Modeling Travel and Infectious Diseases
As the school year draws to a close, most families start thinking about what to do over
the summer  and for many people this means travel. It could be as simple as a roadtrip
to see family, or as extravagant as flying to Europe for a month, but the one thing all
travel scenarios have in common, from a modeling perspective, is the sudden expansion of
contact matrices.
A contact matrix can be defined in many ways, but one direct interpretation for
its contents is the probability of at least one effective contact between a susceptible
and an infected (and infectious) individual per unit time. Breaking that definition
down:
"At least one"  Which means there could be more than one effective contact in
that period of time, but no less. Increasing the number of infective contacts over a
single unit of time may increase the probability of infection above this baseline level.
"Effective contact"  A contact is considered effective if it is a sufficient
interaction such that infectious material can be transferred from one individual to
another. This is, of course, highly dependent upon the disease under consideration: a
sneeze from three feet away won’t transmit cholera, but it could transmit chickenpox.
(See issue #2 for a more indepth discussion of infectious disease contacts.)
"Susceptible"  A susceptible individual is able to become infected, meaning they
lack full protection from vaccination or previous infection, or any other means of
protection against disease transmission. For example, an individual wearing an
effective medical face mask may be less susceptible to transmission of certain
respiratory viruses, and an individual wearing long sleeves, long pants, and insect
repellant is less susceptible to transmission of vectorborne viruses.
"Infected (and infectious)"  Not all individuals who are infected with a virus
are able to transmit it to other individuals. In many mathematical models, those who are
infected but not yet infectious are sometimes defined as "exposed" or
"incubating", with the appearance of clinical symptoms and the ability to transmit the
virus marking the start of the "infected" state. In order for an effective contact to
occur, one of the individuals must be infectious.
"Per unit time"  The selection of the size of the unit of time can have a
dramatic impact on the realism of a mathematical model for the transmission of
infectious disease, and must be based on the epidemiological dynamics of the disease
itself. For example, if the infectious period for measles lasts two weeks and a model
selects a time unit of one month, many epidemiological events could be happening within
that single time unit that would be obscured in the model  multiple new infections and
recoveries that might never be picked up by the mathematics. On the other hand,
selecting a time unit that is too small (for example, setting the unit of time to a
single hour in a measles model), while allowing for a great deal of realism in terms of
tracking individual behaviors throughout the day, may not greatly enhance the
predictive ability of the model  and may instead make running computer simulations so
timeconsuming and intense that results are unobtainable.
The infectious disease contact matrix is best employed by mathematical models
with some kind of structure to the population  age structure, geographical structure,
structure based on jobduties, etc.  so that the elements of the matrix refer to
interactions either within a given subgroup, or between two different subgroups, per
unit time. When travel events suddenly increase  as with summer holidays, for example 
normal contact rates may change abruptly with different subpopulations interacting at
different rates, or with contacts occurring with outside groups not normally included in
the modeled population.
A travelassociated shift in the contact matrix can easily upset what was
previously a stable system, suddenly increasing transmission rates of an already
circulating disease, altering normal seasonal transmission patterns, or introducing
brandnew diseases and mutations. In the case of measles, the Americas eliminated local
measles transmission within the region soon after the start of the new millennium, and
the last endemic case of measles in the Americas was reported in 2002 (see
PAHO.org for more info). However, every year measles cases are imported into at least
one country in the region from elsewhere via global travel, occasionally sparking
additional transmission among small local populations with lessthanoptimal vaccination
coverage. Cases of infectious disease introduced by travelers can be seen throughout
history and throughout the world, and are particularly challenging for countries with
large migrant populations and those focusing on disease elimination within their
regions.
Modeling the impact of global travel on contact matrices is a significant
challenge, however. While air travel data can be obtained which describes commercial
airline traffic between given regions, this represents only a small fraction of the
human movements occurring on a daily basis  many of which are onfoot and unmonitored.
On a smaller geographic scale, commuter surveys can provide data on the frequency of
travel between given metropolitan areas, which can help inform contact parameters for
infectious disease models. But quantifying more stochastic travel events like vacations
(or even evacuations) requires a different approach. In many cases the best option is to
introduce a random parameter drawn from given distributions to represent the probability
of an infectious individual entering the population from outside the system, and then to
perform thorough sensitivity and uncertainty analyses on this parameter to explore its
overall impact on the system and its outcomes.
If your organization is interested in examining the impact of travel on
infectious disease transmission, contact MathEcology today to learn more about how
mathematical modeling can help!
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