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# Model Indicates Travel Frequency Affects Disease Spread

According to empirical and statistical data the number of trips that individuals take each year is strongly dependent on a host of determining factors, including age, gender, income, occupation, ethnicity, and home region. While a small fraction of the overall population travels frequently, the vast majority of people are infrequent or non-travelers. Mathematical models allow researchers to characterize, predict, and control spatial and temporal spread of infectious diseases.

In an article publishing today in SIAM Journal on Applied Mathematics, Daozhou Gao, an applied mathematician at Shanghai Normal University, proposes a multipatch epidemic model that separates the humans in each patch into susceptible unfrequent, infectious unfrequent, susceptible frequent, and infectious frequent classes. The isolated dynamics of each individual patch are described by a simple two-group susceptible-infectious-susceptible (SIS) model incorporating both frequent and infrequent travelers.

Gao’s work is a step toward a better comprehension of the influence of heterogeneity in travel frequency on spatial and temporal spread of infectious diseases.
Although multiple previous models have studied how traveler’s relation to and impact on infectious  disease spread, these models assume that individuals with the same disease status also share the same rate of travel. However, travel frequency varies from person to person – based on the aforementioned factors – and thus cannot be generalized in this way. Additionally, the expansion of public transportation, possession of a driver’s license and/or private vehicle, industry structure, and even climate conditions can all impact the average travel frequency of a population as a whole.

Gao derived the basic reproduction number ($$R_0$$), which acts as a threshold for disease persistence or eradication. It is then estimated in terms of the group reproduction numbers of all patches. So long as the disease is not severe enough to impact one’s mobility, the disease-free equilibrium is globally asymptomatically stable if $$R_0 \le 1$$; the model reaches a unique endemic equilibrium that is globally asymptomatically stable if $$R_0 >1$$. The proportion of frequent travelers is shown to be determined mainly by the frequency exchange rates.

To further his study, Gao compares his new multipatch model with traditional models. Specifically, he numerically analyzes the way in which differences in travel habits and variations in global travel patterns affect the geographic spread of infectious diseases for a two-patch instance of his new model. Gao finds that the traditional models mostly underestimate the transmission potential. Travel habits and the tourism industry are currently undergoing substantial changes based on technological advancements and economic development. Specifically, he acknowledges that travelers are changing locations more frequently than ever before, meaning that the length of stay within one given “patch” of the model is much shorter than expected. Increasing the infected subpopulation’s diffusion coefficient can increasingly or decreasingly or non-monotonically change the reproduction number of the new model but always decrease that of the traditional models. More people are also transitioning from infrequent traveler status to frequent traveler status, thus yielding a higher capacity of frequent travelers. $$R_0$$ can also increasingly or decreasingly or non-monotonically depend on such a change. Gao hopes that his findings can help public health officials in identifying individuals at high infection risk much easier and allocating medical resources more efficient.

While Gao’s work is certainly a step toward a better comprehension of the influence of heterogeneity in travel frequency on spatial and temporal spread of infectious diseases, there is still more to be done. Gao and his team plan to conduct a systematic survey on contact pattern and travel behavior to estimate the matrices for movement rates, transmission rates, and frequency exchange rates, and ultimately determine the model’s validity.

Read the full article, which just published in SIAM Journal on Applied Mathematics, here.