Discrete and Continuous Dynamics is the first in a series of articles on Network Models for Epidemiology. This project began in the Fall quarter of 2014 in my continuous modeling course. Since then, it has taken off and turned into a series of articles, which I hope to compile into a single report. The purpose of the report is to explore mathematical epidemiology. In this article, we discuss the historical approach to disease modeling with compartmental models. We discuss the issues and benefits of using network models. We build a discrete dynamical system to describe infection and recovery of individuals in the population. Lastly, we detail the computational scheme for iterating this model.