# Student Feature – John C. Lang

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John Lang is currently working with Dr. P.J. Lamberson as a Postdoctoral Fellow in the Department of Communication Studies at the University of California Los Angeles. He recently completed a Ph.D. in Applied Mathematics at the University of Waterloo under the supervision of Dr. Hans de Sterck (now at Monash University) and in collaboration with Dr. Daniel Abrams (Northwestern University). His Ph.D. thesis focused on modeling social processes at both the individual and population levels. He had previously completed an M.Sc. in Econometrics and Mathematical Economics at the London School of Economics and Political Science, an M.Sc. in Applied Mathematics at the University of Alberta, and an M.Sc. (Honors) in Mathematics at the University of British Columbia.

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In the first year of my undergraduate degree I enrolled in the University of British Columbia’s Science One Program, a team-taught program that emphasizes interdisciplinary studies and research. I quickly fell in love with how an interdisciplinary approach can combine multiple diverse perspectives to reveal interesting insights that might otherwise go unnoticed. I eventually decided to major in mathematics – a decision that has provided me with modeling tools that have allowed me to contribute meaningfully to many interesting projects.

I like to showcase the following project because it allows me to illustrate some important aspects of what makes a project compelling. We start with a problem of significant interest to a wide and general audience and identify some outstanding questions that could be addressed through mathematical modeling techniques. We identify potentially useful data sets and attempt to manipulate the data in order to gain some preliminary insights into the modeling problem. We are then able to form a mathematical model whose analysis allows us to address the motivating questions.

Most recently, I completed a mathematical modeling project that addressed the question of why the body mass distribution (BMI – ratio of weight to height squared) takes the shape it does. An individual’s BMI is considered to be a proxy for the percentage of fat in an individual’s body composition, and, hence, understanding the shape of the BMI distribution and how it changes over time may have important implications in the development of public health policy that addresses the current obesity epidemic. For example, consider that the mean, variance, and right-skewness (i.e. skewness towards high BMI) of the BMI distribution have all been increasing steadily over at least the past thirty or so years. Also, an increase in any one of these moments could result in individuals originally classified with “normal” BMI to become “obese”, and could result in more severe obesity in individuals already classified with “obese” BMI. Given these observations, it is natural to wonder, therefore, why is the BMI distribution right skewed in the first place? What determines the extent of the right-skewness? What, if any, is the relationship between the mean, variance, and skewness of the distribution, i.e., are they increasing steadily together by coincidence or is there some mechanism that determines their relationships to each other?

Since the shape of the BMI distribution is determined by multitudes of small contributions made from millions of individuals, a logical starting place for studying the BMI distribution would be to consider how individuals’ BMI values change over time. After some searching, we found two data sets that allowed us to investigate individuals’ year-over-year change in BMI: the National Health and Nutrition Examination Survey (NHANES) and anonymized medical records from the Northwestern Medicine group of hospitals. When we plot the average and standard deviation of individuals’ year-over-year change in BMI versus their initial BMI, we find two strong linear relationships: average change in year-over-year BMI was found to decrease linearly with initial BMI (a(x)) and standard deviation in change in year-over-year BMI was found to increase linearly with BMI (b(x)). This finding is interesting in and of itself, because a priori it was not clear at all whether there would be any pattern at all in individuals’ average year-over-year change in BMI. Indeed, the pattern we find is initially at odds with many people’s intuition about weight gain. Whereas we find that high BMI individuals on average lose weight, conventional wisdom has it that people who are susceptible to weight gain enter into a feedback loop where they gain more and more weight and become more and more severely obese (the so-called runaway train argument). What makes this finding even more interesting (at least in my opinion), however, is that from the perspective of an applied mathematician this finding yields a highly intuitive result: the distribution of a population of dynamic individuals is determined by the balance between (a) a drift force a(x) that attempts to squeeze the distribution towards the mean and (b) a diffusion force b(x) that attempts to spread out the distribution. This is analogous to well studied phenomena, for example, the distribution of the velocity of a Brownian particle subject to friction.

We use the above observation to inform a mathematical model where the change in an individual’s BMI (dx/dt) is described by the Langevin Equation

$$\frac{dx}{dt}=a(x) + b(x) \eta(t),$$

with linear drift a(x) and diffusion b(x) terms and with Gaussian white noise η(t). A population of individuals behaving according to the above Langevin Equation then induces a Fokker Planck partial differential equation for the probability density function of BMI p(x,t)

$$\frac{\partial p}{\partial t}(x,t) = - \frac{\partial }{\partial x} [a(x) p(x,t)] + \frac{1}{2} \frac{\partial^2 }{\partial x^2} [b^2(x) p(x,t)],$$

where a(x) and b(x) are as in the Langevin Equation. Solving the Fokker Planck equation for the equilibrium distribution gives us an equation for the distribution of BMI that fits empirical BMI distributions significantly better than standard distribution functions currently used in the literature (log normal and skew normal distributions). In addition, we are able to analyze the equilibrium distribution predicted by our model to give us insight into why the BMI distribution takes the shape it does and into the relationship between the mean, standard deviation, and skewness of the distribution.

What I enjoy most about applied mathematics research is that it allows us to tell interesting stories that might otherwise get overlooked. The above project modeling the distribution of BMIs is a favorite example of mine because we are able to apply well known mathematical techniques to a well established and publicly available data set in order to uncover some fundamental insights into the modeling problem. I love that the story was already in the data far before I came along, just waiting for someone to tell its tale. The irony is that although such stories are all around us, finding them nevertheless remains a needle-in-haystack problem. I have been very fortunate to work under the supervision of Dr. Hans de Sterck and Dr. Daniel Abrams, who always encouraged me to take risks and pursue problems I felt were interesting and had potential to become interesting stories.

I am currently working with Dr. P.J. Lamberson of the University of California Los Angeles. We are applying mathematical modeling techniques to study teamwork and collaboration in the context of team problem solving. We are looking forward to having many more interesting stories to tell.

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