The outbreak of COVID-19 on the "Diamond Princess" cruise ship (hereinafter referred to as Diamond Princess), which was one of the serious cases in the early outbreak, attracted extensive worldwide attention. This case provides ideal samples with little interference to study coronavirus epidemiology, considering that the infected origin is known, the number of susceptible individuals is constant and the prevention and control work are timely deployed after the first case is confirmed. A modeling analysis of the spread on the "Diamond Princess" identifies a more reasonable spread of the disease and further provides implications for global, national and regional control and prevention measures.
Mathematical modeling has long been an epidemiological tool [1] . Typical methods include regression [2] and compartmental models such as SIR (susceptible, infected and recovered) [3] and its variants are frequently used to model disease spread, such as SARS (Severe Acute Respiratory Syndrome) [4] [5] [6] and COVID-19 [7] [8] [9] [10] [11] [12] . However, these models assume that epidemics transmit in a homogeneous population, i.e., new infectees are from the group of susceptible individuals who can be infected with the same transmission probability. This assumption is far from the social contact patterns, especially when population structure is a spatial-temporal variant with quarantine, and socioeconomic activities [13] . Therefore, a more reasonable epidemic model considers social contact patterns.
The advent of complex network theory [14, 15] provides a method for studying heterogenous epidemiological dynamics [16] . The contact network model extends SIR-type models to fit more complex epidemic situations. It can respond to prevention and control work, such as a decline in the number of nodes indicates the rigorous quarantine of close contacts and the infector is isolated, and the restrictions on movement and public gatherings lead to less contact frequency (reduced transmissibility) between individuals.
Another potential contribution of the contact network model is that it provides an alternative statistic, the average transmissibility, for the basic reproductive number 0 to determine the severity of an epidemic [17] .
varies with the population and 0 presents substantial heterogeneity [18] ; for example, the of COVID-19 ranges 0 from 6.47 [8] to 3.6−4.0 [9] and 2−2.5 [19] . To more efficiently estimate the transmission rate, it is urgent to define new statistics, such as a data-driven rate that is easy to obtain [20] and a time-dependent rate [21] . The average transmissibility, however, denotes the mean probability of disease transmission among individuals and is free of the population; therefore, it is theoretically robust to benefit the public health strategy decision making.
However, the contact network model is still limited and fails to meet the scenario without contact, such as the spread in the presence of dense virus-laden aerosols.
Study [22] found that COVID-19 can deposit on air exhaust outlets, which makes airborne spread possible. Therefore, a new mathematical model for no-contact between individuals is required.
In this study, we develop contact network and no-contact models to keep track of the disease spread in heterogeneous populations and contact patterns, such as unprotected and protected contact and airborne spread. We expect that epidemic reconstruction can provide useful implications for implementing control and prevention measures in larger regions.
The Diamond Princess set sail from Yokohama on 20 January (hereinafter omit 2020), via Hong Kong (25 January) , and returned to Yokohama on 3 February. At this time, the ship had 3711 members including 2666 passengers and 1045 crew, and it is not clear whether and how many people went ashore during the stopover.
As shown in Fig. 1 passengers remained on 19 February.
The aim of this study is to model the epidemic dynamics in these stages and then realistically reconstruct the epidemic curve on the Diamond Princess realistically. We determine the unknown figures including , and the models in the following 1 1 text.
We first estimated the unknown number presented in the above section.
It was assumed that all 705 cases were infected on Diamond Princess before 19
February. An investigation from the Ministry of Health, Labor and Welfare of Japan respectively.
Since there are no available data to directly calculate the details of the infected passengers and crews, we have to refer to the indirect information of the SARS outbreak in Amoy Gardens of Hong Kong, cases of which were 331 during the incubation and infectious period [25] . This epidemic typically provided a confirmation of airborne spread of coronavirus [26] ; we consequently assume that all the cases were infected through viral aerosol, and based on the population census from Hong Kong [27] , the corresponding infection rate was 3.01%.
=80 leaving out the infected in stage 1. We now have and the 1 = 539 corresponding infection rate for passengers through unprotected contact 87.08%. If the ratio of infected crew is the same as passengers, then all figures are clear (see Table 1 ).
We designed a combination of epidemic models to understand the two distinct stages.
For the typical properties of economic and social operations, the epidemiologically relevant contact network regards one node as an individual, the degree of a node as the possible contacts of an individual, and transmissibility as the average probability that the disease transmits between two individuals. A brief depiction of these three components is shown in Fig. 2a . The degree for each node is 4
and if define the transmissibility is defined as 0.3, then in the next time interval (Fig.   2b , ), at least one neighbor of the white infected node will probably be infected. passengers. This contact pattern implies that the connectivity among population on the ship is high local clustering, but a fraction of the relationship is randomly changed.
The chain-binomial model [28] is a classic network models. The chain-binomial The basic reproductive number in chain-binomial model is calculated [17] 0 by
Here and denote the probability of transmission
and critical transmissibility between infected and susceptible individuals, where is infected period, is the average transmissibility for each contact, and and 〈 〉 〈 2 〉 present the first and second moments of degree .
is regarded as an epidemic C threshold, i.e., implies and outbreak will continuously expand; > C 0 > 1 otherwise, the disease only spreads through a finite fraction of the population.
Chain-binomial model is applied to simulate the transmission dynamics of the small-world networks in stage 1. The detailed parameters of the small-world network-based chain-binomial model are listed in Table 2. 2. Contact network epidemic model for stage 2a For the quarantine, we have
where is the number of isolated infected.
We use Bayesian inference with Monte Carlo Markov Chain (MCMC) sampling algorithms such as Metropolis-Hastings (MH) [31] to discover the stationary parameter estimation for the above epidemic models in a multidimensional probability space. This technique has been widely applied in various fields including ecological modeling [32, 33] hydrology [34] and geology [35] .
Assume that is the available observation set where the corresponding parameters are unknown; then, the posterior distribution of parameters conditioned on is , where is the prior of parameters, and
is the likelihood function. Generally to directly obtain a full posterior ( │ ) probability distribution is intractable, an alternative is to generate ( │ ) approximate posterior samples according to a proposal ordinary distribution (for example, uniform or normal distribution) by MCMC algorithms such as MH. By comparing the differences between the observations and the outputs of the model that receives possible samples of parameters , MH algorithm determines which sample can be accepted as a candidate for a parameter.
The detailed procedure of parameter estimation using MH algorithm is described below. The total number of confirmed cases is regarded as the observation and further compared with the model outputs conditioned on the candidate sample of parameters.
Accordingly, MH algorithm calculates an acceptance rate to determine which sample can be accepted. We implement an initial test times by using a uniform proposal Contact network epidemic simulations perform differently when using the same parameters and initial values. We therefore employ an ensemble simulation technique to make the simulations more reasonable and stable. We implement an ensemble of models that are driven in a parameter field, and produce the results by the ensemble averages. The ensemble simulation provides probabilistic solutions for stochastic nonlinear models and the uncertainty ranges of outputs. In this study, a 10% perturbation of the modes of the samples is introduced as the parameter field, and 100 epidemic models are created to fulfill the ensemble simulation (Fig. 4) . Note that the configuration is arbitrary due to the lack of knowledge of ensemble simulation in epidemiology, which highlights the need for analyzing the sensitivity of parameters and initial values in epidemic models.
We implement epidemic reconstruction based on a common software for data assimilation development (ComDA [36] ). This software is used to fuse the available information into dynamics and produce more reliable predictions or simulations.
Typical MCMC techniques such as MH and dynamic models in Table 2 , and the configurations and initial values for epidemic models are provided in Table 1 . 
The daily epidemic curves for each stage are shown in Fig. 6 .
Considering the strong randomness in parameter estimation, the simulations of unprotected and protected contact network models are inferior to that of airborne spread with the differential equation model, i.e., the ensemble averages of the network-based simulations slightly deviate from the true infected, while NCSI reconstructs the epidemic exactly. Additionally, the uncertainty is not obvious in stage 2b. This is because, on the one hand, NCSI is not as sensitive as the contact network epidemic model to the parameter, resulting in a more stable model trajectory even when the parameter is perturbed. On the other hand, the estimated parameter for NCSI is very small ( 9.63 × 10 −8 ), implying that an additional 10% perturbation is = insufficient to generate the obvious uncertainties.
Scenario-based modeling without any quarantine is also provided (Fig. 6(a) ), which predicts that almost all the people on the ship will become infected in one month. Defining the infected period as 15 and 31, we found that compared with the epidemic curve and average transmissibility , the basic reproductive number is 0 more sensitive to the infected period, which is also confirmed in Eq. (1).
The modeling strategy using parameter perturbation and ensemble simulation facilitates the understanding of strong random network epidemic models by quantifying their uncertainty. Uncertainty ranges imply the potential ceiling (the worst outcome) and floor (the best outcome) for the spread of the disease. Simulations are acceptable if the reference observation falls in the uncertainty ranges (for example,
The transmission dynamics of COVID-19 on the Diamond Princess are studied.
However, the designed scenarios are simplified because COVID-19 can still transmit via air in stage 1 and 2a. We only consider the major driver because introducing extra transmission makes the simulations more complex, and provides limited knowledge.
A major uncertainty comes from the data, i.e., except for the population and total infected individuals, the available data are mainly deduced from direct or indirect information. Assume that the indirect data has limited reliabilities, considerable errors are consequently introduced into simulations. A promising method for eliminating this type of uncertainty is formulating inference processes and mapping indirect information into epidemic modeling, i.e., using data assimilation [38] [39] [40] [41] .
Quarantine is productive since dramatically drops from 6.94 for unprotected The other containment efforts are for the intensive social contact scenario.
Considering the unprotected and protected contact tests, both and decrease 0 when quarantine is implemented, which proves that personal protective materials such as coveralls, masks, and eye covers are useful for infection prevention. We further notice that in the unprotected contact test is much larger than 2.5, which was 0 provided by the WHO report. This implies that if the intensive unprotected contact pattern is maintained for a couple of days, the epidemic curve may climb dramatically. As shown in the protected contact test, falls far below 2.5.
However, this tiny does not represent the true basic reproductive number when 0 using protective materials; we are inclined to assume that protective measures are not fully implemented and some crews were exposed. This highlights that high-risk populations such as medical staff or healthcare workers, should rigorously practice prevention measures during epidemics.
This study reconstructs the epidemic curve on the "Diamond Princess" cruise ship based on two stages (1, 2a and 2b), i.e., unprotected, protected contact and a) Trial test means the result is from the explorative simulation by using the same data. A simulation will be markedly different from the truth if the parameter is beyond this range.