Heterogeneous epidemic modelling within an enclosed space and corresponding Bayesian estimation.

Since March 11th, 2020, COVID-19 has been a global pandemic for more than one years due to a long and infectious incubation period. This paper establishes a heterogeneous epidemic model that divides the incubation period into infectious and non-infectious and employs the Bayesian framework to model the 'Diamond Princess' enclosed space incident. The heterogeneity includes two different identities, two transmission methods, two different-size rooms, and six transmission stages. This model is also applicable to similar mixed structures, including closed schools, hospitals, and communities. As the COVID-19 pandemic continues, our mathematical modeling can provide management insights to the governments and policymakers on how the COVID-19 disease has spread and what prevention strategies still need to be taken.

Introduction

Since March 11th, 2020, the coronavirus disease 2019 (COVID-19) caused by a novel severe acute respiratory syndrome coronavirus has risen to a global pandemic. As of Nov 2021, over 258 million cases and 5.16 million deaths are reported worldwide (WHO, 2020).

Understanding the epidemic spreading of the coronavirus and the effectiveness of various countermeasures is of high interest for public health and society. Among the mathematical epidemiology studies, the burst of breakout in confined spaces, including prisons, schools, churches, and hospitals, has raised concerns widely due to its important influences in surveillance (Chu et al., 2020; Emery et al., 2020; Gan et al., 2020; Kim, 2020). One typical dataset on the cruise ship ‘Diamond Princess’ has been studied intensely (Azimi et al., 2021; Huang et al., 2021; Mizumoto et al., 2020).

The COVID-19 outbreak on the cruise ship 'Diamond Princess' captures several researchers' attention due to its enclosed environment and relatively complete data (Azimi et al., 2021; Huang et al., 2021; Mizumoto et al., 2020). However, most of these studies did not consider some critical parameters in their corresponding models (Emery et al., 2020; Ivorra et al., 2020; Lin et al., 2020; Liu et al., 2020a, 2020b). For example, a study by Liu et al. used a heterogenous susceptible-infectious-removed (SIR) model to fit the data on the 'Diamond Princess' cruise ship without taking into account the incubation period, which contains contact and non-contact (airborne) transmission (Liu et al., 2020b). On the other hand, another study by Emery et al. added more compartments to the SIR model, covering E (exposed), P (pre-symptomatic), and A (asymptomatic) (Emery et al., 2020). The SEPAIR model fixing the incubation period aims at inferring the spread contribution of asymptomatic cases. Another common practice is to fix one or more parameters (such as incubation period, the number of close contact people, etc.) to infer other parameters (mainly containing infectious rates) in the corresponding model, such as (Huang et al., 2021; Rocklöv, Sjödin, & Wilder-Smith, 2020a, 2020b). However, such assumptions may lead to bias in the estimation, as Yao et al. (Yao et al., 2020) indicated that the SARS-CoV-2 mutated at least 30 distinct strains, whose pathogenicity could vary by 270 times by April 2020.

This paper introduces a heterogeneous susceptible-exposed-asymptomatic-infectious-diagnosed (SEAIJ) model containing two infection sources (asymptomatic and symptomatic patients) based on the data collected from the cruise ship 'Diamond Princess’. Moreover, our model takes into account heterogeneous identities (agent-based), heterogeneous transmission schemes (in-the-room and out-of-the-room), heterogeneous room mixture (double rooms and triple rooms), and six stages of transmission (based on isolation, etc.). The proposed model and corresponding estimation method are completely data-driven. They would provide useful implications for the public health policymakers who implement control and prevention measures in enclosed communities, such as campuses and hospitals.

Materials and methods

The COVID-19 outbreak on the cruise ship 'Diamond Princess'

The cruise ship 'Diamond Princess' began sailing on January 20th, 2020, from Yokohama with one patient in the incubation period. There were 2666 passengers and 1045 crew members on the cruise ship, respectfully staying in double and triple rooms. The original patient was an asymptomatic passenger boarded on January 20th. He became symptomatic on January 23rd and disembarked with his two healthy daughters two days later. The cruise ship was quarantined at sea after finding more diagnoses. All passengers were quarantined in their rooms and were allowed a short time to go out daily, while all crew members wore masks to continue providing services. The passengers and the crew began to disembark on February 17th and complete disembarkation on March 1st. A total of 712 people (excluded the original patient) were confirmed on board, excluding fifty-six who became ill after disembarking. Of these, 331 were asymptomatic.

We divide the outbreak on the 'Diamond Princess' into six stages (Table 1). At the first three stages, people walked around the cruise ship freely. The first stage took place while the original patient was in the infectious incubation period. The primary infection source was the original patient. The second stage involved the original patient in the symptomatic period until he disembarked. The original patient was more contagious than he was at the first stage due to his onset. The potential infection sources were the people in the infectious incubation period who the original patient infected. The third stage was from when the original patient disembarked to the time of quarantine at sea. The infection sources were the infected people who stayed on the cruise ship. A few patients became symptomatic at the end of this stage. The fourth stage was from the time of quarantine to the time of disembarkation. All the passengers were restricted in movement while the masked crew served passengers. So the main spread scheme was the in-the-room transmission. The fifth stage was the disembarkation period. The people whose nucleic acid test came back negative were permitted to disembark. The cases reported at this stage were mainly the close contacts of the isolated patients. The sixth stage was the observation period after all the people left. Since there was no transmission on the cruise ship, the cases reported at this stage were infected at the fifth stage and tested positive at this stage (see Table 2).

Table 1

The stages of outbreak.

Stage	Period	Description
1	Jan. 20th - Jan. 22nd	Patient O in the incubation period
2	Jan. 23rd - Jan. 24th	Patient O in symptomatic period and disembarked on Jan. 25th
3	Jan. 25th - Feb. 4th	Before quarantining
4	Feb. 5th - Feb. 16th	Quarantining before disembarkation of all people
5	Feb. 17th - Mar. 1st	Disembarkation period
6	Mar. 2nd - Mar. 16th	Observation period of the last people disembarked from the ship

Table 2

Notations of models.

S	The number of susceptible people who may be infected and become exposed after close contact with asymptomatic people or infectious people. The nucleic acid test results of susceptible people are negative.
E	The number of exposed people who are in the incubation period and without infectiousness. The nucleic acid test results of the exposed people are negative.
A	The number of asymptomatic people who are able to infect susceptible people during the incubation period. The nucleic acid test results of asymptomatic people are positive.
I	The number of infectious people who have symptoms of COVID-19 and are more contagious than asymptomatic people. The nucleic acid test results of infectious people are positive.
J	The number of diagonal people whose nucleic acid test results are positive and isolated from the population.
N	The total number of people onboard.
c	The number of people that each person close contacts with on average.
cpp	The number of passengers that each passenger close contacts with on average.
cpw	The number of crew members that each passenger close contacts with on average.
cww	The number of crew members that each crew member close contacts with on average.
cwp	The number of passengers that each crew member close contacts with on average.
μ	The mobility (or disembarkation) rate.
β1	The (out-of-room) infectious rate of asymptomatic people.
β2	The (out-of-room) infectious rate of infectious people.
βr1	The (in-the-room) infectious rate of asymptomatic people.
βr2	The (in-the-room) infectious rate of infectious people.
α1	The transformation rate from the exposed to be asymptomatic.
α2	The transformation rate from the asymptomatic to the symptomatic.
γ1	The isolate rate of asymptomatic people.
γ2	The isolate rate of infectious people.

The SEAIJ epidemic model on a cruise ship

The SEAIJ epidemic model divides the population (N) into 5 groups, including the susceptible people (S), the exposed people (E), the asymptomatic people (A), the infectious people (I), and the diagnosed people (J). The basic SEAIJ model is expressed in the following differential equations:

The basic model assumes that all people have the same probability of infection after close contact with infection sources. The infectious rate of asymptomatic (A) people β1 is different from infectious (I) people's ß2, and there is movement μ during the outbreak. A susceptible (S) person may be infected after close contact with an infection resource (an asymptomatic (A) or infectious (I) person) and becomes exposed (E). He is non-infectious in the first stage during the incubation period (1/α1). An exposed (E) person becomes asymptomatic (A) after 1/α1 days, and he is able to infect susceptible (S) people in the second stage during incubation period 1/α2. 1/α2 days later, he becomes infectious (I) and develops symptoms. We supposed that the nucleic acid results of asymptomatic (A) and infectious (I) people are positive, so they have isolate rates γ1 and γ2, respectively. The model can be generated into other forms when fitting, and the derivation is well-formatted in Appendix 6.

The basic reproduction number under anthropogenic intervention

In combination with the basic reproduction number first introduced by Ross (Ross, 1910) and the situation on the cruise ship 'Diamond Princess', patients were isolated from the population before the end of the infection. We redefine the basic reproduction number in our model as:where N is the total number of infected patients, β[i], c[i], T[i] are the infectious rate, the number of susceptible close contacts, and the duration of staying on board after being affected by the patient i, respectively. Moreover, T of asymptomatic patients has two possible calculations. If the asymptomatic patient was isolated during his incubation period, T equals his isolated date minus the estimated beginning date of his infectious incubation period. Meanwhile, T equals the infectious incubation period for his asymptomatic stages if the patient was isolated after symptoms.

A Bayesian estimation method

Estimation of the implementation of epidemic models is based on the Bayesian framework. Details explanation and the pseudo-code can be found in Appendix 2 (see Table 3).

Table 3

The meaning of subscripts of notations.

2	The people living in the double room.
3	The people living in the triple room.
B	The people are onboard.
ci	The change in the number of people takes place in the room.
co	The change in the number of people takes place outside the room.
d	The people disembark.
j	The people are isolated.
p	Passengers.
w	The crew.
r	The people are inside the room.
Sr	The people whose roommate is susceptible in a double room.
Er	The people whose roommate is exposed in a double room.
SSr	The people whose roommates are both susceptible in a triple room.
SEr	The people whose roommates are susceptible and exposed respectively in a triple room.
EEr	The people whose roommates are both exposed in a triple room.

Results

This section shows the estimation results and interpretations are explored in detail. The fitted parameters are listed in Table 4, Table 5.

Table 4

The descriptions of parameters.

Parameter	Description
α1	The transformation rate from the exposed to be the asymptomatic
α2	The transformation rate from the asymptomatic to be the symptomatic
β1	The (out-of-room) infectious rate of the asymptomatic people
β2	The (out-of-room) infectious rate of the infectious people
βr1	The infectious rate (in the room) of the asymptomatic people
βr2	The infectious rate (in the room) of the infectious people
cpp1	The number of passengers that each passenger close contacts with before quarantining
cpp2	The number of passengers that each passenger close contacts with after quarantining
cpw1	The number of the crew that each passenger close contacts with before quarantining
cpw2	The number of the crew that each passenger close contacts with after quarantining
cww1	The number of the crew that each crew member close contacts with before quarantining
cww2	The number of the crew that each crew member close contacts with after quarantining
β∙c	The product of infectious rates and close contacts
CEd	Cumulative number of the exposed people disembarkation
ρ(X,Y)	The summary statistics, or the target error for the simulation method
M	The number of people infected inside room
R0Ap1	The basic reproduction number of the asymptomatic passengers before quarantining
R0Ap2	The basic reproduction number of the asymptomatic passengers after quarantining
R0Aw1	The basic reproduction number of the asymptomatic crew before quarantining
R0Aw2	The basic reproduction number of the asymptomatic crew after quarantining
R0Ip1	The basic reproduction number of the infectious passengers before quarantining
R0Ip2	The basic reproduction number of the infectious passengers after quarantining
R0Iw1	The basic reproduction number of the infectious crew before quarantining
R0Iw2	The basic reproduction number of the infectious crew after quarantining
R0PatientO1	The basic reproduction number of the patient O before quarantining
R0PatientO2	The basic reproduction number of the patient O after quarantining

Table 5

The priors of parameters.

Parameters	Prior	Explanations
α1	1/U(1, 4)	Non-informative
α2	1/(U(4,14)-1/α1)	Derived from the official quarantine measure, a full quarantine period lasts up to 14 days
β1	U(0.0115,0.4551)	Non-informative
β2	U(0.0115,0.4551)	Non-informative
βr1	U(0.0115,0.4551)	Non-informative
βr2	U(0.0115,0.4551)	Non-informative
cpp1	U(0,50)	Non-informative
cpp2	U(0,1)	Limited activity
cpw1	U(0,50)	Non-informative
cpw2	U(0,1)	Limited activity
cww1	U(0,50)	Non-informative
cww2	U(0,1)	Medical protection

The number of passengers that each crew member close contacts with before and after quarantining is dynamic throughout. It depends on the number of passengers (Np) and crew members (Nc) on board and the number of crews that each passenger had close contact with (cpw) at time t.

The summary statistics ρ(X,Y) are calculated by the summation of the difference between the actually and simulated accumulative reported cases, the number of infectious cases, the number of exposed people, and the number of asymptomatic people on board on the day before all the people disembarked. The tolerance set ε was set at {300, 150, 100, 60, 35, 25, 19}. Our target is to find 1000 groups of parameters for the SEAIJ model which satisfy ρ(X,Y)≤19, and the results are shown in Table 6.

Table 6

The descriptive statistics of parameters.

Type	Parameters	Mean	Maximum	Minimum	Median	Variance
α	α1	0.3820	0.6725	0.2817	0.3766	0.0024
	α2	0.0917	0.1023	0.0812	0.0919	0.0000
β	β1	0.0143	0.0347	0.0044	0.0141	0.0000
	β2	0.0520	0.0971	0.0181	0.0523	0.0001
	βr1	0.0631	0.2467	0.0102	0.0583	0.0007
	βr2	0.4925	0.7160	0.1137	0.4963	0.0075
c	cpp1	48.0782	56.7553	21.1747	48.3549	10.2650
	cpp2	0.5076	1.0647	0.2036	0.4976	0.0171
	cpw1	6.8513	8.3091	2.3675	6.9092	0.3485
	cpw2	0.1219	0.3379	0.0007	0.1196	0.0026
	cww1	2.9598	8.4711	0.0023	2.8500	2.5916
	cww2	0.1803	1.4450	0.0002	0.1492	0.0231
β ∙ c	β1∙ cpp1	0.6812	1.0856	0.2241	0.6727	0.0143
	β1∙ cpw1	0.0975	0.1642	0.0311	0.0959	0.0004
	β1∙ cww1	0.0416	0.1461	0.0000	0.0393	0.0005
	β2∙ cpp1	2.4955	4.2872	0.8774	2.5189	0.3331
	β2∙ cpw1	0.3562	0.6616	0.1250	0.3589	0.0075
	β2∙ cww1	0.1534	0.5601	0.0002	0.1435	0.0085
	β1∙ cpp2	0.0072	0.0212	0.0027	0.0069	0.0000
	β1∙ cpw2	0.0017	0.0054	0.0000	0.0016	0.0000
	β1∙ cww2	0.0025	0.0182	0.0000	0.0021	0.0000
	β2∙ cpp2	0.0261	0.0753	0.0080	0.0252	0.0001
	β2∙ cpw2	0.0062	0.0177	0.0000	0.0061	0.0000
	β2∙ cww2	0.0092	0.0839	0.0000	0.0075	0.0001
cumulative	CEd	31.3446	49.2728	14.0374	31.2540	31.2898
	M	297.7655	352.3853	213.2619	297.8613	306.4008
discrepancy	ρ(X,Y)	18.7126	18.9994	18.1945	18.7316	0.0357
R0	R0Ap1	2.0503	3.1666	0.7855	2.0371	0.1099
	R0Ap2	0.2039	0.3842	0.0611	0.2001	0.0017
	R0Aw1	0.4986	0.7763	0.2319	0.4968	0.0048
	R0Aw2	0.1630	0.2829	0.0504	0.1604	0.0011
	R0Ip1	8.5524	14.3709	3.1933	8.6342	3.3652
	R0Ip2	0.3652	0.5637	0.1990	0.3686	0.0025
	R0Iw1	2.1248	3.3162	0.9184	2.1337	0.1372
	R0Iw2	0.3553	0.6108	0.1589	0.3587	0.0038
	R0PatientO1	2.3349	3.7372	0.7651	2.3089	0.1729
	R0PatientO2	5.6942	9.8813	2.0017	5.7512	1.7421

The susceptible people are infected after close contact with an asymptomatic person outside the room with an average probability of 0.0143, infectious people outside room 0.0520, asymptomatic people inside room 0.0631, and infectious people inside rooms with the average probability of 0.4925.

Once the susceptible people are infected, they can not infect the susceptible people in ≈2.6178 days on average. They can infect the susceptible people in ≈10.9051 days before onset. After around 2.6178 + 10.9051 = 13.5229-day incubation period, people developed symptoms. The infectious rates and the periods are based on the close contact population.

Before quarantining, the number of passengers who one passenger close contacts with is 48.0782 on average. On average, the number of crew members that one passenger close contacts with is 6.8513. The number of crew members that one crew member close contacts with is 2.9598 on average. The number of passengers that one crew member close contact with is calculated by the equation and shown in Fig. 2. The mean value is around 16.

Fig. 2

The daily number of passengers that each crew member close contacts with on average.

After quarantine, the number of passengers that one passenger close contacts with is 0.5076 on average. The number of crew members that one passenger close contacts with is 0.1219 on average. The number of crew members that one crew close contacts with is 0.1803 on average. The number of passengers that one crew close contact with is around 0.28 (Fig. 2).

Given the same daily number of infected people, the more people infected outside the room, the less infected inside the room. β∙c represents infectious efficiency outside the room. It is the maximum number of susceptible people infected by an infected person per day. The difference of this product affects the difference of in-the-door infectious rate since we specify the number of people in a room.

There are 31.3446 exposed people disembarked on average. Averagely 297.7655 of 712 people are infected inside the room. The summary statistics ρ(X,Y) are small enough to fit the actual scenario. It is inefficient to find the parameters that make the ρ(X,Y) less than 18.

The basic reproduction number of asymptomatic passengers before quarantining is 2.0503 on average, which is 0.2039 after quarantining. The basic reproduction number of the infectious passengers before quarantining is 8.5524 on average, which is 0.3652 after quarantining.

The basic reproduction number of the asymptomatic crew before quarantining is 0.4986 on average, which is 0.1630 after quarantining. The basic reproduction number of the infectious crew before quarantining is 2.1248 on average, which is 0.3553 after quarantining.

The cumulative number of people infected by the original patient is 2.3349 on average before his symptoms occur, and which is 5.6942 after his symptoms occurred.

In general, the proposed model indicates several characteristics of the COVID-19 transmission within an enclosed space with an application in the ’Diamond Princess’ dataset.

Firstly, the infectious rate of asymptomatic people is less than the infectious people if the transmission schemes are the same. It indicated that the in-room transmission is faster than the out-of-room transmission if the infection sources are the same. However, the asymptomatic people inside the room are more contagious than the infectious people outside the room.

Secondly, people without symptoms transmit COVID-19 with infectious rate 0.0044–0.0143 around 10.9 days during the incubation period. If the daily number of close contacts is 21.2–56.8, an asymptomatic patient is expected to be able to infect 2.4–11.8 people. That means it is necessary to find asymptomatic infected people and the people who close contact with infection sources.

Thirdly, the cumulative number of exposed people disembarking averages around 31.3. Since around 71 people's nucleic acid results were positive after disembarkation, 31.3 is too small. One of the main reasons was that asymptomatic people infected some people during their disembarkation. The other main reason was that we assumed the people permitted to disembark were randomly picked, which were actually specified.

Fourthly, around 298 of 712 people are infected inside rooms. We concluded that out-of-room transmission is the main scheme for the COVID-19 outbreak based on the percentage of people infected inside the room. However, as the number of people in a room increases, the inside-room transmission may become the main scheme for transmission due to the higher inside-the-room infectious rate. To keep the air flowing in the room and decrease the number of people living in one room is also helpful to avoid the COVID-19 outbreak.

Fifthly, the infected passengers infected more susceptible people than the infected crew members. We assumed that asymptomatic people have the same infectious rates, and infectious people have the same infectious rates. Then, the number of close contacts becomes the variable that affects the number of people that passengers and the crew infected. In experiments, the number of close contacts of passengers is greater than that of crew members, so we conclude that the infected passengers were more likely to infect susceptible people outside rooms.

Lastly, the original patient infected more susceptible people during his onset period than during his incubation period. However, he was asymptomatic in 3 days and symptomatic in 2 days on the cruise ship.

Discussion

There are quite a few works on mathematical models to fit COVID-19 outbreak data on the cruise ship 'Diamond Princess' and a comparison is prepared. Table 7a, Table 7b, Table 7c, Table 7d, Table 7e, Table 7f shows the differences of our model with these six models ((Emery et al., 2020), (Huang et al., 2021), (Liu et al., 2020b), (Rocklöv et al., 2020a, 2020b), (Rocklöv et al., 2020a, 2020b), (Morton et al., 2021)) in eleven areas.

Table 7a

Comparison to other studies.

Different Areas	Rocklöv et al. (Rocklöv et al., 2020a, 2020b)	Our study
Publish Date	• Feb. 2020
Epidemic model	• SEIR	• SEAIJ
Inference Method		• Approximate Bayesian method • Population Monte Carlo
Heterogeneities	• Identities • Transmission schemes • Mixed Structures	• Identities • Transmission schemes • Infection sources • Outbreak stages • Mixed Structures
Transmission Schemes	• Protected vs. Nonprotected	• Protected vs. Nonprotected • Close contact • Inside vs. outside the room
Infection sources	• Infectious patients	• Infectious patients • Asymptomatic patients (covered presymptomatic)
Outbreak stages	• Before isolation • After isolation	• Before isolation • After isolation • Duration of disembarkation • After duration of disembarkation
Data period	• Jan. 21st – Feb. 19th	• Jan. 20th - Mar. 16th
Fixed parameters	• transmissibility and contact rate (population, crew, passengers) • Incubation period • Infectious period or time to removal
Sampling Parameters		• Number of close contact • A,I infectious rates inside rooms • A,I infectious rates outside rooms • Non-infectious talent period • Infectious talent period
Conclusions	• Before isolation: • R0: 14.8	• Before isolation: • Mean R0Ap: 2.0503 • Mean R0Aw: 0.4986 • Mean R0Ip: 8.5524 • Mean R0Iw: 2.1248
	• After isolation: • R0: 1.78	• After isolation: • Mean R0Ap: 0.2039 • Mean R0Aw: 0.1630 • Mean R0Ip: 0.3652 • Mean R0Iw: 0.3553
	• Throughout outbreak:	• Throughout outbreak: • Iutside-the-room infectious rate: 0.0143 • I outside-the-room infectious rate: 0.0520 • Inside-the-room infectious rate: 0.0631 • I inside-the-room infectious rate: 0.4925

Table 7b

Comparison to other studies.

Different Areas	Nishiura (Rocklöv et al., 2020a, 2020b)	Our study
Publish Date	• Feb. 2020
Epidemic model	• Richard model	• SEAIJ
Inference Method		• Approximate Bayesian method • Population Monte Carlo
Heterogeneities	• Identities • Transmission schemes	• Identities • Transmission schemes • Infection sources • Outbreak stages • Mixed Structures
Transmission Schemes	• Protected vs. Nonprotected	• Protected vs. Nonprotected • Close contact • Inside vs. outside the room
Infection sources	• Infectious patients	• Symptomatic patients • Asymptomatic patients (covered presymptomatic)
Outbreak stages	• Before isolation • After isolation	• Before isolation • After isolation • Duration of disembarkation • After duration of disembarkation
Data period	• Jan. 20th – Feb. 19th	• Jan. 20th - Mar. 16th
Fixed parameters
Sampling Parameters	• Incubation period	• Number of close contact • A,I infectious rates inside rooms • A,I infectious rates outside rooms • Non-infectious talent period • Infectious talent period
Conclusions	• Before isolation: • Peak time of infection: Feb. 2nd – Feb. 4th	• Before isolation: • Mean R0Ap: 2.0503 • Mean R0Aw: 0.4986 • Mean R0Ip: 8.5524 • Mean R0Iw: 2.1248
	• After isolation: • Incidence abruptly declined • Daily 0.98 passenger infected • The cumulative incidence (as of Feb. 24th): 102 passengers with close contact• 47 passengers without close contact • 48 crew members	• After isolation: • Mean R0Ap: 0.2039 • Mean R0Aw: 0.1630 • Mean R0Ip: 0.3652 • Mean R0Iw: 0.3553
	• Throughout outbreak:	• Throughout outbreak: • Outside-the-room infectious rate: 0.0143 • I outside-the-room infectious rate: 0.0520 • Inside-the-room infectious rate: 0.0631 • I inside-the-room infectious rate: 0.4925

Table 7c

Comparison to other studies.

Different Areas	Liu et al. (Liu et al., 2020b)	Our study
Publish Date	• Apr. 2020
Epidemic model	• SIR	• SEAIJ
Inference Method	• Bayesian framework • Metropolis-Hastings sampling	• Approximate Bayesian method • Population Monte Carlo
Heterogeneities	• Identities • Transmission schemes • Infection sources • Outbreak stages	• Identities • Transmission schemes • Infection sources • Outbreak stages • Mixed Structures
Transmission Schemes	• Protected vs. Nonprotected • Contact vs. Airborne	• Protected vs. Nonprotected • Close contact • Inside vs. outside the room
Infection sources	• Infectious patients • Airborne	• Symptomatic patients • Asymptomatic patients (covered presymptomatic)
Outbreak stages	• Before isolation • After isolation	• Before isolation • After isolation • Duration of disembarkation • After duration of disembarkation
Data period	• Jan. 20th - Feb. 19th	• Jan. 20th - Mar. 16th
Fixed parameters	• Infected period • Viable period of virus in the air
Sampling Parameters	• Number of close contact • I infectious rate • Infectious rate of airborne	• Number of close contact • A,I infectious rates inside rooms • A,I infectious rates outside rooms • Non-infectious talent period • Infectious talent period
Conclusions	• Before isolation: • Mean R0: 6.94 • I infectious rate: 0.026	• Before isolation: • Mean R0Ap: 2.0503 • Mean R0Aw: 0.4986 • Mean R0Ip: 8.5524 • Mean R0Iw: 2.1248
	• After isolation: • Mean R0: 0.2 • I infectious rate: 0.0007	• After isolation: • Mean R0Ap: 0.2039 • Mean R0Aw: 0.1630 • Mean R0Ip: 0.3652 • Mean R0Iw: 0.3553
	• Throughout outbreak:	• Throughout outbreak: • Outside-the-room infectious rate: 0.0143 • I outside-the-room infectious rate: 0.0520 • Inside-the-room infectious rate: 0.0631 • I inside-the-room infectious rate: 0.4925

Table 7d

Comparison to other studies.

Different Areas	Emery et al. (Emery et al., 2020)	Our study
Publish Date	• Aug. 2020
Epidemic model	• SEPAIR	• SEAIJ
Inference Method	• Bayesian framework • Markov Chain Monte Carlo	• Approximate Bayesian method • Population Monte Carlo
Heterogeneities	• Identities • Transmission schemes • Infection sources • Outbreak stages	• Identities • Transmission schemes • Infection sources • Outbreak stages • Mixed Structures
Transmission Schemes	• Protected vs. Nonprotected • Close contact	• Protected vs. Nonprotected • Close contact • Inside vs. outside the room
Infection sources	• Infectious patients • Asymptomatic patients • Presymptomatic patients	• Symptomatic patients • Asymptomatic patients (covered presymptomatic)
Outbreak stages	• Before isolation • After isolation	• Before isolation • After isolation • Duration of disembarkation • After duration of disembarkation
Data period	• Jan. 20th - Feb. 20th	• Jan. 20th - Mar. 16th
Fixed parameters	• Talent period • Asymptomatic period • Pre-symptomatic period • Symptomatic period
Sampling Parameters	• Number of close contact • P,A,I infectious rates • Percentage of A patients	• Number of close contact • A,I infectious rates inside rooms • A,I infectious rates outside rooms • Non-infectious talent period • Infectious talent period
Conclusions	• Before isolation: • R0 range: 6.7–10.9 depends on percentages of A	• Before isolation: • Mean R0Ap: 2.0503 • Mean R0Aw: 0.4986 • Mean R0Ip: 8.5524 • Mean R0Iw: 2.1248
	• After isolation:	• After isolation: • Mean R0Ap: 0.2039 • Mean R0Aw: 0.1630 • Mean R0Ip: 0.3652 • Mean R0Iw: 0.3553
	• Throughout outbreak:	• Throughout outbreak: • Outside-the-room infectious rate: 0.0143 • I outside-the-room infectious rate: 0.0520 • Inside-the-room infectious rate: 0.0631 • I inside-the-room infectious rate: 0.4925

Table 7e

Comparison to other studies.

Different Areas	Lai CC. et al. (Morton et al., 2021)	Our study
Publish Date	• Jan. 2021
Epidemic model	• SEIR	• SEAIJ
Inference Method	• Deterministic • Bayesian Markov Chain Monte Carlo • Likelihood theory	• Approximate Bayesian method • Population Monte Carlo
Heterogeneities	• Identities • Transmission schemes • Mixed Structure	• Identities • Transmission schemes • Infection sources • Outbreak stages • Mixed Structures
Transmission Schemes	• within-deck vs. between-deck transmission	• Protected vs. Nonprotected • Close contact • Inside vs. outside the room
Infection sources	• Infectious patients	• Symptomatic patients • Asymptomatic patients (covered presymptomatic)
Outbreak stages	• Before isolation (Jan. 20th – Feb. 4th) • Early period of daily symptom-reported (Feb.5- Feb. 10th) • Before systemic test (Feb. 11th- Feb. 13th) • Before the evacuation of passengers from USA (Feb. 14th- Feb. 16th) • Before the duration of disembarkation (Feb. 17th – Feb. 19th)	• Before isolation • After isolation • Duration of disembarkation • After duration of disembarkation
Data period	• Jan. 20th – Feb. 19th	• Jan. 20th - Mar. 16th
Fixed parameters	• incubation period (Deterministic, and Likelihood theory) • recovery rate (Deterministic, and Likelihood theory)
Sampling Parameters	• transmission coefficients (Deterministic, Bayesian MCMC, and Likelihood theory) • incubation period (Bayesian MCMC) • recovery rate (Bayesian MCMC) • number of unknown infected status (Bayesian MCMC) • the daily confirmed cases (Bayesian MCMC)	• Number of close contact • A,I infectious rates inside rooms • A,I infectious rates outside rooms • Non-infectious talent period • Infectious talent period
Conclusions	• Before isolation	• Before isolation: • Mean R0Ap: 2.0503 • Mean R0Aw: 0.4986 • Mean R0Ip: 8.5524 • Mean R0Iw: 2.1248
	• After isolation	• After isolation: • Mean R0Ap: 0.2039 • Mean R0Aw: 0.1630 • Mean R0Ip: 0.3652 • Mean R0Iw: 0.3553
	• Throughout outbreak: • Overall R0: 5.70 (Bayesian MCMC) • Overall R0: 5.27 (Deterministic) • Overall R0: 5.43 (Maximum Likelihood) • R0p: 5.18 (Bayesian MCMC) • R0w: 2.46 (Bayesian MCMC)	• Throughout outbreak: • Outside-the-room infectious rate: 0.0143 • I outside-the-room infectious rate: 0.0520 • Inside-the-room infectious rate: 0.0631 • I inside-the-room infectious rate: 0.4925

Table 7f

Comparison to other studies.

Different Areas	Huang, LS et al. (Huang et al., 2021)	Our study
Publish Date	• Mar. 2021
Epidemic model	• Chain binomial model	• SEAIJ
Inference Method	• Likelihood theory	• Approximate Bayesian method • Population Monte Carlo
Heterogeneities	• Identities • Transmission schemes • Infection sources	• Identities • Transmission schemes • Infection sources • Outbreak stages • Mixed Structures
Transmission Schemes	• Close contact • No quarantine vs. quarantine	• Protected vs. Nonprotected • Close contact • Inside vs. outside the room
Infection sources	• Infectious patients • Asymptomatic patients	• Symptomatic patients • Asymptomatic patients (covered presymptomatic)
Outbreak stages	• Before isolation • After isolation	• Before isolation • After isolation • Duration of disembarkation • After duration of disembarkation
Data period	• Jan. 21st – Feb. 19th	• Jan. 20th - Mar. 16th
Fixed parameters	• Serial interval • Proportion of infection that occurred in cabins • Asymptomatic ratio • Proportion of passengers and crew
Sampling Parameters	• Transmission rate	• Number of close contact • A,I infectious rates inside rooms • A,I infectious rates outside rooms • Non-infectious talent period • Infectious talent period
Conclusions	• Before isolation: • serial intervals: • 5 days R0: 3.27 • 6 days R0: 3.78	• Before isolation: • Mean R0Ap: 2.0503 • Mean R0Aw: 0.4986 • Mean R0Ip: 8.5524 • Mean R0Iw: 2.1248
	• After isolation: • serial intervals: • 5 days R0p: 4.18 • 6 days R0p: 4.73 • 5 days R0w: 0.91 • 6 days R0w: 1.06	• After isolation: • Mean R0Ap: 0.2039 • Mean R0Aw: 0.1630 • Mean R0Ip: 0.3652 • Mean R0Iw: 0.3553
	• Throughout outbreak	• Throughout outbreak: • Outside-the-room infectious rate: 0.0143 • I outside-the-room infectious rate: 0.0520 • Inside-the-room infectious rate: 0.0631 • I inside-the-room infectious rate: 0.4925

Some limitations should be noted in the methodology. For example, the data we used were not from a random sample. Since only symptomatic cases were investigated during the early stage of the quarantine, it is possible that asymptomatic cases were missed out, which would affect the overall proportion of patients who tested positive (Rocklöv et al., 2020a, 2020b). In addition, since a majority of the passengers were older adults and it was unclear if older adults would develop more symptoms due to the underlying chronic diseases, including diabetes and cardiovascular disease (CVD), our study might underestimate the number of individuals who develop symptoms. Therefore, more detailed data regarding the passengers' health status, including the comorbidities, would improve the construction of our model.

With the increasing knowledge from clinical and epidemiological studies on the COVID-19 disease, the governments and policymakers can now design better mitigation strategies to control the spread of the COVID-19 pandemic. Some preventive measures, including the COVID-19 vaccination, social distancing, and wearing face masks, have been proven to control the pandemic successfully. Still, it is unclear about the timing and effectiveness of these measures. Also, the populations’ response, psychological tolerance, and willingness to follow these preventive measures can vary significantly depending on how severe the COVID-19 pandemic affected the area (Morton et al., 2021). Therefore, the construction of a mathematical epidemic model can answer some critical questions that cannot be extrapolated from clinical and epidemiological studies. Under these models, the parameters can be incorporated, and the impact of the preventive measures on the COVID-19 community spread can be stimulated.

Moreover, these epidemic models can be used to predict the severity degree of the COVID-19 pandemic in populations under different risk exposures to the COVID-19. The findings from these models can then be used to generate informed predictions as to the most effective and sustainable mitigation strategies, including whether single or multiple preventive measures may offer the optimum protection while relaxing the mobility restriction in populations (Bertsimas et al., 2021). In addition, these models can be used to confirm the benefits of the implemented preventive measures by the government and policymakers.

Conclusion

This study demonstrated how a data-driven epidemic model could model virus transmission within an enclosed space. We discussed the heterogeneity of the model in five aspects: identities, infection sources, transmission methods, size of rooms, and transmission stages. The basic reproduction number R0 for different infection sources is calculated, and an improved approximate Bayesian updating computation method is proposed. This work analyses the 'Diamond Princess’ data in detail and provides some valuable management insights to the governments and policymakers for handling the COVID-19 pandemic spread.

Table A1

Daily Disembarkation without Isolation.

Date	Daily disembarkation without isolation	Comments
2020/2/1	22
2020/2/14	11
2020/2/17	400	Evacuation of U.S.
2020/2/19	450	Beginning of disembarkation
2020/2/20	544
2020/2/21	461
2020/2/22	15
2020/2/23	230
2020/2/24	320	Estimated Disembarkation
2020/2/25	18
2020/2/26	321	Estimated Disembarkation
2020/2/27	92
2020/3/1	131	End of disembarkation

Table A2

Daily isolated passengers and crew members.

Date	Isolated passengers	Isolated crew members	Comments
2020/2/5	10	0
2020/2/6	10	0
2020/2/7	37	4
2020/2/8	3	0
2020/2/9	5	1
2020/2/10	53	12
2020/2/12	31	8
2020/2/13	34	10
2020/2/15	48	19
2020/2/16	53	17
2020/2/17	74	25
2020/2/18	66	22
2020/2/19	61	18
2020/2/20	10	2
2020/2/21	0	1
2020/2/23	43	4	Estimated. Adjust due to double counting
2020/2/26	9	5
2020/2/2	0	1
2020/3/12	0	1
2020/3/16	15	0	Adjust by JMHLW

Table A3

Daily totally isolated people

Date	Isolated people	Cumulative isolated people	Comments
2020/2/5	10	10
2020/2/6	10	20
2020/2/7	41	61
2020/2/8	3	64
2020/2/9	6	70
2020/2/10	65	135
2020/2/12	39	174
2020/2/13	44	218
2020/2/15	67	285
2020/2/16	70	355
2020/2/17	99	454
2020/2/18	88	542
2020/2/19	79	621
2020/2/20	12	633
2020/2/21	1	634
2020/2/23	47	681	Estimated. Adjust due to double counting
2020/2/26	14	695
2020/2/2	1	696
2020/3/12	1	697
2020/3/16	15	712	Adjust by JMHLW

Table A4

Daily isolated asymptomatic people

Data	Asymptomatic People
2020/2/13	32
2020/2/15	38
2020/2/16	38
2020/2/17	70
2020/2/18	65
2020/2/19	68
2020/2/20	8
2020/2/26	12