Abstract by Jacob Stern
The Stability of Traveling Wave Solutions to a Two-group Epidemic Model with Time Delay
Partial differential equations can be used to model many natural phenomena. One prominent field in which PDE's have been found to be effective is epidemiology -- the study of disease. The most prominent model is known as the SIR model, which is a system of 3 equations representing susceptible, infected, and recovered individuals in a population. In more simple models, and individual moves from one class to the next. But more realisitically, there is often a latent period in which individuals become infected before they are infective. This turns the model from a PDE to a DPDE -- a Delay Partial Differential Equation. We are studying a model proposed in 2017 by Zhao, Wang, and Ruan that takes into account the latency of the disease, the fact that individuals can disperse in space during that latent period, and differential susceptibility and infectivity between two groups. Zhao, Wang, and Ruan proved the existence of traveling wave solutions to their system. We undertake to determine under what conditions the waves will remain stable. To do so, we solve for the profile of those traveling waves and determine their spectral stability using the Evans Function.