Minimizing disease spread on a quarantined cruise ship: A model of COVID-19 with asymptomatic infections

Abstract

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.

Keywords: Asymptomatic; COVID-19; Cruise ship; Flu; Isolation; Quarantine; Reproduction number; SEIR; Sensitivity.

Fig. 1

Disease transmission routes for susceptible passengers and crew. Dashed lines indicate infection routes. Passengers are infected by asymptomatic crew, asymptomatic passengers, and symptomatic passengers through a shared cabin. Since the crew interacts with all people on the ship, they can get the disease from all infected individuals.

Fig. 2

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 ${\lambda }_P$ and ${\lambda }_C$ combine the disease transmissions shown in Fig. 1. Exposed passengers and crew become infectious at the rate $\ell$ with or without symptoms. Initially asymptomatic individuals may develop symptoms at the rate ${\alpha }_C$, ${\alpha }_P$, or recover at the rate $\gamma$. Symptomatic individuals either recover at the rate $\rho$ or are evacuated from the ship at a rate $\mu$.

Fig. 3

Sensitivity Analysis for reproduction number for (a) influenza and (b) COVID-19 evaluated at the baseline parameter values in Table 4 and pre-quarantine values in Table 3.

Fig. 4

Sensitivity Analysis for the effective reproduction number for (a) influenza and (b) COVID-19 evaluated at the baseline parameter values in Table 4 and quarantine values in Table 3.

Fig. 5

This figure illustrates ${\mathcal{R}}_e$ values as a function of $\beta$ and $d$. Figure (a) uses influenza parameter values and Figure (b) uses COVID-19 parameter values as listed in Table 3, Table 4. When $d$ = 0, all non-symptomatic passengers are free to move about the ship. When $d$ = 1, all the passengers are restricted to their quarters. The density shown is ${\mathcal{R}}_e$. No immunity is assumed to influenza or COVID-19.

Fig. 6

Comparison of influenza and COVID-19 infections on the ship without immunity: Figures (a–c) are for an infection matching the flu, but without any immunity on the part of the crew or passengers. Figures (d–f) are for COVID-19 infections. Figures (a) and (d) show the total number of infected crew members and passengers. Figures (b) and (e) show the rate of evacuation of individuals with serious infections. Figures (c) and (f) capture the infection type of the all the individuals remaining on the ship.

Fig. 7

Comparison of influenza and COVID-19 infections on the ship with immunity: 70% of passengers are vaccinated/immune and 50% of the crew are vaccinated/immune. Figures (a–c) are for an infection matching the flu with immunity. Figures (d–f) are for COVID-19 infections with immunity. Figures (a) and (d) show the total number of infected crew members and passengers. Figures (b) and (e) show the rate of evacuation of individuals with serious infections. Figures (c) and (f) capture the infection type of the all the individuals remaining on the ship.