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 road-trip 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 in-depth 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 vector-borne 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 time-consuming 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 job-duties, 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 travel-associated 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 brand-new 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 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 less-than-optimal 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 on-foot and un-monitored. 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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