## 2015 Summer Undergraduate REU Program

### Penn State University

**PSU- Department of Mathematics**

**PSU- Department of Mathematics**

#### Topic 1:

* Evaluating vaccination strategies leading to herd immunity in models of disease dynamics with application to measles - John Fricks*

#### Project Description:

Measles is a disease that is both highly communicable and highly responsive to immunization, with worldwide immunization efforts aiming to end measles in the not-too-distant future. An important aspect of this effort is the efficient distribution and administration of vaccines, especially in regions of the world that are difficult to access. In regions of high-vaccination rates, herd immunity is observed, where unvaccinated individuals do not contract the disease, since most everyone near them has been vaccinated and is therefore immune, though historical data concerning this shift is limited. The question then is whether we can observe the shift from no herd immunity to herd immunity and whether we can characterize the infection dynamics of a country or region that is undergoing such a shift. Using only enough vaccination to shift a region to such a herd immunity regime would represent an efficient use of the health resources available. Students will implement stochastic simulations of SIR-type and extensions providing practical epidemiology models. These will be used to study the shift from no herd immunity to herd immunity under various vaccination strategies. Students will also apply techniques of statistical inference to analyze output from stochastic simulations and relate them to real data.* *

#### Topic 2:

* Non-stationary models for estimating evolutionary relationships - Dennis Pearl*

#### Project Description:

Stochastic models of molecular evolution used in calculating the likelihood of phylogenetic trees assume that the distribution of characters has reached an equilibrium state at a time prior to the root of the tree. However, when reconstructing a total “tree of life” from highly conserved sequences in all domains of life, recent evidence in several ancient enzymes shows that this “stationarity” assumption is often far from true. In particular, the distribution of amino acids is different - more heavily weighted toward those thought to be associated with a pre-biotic era - in highly conserved areas of the enzymes than in less conserved areas (Pollack et al., 2013). In this project students will develop and analyze a new test for the validity of the stationarity assumption in phylogenetics. They will also develop and apply a new model of molecular evolution that allows for the amino acid distribution to vary with the level of site conservation. Students will advance their knowledge of probability, statistical inference, and modeling of evolutionary processes, and will become skilled in programming using the R language.