Mechanistic transmission modeling of COVID-19 on the Diamond Princess cruise ship demonstrates the importance of aerosol transmission

Significance We find that airborne transmission likely accounted for >50% of disease transmission on the Diamond Princess cruise ship, which includes inhalation of aerosols during close contact as well as longer range. These findings underscore the importance of implementing public health measures that target the control of inhalation of aerosols in addition to ongoing measures targeting control of large-droplet and fomite transmission, not only aboard cruise ships but in other indoor environments as well. Guidance from health organizations should include a greater emphasis on controls for reducing spread by airborne transmission. Last, although our work is based on a cruise ship outbreak of COVID-19, the model approach can be applied to other indoor environments and other infectious diseases.

A combination of mechanistic infection transmission, dose-response model and epidemic models was adopted to estimate the contributions of specific infection transmission modes to the number of COVID-19 cases among individuals aboard the Diamond Princess Cruise Ship. Four main transmission pathways, including long-range inhalation, short-range inhalation, direct deposition, and fomite contact, were considered as the infection transmission mechanisms from infected individuals to others under a wide range of possible scenarios. Daily cumulative case counts from passengers aboard the ship are used as the primary model outcome. The model approach is designed to identify the most likely values of several unknown or uncertain parameters by analyzing only those model results that yield acceptable coefficients of determination (R 2 ) between reported and modeled daily cumulative and daily case numbers, and thereby providing insight into the likely importance of the various modes of transmission included in the framework.
Here we describe in detail: (1) the mechanistic infection transmission and dose-response model, (2) the epidemic model, (3) processes for identifying key unknown or uncertain model parameters, and (4) processes for analyzing model outcomes and conducting sensitivity analyses.

Mechanistic Infection Transmission and Dose-Response Model Framework
The mechanistic transmission model uses a Markov chain process to estimate the number of SARS-CoV-2 copies present in numerous physical states, as well as the probability of transmission of SARS-CoV-2 viruses between each defined state, aboard the ship over time. The transmission model is coupled with a dose-response model to predict the probability of infection to susceptible individuals onboard the ship over time.

Markov chain framework
A Markov chain is a random process that undergoes transitions from one state to another in a state space. Physical elements (e.g., room air and surfaces, human skin and mucus membranes, etc.) and pathogen removal mechanisms (e.g., loss of viability, ventilation, and filtration) in the source environment-receptor pathways are represented as "states" in a discrete-time Markov chain model. Pathogens can be transferred and exchanged between states due to physical mechanisms such as emission, deposition, resuspension, filtration, and ventilation. Markov chain models have been used previously for estimating doses of influenza virus in several environments including healthcare facilities and airplanes.(1- 6) Markov chain process consist of a Markov chain matrix (MCM), a distribution array showing the number of pathogens in considered "states" after a certain time, and an injection array showing the new number of pathogens injected to the system in each time step. To generate a MCM for a state space with "states", first, we need to generate an × transmission rate matrix demonstrating the transmission or removal rates of pathogens between two considered "states" of and ( ) in the state space. In the transmission rate matrix, the values are considered equal to zero, as demonstrated in Figure S1. The overall rate at which a pathogen can leave state ( ) is the sum of the rate constants for removal from that state.

Equation S1
MCM is an × probability matrix demonstrating if a pathogen is in state in time , what the probabilities of remaining in the same state ( ) or moving to state ( ) are in time + ∆ . and values can be estimated from Equation S2, and S3, respectively.

Equation S3
We considered time steps (∆ ) of one second for the Markov chain process. If there are time steps between two pathogen injections into the state space, the number of pathogens in each state after injections can be estimated from the Equation S4. In this model, the Markov chain process was repeated for a one-day period and then a new MCM was generated for the next day until the end of the simulation period. We used PyCharm 2019.1.1 (Copyright © 2010-2020 JetBrains) with Python interpreter to deploy the process.
We considered 12 states for the Markov chain process as demonstrated in Figure S1. We considered two types of susceptible individuals aboard the ship: (i) uninfected individuals who were cabinmates of, and thus spent a significant amount of their time with, infected individuals (particularly after the passenger quarantine started), and (ii) uninfected individuals who were not cabinmates of infected individuals before they became infected. The list of states in Figure S1 excludes the air and surfaces of cabins with only uninfected individuals in them because no infectious virus is assumed to be present in cabins with uninfected individuals in them and defining these two additional states of indoor air and surfaces of these cabins would not change our calculations. The model then adjusts the number of cabins with infected individuals present at the end of each simulation day based on the number of new infected cases stemming from interactions in the common areas.
As reported, the COVID-19 outbreak was traced to a single passenger from Hong Kong who boarded the cruise ship in Yokohama on January 20 and then disembarked in Hong Kong on January 25. The first 10 cases were confirmed on February 4 after the ship arrived in the Yokohama port. The laboratory-confirmed cases of COVID-19 led to the quarantine of the Diamond Princess for 14 days beginning on February 5 at 7 am, with most passengers required to remain in their cabins. During February 16-23, nearly 1,000 persons were repatriated by air to their home countries, including 329 persons who returned to the United States and entered quarantine or isolation. During February 24 -March 1, the remaining crewmembers disembarked from the cruise ship.
Using the well-documented case information aboard the Diamond Princess Cruise Ship,(7) the transmission patterns of SARS-CoV-2 aboard the ship are divided into four distinct time periods: 1. January 20-25, 2020: when there was only one index case aboard the ship 2. January 25-February 5, 2020: the time between when the index case disembarked and before the passenger quarantine began 3. February 5-24, 2020: the time between the beginning of the passenger quarantine and the time when all passengers disembarked 4. February 24-March 1, 2020: the time between the disembarking of all passengers and the time when the remaining crew members disembarked We applied the model only to the first three periods (January 20 through February 24, 2020) because detailed information was available on the daily counts of the number of people onboard and their infection status. We generated a new MCM for each day in this period to model mechanistic transmission and infection risk based on a number of assumptions for built environment parameters, crew and passengers' interactions, adopted infection control strategies, and the number of infectors and susceptible individuals estimated from application of the transmission risk model to the previous days. Figure S1 shows the transmission rate matrix that we used to generate the MCM for the first period of the outbreak as an example of the transmission rate matrices used in this modeling work. The gray cells demonstrate the possible transmission routes for SARS-CoV-2 between two states and the cell values are the transmission rates in units of inverse time.
The majority of epidemiological characteristics of the COVID-19 outbreak and built environment factors of the Diamond Princess Cruise Ship was culled from peer-reviewed journal articles, information provided by the cruise ship owner's website, (8) and CDC notifications and reports.(9, 10) However, some of the required parameters for the mechanistic infection transmission model, such as the rates of interactions among passengers and crew before the quarantine began and the built environment factors that were not reported in existing resources (e.g., the ventilation rate of cabins and public areas, inter-zonal airflow between cabins and public areas, etc.) were assumed to be similar to a typical cruise ship environment. Because we ultimately calibrate the model with the range of estimated effective reproduction numbers during the first distinct period, the impacts of our assumptions for unknown modeling parameters are limited. Most of our assumed model parameters related to the characteristics of a typical cruise ship were culled from two prior studies. The first study we relied upon is Zheng et al., which modeled the risk of influenza on a typical cruise ship with 2000 passengers and 800 crew members using a Wells-Riley model.(11) Their model was validated using data from a previous influenza outbreak aboard a cruise ship from New York City in 1997 and was able to simulate the spread of the infection. (11) The second study we relied upon is Zhang et al., which modeled the transmission of infectious diseases via fomite contact in a typical cruise ship by considering seven functional areas including bench seats, restroom, retail counter, stair rails, dining table, dining chair, and handrail of viewing deck, as well as several types of contact surfaces. (12) The following sub-sections describe several processes and assumptions for parameters used in the Markov chain model that relate to virus transmission and/or removal, primarily culled from a variety of literature sources relevant to cruise ship environments, human activity patterns, and virus viability.

Volume of surface area of various zones on the Diamond Princess Cruise Ship
We categorized the passenger-accessible areas of the cruise as either (i) cabins or (ii) common (public) areas such as restaurants, hallways, bars, galleries, gyms, casinos, and sport courts. The areas and volumes of public areas and cabins were estimated based on the information provided on the Princess Plus website. ( Cabin Air -Infected Cabins Transmission/Removal Rates Per Hour using floor plans from the website. We estimated the floor area of indoor public areas to be ~18,000 m 2 , located on Decks 5, 6, and 7. The floor area of hallways between cabins and outdoor zones accessible to passengers were estimated to be ~3,700 m 2 and ~17,000 m 2 , respectively. The Princess Plus website did not provide the floor plan of other parts of the cruise ship where passengers did not have access, such as kitchens, control rooms, and the crew members' sleeping cabins. However, we assumed the floor area of those zones (except the sleeping cabins of crew members) was ~8000 m 2 (equal to the floor area of one deck). We also assumed that the indoor public areas were completely closed during the quarantine period, and that only half o f the other indoor spaces were operating, with a total floor area of ~4000 m 2 . To estimate space volumes, we assumed a ceiling height of 2.5 meters for cabins and hallways and 4 meters for other indoor public areas.
As the indoor public areas in the cruise ship were connected via hallways and stairways and the HVAC system was mixing the air in those spaces, we assumed the public area indoor spaces as a well-mixed single compartment. Although this assumption has limitations, in the absence of more detailed information on airflow pathways and interactions among individuals on the cruise ship, we found it a reasonable simplification. On the other hand, while each stateroom air was assumed well-mixed, we assumed different average exposure concentrations for non-infected and infected staterooms during one simulation day. The number of infected cabins was changed at the end of each simulation day based on the number of new infected cases in the public areas, as described in more detail in Section 2.1.
The floor plan of the public-accessible spaces in the Diamond Princess Cruise Ship including 14 decks out of 18 decks of the cruise ship was downloaded from the Princess Plus Company webpage and demonstrated in Figure S2.(8) Figure S2. Floor plan of public-accessible decks of the Diamond Princess Cruise Ship

Ventilation and inter-zonal airflow rates in cabins and public areas
Our assumptions for the ventilation rate in the cruise ship were similar to Zheng et al., in which the air change rate was assumed to be 9 per hour in passenger and crew cabins and 12 per hour in other indoor locations (11). We also made a conservative assumption against the contribution of long-range transmission of infectious aerosols in that there was no air re-circulation in the cruise ship spaces and the supply air to the cabins and public area were 100% outdoor air (13). We assumed the cabins in the Diamond Princess Cruise Ship were positively pressurized and the cabin air were forced outside into hallways, which were considered as part of the public areas, similar to a cruise ship studied by Vivancos et al. (13) Unfortunately, we are not aware of the interzonal airflow between the cabins and the hallways; therefore, for simplicity we assumed it was equal to 5% of the ventilation rate of the cabins.

Inhalation rate of susceptible individuals
The average inhalation rate of humans varies based on their age. Moriarty et al. reported the median ages of the 1045 crewmembers and 2666 passengers in the cruise ship were 36 and 69 years old, respectively(7); therefore, the average age of all passengers and crew in the cruise ship was ~60 years old. We estimated the average inhalation rate of 15.7 m 3 /day for passengers and crew of the Diamond Princess Cruise Ship based on their average age and the EPA recommended long-term exposure values for inhalation rates. (14) The inhalation rate of SARS-CoV-2 RNA copies for one susceptible individual in zone i ( ℎ , ) was estimated based on Equation S5. We assumed that before the quarantine started, passengers and crewmembers did not use surgical facemasks (i.e., , = 0) and subsequently started using surgical masks during the quarantine only when they were in public area (assumptions for , are described in detail in Section 3.8).

Time spent in cabins and public areas
Similar to the daily schedule of passengers on a typical cruise ship used by Zheng et al., (11) we assumed passengers and crewmembers spent an average of 9 hours per day in their cabins and spent rest of their time in public areas before the quarantine began on February 5. After the start of the quarantine, we assumed the majority of passengers stayed in their cabins all of the time and only contacted with their cabin-mates and the crew who provided daily services. It was reported that the passengers in the interior cabins were allowed to have outdoor air breaks once a day during the quarantine period, (15) but the duration of the breaks was not reported. Therefore, we assumed a daily outdoor air break of one hour for those passengers. We assumed that one crewmember provided necessary services to each cabin for 15 minutes per day, similar to Nicas and Jones assumption for the service time of healthcare workers in healthcare centers per each visit.(1)

Inactivation rate on air, surfaces and hand skin
van Doremalen et al. (16) reported the average half-life of SARS-CoV-2 viable viruses in aerosols as well as on copper, cardboard, stainless steel, and plastic surfaces as approximately 1.0, 0.7, 3.5, 5.7, and 7 hours, respectively. We assumed that the majority of surfaces on the ship that individuals were likely to touch were made of either plastic or stainless steel. Therefore, the halflife of viable SARS-CoV-2 viruses on surfaces was assumed to be ~6 hours. The inactivation rate ( ) of the viruses in units of per hour was estimated from Equation S6.

Transfer efficiencies between surfaces
We considered three transfer efficiencies between surfaces in each zone of the cruise ship to estimate the SARS-CoV-2 surface-to-hand ( We are not aware of any measured data on the transfer efficiency of SARS-CoV-2 between surfaces such as hands, surfaces, fomites, and mucous membranes. Therefore, we relied upon the assumptions used for transfer efficiencies of MERS-CoV and SARS-CoV between indoor surfaces in Xiao et al. and Lei et al. studies. (17,18) Table S1 demonstrates these assumptions. Based on the information provided in Table S1, we assumed the surface-to-hand, hand-tosurfaces, and hand-to-mucus-membrane transfer efficiencies are 19%, 16%, and 34%, respectively. We also assumed hand-to-surface and hand-to-face contact rates to be 1.5 (± 0.34) touches per minute (19) and 15.7 (± 11.3) touches per hour (20), respectively. We assumed a hand skin area of ~140 cm 2 based on Göker and Bozkir. (21) The next section describes key processes and assumptions used in the Markov chain model to track virus injection and transmission across short-and long-range distances on the ship.

Injection rates of SARS-CoV-2 to various states aboard the cruise ship
Similar to the Markov Chain Matrix, we considered 12 'states' for the injection array ( described in Equation S4) aboard the Diamond Princess Cruise Ship. We assumed that SARS-CoV-2 RNA copies could reach to the "HVAC System" and "Inactivation-Removal" states only from other states, not directly after they are injected from the infectors (i.e. direct injection rate to the HVAC System and Inactivation-Removal states were assumed to be zero). We assumed that infected individuals injected infectious viral particles into the defined spaces onboard the cruise ship in two forms: large diameter droplets and smaller diameter inhalable aerosols. Similar to MCM, we generated a new injection array for every day during the simulation period based on the number of infectors and susceptible individuals, passengers and crew interactions, and type of infection control strategy that was adopted (e.g. wearing facemasks, explored in detail in Section 3).

Injection of SARS-CoV-2 RNA copies in the form of large droplets and inhalable aerosols to the surfaces and indoor air
We assumed SARS-CoV-2 RNA viruses injected in the form of large droplets (i.e., > ~10 µm in diameter, as described in Section 3.4) deposit rapidly onto surfaces of the cruise ship. Viruses are tracked in terms of RNA copies because we utilize the only known available empirical data on SARS-CoV-2 aerosol concentrations and deposition rates in the literature, which report viral RNA rather than infectious/viable virus, as described in Section 3.3. The injection rate of SARS-CoV-2 RNA copies on surfaces of zone i in the form of large droplets per breath ( , ) was estimated using Equation S10:

Equation S10
, : Number of infectors in zone i , : Time fraction of presence in zone i ℎ : Number of breaths per hour We assumed that before the quarantine started, passengers and crewmembers did not use surgical facemasks (i.e., , = 0) and subsequently started using surgical masks during the quarantine only when they were in public area (assumptions for , are described in detail in Section 3.8).
The injection rate of SARS-CoV-2 RNA copies in the form of inhalable aerosols (i.e., less than ~10 µm in diameter, based on available data, as explored in Section 3.4) into the indoor air of zone i in the cruise ship ( ℎ , ) was estimated from Equation S11:

Short-range transmission of SARS-CoV-2 RNA copies
We considered three mechanisms for short-range transmission of SARS-CoV-2 between an infector and a susceptible individual, when both are in close proximity to each other, to include: (i) Direct deposition of large diameter respiratory droplets to the upper respiratory tracts of susceptible individuals (i.e. deposition onto their mucus membranes) (ii) Direct deposition of respiratory droplets and aerosols to the hands of susceptible individuals, followed by transmission to their upper respiratory tracts via hand-tomucus-membranes contact (iii) Inhalation of smaller diameter respiratory aerosols that deposit in the lower respiratory tracts of susceptible individuals The probability of direct deposition transmission is a function of the probability of being within a short-range contact zone in front of the infector (i.e., 'close contact') and the probability of large droplet deposition to the individuals' mucus membranes. Similarly, the probability of short-range inhalation transmission is a function of the probability of close contact and the probability of inhalation of infectious aerosols in that zone.
To estimate the probability of close contact with infected individuals, the close-range contact area was defined assuming a conical area in front of an infector with the head angle of 60° and length of 3 meters.  Again, we assumed that before the quarantine started, passengers and crewmembers did not use surgical facemasks (i.e., , & , = 0) and subsequently started using surgical masks during the quarantine only when they were in public area (assumptions for , and , are described in detail in Section 3.8).

Dose Response Model
To estimate the subsequent infection risk of SARS-CoV-2 viruses deposited to different sites of susceptible individuals, we used a negative exponential dose-response model, which implies that a single particle can start an infection and all single particles are independent of each other. The probability of infection for one susceptible individual ( ) in the cruise ship was calculated according to Equation S14:

Equation S14
Where Estimates of ID50 and infectivity for upper and lower respiratory tracts (URT and LRT) play a critical role in understanding the transmission of airborne infectious diseases. However, we are not aware of any clinical studies to date that report these values for SARS-CoV-2 in humans or animals.
Therefore, we rely on our model approach to back-calculate effective ID50 for upper and lower respiratory tracts (on a basis of RNA copies) by the following steps: 1-We selected an effective reproduction number for the COVID-19 outbreak on the cruise ship during the presence of the index case in the range of 1 and 6 based on Mizumoto and Chowell,(23) 2-We estimated the probability of infection based on the number of infected cases and susceptible individuals during the presence of the index case onboard the cruise ship, 3-We estimated the average number of SARS-CoV-2 viruses (i.e. RNA copies) that reached to the upper and lower respiratory tracts of the susceptible individuals during the first period of the analysis, and 4-We defined three scenarios for the proportion of infection dose of URT to LRT (i.e. ID50 URT/LRT = 1, 10, 100) and back-calculated the infection dose accordingly (explained in more detail in Section 3.7).
Step 4 allows us to test three logarithmically spaced assumptions for the ratio between ID50 for URT and LRT without knowing (or needing to know) the actual magnitude of ID50, while also anchoring ID50 to RNA copies rather than infectious units (e.g., PFU). Thus, the utility of the model approach goes beyond estimating transmission modes; the framework also allows for some inference of these important parameters by identifying which model scenarios and assumptions most accurately fit cumulative case counts.

Contribution from each transmission mode
In addition to estimating the number of infected cases with the model framework, we also estimated the contribution of several infection transmission modes, including direct deposition of droplets, fomite, and both close-and long-range inhalation of aerosols, to the estimated number of infected cases in both cabins and public areas using Equation S16:

Equation S16
Where : Four considered scenarios for infection transmission modes including direct deposition, fomite, long-range inhalation, and short-range inhalation : Two considered micro-environments in the cruise ship including cabins and public areas : Three considered simulation periods including during the whole outbreak, before the passenger quarantine began, and after the passenger quarantine began , , , : Infection contribution associated with transmission mode in microenvironment in simulation period : Number of simulation days in simulation period , which was considered to be 36, 16, and 20 for total duration of the outbreak before all passengers disembarked, before the passenger quarantine began, and after the passenger quarantine began, respectively , , : Total number of infected cases in the cruise ship during the simulation period , , , : Number of SARS-CoV-2 RNA copies that reached the target respiratory tract (i.e. LRT for inhalation and URT for direct deposition and fomite) via transmission mode in microenvironment on day of the simulation period : Infectivity of SARS-CoV-2 for the target respiratory tract (i.e. LRT for inhalation and URT for direct deposition and fomite) Direct deposition transmission was assumed to occur only when susceptible individuals were within the close-range contact area (defined in Section 1.2.2) and subject to direct contact with respiratory droplets from infected individuals (defined in Section 3.4). Short-and long-range inhalation transmission was assumed to occur via inhalation of aerosols either inside or outside the close-contact area, respectively. Fomite transmission was assumed to occur when susceptible individuals came in contact with contaminated surfaces, which could be contaminated by infected individuals through direct touching, direct deposition of respiratory droplets, and/or deposition of respiratory aerosols at any time point in the model framework. This approach also allows for summarizing infection contributions only by transmission mode, contact-range, microenvironment, or simulation period independently, as needed.

Combining the Complex Infection Transmission Risk Model with a Developed Epidemic Model
Generally, infection transmission risk models have been applied to the outbreaks during lessthan-a-day events. In these events, the number of infectors and susceptible individuals were assumed to stay constant during the outbreak. However, the COVID-19 outbreak in the Diamond Princess Cruise Ship was a unique situation, where the outbreak happened over a 40-day period. Therefore, we combined the complex infection transmission model with a newly developed version of the Reed-Frost epidemic model to simulate the transmission of COVID-19 in the Diamond Princess Cruise ship.

Developing an adjusted Reed-Frost model for the cruise ship outbreak
The Reed-Frost model is one of the simplest stochastic epidemic models. The model has been used for estimating the transmission of infectious diseases such as tuberculous (24) and measles, (25) although in some of those studies (e.g. Chen et al. (25)) it was shown that the model alone cannot explain the infection transmission within the studied indoor environment. As the basic Reed-Frost model that was described by Abbey (26) could not be used for this modeling work, we developed an advanced version of the Reed-Frost model for the cruise ship outbreak. The assumptions of the newly developed Reed-Frost model are described in the following: (i) The infection is spread from infected individuals to others by four main transmission pathways including long-range inhalation, short-range inhalation, direct deposition, and fomite, (ii) A portion of susceptible individuals in the group, after such contact with an infectious person in a given period, will develop the infection and will be infectious to others (the portion of 'susceptibles' who will develop the infection is estimated by the complex transmission risk model), (iii) The probability of coming into adequate contact with any other specified individual in the group within one time interval depends on the interaction behavior of the individual and is estimated using the Markov chain method, (iv) The susceptible individuals in the cruise ship were isolated from others outside the cruise ship, and (v) These conditions remain constant during one-whole day of the outbreak. For the epidemic model, we took the following steps: 1-The numbers of infected cases among susceptible individuals, some of whom were cabinmates with infected individuals and some were not (described in Section 1.1), were estimated using the complex transmission risk model at the end of each simulation day.
2-The infected cases were assumed to develop infection and become infectors after the latent period, which was estimated by reducing the assumed sub-clinical infectious period from the incubation period.
3-The number of cabins with at least one infected individual (i.e. 'infected cabins') was calculated at the end of each simulation day by assuming the number of newly infected cabins is equal to the number of newly infected cases who were not in one of the previously infected cabins at the beginning of the simulation day. : Cumulative number of detected infected cases or disembarked individuals from the cruise ship 5-We assumed the infected cases could spread infectious particles only one day after the incubation period, when their clinical symptoms began.

Epidemic characteristics of the outbreak during four assumed infection transmission periods
As mentioned previously, we divided the transmission patterns of SARS-CoV-2 on the Diamond Princess Cruise Ship into four periods. Each of these periods had different epidemic characteristics that we considered in our model.
During the first period, when the only infector onboard the cruise ship was the index case, we assumed no one was sharing a cabin with the index case, as none of the sources reported any information regarding the index case cabinmate. (7,15) Therefore, the number of susceptible individuals in the infected cabins was zero. The number of susceptible individuals in the public area during this period was assumed to be 3,188 (i.e. 2,666 guests and 522 crewmembers). It is noticeable that the total number of crewmembers in the cruise ship on February 4 th , when the quarantine started, was 1045, but most likely not all of them were interacting with passengers or would have been able to be in the same indoor environment with the index case (e.g., crewmembers who worked in the engine, kitchen, or control areas). With a lack of reliable sources on the number of crewmembers interacting with passengers during the first period, we simply assumed that half of the crewmembers possibly interacted with any passengers or were able to be in the passenger-accessible areas during the first period. The number of susceptible individuals among the infectors' cabin mates and other individuals during the other infection transmission periods were calculated from the aforementioned equations.
We assumed that after the passenger quarantine began, most of the passengers spent their time in their cabins, except the passengers in interior cabins, who were allowed to have outdoor time in open public areas under the guidance of Japanese Ministry of Health. (15) As the Diamond Princess Cruise Ship had 377 interior cabins (8) and the average number of passengers in each cabin was 1.98 people,(7) we assumed that 754 passengers in the interior cabins used the public areas during the quarantine for one hour per day. We also assumed that during the quarantine the indoor public areas were closed and only half of the crewmembers (i.e., 523 crewmembers) on the cruise ship were considered as 'essential' crew and interacted with other crewmembers and passengers in the hallways and areas such as the kitchen, control rooms, and the health clinic to provide necessary services. The rest of the crewmembers were assumed to remain in their interior cabins and to have followed similar guidelines as the passengers.

Adding checkpoint conditions to the epidemic model
We introduced several checkpoint conditions to the model to avoid unreasonable outcomes, as follows: 1-The cumulative number of infected cases and infectors should be always smaller than the total number of people on board.

Identifying the primary unknown or uncertain epidemic and infection transmission modeling parameters
Using these models requires numerous assumptions or estimates for unknown or uncertain input parameters, which were culled from existing literature where possible and otherwise estimated or assumed using known information about the Diamond Princess Cruise Ship. Because there is high uncertainty around several critical model parameters, especially those related to COVID-19 epidemic and mechanistic transmission characteristics, the interactions among individuals onboard, and the effectiveness of infection control strategies adopted during the quarantine period, we utilized a scenario modeling approach in which values for unknown or uncertain epidemic and transmission modeling parameters were varied over a wide range of possibilities to generate a matrix of possible solutions. This approach resulted in a total of 21,600 model iterations, with daily case counts and daily cumulative case counts among passengers aboard the ship serving as the primary model outcomes. Only those model scenarios with an acceptable coefficient of determination (R 2 ) between reported and modeled daily (i.e., daily R 2 > 0) and daily cumulative (i.e., R 2 > 0.95) case counts were analyzed further to determine the resulting bounds of these unknown or uncertain parameters. We also performed a sensitivity analysis of several important model parameters (discussed in Section 4).
The utility of this approach is that it allowed us to consider a wide variety of possible input parameters, based in part on emerging empirical evidence in the literature and in part on assumptions from other literature, to seek the most plausible solutions that fit the daily cumulative case numbers, which in turn provide insight into the values of input parameters that were associated with successful model runs. In other words, by selecting a range of unknown or uncertain values of key model input parameters and analyzing only those model results that meet acceptable criteria for predicted cases over time, not only do we infer information on the likely modes of transmission, but we also infer information on the likely ranges or bounds of the original unknown or uncertain input parameters.
Eight key model input parameters with unknown or uncertain values included: 1. Effective incubation periodthe time span between infection and detection among infected cases 2. Effective subclinical infectious period -the time span between the onset of an individual's infectious period and the appearance of clinical signs of disease 3. Viral generation rate in aerosols and droplets 4. Viral generation rate among asymptomatic and symptomatic individuals 5. Close interaction frequency 6. Median infectious dose 7. Effective reproduction number for the index case 8. Efficacy of infection control strategies during the quarantine period

Effective incubation period
We define the effective incubation period as the time span between infection and detection among infected cases. Based on the information available on the COVID-19 outbreak on the Diamond Princess Cruise Ship, the first 10 infected cases demonstrated COVID-19-related symptoms on February 4 th , 15 days after the index case boarded the Cruise Ship. A day after that, the quarantine period began, and the majority of passengers and crewmembers had to stay in their cabins. During the quarantine, suspected passengers and crew were tested for COVID-19 infection on a daily basis, and positive cases were then physically separated from other passengers and crewmembers by sending them to local hospitals. (10,27) Laboratory tests by PCR were conducted in the cruise ship focusing on symptomatic cases, especially at the early phase of the quarantine.(23) Therefore, we assumed that infectors, except the index case, infected the susceptible individuals in the cruise ship until they tested positive for COVID-19 during the quarantine period. We also assumed that it took one day for the laboratory to send back the results.
Lauer et al. estimated that 97.5% of infected cases who develop symptoms will do so within 11.5 days (95% CI: 8.2 to 15.6 days) of infection. (28) They also estimated the median incubation period (i.e., the time span before clinical signs show) was 5.1 days (95% CI: 4.5 to 5.8 days) among 181 confirmed cases. (28) Another study estimated the mean incubation period to be 5.2 days (95% CI: 4.1 to 7.0), with the 95 th percentile of the distribution at 12.5 days among patients with a median age of 59 years. (29) As the detection of the infected cases in the cruise ship was mostly based on symptomatic cases at the beginning of the quarantine and gradually all passengers and crew were tested for COVID-19 infection toward the end of the quarantine, we assumed that the average time span between infection and detection of infected cases (i.e. effective incubation period) in the cruise ship was anywhere between 6 and 15 days. Thus, we varied model inputs for this parameter from 6 to 15 days in increments of 1 day. This assumption was not free of limitations, particularly for asymptomatic cases.

Effective subclinical infectious period for infectors in the cruise ship
Another critical parameter for the epidemic modeling is the time span between the onset of the infectious period and the appearance of clinical signs of disease, which is also known as subclinical infectious period.(30) A positive subclinical infectious period demonstrates the time when infected cases could spread pathogens of an infectious disease before the clinical signs appear. Several studies suggested that subclinical infectious period is a key area of uncertainty and should be considered into account in infection transmission models.(31, 32) A description a family cluster of SARS-CoV-2 infection involving 11 patients in Nanjing, China, demonstrated that transmission of the COVID-19 can occur as early as five days before onset of symptoms. (33) Similar to the effective incubation period, because of the unique situation of the outbreak in the Diamond Princess Cruise Ship, we assumed an effective subclinical infectious period between 1 and 5 days for the infected cases in the cruise ship, with model inputs ranging from 1 to 5 days in increments of 1 day. The effective subclinical infectious period demonstrates on average how many days before the infected cases tested positive they spread the SARS-CoV-2 in the cruise ship.

Emission rate of SARS-CoV-2 from symptomatic and asymptomatic (or presymptomatic) cases
We considered the plausibility of different emission rates of SARS-CoV-2 from both symptomatic and asymptomatic (or pre-symptomatic) cases. We estimated the proportion of asymptomatic (or pre-symptomatic) cases to symptomatic cases based on the reported number of asymptomatic and symptomatic cases on the Diamond Princess Cruise Ship between February 5 and 20 provided in Mizumoto et al. (23) For our purposes, asymptomatic and pre-symptomatic are interchangeable because an individual classified as asymptomatic could have also been presymptomatic if they simply developed symptoms later than the simulation time period.
There is very limited information in the current literature on the emission rate of SARS-CoV-2 from asymptomatic (or pre-symptomatic) and symptomatic cases. Chia et al. measured the concentration of SARS-CoV-2 in two hospital rooms with symptomatic and asymptomatic patients. (34) The airborne concentrations of SARS-CoV-2 in the asymptomatic and symptomatic patient rooms were 1843 and 3384 RNA copies m -3 , respectively. (34) In another study, Arons et al. concluded that asymptomatic and pre-symptomatic residents in a nursing facility had the potential for substantial viral shedding, although they were unable to quantify their contributions in transmission of SARS-CoV-2 in the facility.(35) They also reported similar viral loads for asymptomatic, pre-symptomatic, and typical and atypical symptomatic individuals. (35) Therefore, to consider a range of plausible scenarios involving asymptomatic (or presymptomatic) and symptomatic cases, we considered two scenarios for the ratio of SARS-CoV-2 emission rates of asymptomatic to symptomatic cases, including 0.545 and 1 based on Chia et al. (34) and Arons et al. (35), respectively. This remains a highly uncertain parameter, but our selection of inputs that span a factor of two is intended to capture a relatively wide range in this parameter and will ideally offer insight into the most likely ratio based on model outcomes.

Emission rate of SARS-CoV-2 RNA copies in the form of large droplets and inhalable aerosols
We considered the emission rate of large respiratory droplets and small inhalable aerosols separately. To evaluate the transmission modes SARS-CoV-2, we needed to estimate the emission rate of SARS-CoV-2 RNA copies in the form of large droplets (> ~10 µm in diameter) and inhalable aerosols (≤ ~10 µm in diameter). The size of particles containing infectious viruses has a direct impact on the viral transmission mechanisms in indoor environments. Moreover, we assumed that only inhalable aerosols (≤ ~10 µm in diameter) can reach to the lower respiratory tracts of susceptible individuals, similar to prior assumptions by Nicas and Jones in their estimates of the relative contribution of four transmission pathways of influenza viruses in a typical health care center.(1) We defined three scenarios for emission rates based on two studies that measured size-resolved concentrations of SARS-CoV-2 RNA copies in patient rooms, as described below.

Measurements of SARS-CoV-2 RNA copies in patient areas in Fangcang and Renmin hospitals in Wuhan, China
The first scenario was based on Liu et al., which measured the number of SARS-CoV-2 RNA copies deposited on two locations on the floor of a 16-m 2 ICU room in Renmin Hospital, Wuhan, China and concentration of aerosols smaller than 10 µm in nine patient areas in Fangcang and Renmin hospitals in Wuhan, China. (36) The normalized deposition rate of SARS-CoV-2 in the ICU rooms were 31 copies m -2 hour -1 and 113 copies m -2 hour -1 for a location under medical equipment 2 meters from a patient's bed with severe symptoms and another location 3 meters from the same patient's bed without any objects above it, respectively. As the ICU room was relatively small and most probably well-ventilated with a high ventilation rate, we assumed that these measurements occurred under well-mixed conditions. Therefore, we estimated the emission rate of large droplets (> 10 µm in diameter) that rapidly deposit onto the floor (i.e. shedding rate: ℎ ) to be between 496 and 1808 RNA copies per hour per person, with the best estimate of 1152 (i.e. [(113 + 31 copies m -2 hour -1 ) / 2] × 16 m 2 ) RNA copies per hour per person. This is the only study to date of which we are aware that empirically assessed SARS-CoV-2 deposition rates.
The same study also measured airborne SARS-CoV-2 concentrations in several different sites. The measurements of airborne SARS-CoV-2 concentrations in "patient areas" were conducted in three workstation zones, one patient mobile toilet room, one ICU, one CCU, and one ward zone. The floor area of the workstation zones and the toilet room in the Fangcang hospital were approximately 500 m 2 and 1 m 2 , respectively. The toilet room did not have any ventilation and the workstation zones were relied on natural ventilation only. The measured airborne concentrations of SARS-CoV-2 varied between 0 and 9 copies m -3 in the workstation areas and was 19 copies m -3 in the toilet room. The airborne concentration of SARS-CoV-2 was zero in the other patient areas including the ICU, CCU, and ward zones of the Renmin Hospital. Moreover, the number of patients in the workstation areas was between 100 and 200 people. Patients in the workstation zones had mild symptoms, while the patients in the ICU and CCU rooms had severe symptoms. We use these concentrations in Section 3.4.3 to back-calculate emission rates.

Size-resolved measurements of SARS-CoV-2 RNA copies in patient rooms at the National Centre for Infectious Diseases in Singapore
Our second and third scenarios for the SARS-CoV-2 emission ratesor more accurately, the ratio between large droplet and inhalable aerosol emissionswere based on Chia et al., which conducted bioaerosol sampling in three airborne infection isolation rooms at the National Centre for Infectious Diseases, Singapore.(34) The rooms had mechanical ventilation systems delivering 12 air changes per hour, an average temperature of 23°C, relative humidity of 53 -59%, and exhaust flow of 579.6 m 3 /h. They deployed six NIOSH BC 251 bio-aerosol samplers in each room in the general ward to collect air samples. Particles collected with the NIOSH sampler were distributed into three size fractions including particles >4 μm, 1-4 μm, and <1 μm in diameter.
Among the three hospital rooms where patients were treated, no airborne SARS-CoV-2 RNA copy was detected in the room with a patient on 9 th day of illness. The other two patients were on their 5 th day of illness, and one of them was symptomatic and the other one was asymptomatic. The airborne SARS-CoV-2 concentration in the symptomatic patient room was 2000 and 1384 copies per m 3 for particles >4 μm, 1-4 μm in diameter, respectively and in the asymptomatic patient room was 927 and 916 copies per m 3 for particles >4 μm, 1-4 μm in diameter, respectively. No virus was detected in the <1 µm size ranges, although the authors mention that this could be due to low viral extraction efficiencies from the filter used on this stage (the other two stages deliver bioaerosols directly into fluid media for analysis). Regardless, this work considers only the 1-4 µm and <4 µm results from this study.
Because we consider ~10 μm as the cut off size between large droplets and inhalable aerosols (which we recognize is a somewhat arbitrary, albeit practical and useful, definition), we defined two scenarios based on the measurements from Chia et al. In one scenario, we assumed that half of the SARS-CoV-2 RNA copies collected in particles >4 μm in diameter were between 4 and 10 μm in diameter and the rest were in particles >10 μm in diameter (the NIOSH sampler does not have an upper size cut-off to our knowledge). In the other scenario, we assumed that half of the total measured SARS-CoV-2 RNA copies were in particles ≤10 μm and the other half were in particles >10 μm in diameter, primarily for simplicity and to provide another plausible value for this input parameter.

Applying well-mixed number balance model to estimate the emission rates of SARS-CoV-2 RNA copies in forms of large droplets and inhalable aerosols
We assumed steady-state well-mixed conditions for the workstation zones and the mobile toilet room in Liu et al. (36) and in the airborne infection isolation rooms in Chia et al. (34) to backcalculate emission rates for inhalable aerosols. For all of the rooms, we assumed that the resuspension rate is negligible compared to other transmission mechanisms of SARS-CoV-2. The emission rate of SARS-CoV-2 RNA copies in the form of inhalable aerosols smaller than ~10 µm in diameter ( ℎ − , copies per hour per person) was estimated using Equation S19: Deposition rate of inhalable aerosols: Liu et al. measured size resolved distributions of SARS-CoV-2 RNA copies in aerosols less than 10 µm in diameter in three locations in the Fangcang Hospital, Wuhan, China. We calculated the average proportion of SARS-CoV-2 RNA copies in the five particle size bins reported in the study.(36) Next, we mapped the size distribution of SARS-CoV-2 RNA copies on the size-resolve distribution of particle deposition rate reported in Riley et al. (40) to estimate the bulk deposition rate of SARS-CoV-2 RNA copies in aerosols, which was estimated to be about 0.64 per hour. A similar approach was used previously in Azimi and Stephens (41) for estimating the bulk deposition rate of influenza viruses. Table S2 demonstrates the relative size distribution of SARS-CoV-2 RNA copies, the estimated deposition rate for each particle size bin, and the estimated bulk deposition rate for SARS-CoV-2 RNA copies in inhalable aerosols less than 10 µm.  Table S2 [e] Assumption based on the air exchange rate of a typical sport area (39) Based on the estimates of the deposition rate and patient area characteristics, we estimated the emission rate of SARS-CoV-2 RNA copies in inhalable aerosols smaller than 10 µm in diameter to be ~315 RNA copies per hour, with a likely range between ~50 and ~1100 copies per hour for the Fangcang Hospital scenario (Emission Scenario 1). Similarly, the emission rate of SARS-CoV-2 RNA copies in the form of inhalable aerosols (≤10 µm in diameter) and large droplets (>10 µm) in isolation patent rooms in the National Centre for Infectious Diseases in Singapore was It is worth noting that our estimation of SARS-CoV-2 RNA copy exhalation rates are within the realm of plausibility in comparison to other studies that have measured the number of RNA copies of other respiratory pathogens in the exhaled breath. For example, Milton et al. measured the median Influenza virus copy number in aerosol particles (< 5 µm) exhaled by patients equal to ~200 copies per hour with 1 st and 3 rd quartiles of ~80 and ~400 copies per hour and a range from lower than detection level to 2.6 x 10 5 copies per hour. (42) In another study with a similar approach, Yan et al. measured the geometric mean of emission rate of influenza virus RNA copies in exhaled fine aerosols (< 5 µm) equal to 7.6 x 10 4 copies per hour with a range from not detectable to 8.8 x 10 7 copies per hour. (43) It is also worth noting that this model was designed to back-calculate 50% infective dose for upper and lower respiratory tracts based on our assumptions for emission rate of SARS-CoV-2 RNA copies (Section 1.3); therefore, the absolute magnitude of these values are not crucial in the model, but rather the ratio between large respiratory droplets and small inhalable aerosols is most useful.

Minimum close-range interaction time in the cabins
Close-range interactions between people plays a critical role in the transmission of infectious airborne diseases. In this model, we defined a close-range interaction as any interaction between an infector and a susceptible individual that happens within a 3-meters-length hypothetical zone in front of the infector, as described in Section 1.2.2. In the public areas aboard the Diamond Princess Cruise Ship, we assumed that infectors and susceptible individuals interact with each other randomly. However, to estimate the close interaction time inside the cabins, particularly after the passenger quarantine was enacted, we considered two scenarios. In the "Standard" scenario, we assumed that the infector and susceptible individual had a minimum close interaction of 8 hours; in the "Extended" scenario, we assumed that the infector and susceptible individual had a minimum close interaction of 16 hours. We assumed that the passengers spent 9 hours on average in their cabins before the quarantine started; therefore, in the "Extended" scenario, passengers' close interaction time in cabins before the quarantine began was limited to 9 hours.

COVID-19 effective reproduction number for the index case
The effective reproduction number is defined as the average number of secondary cases per infectious cases in a population. Estimating the effective reproduction number of COVID-19 on the Diamond Princess Cruise Ship has been used for investigating the result of interventions strategies that were imposed on travelers and crew aboard the cruise ship.(23) Herein, we used the estimates of COVID-19 effective reproduction number for the index case to back-calculate the ID50 and infectivity of SARS-CoV-2 for upper and lower respiratory tracts as describe in detail in Section 1.3. For the effective reproduction number, we relied on Mizumoto and Chowell, which estimated the range of effective reproduction number to be between 1 and 6 during the five days that the index case was onboard the cruise ship.(23)

SARS-CoV-2 median infectious dose
Our knowledge on the transmission mechanisms of SARS-CoV-2 is still very limited. For example, still there is no information about the 50% infective dose (ID50) of SARS-CoV-2 for upper and lower respiratory tracts (URT and LRT). (44) Moreover, the proportions of SARS-CoV-2 depositing in the LRT and URT of a susceptible individual when they inhale infectious aerosols are not characterized. To circumvent this gap, we (1) assumed different effective ID50 of SARS-CoV-2 for (i) inhalation and (ii) fomite and direct deposition transmission pathways, and called them ID50 for (i) LRT and (ii) URT, respectively, and (2) estimated the ID50 of SARS-CoV-2 for upper and lower respiratory tracts by (i) using estimates of the effective reproduction number of COVID-19 in the Diamond Princess Cruise Ship from the existing literature during the time that the index case was onboard the cruise ship (Section 3.6); and (ii) considering a variety of scenarios for the proportion of ID50 of SARS-CoV-2 for URT to the LRT (i.e. 1:1, 10:1, and 100:1). In Section 1.3, we explained our approach for estimating the ID50 of SARS-CoV-2 in more detail. Here, we describe how we arrived at these three logarithmically spaced assumptions.
In this model, we estimated the number of virus copies that were predicted to reach the upper and lower respiratory tracts of susceptible individuals separately because the site and efficiency of respiratory deposition would be different for different droplet sizes. For example, larger droplets would deposit more efficiently in the upper respiratory tract (45), while the pathogens carried by inhalable aerosols (≤ 10 µm in diameter) would be able to reach to the lower respiratory tract. (46) It is also shown that for some viral airborne diseases such influenza the infectivity of viruses reaching to LRTs of an susceptible individual is about two orders of magnitude higher than the viruses deposited in the URTs(47) (i.e. influenza human ID50 is estimated between 0.6 and 3 TCID50 for lower respiratory tracts (48) and between 127 and 320 TCID50 for upper respiratory tracts (49)). Therefore, our three assumed ranges of the ratio between ID50 for URT and LRT of 1, 10, and 100 represent three theoretically possible values across a wide range in magnitude.

Efficacy of infection control strategies during the quarantine period
Based on the information provided in various sources, we know that the Diamond Princess Cruise Ship was quarantined at Yokohama on February 3 rd , 2020, and then a quarantine of all passengers began on February 5 th , 2020. (7,15,23,50) During the quarantine, the passengers and crew had to follow strict guidelines, including staying in their cabins all of the time, except for individuals who were in interior cabins, as well as essential crew, and all passengers and crew in the public area had to use facemasks. However, the effectiveness of wearing facemasks for reducing COVID-19 transmission risk in the cruise ship was not measured or reported; therefore, we relied on other studies that have reported particle/viral removal efficiencies of facemasks. We also believe it is reasonable to assume people during the quarantine washed their hand more frequently and effectively amid the safety guidelines that was provided for passengers and crew.
Therefore, we considered two scenarios for the efficacy of infection control strategies adopted on the cruise ship. In the "moderate efficacy" scenario, we assumed that before the passenger quarantine began, the viral removal effectiveness of hand washing ( ℎ − ℎ ) was ~50% by considering a 70% probability of washing hands after touching fomite and a 72.5% viral removal efficiency for washing hands similar to the values used for a typical cruise ship model in Zhang et al. (12) Then, we assumed the viral removal effectiveness of hand washing increased to ~80% after the passenger quarantine began (i.e. probability and viral removal efficiency of 90% for washing hands). We also assumed that before the quarantine, complete surface disinfection was performed on a daily basis at the end of the day. We assumed that after the quarantine started, in addition to the complete end-of-day disinfection, the public area and cabin surfaces were disinfected by crewmembers and passengers several times during the day. Unfortunately, we are not aware of how often the regularly-touched surfaces in the cruise ship were cleaned during this time, particularly inside the cabins, as the housekeepers were not allowed to enter cabins and cleaning supply was provided for passengers to clean their own rooms. (15,51) In the lack of reliable sources on surface disinfection frequency and efficiency, we simply assumed the additional surface disinfection efforts reduced the transfer rate of SARS-CoV-2 between fomite and hand by 72.5% ( in Equation S6) in comparison to a scenario where the cruise ship surfaces were not disinfected during the day, similar to the surface disinfection efficiency assumed in Zhang et al. (12) We assumed that the majority of passengers and crew were not wearing facemasks before the quarantine started, while they all wore facemasks in public areas after the quarantine began. The facemasks were assumed to have an in-vivo filtration protection ( ) of 95%, similar to protection of surgical masks against SARS-CoV,(52) and a 54% coverage of human facial features ( ) assuming they only covered mouth and nasal areas.(53) Our approach for deploying these parameters into the mechanistic infection transmission model was described in detail in Sections 1.1 and 1.2.
For the "high efficacy" scenario, we considered the same model parameters for the before quarantine time period; however, we assumed higher efficacy infection control strategies after the quarantine began. In this scenario, we assumed the probability and effectiveness of handwashing, removal efficiency of facemasks, and during-the-day surface viral disinfection efficiency were 99% after the quarantine started. We kept the coverage proportion of the facemasks equal to 54% similar to the previous scenario.

Analyzing model outcomes and conducting sensitivity analyses
In this section, we detail how we analyzed model outcomes, defined acceptable model iterations, and conducted a sensitivity analysis to further explore transmission modes and the importance of several key model parameters.

Selecting acceptable model iterations
The model approach resulted in a total of 21,600 model iterations, with each iteration representing a scenario with distinct combinations of assumptions for unknown or uncertain model input parameters (e.g., incubation period, sub-clinical infectious period, effective reproduction number, viral shedding rates in aerosols and droplets, close-range interaction times), as shown in Table  S4. We ran the model with each possible combination of the eight input parameters shown in Table S4 (10×5×6×3×3×2×2×2=21,600) in order to search a wide range of possible parameters and combinations of parameters. With this approach, many of the scenarios are expected to yield unacceptable results because they combine multiple unlikely parameter assumptions (e.g., poor assumptions for incubation period, effective reproduction number, and URT/LRT ratios).   Coefficients of determination (R 2 ) were calculated between model predictions and reported case numbers for both daily cases and daily cumulative cases for each of the 21,600 model scenarios.
Only those model scenarios that yielded an acceptable coefficient of determination (R 2 ) between reported and modeled daily and daily cumulative cases were analyzed further to explore the likely ranges and bounds of the unknown or uncertain model input parameters. We considered the model scenarios with R 2 > 0.95 for daily cumulative cases and a positive R 2 for daily cases as 'acceptable.' The weaker criterion for daily R 2 values was used because reported daily infection numbers likely suffered from delays in reporting and detection that the model does not predict; however, the cumulative case numbers smooth out these daily variations and warrant more stringent criteria. Results are shown in the main text.

Model sensitivity to eliminating transmission modes using 'best estimates' of model parameters
In order to explore the importance of each transmission mode, we used a transmission mode elimination process combined with resulting 'best estimates' of primary model parameters, in which we considered only one transmission mode at a time (e.g., long-range inhalation only, close-range inhalation only, direct deposition, only), as well as combining two transmission modes at a time (e.g., long-and short-range inhalation, direct deposition and fomite).
In this analysis, we assumed the following: (i) we only used our 'best estimates' of model parameters, (ii) we rounded the best estimates of incubation period and subclinical infectious period to integer numbers, (iii) instead of using the effective reproduction numbers to back-calculate the infectious dose for URT and LRT from the first period of the outbreak simulation for each transmission mode separately, we used the calculated infection doses, calculated using the 'best estimates' of model parameters deployed in the model, and (iv) the total number of emitted SARS-CoV-2 RNA copies per hour was assumed to be equal to the average of reported emission rates across Liu et al. (36) and Chia et al. (34) (i.e., ~10 6 per hour), with the proportion of emitted inhalable aerosols to the large droplets assumed to be 1.3 (i.e., the 'best estimate' resulting from acceptable model iterations, as shown in Table S5). Table S5 summarizes the model outcomes when only one or two transmission modes were considered with 'best estimates' of primary model parameters. Results in Table S5 demonstrate the importance of considering all transmission modes, as the R 2 value between predicted and reported daily cumulative case numbers was 0.98 considering all modes. Limiting to any single transmission mode (e.g., only short-range inhalation of aerosols, only long-range inhalation of aerosols, only direct droplet deposition, or only fomite transmission) resulted in R 2 values less than 0.15 and severe undercounting of cases in cabins and public areas.
In other words, using our best estimates of several key model parameters, no individual transmission mode can explain reported cases. However, including both long-and short-range inhalation of inhalable aerosols improved model performance (R 2 = 0.75), while including direct droplet deposition and fomite together (but ignoring inhalation of aerosols) did not improve model performance. These results suggest that inhalation of smaller diameter aerosols (< 10 µm), across both short-range and long-range distances, were likely the dominant contributor to the transmission of COVID-19 aboard the Diamond Princess Cruise Ship, and that fomite and direct droplet deposition likely played a smaller role. Table S5 also demonstrates the potential effects of indirectly considered parameters and processes on the estimated contribution of various infection transmission pathways for extreme scenarios. For example, there is some evidence supporting the fact that relative humidity has a critical effect on risk and contribution of different infection transmission pathways (57,58). We can indirectly test this if we consider a hypothetical extreme scenario, where for example, only large liquid droplets are assumed to carry SARS-CoV-2 because the coronaviruses could lose their viability when droplets reduce to their dry nucleus, the sensitivity analysis shows that the total number of infected cases under this hypothetical direct droplet deposition and fomite only scenario would be only 104 cases (a severe underprediction of cases versus actual cases). As another example, Table S5 shows if we consider another hypothetical extreme scenario, where all SARS-CoV-2 RNA copies were carried by inhalable aerosols smaller than 10 µm in diameter, the number of infected cases on the cruise ship would reduce to 479 cases (a less severe underprediction of cases).

Model sensitivity to primary epidemiological inputs
Among the 8 unknown or uncertain primary inputs of the developed transmission risk model demonstrated in Table S4, the effective incubation period, effective sub-clinical infectious period, and effective reproduction number for the index case were considered as epidemiological inputs of the model. Table S6 summarizes the coefficients of determination (R 2 ) for all 21,600 explored model iterations when various combinations of primary epidemiological inputs were adopted in the risk model. Each cell demonstrates the average R 2 value of 72 explored model iterations (i.e. 2 asymptomatic/symptomatic emission scenarios x 3 emission rate scenarios x 2 minimum close interaction time in the cabins scenarios x 3 ID50,URT/ID50,LRT scenarios x 2 infection control efficiency scenarios) combined with a distinct combination of primary epidemic input values used in the risk model.
The vast majority of epidemic model input combinations yielded negative R 2 values, on average, suggesting they were implausible combinations. Table S6 shows several diagonal series of combinations of input values that yielded greater numbers of acceptable iterations, most commonly clustered around effective sub-clinical infection periods of 5 days (with some 2-3 days) and effective incubation periods of 11-13 days. Table S6. Average coefficients of determination associate with various combinations of primary epidemiological model inputs Reff = 1 C = 6 C = 7 C = 8 C = 9 C = 10 C = 11 C = 12 C = 13 C = 14 Reff = 2 C = 6 C = 7 C = 8 C = 9 C = 10 C = 11 C = 12 C = 13 C = 14

Model sensitivity to changes in the ratio between infectious dose for URT and LRT
The model results demonstrate that the ratio between SARS-CoV-2 infectious dose for upper and lower respiratory tracts plays a critical role in estimating the modes of transmission of COVID-19 aboard the ship. Figures S5 and S6 show the impacts of our three logarithmically spaced assumptions for the ratio between SARS-CoV-2 infectious doses for URT and LRT on the estimates of infection contribution of various transmission modes and viral sources among the 132 acceptable model iterations.  Figure S5 demonstrates that when the SARS-CoV-2 infectious doses for URT and LRT are assumed to be equal (1:1 URT/LRT ID50), the contribution of fomites to infection transmission in the cruise ship was significantly higher than the other transmission modes (i.e., long-and shortrange). This means that ~60% of the total number of infectious SARS-CoV-2 that reached to the respiratory tracts of susceptible individuals was estimated to reach to URT via fomite pathways. However, when we considered a lower infectious dose for LRT of susceptible individuals, our estimates of contribution of long-and short-range transmission modes in number of infected cases was increased. For an assumption of URT infectious dose 10 times higher than the LRT infectious dose, the estimated infection contributions of all transmission modes were approximately similar. For an assumption of URT infectious dose 100 times higher than the LRT infectious dose, more than 90% of infection transmission was estimated to be via short-and longrange transmission modes. Clearly, this ratio is a critical factor in the model and remains to be better understood from clinical investigations. However, recall that a ratio of 100:1 had the largest number of acceptable model iterations associated with the assumption (i.e., 58 compared to 35 for 1:1 and 39 for 10:1), with an average ratio of ~47:1 ( Table 2 in the main text), which provides some guidance on where this value may reasonably lie. Similarly, our assumptions for the ratio between infectious dose for URT and LRT affected our estimates of infection contribution of different viral sources. In the 1:1 URT/LRT ID50 scenario, transmission trough larger droplets (> ~ 10 µm in diameter) was ~ 4 times higher than the transmission through aerosols. Conversely, in the 100:1 URT/LRT ID50 scenario, the infection transmission via smaller inhalable aerosols (< ~ 10 µm in diameter) was ~9 times higher than of large droplets. The infection contributions of droplets and aerosols were approximately similar for the 10:1 URT/LRT ID50 scenario. One thing to notice is because we assumed inhalable aerosols and large droplets deposit on LRT and URT, respectively, the infection contribution of aerosols and droplets could also be considered as the infection contribution of LRT and URT, respectively.

Model sensitivity to changes in emission rate scenarios
Finally, we explored the sensitivity of our model results to the changes in our assumptions for the emission rates of SARS-CoV-2 in the form of droplets and aerosols. Figures S7 and S8 demonstrate the impacts of ratio of aerosol versus droplet emissions on estimated infection contributions of various transmission modes and viral sources, respectively. Figure S7. Estimated infection contributions of various transmission modes corresponding to three different assumptions for the ratio between aerosol and droplet emissions among acceptable model iterations (i.e., aerosol/droplet ratios of ~1:4, ~2:1, and 1:1) Figure S7 shows that the median estimated infection contribution from fomite transmission mode is ~50%, and is higher than both short-and long-range transmission modes, when we assumed that the ratio of SARS-CoV-2 emitted in the form of droplets is about 4 times higher than the number of viruses emitted in inhalable aerosols (based on approximations from data reported in Liu et al. (36)). The median estimated infection contribution of long-range transmission during this emission scenario was only ~20%, but then increased to ~45% when the ratio of aerosol versus droplet emissions increased to 2:1 (based on approximations from data reported in Chia et al. (34)). The estimated infection contribution of short-range transmission, which was a combination of deposition of larger droplets on URT and inhalable aerosols on LRT both within close-contact range, were approximately similar for all three emission ratio scenarios. When we assumed a similar emission rate of SARS-CoV-2 in forms of droplet and aerosols (i.e., 1:1), the median infection contribution of short-range, long-range, and fomite transmission modes were estimated to be ~35%, ~45%, and ~20%, respectively. This remains an uncertain parameter, with approximately equal numbers of acceptable model iterations associated with each assumption (i.e., Table 2 in the main text). Similarly, Figure S8 shows the infection contribution via droplets was higher using estimates of aerosol/droplet ratios based on Liu et al. (36) (~1:4), with a median value of ~65%, and it decreased to ~30% when we assumed aerosol/droplet ratios of ~2:1 based on Chia et al. (34) One thing to notice is the large variation in the estimated infection contributions based on the emission scenarios observed in Figures S7 and S8 is due primarily to changes in our assumptions for the infectious doses of upper and lower respiratory tracts, as explained in Section 4.4.

Statistical significance testing on the model results
We performed a Mann-Whitney U-test on the model results to evaluate the statistical significance of each mode comparison. For the main analysis, Table S7 indicates that the estimated infection contributions by droplets and aerosols, as well as between all transmission modes, are significantly different after the passenger quarantine started (p < 0.0001). However, differences in the estimated infection contributions between droplets and aerosols, and between short-range and fomite transmission, before isolation started, and between long-range and both short-range and fomite transmission modes for the entire modeling duration are not statistically different (p > 0.05).  Table S8 shows the results of Mann-Whitney U-tests comparing the infection contributions of various viral sources and transmission modes when different assumptions for several mechanistic transmission factors are adopted for the entire modeling duration. The estimated infection contributions of various viral sources and transmission modes when different URT/LRT infectious dose scenarios are adopted are statistically different with p-values lower than ~0.01. They also show that the adopted scenarios for symptomatic versus asymptomatic emissions do not yield a significant difference in the estimates of infection contributions. The statistical significance testing on estimates of infection contributions of various viral sources and transmission modes when different mechanistic transmission factor scenarios are adopted demonstrates a mixed result as shown in Table S8. For example, it is shown that infection contribution of short-range transmission mode is statistically different when the minimum close interaction time in cabins was assumed to be 8 hours or 16 hours (p < 0.0001), but the infection contribution of droplets with similar assumptions is not significantly different.