Continuum Models for the Spread of Alcohol Abuse

This thesis introduces mathematical models to understand the role of peer influence on the spread of alcohol abuse. Our first model involves an integro-differential equation (IDE) model and is based on the Braun et al [BWPBG] network model. IDE problems often arise in studying non-local interactions between populations [M]. We provide a mathematical formula to explain the relationship between a constant population resilience level and the initial average peer influence, a detailed analysis of a regularized partial differential equation approximation to the IDE problem traveling wave solutions, and a mathematical model for treatment that would be available to severe alcoholics.

We also propose a new network model that shows similar dynamics with the one in Braun et al (2006). However, this new approach does not have a certain degeneracy we observed in the Braun model. This work is the first step toward the development of continuum models for the dynamics of social networks. There are many exciting new directions to pursue including applications in obesity and gang violence as well as evolution of network connection dynamics.