# Research

My doctoral dissertation focused on regularity of solutions to problems arising in the calculus of variations and partial differential equations. I am now also interested in seeing how mathematics can be applied to biological problems. In particular, I am interested in mathematical epidemiology.

### Research Involving Students

Much of my research at Luther involves students.  Below are the research projects I have worked on with students over the course of a summer.

John Doorebos (‘16) and Erin Ellefsen (‘17) - “The effect of population structure on vaccination thresholds for herd immunity”

For some diseases, there exists a vaccine which confers immunity to an individual.  This immunity benefits the vaccinated individual directly, but also provides indirect benefits to others in the population by interrupting possible transmission routes for the disease to spread.  For example, if B is unvaccinated, then A can infect B, who in turn infects C.  But if B is vaccinated, then this transmission route from A to C is interrupted (though there may be other routes from A to C).  However, when a large enough proportion of the population is vaccinated, so that nearly all transmission routes between unvaccinated individuals are cut off, then the population has what is known as “herd immunity.”  In this situation, the unvaccinated individuals still are protected from the disease because of the lack of possible transmission routes.  What proportion of the population needs to be vaccinated in order for the population to possess this herd immunity depends on a variety of things, including how easily the disease spreads, and on the structure of the contact network in the population.

The goal for our research was to explore how the threshold vaccination level required for herd immunity depended on the structure of the contact network in the population.  We first looked at “random networks,” where the probability that any pair of individuals could spread the disease to each other did not depend on who else they were “connected” to.  So if A is connected to B, and B is connected to C, that does not influence to probability that A is connected to C.  We then also explored how the herd immunity threshold changes as the contact network becomes more “clustered,” consisting of subgroups of individuals that are highly connected. (Think of individuals in the same family, same soccer team, or same church congregation, for example).  In this setting, if A is connected to B, and B is connected to C, that makes it more likely that A is connected to C too.  The main thing we explored was how the random networks and clustered networks differed in the herd immunity threshold, and also how this threshold compared to that which is predicted by a commonly used SIR model involving differential equations, which essentially makes the assumption that everyone is connected to everyone else in a uniform way.

Megan Gelsinger (‘15) and Scott Mittman (‘14) - “B-cell chronic lymphocytic leukemia - a model with immune response of genetically modified anti-cd19 CAR-targeted T cells”

Megan and Scott developed a mathematical model using differential equations for a relatively new treatment for chronic lympocytic leukemia.  This treatment involves taking T cells from the patient, genetically modifying them so that they target and destroy any cell expressing the CD19 antigen.  Since B cells express CD19, the T cells, a key component in the immune system, destroy the cancerous cells.  One unfortunate side effect of this treatment is that these T cells do not discriminate between healthy and cancerous B cells.  This treatment is often successful, but not always.  Our goal was to use the model to help determine which characteristics about a patient might make them a good candidate for the treatment.

According to the results we obtained from the mathematical model, the most influential parameters are the initial tumor burden and the rate the cancerous cells are produced in the bone marrow.  The fact that initial tumor burden is significant has also been observed empirically, as this treatment is much more successful when combined with pre-treatment chemotherapy.  One somewhat surprising result predicted by the model was that the division rate of the cancerous cells was not an important factor in determining the success or failure of the treatment.

### Publications

• Morrey Regularity for Almost Minimizers of Asymptotically Convex Functionals with Nonstandard Growth, with Mikil Foss, Forum Mathematicum 25(5): 887-929, 2013.  (View preprint or see final version)
• Young Measures Generated by Sequences in Morrey Spaces, Calculus of Variations and Partial Differential Equations 39(1):183-201, 2010.  (View preprint or see final version)
• Morrey Regularity Results for Asymptotically Convex Variational Problems with (p,q) Growth, with Mikil Foss, Journal of Differential Equations 246(12):4519-4551,2009.  (View preprint or see final version)
• On Morrey Spaces in the Calculus of Variations (doctoral dissertation), advised by Dr. Mikil Foss.