Network Models in Epidemiology: Considering Discrete and Continuous Dynamics

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. 1 Mathematical Epidemiology and the SIR Model 1.1 Mathematical Epidemiology Diseases are studied in many areas of research, such as virology and pathology. Epidemiology concerns itself with the onset and spread of a disease through a population. It considers disease characteristics, such as contagiousness, reproduction, and lethality; as well as population characteristics, such as susceptibility and connectivity. Furthermore, epidemiologists work to contain and eradicate the disease. It is not enough to only know how a disease spreads. Epidemiology also asks the question, “How can we best stop the disease?” In this, we consider control protocols, quarantine, and vaccination. Mathematical Epidemiology applies the language of mathematics to the study of epidemiology. We apply mathematical theory to describe the disease within the population. We analyze for qualitative understanding, and we build models that give quantitative results. One such model is the classical SIR model, which provides us with the notion of the basic reproductive number. Besides this, mathematical modeling provides us equations that describe the dynamics of infection and provide insight for effective control protocols. Mathematical modeling is an essential tool for framing the world around us. It provides us with helpful insights and accurate guidance. We will examine the SIR model, and then provide a network model to examine the spatial constraints of social structures.