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Mathematically Modeling Asthma

By Fred Adler

Asthmalike diabetes, heart disease, and cancer—does not have a single initiating cause, making the development of models that start with cause and end with effect impossible to formulate. It doesn’t have well-defined triggers like the human immunodeficiency virus (HIV) for AIDS or the Cystic fibrosis transmembrane conductance regulator (CFTR) gene for cystic fibrosis.Growing rates of asthma, particularly in urban areas, increase the urgency of finding ways to unravel the causal pathways underlying this condition in order to find places to effectively intervene. Mathematical models provide one of the most powerful approaches to analyzing such complex problems.

The diagram shows just some of the connections leading to asthma, with arrows in green indicating the ones most amenable to mathematical modeling. To illustrate the thinking, we trace two paths from urbanization to asthma. First, as we all know, urbanization leads to crowding and thus increased chances for viral transmission. But modeling this process is complicated, because it depends not only on population density, but also on how people move and interact, which in turn depends on social and economic factors that are difficult to pin down.  

From there, the path from virus infection to asthma, although far from well-understood, involves the better-defined dynamics of the immune system, including how a severe early childhood infection, often coupled with exposure to an allergen, can switch the immune system to an asthma-prone state, or can damage the airway leading to a destructive dynamic of repair. Certain genetic predispositions, such as impaired ability to fight viruses, can be understood and mathematically modeled in this context. The more speculative link between cleanliness, the so-called hygiene hypothesis, and improper tuning of the immune system in early life, will be ripe for mathematical modeling as the relevant components are identified and measured.

Various factors contributing to immune dysfunction and asthma. Image credit: Fred Adler

Other pathways are much further from being ready for mathematical approaches. Urbanization leads to stress of many sorts, mediated through crowding, social isolation, lack of quiet, or access to green space, and of course, economic uncertainty.  Although mathematical sociology is a growing field—with its own journal--linking what we know to the biological effects of stress remains highly speculative.

In the long run, then, what does mathematical modeling have to contribute to the understanding of asthma?  Our work has focused on three particular elements of the story, centered around rhinoviruses, the most common cause of the common cold. Despite their generally benign presentation, severe rhinovirus infections in infants lead to many hospitalizations, and those who suffer from wheezing are predisposed to asthma many years later.  

First, the frequency of infections in infants depends on transmission at home and at daycare centers, and we have modeled how the distribution of family size and age can affect early exposure and infection. One key result is that transmission of rhinovirus is strongly dependent on symptoms. Given the general consensus that most symptoms are a consequence of the immune response, which in turn depends on each individual's history of infection, embedding the transmission dynamics for a given rhinovirus serotype (of which there are well over 100) into a long-term high-dimensional system will be crucial.  Modeling the immune response is thus the second and most important component of a comprehensive modeling approach to asthma.

Third, the triggers for asthma, as noted above, center around the immune response, whether to severe early infections, allergens, or lack of exposure to the right kind of dirt. Our models have examined how a viral infection can tip the immune system into a different state, with the potential to predispose an individual to future asthma.

We are far from knowing enough to build a comprehensive mathematical, or even statistical, model of asthma. Parameter values for even the best-known parts of the pathway are largely unavailable. Even though quantitative prediction remains elusive, models provide the logical framework needed to break up a complex system into more manageable pieces and identify key loci for future research.

The author presented this research during a minisymposium at the 2017 SIAM Conference on Applications of Dynamical Systems, held in Snowbird, Utah, this May.

Fred Adler is a professor in the Department of Mathematics and the Department of Biology at the University of Utah. His interests include ecology, epidemiology, immunology, and many fields of molecular and biomedical biology. He is past president of the Society for Mathematical Biology and director of the Center for Quantitative Biology. 
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