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On the Treatment of Airline Travelers in Mathematical

Models

Michael A. Johansson1*, Neysarı ´ Arana-Vizcarrondo1, Brad J. Biggerstaff2, J. Erin Staples2, Nancy

Gallagher3, Nina Marano3

1Division of Vector-Borne Diseases, Centers for Disease Control and Prevention, San Juan, Puerto Rico, United States of America, 2Division of Vector-Borne Diseases,

Centers for Disease Control and Prevention, Fort Collins, Colorado, United States of America, 3Division of Global Migration and Quarantine, Centers for Disease Control

and Prevention, Atlanta, Georgia, United States of America

Abstract

The global spread of infectious diseases is facilitated by the ability of infected humans to travel thousands of miles in short

time spans, rapidly transporting pathogens to distant locations. Mathematical models of the actual and potential spread of

specific pathogens can assist public health planning in the case of such an event. Models should generally be parsimonious,

but must consider all potentially important components of the system to the greatest extent possible. We demonstrate and

discuss important assumptions relative to the parameterization and structural treatment of airline travel in mathematical

models. Among other findings, we show that the most common structural treatment of travelers leads to underestimation

of the speed of spread and that connecting travel is critical to a realistic spread pattern. Models involving travelers can be

improved significantly by relatively simple structural changes but also may require further attention to details of

parameterization.

Citation: Johansson MA, Arana-Vizcarrondo N, Biggerstaff BJ, Staples JE, Gallagher N, et al. (2011) On the Treatment of Airline Travelers in Mathematical

Models. PLoS ONE 6(7): e22151. doi:10.1371/journal.pone.0022151

Editor: Michael George Roberts, Massey University, New Zealand

Received November 12, 2010; Accepted June 19, 2011; Published July 25, 2011

This is an open-access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for

any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

Funding: These authors have no support or funding to report.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: mjohansson@cdc.gov

Introduction

The global airline network has brought the entire world closer

together than ever before, creating an environment in which

pathogens can readily spread to distant locations. Influenza viruses

are perhaps the most widely recognized example of this

phenomena [1,2], but by no means the only one [3,4,5]. Study

of past spread and discussion about potential mitigation efforts has

lead to a significant body of literature investigating the use of

mathematical models to predict global pathogen spread and to

assess the potential effectiveness of various interventions [3,6,

7,8,9,10,11,12,13].

These models are generally framed as metapopulation models,

with spatially discrete populations connected by transportation

networks. The discrete populations themselves are subdivided

in terms of infection status with compartments for susceptible,

incubating, infectious, and recovered individuals. Development of

these models presents many challenges related to the cha-

racterization of the populations, the disease, and relevant travel

patterns. We focus here on oft-neglected assumptions related to

the characterization of travel which may affect the speed and

pattern of epidemic growth and spread.

One fundamental assumption in many models is that travelers

are, in essence, migrants; people who move permanently or semi-

permanently from one geographic population to another [3,6,

7,8,9,10,11,12,13]. In the real world, some travelers are migrating,

but for most, travel is temporary. For example, each year there are

approximately 175 million total admissions to the United States

and approximately 1.1 million people obtain legal permanent

resident status [14]. Thus, less than 1% of travelers entering the

U.S. are migrating. An alternative to adopting this approach is to

segregate travelers from local residents structurally, maintaining

them as separate groups inhabiting the same area as a resident

population (Figure 1). Separating travelers from residents this way

allows them to return to their place of origin after a temporary stay

in their destination.

It is also generally assumed that exposure occurs via mass

action, i.e., that all susceptible individuals in a given location are

exposed with the same probability to the infectious agent. Even

without considering travel, however, there is significant stratifica-

tion within populations and people do not all interact equally [15].

For travelers, exposure is often related to their purpose for travel.

The majority of outbound US traveler purposes can be classified

as leisure/recreation/holidays (40%), visiting friends and relatives

(34%), or business (18%) [16]. Those visiting friends and relatives

are more likely to stay in private homes, to stay longer, to interact

more closely with the local population, and to have higher

exposure risks [16,17,18,19,20,21].

Infection with a pathogen may also have a direct impact on

travel. Most models assume that infectious travelers are ill and thus

are unlikely to travel [6,7,8,9,12]. However, infectious people are

not necessarily ill and even if they are, they may travel anyway if

they can.

Characterization of a travel network itself is another challenge.

Many models quantify travel volumes or frequencies in terms of

the number of seats available on direct flights between each loca-

tion and average occupancy on those flights [6,7,8,9,10]. While

this may accurately classify non-connecting travel (,80% of travel

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itineraries based on U.S Department of Transportation sampled

itinerary data), it completely misses travelers who fly on itineraries

containing one or more connections. As a result, this simplification

underestimates long-distance travel, because no one can fly

between cities that are not directly connected, and overestimates

short-distance travel, because it assumes that all travelers are

traveling direct. Limiting travel in this way may lead to alteration

and truncation of the pattern of spread, especially when trans-

mission risk is heterogeneous across the network.

We used mathematical models of two generalized pathogens,

one directly transmitted (modeled on severe acute respiratory

syndrome-associated coronavirus (SARS-CoV)) and one vector-

borne (modeled on dengue viruses (DENV)) to evaluate the effects

of different structural model designs and characterizations of

airline travel on the speed and pattern of epidemic spread. In

addition to evaluating directly transmitted pathogens, we also

include a vector-borne pathogen because the dependence on a

vector increases the geographical heterogeneity in transmission

dynamics and may alter the effects of travel-related assumptions

for a pathogen.

Methods

Model description

Two base models were constructed, one for the directly

transmitted pathogen and one for the vector-borne pathogen.

Described in detail in Text S1, both models are stochastic

metapopulation models with discrete daily time steps. Each model

includes 141 cities selected to represent a realistic sample network

with varied connectivity and population sizes. Travel and infection

are assumed to be independent. Travel rates and stay durations

were estimated using sampled travel data (Official Airline Guide

(OAG), www.oagaviation.com/Solutions/AnalysisTools/Traffic/

t100inet.html and US Department of Transportation, www.

transtats.bts.gov/Tables.asp?DB_ID=125). Each city in the model

had susceptible, incubating, infectious, and recovered/immune

compartments, with separate compartments for residents and

travelers. Though maintained separately, residents and travelers

contribute equally to the local force of infection and are exposed to

infection at the same rate. The incubating and infectious com-

partments all included sub-compartments to allow more realistic

incorporation of the time periods spent in each [22]. For models of

the directly transmitted pathogen, the rates of effective contact, time

to progression to the infectious compartment, and time to recovery

are based on previous studies [23,24]. In contrast to previous

efforts, our model has been simplified to specifically address the

influence of travel in the absence of any interventions.

To introduce straightforward geographic variability in the

models of the vector-borne pathogen, we randomly selected 10

cities to have Aedes aegypti DENV-vector mosquito populations.

Each of these cities was seeded with two susceptible mosquitoes

per human. Upon effective exposure to infectious humans, vectors

moved through incubating sub-compartments before becoming

infectious at rates estimated based on previous models [25].

Vectors from all compartments were susceptible to nominal

mortality (defined globally based on [26]) and were replaced by

new susceptible vectors at the same rate. Human infection is

dependent on estimated effective contact rates [27,28] and the

prevalence of infectious vectors.

Model alterations to assess specific assumptions.

To test the effect of treating travelers as migrants rather than as

temporary travelers, we compared the base, ‘‘traveler’’ models

with ‘‘migration’’ models in which we eliminated the traveler

compartments and assumed that all travelers immediately mixed

with local populations upon arrival in a new city.

To test the role of heterogeneous travel patterns, we divided

travelers into two compartments based on approximations of the

duration of stay for those residing in hotels versus private homes.

In the base models, travelers stay an average of 18 days in their

destination (exponentially distributed) [16]. In the modified

models, we assume that approximately 60% of travelers stay in

hotels [16]. These travelers are also less likely to stay as long, so we

modified the length of stay to average 14 days for those staying in

hotels, and 24 days for those staying in homes. We therefore

maintained the average length of stay of approximately 18 days

used in the base models. We then prevented transmission

Figure 1. Alternative travel model structures. Two different model

structures are shown using a simplified, two-city, Susceptible (S)-Exposed

(E)-Infected (I)-Recovered (R) model. In the migration model, travelers

representmigrants,mixingthepopulationsofthetwocities,CityAandB.

Inthetravelermodel,someresidents from CityA (blue)andCityB (green)

travel temporarily to the other city, but eventually return to their original

city. In each model, individuals may progress through the infection

stages (red arrows), the rates of which may be city-dependent.

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associated with hotel or home stays alternatively by eliminating

interactions for the group of interest so they could neither

contribute to the force of infection nor be exposed.

To test the importance of infectious travelers, we compared the

base models, which allow individuals to travel regardless of infection

status, with models in which infectious individuals do not travel.

We also assessed three different travel network parameteriza-

tions: seat-based direct-travel only, connecting-travel-inclusive,

and a skewed version of the connecting-travel. Connecting travel

was estimated using a regression model of sampled itinerary data

and covariates based on travel network characteristics (described

in detail in Text S1). For the models with skewed travel, we used

the same overall passenger flow as the connecting-travel-inclusive

models but changed directionality so that 2/3 of travelers between

tropical and non-tropical cities are residents of non-tropical cities,

leaving 1/3 as residents of the tropical cities.

Model comparison.

To analyze the effect of different assumptions, we ran simu-

lations of each model under the different sets of assumptions. In

each simulation, it was assumed that the global population was

100% susceptible and 10 infectious individuals were introduced

into a single city. The origin city was selected from the subset of

cities containing vector populations and was used for simulated

epidemics of both the directly transmitted and the vector-borne

pathogens. The same city was used as the origin city for the

directly transmitted model.

Different models were compared by performing 100 simulations

of each model and comparing quantitative outcomes via regression

analysis. The timing of the first autochthonous transmission event

in each city and the final epidemic size in each city were compared

using a linear Gaussian regression with city as a covariate to

control for intra-city correlation across simulations. Intra-simula-

tion correlation was not found, i.e., earlier or later introduction

was related to the city and the model, but not to the simulation.

Statistical analyses were performed in R version 2.11.1 [29].

Results

Temporary travel versus migration.

We contrasted two different model designs, one treating all

travelers as migrants (migration models) and one treating them as

temporary travelers (traveler models, the base models). For the

directly transmitted pathogen, the pathogen introduction caused a

pandemic. Figure 2a shows representative cumulative distributions

of the time of the first autochthonous human infection in two

cities. Controlling for the inter-city variation, the average onset of

autochthonous transmission in the traveler model was 3.3 days

earlier (95% CI: 3.0–3.6 days) than in the migration model. For

the vector-borne pathogen, this difference was more pronounced

(Figure 2b). In the nine cities with vector populations, introduction

occurred an average of 6.7 days earlier (95% CI: 5.0–8.3 days) in

the traveler model than the migration model.

The final epidemic size was slightly decreased in the migration

model (0.079%, 95% CI: 0.088–0.069%) compared to the traveler

model for the directly transmitted pathogen. However, long-term

dynamics were generally consistent with the pandemic ending in

global extinction of the pathogen within 1.5 years. In contrast,

there were long-term differences in the models of the vector-borne

pathogen. In the traveler model, initial epidemics were followed by

global extinction of the pathogen within two years (Figure 3A). In

contrast, the migration model allowed continued mixing of the

population leading to susceptible replenishment in affected cities.

Once the initial outbreak in the initial city subsided, 18.0%

(95%CI: 17.6–18.3%) of that population was replaced by im-

migrating susceptibles within one year. Because of reintroduction

of the pathogen or residual transmission from the previous epide-

mic, this replenishment led to recurrent outbreaks (Figure 3B).

The continual migration also slowly homogenizes all model

populations such that, after many years, a global equilibrium is

reached. In comparison, the traveler model results in a stable,

geographically heterogeneous distribution as soon as extinction of

the pathogen occurs.

Heterogeneity among travelers.

We designed additional traveler models (heterogeneous-traveler

models), which had different periods of stay for travelers staying in

hotels compared to travelers staying in homes. We investigated the

effects of the traveler heterogeneity in the model with the directly

transmitted pathogen by blocking the interaction of travelers

staying in hotels with the rest of the population (i.e. residents or

travelers staying in homes). This delayed spread to other cities by

an average of 9.2 days (95% CI: 8.9–9.5 days) and reduced

ultimate epidemic size by 0.36% (95% CI: 0.34–0.37%). Blocking

Figure 2. Timing of first autochthonous human infection: migration vs. traveler model. Each line is the empirical cumulative probability

(for 100 simulations) of the first autochthonous transmission event in a single city in a single model. The dotted lines are for a city closely connected

to the origin city for the traveler (black) and migration (red) models. The solid lines are for a more distal city. A. is the directly transmitted pathogen

and B. is the vector-borne pathogen.

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transmission associated only with travelers staying in homes

delayed spread by 5.2 days (95% CI: 4.9–5.5 days) and decreased

final epidemic size by 0.12% (95% CI: 0.11–0.13%).

For the heterogeneous-traveler model with the vector-borne

pathogen, blocking the interaction of travelers staying in hotels

with mosquitoes reduced the speed of spread by an average of 11

days (95% CI: 9–13 days) and decreased the ultimate size of the

epidemics by 0.40% (95% CI: 0.39–0.41%). In contrast, blocking

traveler-associated transmission in homes slowed spread by 5.5

days (95% CI: 3.8–7.3 days) and decreased final epidemic size by

0.36% (95% CI: 0.35–0.37%).

Infectious travelers.

In the base model, infectious individuals travel with the same

frequency as any other individual. We constructed additional

models (no-infectious-travel models), in which infectious individ-

uals did not travel, i.e., infectious individuals neither leave their

city of residence nor do they return home from another city until

they have recovered. Eliminating infectious travel resulted in

spread to other locations occurring approximately 3.8 days (95%

CI: 2.6–5.0 days) and 3.9 days (95% CI: 2.2–5.5 days) later in the

models with directly transmitted and vector-borne pathogens,

respectively. In the no-infectious-travel models, the final epidemic

size was slightly decreased (mean difference: 20.018%, 95% CI:

20.028–20.007%) for the directly transmitted pathogen, and

was unchanged (mean difference: 0.0007%, 95% CI: 20.0006–

0.0020%) for the vector-borne pathogen.

Travel network characterization.

We first assessed the role of different assumptions regarding the

structure of the airline travel network using the base models (with

connecting travel) compared to models with travel restricted to

single-leg, direct flights (direct-travel-only models). Overall, for the

directly transmitted pathogen, the onset of autochthonous

transmission in secondary cities in the direct-travel model occurred

on average 10.5 days (95% CI: 10.1–11.0 days) later (Figure 4).

Although spread was generally slower, this was not universally the

case. For 42 (30%) of the 141 cities, spread was faster (Figure 4A),

for 90 cities (64%), slower (Figures 4B and 4C), and for 9 cities

(6%), there was no significant difference. Average epidemic size

was unchanged (95% CI: 20.040–0.008%).

Models of the vector-borne pathogen had significant geograph-

ical heterogeneity and showed more drastic effects when travel

characteristics were modified (Figure 5). Under the direct-travel-

only model, spread to the nine cities with vector populations was

delayed by an average of 188 days (95% CI: 181–196 days).

However, there was significant variation between cities with

average introduction occurring as early as 48 days (95% CI: 33–62

Figure 3. Epidemic recurrence in the city where the epidemic originates. A. The mean proportion of the population newly infected per day

for the traveler (black) and migration (red) models. The pathogen only persists in the migration model and causes a second outbreak approximately

two years after the first. B. The mean proportion of the population susceptible (black), incubating and infectious (red), and recovered (blue) in the

migration model over time. Approximately 20% of the population is replaced by incoming susceptibles between the two epidemics.

doi:10.1371/journal.pone.0022151.g003

Figure 4. Timing of first autochthonous human infection: direct- vs. connecting-travel network model with the directly transmitted

pathogen. Each line is the empirical cumulative probability (for 100 simulations) of the first autochthonous transmission for the connecting-travel

model (black) and direct-travel model (red) for 3 cities: A. directly connected to the origin city; B. non-directly connected, intermediate distance city;

and C. a distant city.

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days) earlier for the only directly connected city (Figure 5A), to 330

days (95% CI: 317–345 days) later for a distant city. For one city,

introduction was sporadic, occurring in only 40 of the 100

simulations (Figure 5C). Compared to the connecting-travel

model, the epidemics in the direct-travel model affected 1.9%

(95% CI: 1.6–2.1%) more people.

The connecting-travel network was then modified to skew traffic

such that more of the travelers would be residents of non-tropical

areas visiting tropical areas than vice versa. Spread occurred

approximately 1.6 days (95% CI: 1.3–1.9 days) earlier in the

skewed-travel model for the directly transmitted pathogen. In non-

tropical cities, epidemic size did not change (95% CI: 20.006–

0.015%), but in tropical cities, average epidemic size increased by

0.57% (95% CI: 0.52–0.62%). The skewed-travel model of the

vector-borne pathogen behaved similarly with a non-significant

decrease in the speed of spread (95% CI: 20.3–3.0 days).

Epidemic size decreased in the non-tropical cities by 0.040% (95%

CI: 0.028–0.052%) and increased in the tropical cities by 0.26%

(0.23–0.29%).

Discussion

The global connectivity of air travel presents a significant public

health issue as infected travelers may move pathogens long

distances in short time periods. We explored various assumptions

and considerations related to the incorporation of travel in models,

including the structural treatment of travelers, the role of infectious

travelers, the interaction of travelers with local communities, and

the characterization of the travel network. Analyzed outcomes

included both local epidemic characteristics (epidemic size) and

global characteristics (the speed of spread).

Model outcomes.

Most of the assumptions we tested resulted in relatively small

modifications to local transmission, affecting the size of epidemics

by less than one percent. For example, introducing heterogeneity

among travelers reduced the average epidemic size in all models

by a maximum of approximately 0.40%. These reductions occur

because a proportion of travelers are never exposed, thus reducing

the effective susceptible population size. A greater change in local

epidemic size was seen in the direct-travel-only model for the

vector-borne pathogen. The direct-travel-only model assumes that

every seat is potentially used by a direct traveler as opposed to a

seat potentially used by a connecting traveler. This overestimation

of direct travel increases the overall volume of travel and thus leads

to greater local replenishment of susceptibles, a phenomenon

which takes on added importance when transmission is spatially

heterogeneous. Though the mechanism is different, the effect of

susceptible replenishment is also seen in the migration model. In

this case, the replenishment of approximately 18% of the popu-

lation within a single year led to recurrent epidemics, something

completely absent from the traveler model.

Another critical outcome of interest from these models is the

speed of spread across the network. Treating travelers as migrants

reduced the speed of spread to other areas by approximately 3.3

and 6.7 days in models of the directly transmitted and vector-

borne pathogens, respectively. This difference is likely due to

proportional changes in the travelling population. Once an epide-

mic begins in a given city, travelers leaving that city will contain a

mix of previously exposed (i.e. immune) individuals proportional

to the general population. In the traveler model, this proportion is

reduced by half because only half of the travelers are outgoing

residents; the other half are visitors returning to their home.

Because there are more susceptible travelers leaving and returning

to cities where the pandemic has yet to arrive, the likelihood of a

newly exposed individual arriving is slightly increased in the

traveler model.

Preventing infectious individuals from travelling also led to

delayed introduction by an average of 3.8 and 3.9 days for the

models of the directly transmitted and vector-borne pathogens,

respectively. Adding heterogeneity to exposure among travelers

eliminates some potential infections among travelers and thus has

the same effect.

Using different travel-network designs had an even greater effect

on spread. Assuming that no travelers travel beyond the cities

directly connected to their residential city delayed spread of the

directly transmitted pathogen by an average of 10.5 days and the

vector-borne pathogen by an average of 188 days. For both, that

effect was spatially heterogeneous. The cities with high connec-

tivity to the origin city experienced early introduction and distant

cities, delayed introduction. For the vector-borne pathogen, the

effects are more evident because cities with vector populations are

not necessarily directly connected to each other. Of the ten

randomly selected cities with vectors, a single city was directly

connected to the origin city and four cities were not directly

connected to any other city with a vector population. For these

isolated cities, the pathogen could only arrive via an individual

who is exposed in one city, returns to his/her vector-free home city

and later travels to another city before recovering. With the large,

intervention-free epidemics produced in our model, spread to

these isolated cities did reliably occur except for in the city where

an epidemic only occurred in 40% of the simulations (Figure 5C).

Figure 5. Timing of first autochthonous human infection: direct- vs. connecting-travel network model with the vector-borne

pathogen. Each line is the empirical cumulative probability (for 100 simulations) of the first autochthonous transmission for the connecting-travel

model (black) and direct-travel model (red) for 3 cities: A. directly connected to the origin city; B. non-directly connected, intermediate distance city;

and C. a distant city.

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