Markov Chain Model.

The Markov model adopted in this study uses a discrete-time discrete-space Markov chain 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 between each defined state, aboard the ship over time (SI Appendix, section 1). 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 mucous 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.

We chose the Markov Chain model over other existing infection transmission risk models because of its ability to stochastically track all modes of transmission under a wide variety of assumptions and with high computational efficiency. This approach offers advantages over the extensively used Wells–Riley model for quantitative infection risk assessment of respiratory infectious diseases in indoor environments, which does not consider all disease transmission routes (38). The latter reasoning offers advantages over infection transmission models that are based on more complex and computationally intensive computational fluid dynamics (CFD) simulations (39), as the Markov chain model does not require solving the partial differential equations for governing the particle transport. Moreover, others have shown that the Markov chain model can predict transient particle transport in enclosed environments with similar accuracy to Eulerian and Lagrangian models, which have been widely used for particle modeling in CFD tools (40). Many scientists also have used Markov model for predicting the transmission risk of infectious disease indoors (33, 41⇓⇓⇓–45).

We considered the following 12 states for the Markov chain process, with state numbers corresponding to SI Appendix, Fig. S1: (#1 and 3) indoor air and surfaces of cabins occupied by infectors; (#2 and 4) public area indoor air and surfaces; (#5 to 7) hands (palms), upper respiratory tract (URT), and lower respiratory tracts (LRT) of uninfected individuals who were cabinmates of infected individuals before they became infected; (#8 to 10) palms, URT, and LRT of uninfected individuals who were not cabinmates of infected individuals before they became infected; (#11) heating, ventilation, and air-conditioning systems; and (#12) inactivation of viable virus. We conservatively (i.e., favoring against long-range airborne transmission) assumed cabins are positively pressurized and there was no air recirculation (SI Appendix, section 1.1.2); therefore, infectious particle could reach to a “clean” cabin only if an infector occupies it. We generated a new Markov chain matrix for each day in the simulation period to model mechanistic transmission and infection probability 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. 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.

The modeling framework incorporates available empirical data on key mechanisms of SARS-CoV-2 dynamics culled from recent literature, including 1) viral RNA emission rates in large droplets (assumed greater than ∼10 µm, consistent with the conventional WHO definition) and inhalable aerosols (assumed less than ∼10 µm) from infected individuals, which were back-calculated from recent reports of air and surface sampling in health care settings and were assumed to be the same ratio for all infected individuals; 2) viability loss in air and on surfaces reported in controlled studies; and 3) estimates of aerosol deposition rates to surfaces based on typical assumptions for aerosol dynamics.

The framework also leverages estimates and assumptions for several human and built environment transmission factors, culled from prior literature where possible, including average rates of face and surface touching, inhalation rates, the shape and size of close-contact zones, time spent in various environments (e.g., public areas and cabins), floor areas and volumes of cabins and public areas, the probability of uninfected individuals within close proximity of an infected individual, and the impact of infection control strategies that were implemented during the quarantine period (e.g., mask wearing, hand washing, and surface disinfection). Detailed descriptions of all model inputs are provided in SI Appendix, section 1 (for relatively certain parameters) and SI Appendix, section 3 (for relatively unknown or uncertain parameters).

Dose–Response Model.

To estimate the infection probability of SARS-CoV-2 viruses deposited to different body 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 (Pinfection) in the cruise ship was calculated using Eq. 1:Pinfection=Number of infected casesNumber of susceptibles=1−exp[−(αURT×NURT+ αLRT×NLRT)],[1]

where NURT and NLRT represent numbers of viable SARS-CoV-2 RNA copies in URT and LRT of one susceptible individual; and αURT and αLRT, infectivity of SARS-CoV-2 for URT and LRT.

The 50% infectious dose (ID50), or the number of viruses necessary to infect a susceptible individual in 50% of a sample population, of SARS-CoV-2 for URT and LRT can be estimated from Eq. 2 (41, 46):ID50=ln(2)α, where ID50≥ln(2).[2]

Estimates of ID₅₀ and infectivity for 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. Moreover, the proportions of SARS-CoV-2 depositing in the LRT and URT of a susceptible individual when they inhale infectious aerosols are not yet characterized. Therefore, we tested three logarithmically spaced assumptions for the ratio of the effective ID₅₀ for SARS-CoV-2 for aerosol inhalation (assuming deposition in the LRT) and fomite and droplet deposition (assuming deposition in the URT): ID₅₀ URT:LRT = 1:1, 10:1, and 100:1. Our assumptions for equal or higher SARS-CoV-2 median infectious doses for fomite and droplet deposition in comparison to aerosol inhalation are generally in line with existing studies showing SARS-CoV-2 preferentially replicates deeper in the lungs (15) and leads to clinical symptoms at lower doses of aerosol exposures compared to ocular or intranasal routes in animal models such as African green monkeys (47, 48) and golden hamsters (49). We rely on our model approach to back-calculate effective ID₅₀ values (using a basis of RNA copies) by analyzing successful model results, as described in SI Appendix, section 1.3. This approach allows us to test scenarios with these uncertain parameters without knowing (or needing to know) the actual magnitude of ID₅₀, which can then be used to infer the likely magnitude of this ratio based on successful model iterations.

Transmission Mode Contribution to Infection.

In addition to estimating the number of infected cases with the model framework, we also estimated the contribution of multiple infection transmission modes to the estimated number of infected cases in both cabins and public areas, including 1) direct deposition of respiratory droplets (within close range only), 2) contact with fomites, and 3) inhalation of aerosols (with both close- and long-range transmission traced separately) (Eq. 3):Cinfection, k,r,p=∑l=0Dp{Ninfected,r,lNinfected,total,p×1−exp(−Nvirus,k,r,l×αk)∑k,r[1−exp(−Nvirus,k,r,l×αk)]} ,[3]

where k represents four considered scenarios for infection transmission modes, including direct droplet deposition, fomite, long-range aerosol inhalation, and short-range aerosol inhalation; r, two considered microenvironments in the cruise ship including cabins and public areas; p, three considered simulation periods including during the entire outbreak duration, before the passenger quarantine began, and after the passenger quarantine began; Cinfection, k,r,p, infection contribution associated with transmission mode k in microenvironment r in simulation period p; Dp, number of simulation days in the simulation period p (i.e., 36, 16, and 20 for the entire outbreak duration before all passengers disembarked, before the passenger quarantine began, and after the passenger quarantine began, respectively); Ninfected,r,l, number of infected cases in microenvironment r on day l of the simulation period; Ninfected,total,p, total number of infected cases in the cruise ship during the simulation period p; Nvirus,k,r,l, number of SARS-CoV-2 RNA copies that reached the relevant respiratory tract region (i.e., LRT for inhalation and URT for direct deposition and fomite) via transmission mode k in microenvironment r on day l of the simulation period; and αk, infectivity of SARS-CoV-2 for the target respiratory tract (i.e., LRT for inhalation and URT for direct deposition and fomite).

This approach allows for summarizing estimates of infection contributions by transmission mode, contact range, microenvironment (i.e., public areas or passenger cabins), and/or simulation period independently, as needed.

Short-range transmission occurs by direct deposition of respiratory droplets and inhalation of aerosols only when susceptible individuals were within a defined close-range contact area of 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 m (described in detail in SI Appendix, section 1.2.2) (42, 50). The projected surface area of the cone on the floor was ∼4.7 m², which is equivalent to a surface area of a circle around the infector with a radius of ∼1.2 m. The probability that a susceptible individual was present within the close-contact cone was estimated based on the proportion of the zone surface area to the projected surface area of the cone on the floor (SI Appendix, section 1.2.2).

Long-range inhalation transmission occurs via inhalation of aerosols when susceptible individuals were outside the close-contact area. Fomite transmission occurs 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 and location in the model framework.

Combining the Transmission Risk Model with a Developed Epidemic Model.

The mechanistic infection transmission model was combined with a modified version of the Reed–Frost epidemic model to simulate the transmission of COVID-19 aboard the ship. We assumed that 1) the infection is spread from infected individuals to others by four main transmission pathways (long-range inhalation, short-range inhalation, direct deposition within close range, and fomite contact), 2) a portion of susceptible individuals in the group will develop the infection and will be infectious to others (the portion of “susceptibles” who will develop the infection is estimated by the transmission risk model), 3) 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, 4) the susceptible individuals in the cruise ship were isolated from others outside the cruise ship, and 5) these conditions remain constant during one whole day of the outbreak.

To estimate the spread of the disease, we estimated the number of infected cases among susceptible individuals, some of whom were cabinmates with infected individuals and some were not, at the end of each simulation day using the transmission risk model. The infected cases were assumed to develop infection and become “infectors” after the latent period, which was estimated by reducing the assumed effective subclinical infectious period (i.e., the time span between the onset of the infectious period and the appearance of clinical signs of disease) from the effective incubation period (i.e., the time span between infection and detection among infected cases). 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. The numbers of susceptible individuals who were not cabinmates with an infector (Nsusceptibles−common) and susceptible individuals inside the infected cabins (Nsusceptibles−cabin) at the beginning of each simulation day (d) were estimated using Eqs. 4 and 5 (except for the first period of infection transmission):Nsusceptibles−common(d)=Ntotal−onboard−[Ninfected−cabin(d)×Naverage−cabin],[4]Nsusceptibles−cabin(d)=[Ninfected−cabin(d)×Naverage−cabin]−Ninfector(d)−∑i=0d−1Ndetected−cases(i),[5]

where Ntotal−onboard represents total number of passengers and crew onboard (constant during the outbreak); Ninfected−cabin, estimated number of infected cabins at the beginning of each day; Naverage−cabin, average number of individuals in one cabin; Ninfector, number of infectors; and ∑Ndetected−cases, cumulative number of detected infected cases or disembarked individuals from the cruise ship.

We assumed the infected cases could spread infectious virus until only 1 d after the incubation period because the passengers were screened daily and removed from the cruise ship if they had shown symptoms or positive test results. We divided the transmission patterns into four periods, each of which having different epidemic characteristics, as described in SI Appendix, section 1.1. Several checkpoint conditions were introduced to the epidemic model to ensure reasonable bounds (SI Appendix, section 2.3).

Analysis.

The model framework 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, 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. In this approach, our estimates of ID₅₀ and infectivity for aerosol inhalation (assuming deposition in the LRT) and fomite and droplet deposition (assuming deposition in the URT) play a critical role in understanding the transmission of airborne infectious diseases. As we are not aware of any clinical studies to date that report these values for SARS-CoV-2 in humans or animals, we rely on our model approach to “back-calculate” effective ID₅₀ for upper and lower respiratory tracts (on a basis of RNA copies) for each considered scenario from the first 5 d of the simulation period called the “calibration period.” A total of 21,600 scenarios were modeled across a range of estimates or assumptions for eight critical unknown or uncertain input parameters (Table 1). Estimates and assumptions for these parameters are described in detail in SI Appendix, section 3. We ran the model with each possible combination of the eight unknown or uncertain input parameters shown in Table 1 (10 × 5 × 6 × 3 × 3 × 2 × 2 × 2 = 21,600) in order to search a wide range of possible parameter values and combinations of parameter values.