ship is around one month. The natural dynamic curve also provides the control group for the evaluation of the effectiveness of containment measures on quarantine and isolation measures. The partial containment measure on quarantine and isolation in the current scenario of the Diamond Princess led to the reduction of infected passengers by only 34% (95% CI: 31-36%). Had the complete containment measures been implemented on Feb. 14 th , the total number of cases would have been reduced from 761 to 410 cases, resulting in the effectiveness of 54% (95% CI: 46-64%, S- Table 4 in the Supplementary Material) and an even larger effectiveness would have been achieved given an early implementation on Feb. 10 th suggested from the estimated dynamic curve (Figure 4(b) ) while the testing capacity is limited. Moreover, even when the implementation was delayed to Feb. 14 th , the measure adopted to guarantee complete containment as evacuation with adequate deposition should be a high priority. However, timely evacuation is also important as those who were arranged to be evacuated also bore a high risk due to the mixing up of subjects already exposed not yet confirmed as infected cases. The timely evacuation would be expected after the peak of the transmission dynamic on Feb. 26 th rather than the real date of Feb. 19 th . The decision on this early evacuation thus results in the subsequent occurrence of cases among these exposed subjects after evacuation returning to each country that may invoke cluster infection of public gathering and the epidemic in the community. These findings derived from the Diamond Princess can be applied to military . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. .
March. In the outbreak occurring on the USS Theodore Roosevelt, 1156 COVID-19 cases out of 5000 onboard Navy sailors were reported (Wikipedia, 2020c) . Given the similar situation of clustering and close contact in a confined setting, the timely containment measures and evacuation should be implemented as suggested by the transmission dynamic derived from the Diamond Princess outbreak.
The proposed Bayesian ODE SEIR model provides a flexible approach to elucidate the transmission of SARS-CoV-2 by using aggregate data on observed cases with limited resolution. Thanks to the Bayesian framework, the information on disease characteristics such as incubation and transmission period learned from previous studies (Huang et al., 2020; Guan et al., 2020) can be incorporated into our models with their uncertainty being considered. The parameters estimated from the integration of observed data and state-of-the-art knowledge on disease pattern can shed light on the transformation of disease pattern which is crucial for the allocation of resources and the implementation of strategies for us to contain the coming cruise ships that may be attacked by COVID-19.
. CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . 
Source data are available in the supplementary materials. The code for analysis is available from author upon request.
. CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . Taiwan (MOST 108-2118-M-002-002 -MY3; MOST 108-2811-M-002 -640; MOST 108-2118-M-038-001-MY3), Ministry of Education, Taiwan (NTU-107L9003).
The funders had no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review, or approval of the manuscript; and decision to submit the manuscript for publication.
. CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020. . . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 23, 2020 . . https://doi.org/10.1101 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 June 23, 2020. 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 June 23, 2020. 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 June 23, 2020. 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 June 23, 2020. Figure 4 . Predicted COVID-19 cases. Figure 4 (a) shows the predicted number of daily cumulated COVID-19 cases that occurred on the Diamond Princess Cruise Ship given the scenario of no containment measures till Mar. 7 th . Figure 3(b) shows the dynamic curves of the SEIR model, including the frequencies of the components of susceptible (S, green line), exposed (E, yellow line), infectious (I, blue line), and removed (R, red line) along with the estimated 95% credible intervals on the dynamics of each component. Jan-20 Jan-21 Jan-22 Jan-23 Jan-24 Jan-25 Jan-26 Jan-27 Jan-28 Jan-29 Jan-30 Jan- 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 June 23, 2020. 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 June 23, 2020. The blue line shows the expected cumulative cases without containment measure on the Diamond Princess Cruise Ship. The red line is the observed daily cumulative case with containment measure. The total number of COVID-19 cases associated with the outbreak on the Diamond Princess Cruise Ship at Feb. 29 th was 761. Compared with the scenario of no intervention (2262.4, 95% CI: 2088.4-2431.1) the effectiveness of quarantine and isolation in reducing the infected passenger was 34% based on the estimate of relative risk 0.66 (95% CI: 0.64-0.69) between the observed and the expected. Jan-20 Jan-21 Jan-22 Jan-23 Jan-24 Jan-25 Jan-26 Jan-27 Jan-28 Jan-29 Jan-30 Jan- 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 June 23, 2020. 