An introduction to epidemic modeling

Abstract

To begin, I discuss the basic ideas behind the theoretical modeling of epidemics. Briefly, in constructing a model of the spread of an infectious disease we first identify a set of categories or states that individuals may be in that are important in describing the course of an epidemic. For instance individuals may be infected with a given disease, they may be susceptible to acquiring it, or they may be vaccinated and protected against acquiring it. Once the categories are identified we then group individuals in our population of interest into these categories according to their status. The important dynamics of the course of an epidemic occur because of transitions individuals make between these states. Individuals may go from being susceptible to being infected (and infecting others) to being recovered. Or they may go from being susceptible, to receiving a vaccination, to being immune from becoming infected. The description of an epidemic states how these transitions between categories occur across time and answers how many individuals are in a given category at each point in time? To do this within a model we need to calculate the rates at which these various transitions occur (infection, recovery, death, etc.) Initially I describe these transition rates between states abstractly with generic parameters, like $$\alpha $$ and $$\kappa $$. When one moves to empirical consideration one can use statistics to estimate these transition rates or use underlying knowledge of biology to fix values to these transition rates. As these transitions occur individuals move between categories and thereby change the population sizes in each of these categories. Once this modeling process is specified one can write down a set of equations as a formal description of the model. The equations can then be used to describe the system outcomes either analytically, statistically, or computationally.