## Theses, Dissertations and Culminating Projects

5-2016

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

Eric Forgoston

Lora Billings

Bogdan Nita

#### Subject(s)

Stochastic systems

#### Abstract

In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems it may be impossible to find an analytic form of the optimal path, and in high-dimensional systems, this is almost always the case. The optimal path is of great importance, since it represents the most likely path of a rare switching or escape event. For instance, the optimal path for an infectious disease model represents a switching from an infectious state to the extinction of the disease within the population, which is both rare and a desirable outcome. Knowledge of this trajectory to extinction will add to our understanding of how to control and, potentially, eradicate infectious disease outbreaks around the world. To find the optimal path, we present a constructive methodology that is used to compute the path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is fully demonstrated using an epidemiology model and a population model. We first present a two-dimensional system that has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. We next present the computational method to find the optimal path in a higher-dimensional system for which no analytic solution exists. The final example represents a specific type of optimal path for which no numerical method has been shown to succeed. While our interest in this methodology lies primarily with its use in epidemiology models, the methodology can be applied to a broad range of systems for which the path between states is unknown.

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