# Past Projects

## Persistence and Permanence of Reaction Networks

In order to understand global long term behavior of solutions of polynomial dynamical systems, I sought to determine whether or not solutions with positive initial conditions remain bounded, and bounded away from zero. A dynamical system on is said to be *persistent *if for any solution which starts in positive space, the system remains in positive space as time approaches infinity. In the context of population modeling, this means that no species becomes extinct. The stronger property *permanence *means that there exists a compact region which does not intersect the border of positive space such that any solution which begins in positive space ultimately resides inside this region. In other words, there exists a compact attracting region in positive space. Clearly, permanence implies persistence, and additionally it implies that solutions are uniformly bounded.

In our paper (arxiv link), we proved that an easily checked condition, termed *tropically endotactic*, on a differential inclusion in two dimensions was a sufficient condition for permanence of polynomial dynamical systems embedded in it.

This work was in collaboration with Prof. Gheorghe Craciun.

## Equivalence of Reaction Network Models

Many results about reaction networks, including classical results on the existence of unique, stable, equilibrium depend on the structure of a graph underlying the network. In our paper (arxiv link), we investigate the overlap between classes of reaction networks, in the sense of dynamical equivalence.

This work was in collaboration with Prof. David Anderson, Prof. Gheorghe Craciun, and Prof. Matthew Johnston.

## Confidence in Sampling an Epidemic

In response to the COVID-19 outbreak of 2020, I investigated the reliability of sampling during an epidemic by building a continuous stochastic model to simulate testing. In our paper (arxiv link), we introduced a method for simulating sampling of a population during the dynamic spread of a disease, with biased and imperfect testing. Our aim was to demonstrate that it is possible to estimate the reliability of an estimate of a given metric of disease spread using the model. We showed that biased sampling could still reliably estimate the sign of a trend in the spread of a with less than 10,000 tests performed each day, and that an actual estimate of the spread of disease could be made with an additional 1,000 unbiased tests per day.