key: cord-260214-2axc1wn3
authors: Batista, Berlinda; Dickenson, Drew; Gurski, Katharine; Kebe, Malick; Rankin, Naomi
title: Minimizing disease spread on a quarantined cruise ship: A model of COVID-19 with asymptomatic infections()
date: 2020-08-07
journal: Math Biosci
DOI: 10.1016/j.mbs.2020.108442
sha: 
doc_id: 260214
cord_uid: 2axc1wn3

On February 5 the Japanese government ordered the passengers and crew on the Diamond Princess to start a two week quarantine after a former passenger tested positive for COVID-19. During the quarantine the virus spread rapidly throughout the ship. By February 20, there were 651 cases. We model this quarantine with a SEIR model including asymptomatic infections with differentiated shipboard roles for crew and passengers. The study includes the derivation of the basic reproduction number and simulation studies showing the effect of quarantine with COVID-19 or influenza on the total infection numbers. We show that quarantine on a ship with COVID-19 will lead to significant disease spread if asymptomatic infections are not identified. However, if the majority of the crew and passengers are immune or vaccinated to COVID-19, then quarantine would slow the spread. We also show that a disease similar to influenza, even with a ship with a fully susceptible crew and passengers, could be contained through quarantine measures.

14 Disease outbreaks in closed environments like cruise ships, nursing homes, 15 military barracks, and college dormitories (see for example [4] , [5] , a bigger concern is to stop the spread between 22 people than to find the source. Outbreaks in contained environments full of 23 older individuals, such as cruise ships and nursing homes, create a problem 24 of heightened transmission rates and severe cases, and modeling them can 25 show us the best way to mitigate an outbreak in a closed environment. In this paper we show that quarantine on a ship with COVID-19, even 68 with a majority of crew and passengers having immunity, either acquired or 69 through vaccination, will lead to disease spread. However, we also show that 70 a disease similar to influenza, even with a ship with a fully susceptible crew 71 and passengers, could be contained through quarantine measures. The goal 72 of this paper is to highlight preventive and quarantine measures for a disease 73 with a less pervasive spread that may not lead to an explosion of cases. The discussions and further studies on quarantines in closed systems, especially 76 regarding the role of uninfected care givers. We address the idea of reduc-77 tion of disease spread through a minimization of contact between potentially 78 infected passengers and crew members. Appropriate means to reduce this 79 contact without compromising care will still need to be addressed, but we 80 hope to bring forth discussions of airdropping sufficient personal protective 81 gear or following the lead of the quarantine of exposed sailors in Guam with 82 minimal interactions with self sufficient quarantined people. In addition, we Recently, the outbreak of SARS-COV-2 on cruise ships has been modeled 87

[17], but without the separation of asymptomatic and symptomatic cases. 88 Retrospective reviews have been completed to see the comparison of mild 89 to severe cases on the Diamond Princess, as well as statistical analyses to 90 estimate the number of asymptomatic passengers [18] , [19] . There is also ev-91 idence that asymptomatic people can transmit the disease to others [20] and 92 that 17.8% of the infected Diamond Princess passengers and crew members 93 were asymptomatic and did not develop any symptoms [21] . 94 We highlight the importance of a multi tiered model containing two are susceptible (S), exposed (E), asymptomatic infectious (I A ), symptomatic 104 infectious (I S ), and recovered with temporary or permanent immunity (R).

In the model we can assume that the disease behaves nearly the same 106 in the crew members and passengers, modeled with the shared parameter 107 values, in order to capture the difference in disease spread by shipboard 108 role. Since the average passenger, age 70, being older than the average crew hope to examine isolation and quarantine in a closed system in order to find 114 the conditions to reduce transmission rates. In the model we assume that 115 symptomatic passengers and crew members are confined in their quarters, 116 although with a mean of 1.98 passengers per cabin and 1.73 crew members 117 per cabin [3] they are not truly isolated. The system we examine is not 118 entirely closed, since we allow seriously ill passengers to be evacuated.

In this paper we explore the spread of COVID-19 and an influenza-like 120 illness in relation to different isolation and quarantine levels on a cruise ship. 121 We have chosen to compare COVID-19 to H1N1 since the recent pandemics: Table 1 .

While it is true that not all crew members interact with the same fre- Table 5 in Appendix A has the 176 infection data for the Diamond Princess cruise ship with the results for the 177 passengers and crew members combined [1] for February 5-20, 2020. Table   178 6, also in Appendix A, has the test results for February 4-9, 2020 with the [22] . This indicates that the spread was must likely crew-crew in that 186 cluster. However, we do not know more about the primary contact that Since only interactions with infected individuals spread the disease, we 219 model just these contacts. In Figure 1 we illustrate the interactions with 220 infected individuals with dashed lines and the transmission rate listed by the 221 dashed lines. In Table 2 we define the parameters, where β is the trans-222 mission probability per contact, ξ is the reduction for asymptomatic infec-223 tiousness as compared to symptomatic, and κ XY represents the interaction 224 rate of population X with population Y . Figure 1 shows that the suscepti- In Figure 2 we show the progression of the infection from susceptible S, 231 to exposed E, to infectious, either asymptomatic I A or symptomatic I S , and 232 then recovered, R. The infection rates λ P and λ C combine the disease trans-233 missions shown in Figure 1 . These rates are explicitly defined in the system 234 1. Once the passenger or crew member becomes infected at the rate λ P or 235 λ C , they remain in the exposed category until they are infectious. Once 236 the individual becomes infectious, they move out of the exposed category Notation Definition S P (t) Number of susceptible passengers at time t E P (t) Number of exposed passengers at time t I A P (t)

Number of infected asymptomatic passengers at time t I S P (t)

Number of infected symptomatic passengers at time t R P (t) Number of recovered passengers at time t S C (t)

Number of susceptible crew members at time t E C (t) Number of exposed crew members at time t

Number of infected asymptomatic crew members at time t I S C (t)

Number of infected symptomatic crew members at time t R C (t)

Number of recovered crew members at time t rate γ and develop symptoms at the rate α P or α C . We assume that the 241 average passenger, age 70, being older than the average crew member, age 242 40, has a weaker immune system and develops symptoms faster [22] , [23] . 

The progression of infection for the passengers and crew from susceptible S, to exposed E, to infectious, either asymptomatic I A or symptomatic I S , and then to recovered R. The subscript reflects crew or passenger. The infection rates λ P and λ C combine the disease transmissions shown in Figure 1 . Exposed passengers and crew become infectious at the rate with or without symptoms. Initially asymptomatic individuals may develop symptoms at the rate α C , α P , or recover at the rate γ. Symptomatic individuals either recover at the rate ρ or are evacuated from the ship at a rate µ.

J o u r n a l P r e -p r o o f a simple exponential model for the beginning of the pandemic in each location. Since initially no one had immunity to SARS-CoV-2 we can approximate the total number of susceptible individuals to be the total population when using data from initial infections in Guangzhou, China, South Korea, and New York city. With this assumption one can use the exponential model

to calculate βκ by using data values for I n and I n+1 .

First we need to approximate the average number of contacts per day, κ, 

where x represents the symptomatic transmission probability. Therefore our give a false air, so we have rounded those precise numbers to 2 to define 304 κ P P for after quarantine, κ P S , and κ CS . We estimate that aboard a cruise Once the ship went into quarantine, the passenger-passenger contacts 315 were limited to within the cabin (and balconies) as the passengers were iso-316 lated. News reports [41] indicate that thrice daily meal delivery and hospi-317 tality services were continued so we assume that the passenger-crew contact 318 rate remained unchanged. We assume that the crew was able to minimize 319 contact with each other and reduced the crew-crew contact rate from 5 to 320 3. We also assume that the crew-seriously ill individuals contact rate, κ 

Journal Pre-proof [33] and dividing by 3 interactions daily. Table 3 has the values for µ before and after quarantine.

Influenza has a daily hospitalization and death rate of 0.005% = 1/µ [32], 345 [33] .

The baseline parameter values that we use in our analysis of system 1 are 347 given in Table 4 . The ranges reflect either a variation in data values or the 348 uncertainty in the true values for COVID-19. begins an average of 2 days before symptoms and thus calculate an exposed 355 and noninfectious period of 1/ = 3.1 days for COVID-19. Influenza has a 356 shorter exposed and not yet infectious period on the average of 1.9 days [33] . 

The motivation for developing mathematical models of infection in a By using sensitivity analysis, we estimated which parameters had the most impact in affecting the value of our R 0 . We use our results to determine the importance of each parameters in achieving disease free equilibrium on the ship. In our case, we use sensitivity analysis to point out the highest impact on the transmission of COVID-19 in which could be used to help determine strategies for preventative and quarantine measures. Sensitivity indices measure the percentage change of a key quantity, such as the reproduction number, in response to a percentage change of a parameter in that quantity. The normalized forward sensitivity index of R 0 relative to a differentiable parameter p is defined as follows [47] :

The value of the normalized forward sensitivity index determines whether Table 4 and pre-quarantine values in Table 3 . Figure 3 (a) the most sensitive parameters for the reproduc-421 tion number for influenza (H1N1) before quarantine are β, ξ, ρ, and κ CS .

Secondary sensitivity belongs to σ, µ, and κ P S . While β is fixed for each dis-423 ease, measures can be taken to reduce R 0 for each of the other parameters.

When ρ is increased, R 0 decreases. This means that decreasing the recovery Table 4 and quarantine values in Table 3 .

values after quarantine was imposed.

Once quarantine begins, see Figure 4 rate of passenger-passenger interactions, κ P P , once quarantine is in effect.

The key to reducing R e by parameters currently under our control is in- The heat maps in Figure 5 show the dependence of the effective repro- Princess cruise ship. In Figure 6 neither the crew members nor passengers 525 have any immunity to the disease. Figure 6 (a-c) demonstrate the effects of 

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