Norovirus outbreaks on cruise ships

What makes a shipboard norovirus outbreak large? Beta-binomial models of 63 outbreaks reported to the CDC Vessel Sanitation Program, 2014–2019, fit in base R.

Robert M. ShollKansas State University

Summary

Each row is a norovirus outbreak reported to the CDC Vessel Sanitation Program (VSP) from 2014 to 2019: the people who fell ill, everyone on board, the itinerary and the dates. What predicts how large an outbreak gets, measured as the share of passengers and crew who fell ill (the attack rate)?

No voyage characteristic in these data reliably predicts outbreak size. Ship size, the obvious candidate, shows no association: each doubling of people on board multiplies the odds of illness by 1.03 (95% CI 0.89–1.19). The full model (size, duration, season and region) has the lowest AIC, but it predicts left-out voyages worse than the intercept-only model (leave-one-out log score -348.3 vs -342.3). Its apparent effects lean heavily on two extreme outbreaks. Season matters for how often outbreaks happen (December–April), not for how large they get.

Every voyage in these data had a reported outbreak, because VSP only lists voyages where at least 3% of passengers or crew reported illness. The models describe outbreak size, given that an outbreak was reported. They can’t estimate the probability that a voyage has an outbreak.

Data

The data join six yearly VSP outbreak tables to a hand-built table of departure ports. Before modelling:

Season comes from the start month: Dec–Feb winter, Mar–May spring, Jun–Aug summer, Sep–Nov fall.
Cruise line names are unified ("Royal Caribbean", "Royal Caribbean Cruise Line" and "Royal Caribbean International" are one line).
Departure ports are grouped into six regions rather than entered as latitude and longitude.
Two itineraries look implausible for their length: Conquest, Sep 2019 from Callao-Lima in 7 days; Oasis of the Seas, Jan 2019 from Los Angeles in 7 days. A sensitivity check refits the full model without them.

What drives outbreak size

Figure 1. Attack rate against people on board for 63 reported outbreaks. The line is the beta-binomial size model with its 95% confidence band; it is flat. Two outbreaks stand out: Explorer of the Seas (2014, 689 ill) and Balmoral (2016, a 34-day voyage from Southampton).

Figure 2. Left: reported outbreaks by start month, December–April highlighted. Right: attack rate by season; ochre marks the model estimate for a median-sized ship with its 95% CI. Frequency is seasonal; size is not.

Figure 3. Attack rate against voyage length and departure region. The long-voyage and Europe & Asia signals in the full model come mostly from Balmoral, the only Southampton departure and the longest voyage.

Model comparison

A binomial model treats each person on board as an independent coin flip. Its Pearson dispersion here is 30, meaning voyages vary about 30 times more than independence allows: people on one ship share food, water, cabins and crew. A beta-binomial model adds one parameter, ρ, the correlation in illness between two people on the same voyage. That corrects the standard errors, which a binomial fit understates by a factor of about √30.

Six beta-binomial models, from intercept-only to the full model. pp = percentage points of attack rate. LOOCV refits each model 63 times, leaving out one voyage each time. Lower AIC and RMSE are better; a higher log score is better.
Model	Parameters	AIC	ΔAIC	LOOCV RMSE (pp)	LOOCV MAE (pp)	LOOCV log score	rho
Intercept only	2	674	6.58	2.88	1.62	-342	0.0105
Size	3	676	8.41	2.92	1.65	-344	0.0105
Size + duration	4	672	5.03	2.95	1.71	-345	0.00958
Size + season	6	678	11.0	2.91	1.71	-344	0.00990
Size + region	8	678	10.8	2.96	1.77	-347	0.00921
Size + duration + season + region	12	667	0	2.79	1.77	-348	0.00663

AIC prefers the full model (ΔAIC 6.6 over intercept-only). But AIC measures in-sample fit with a fixed penalty per parameter; leave-one-out cross-validation scores each voyage with a model that never saw it. By that test the full model is the worst predictor by log score (-348.3), and intercept-only is the best (-342.3). Its slightly lower RMSE (2.79 vs 2.88 pp) comes from partly anticipating the two extreme outbreaks, while its MAE is worse. With 63 outbreaks and 12 parameters, the full model is fitting individual voyages.

Full-model coefficients

Full beta-binomial model (ρ = 0.0066). Odds ratios are per doubling of people on board and per week of voyage; reference levels are winter and Florida, Gulf & Caribbean.
Term	Odds ratio	95% CI low	95% CI high	z	p value
(Intercept)	0.0151	0.00257	0.0887	-4.64	0.00000343
Size (per doubling)	1.07	0.922	1.24	0.887	0.375
Duration (per week)	1.21	1.05	1.39	2.65	0.00814
Season: Spring	0.989	0.774	1.26	-0.0865	0.931
Season: Summer	0.596	0.299	1.19	-1.46	0.143
Season: Fall	0.548	0.392	0.764	-3.54	0.000402
Region: US Northeast, Canada & Great Lakes	1.37	1.00	1.88	1.97	0.0492
Region: US West Coast	1.15	0.860	1.54	0.938	0.348
Region: Alaska & Pacific Northwest	1.60	0.766	3.33	1.25	0.211
Region: South & Central America	0.925	0.642	1.33	-0.415	0.678
Region: Europe & Asia	2.34	1.52	3.60	3.86	0.000114

Without the two largest outbreaks

Full-model odds ratios (95% CI) refit without the two implausible itineraries, and without the two largest outbreaks (Explorer of the Seas 2014, Balmoral 2016).
Term	All 63 voyages	Without 2 flagged itineraries	Without 2 largest outbreaks
Size (per doubling)	1.07 (0.92–1.24)	1.04 (0.89–1.22)	1.04 (0.92–1.17)
Duration (per week)	1.21 (1.05–1.39)	1.21 (1.05–1.40)	1.04 (0.91–1.19)
Season: Spring	0.99 (0.77–1.26)	1.00 (0.78–1.29)	1.05 (0.86–1.28)
Season: Summer	0.60 (0.30–1.19)	0.64 (0.32–1.27)	0.99 (0.54–1.83)
Season: Fall	0.55 (0.39–0.76)	0.58 (0.41–0.82)	0.78 (0.59–1.02)
Region: US Northeast, Canada & Great Lakes	1.37 (1.00–1.88)	1.37 (1.00–1.89)	1.06 (0.81–1.38)
Region: US West Coast	1.15 (0.86–1.54)	1.09 (0.80–1.47)	1.17 (0.93–1.47)
Region: Alaska & Pacific Northwest	1.60 (0.77–3.33)	1.49 (0.72–3.07)	0.91 (0.47–1.75)
Region: South & Central America	0.93 (0.64–1.33)	0.94 (0.64–1.38)	0.97 (0.72–1.29)
Region: Europe & Asia	2.34 (1.52–3.60)	2.23 (1.45–3.44)	1.47 (0.97–2.24)

The Europe & Asia and voyage-length effects shrink sharply without the two largest outbreaks; Balmoral alone is the only Southampton departure and the longest voyage. Effects that depend on one or two voyages are not a basis for policy. The defensible conclusion is that attack rates cluster tightly around 4% regardless of ship size, region or season, with occasional extreme outbreaks that these variables do not anticipate.