Metric Graphs: Numerical Methods, Localization, and the Spread of Epidemics

Abstract

Metric graphs may be used to describe a variety of physical occurrences from motor vehicle traffic to random network lasers. Partial differential equations (PDEs) are a useful tool for describing these phenomena. PDEs are defined on the edges of the metric graph and are coupled to the other edges through junction conditions. In this dissertation, some introductory metric graph terminology is defined and the construction of three example metric graphs is discussed. Methods for solving the wave equation on a metric graph are then presented, in particular a spectral method and a finite difference method are developed. Splitting solutions by their frequencies using the spectral method reveals that for some frequencies solutions get trapped in specific shapes. This process is known as localization. Conditions are derived for exact localization of a selection of shapes, and a criterion for localization is developed. Finally, the metric graph model is extended and applied to a susceptible-infected-removed (SIR) model for infectious disease. In this model the edges of the metric graph represent 1D approximations of travel routes and are coupled to vertices representing population centers. The metric graph is then embedded in and coupled to a 2D region with exchange occurring along the edges. This model is run on two example regions.