# Keoni Castellano

460 McBryde Hall, Virginia Tech

225 Stanger Street

Blacksburg, VA 24061-1026

My research involves the mathematical analysis of infectious disease models. The models that I study are part of a class of epidemic models known as susceptible-infected-susceptible models (SIS models) in which the population is split into two groups: those that do not have the disease (the susceptible) and those that do have the disease (the infected). The name “susceptible-infected-susceptible” refers to the pathway given by the model, that a susceptible person can be infected with disease and then possibly recover from the disease, going back into the susceptible group.

The particular models that I work with come in the form of a system of reaction-diffusion partial differential equations. These models are notable because they take into account the movement of individual populations and how this movement affects the dynamics of the disease. Furthermore, unlike in earlier infectious disease models, the transmission and recovery rates are taken to be spatially-dependent. Making these rates spatially-dependent serves to create an environment where certain regions are at greater risk for the disease than other regions. Lastly, the term that represents infection in the model is written as a density-dependent term. This allows the model to better represent how a disease would behave in an urban population. Previous, but similar, work often made use of a frequency-dependent term instead.

The ultimate goal of my research is to find an optimal control strategy for tackling an infectious disease by determining the asymptotic profiles of solutions to this model as the diffusion rates of the populations change. At the very least, I would like to better understand how diffusion rates affect the behavior of a given infectious disease in a spatially heterogeneous environment.