RUI: An Analysis of Infectious Disease Dynamics

Billings

The overall objective of this work is to develop a thorough

understanding of emergent dynamics in epidemiological models as a practical

way to prevent disease outbreaks. Theproject has four parts: 1) developing

new, detailed models, with the advantage of analysis by dynamical systems;

2) incorporating time-dependent parameters that more accurately capture

climate variability and other nonstationary behavior; 3) adding stochastic

perturbations to analyze emergent dynamics; and 4) controlling the system.

These problems are studied using theory from dynamical systems, topology,

numerical analysis, asymptotics, and stochastic perturbations. In this

study, the investigator includes noise, time-dependent parameters, and

spatial networks, which are known to combine to produce unexpected

complexity in systems, or emergent dynamics. In particular, the

investigator builds on recent work developing the Galerkin Transport

Matrix, which allows a rigorous study of stochastic dynamics in the context

of the topology of the system. This tool can identify precursor behavior

to outbreaks, allowing the control of the dynamics by a parameter, such as

vaccination, to avoid these patterns and avert epidemics.

Disease dynamics have been modeled and studied for hundreds of years.

Yet, epidemiologists cannot accurately predict large-scale outbreaks or

epidemics in the most common diseases. Currently, there are fears of new

strains of deadly diseases and calls for a defensive plan. Questions are

being asked about which vaccination or quarantine strategies are necessary

to avert outbreaks. Due to ethical issues of testing vaccination

strategies on people, epidemiologists must rely on data from past epidemics

and corresponding mathematical models to guide these decisions. Models are

powerful tools if their results are interpreted correctly. They can

capture qualitative dynamics that point to patterns and mechanisms that

enable disease persistence in a population. They can also predict the

success of an action taken to avert an outbreak. In collaboration with

epidemiologists, a better mathematical understanding of the possible

dynamics in these models can lead to improved methods of disease control.

This work can also be extended to new models to describe the propagation of

multiple disease strains and certain types of computer viruses.