Difference between revisions of "Introduction to Compartment Models"

Revision as of 13:51, 7 October 2009

Introduction

Transmission of infectious disease in humans is a stochastic process. Whether or not a person susceptible to the flu in a given year actually contracts it is in large part a matter of chance. It is possible to become exposed by an infectious person shopping at the same supermarket – or to escape exposure based on the fortune of gathering goods in a different aisle. How can we do mathematical modeling of a process when its dynamics are subject to random events?

Jakob Bernoulli (1654–1705) was the first mathematician to prove a theorem known as the Law of Large Numbers. In 1713, eight years after his death, his proof was published by another mathematician (his nephew Nikolaus Bernoulli) in a paper entitled The Art of Conjecturing (Ars Conjectandi). Bernoulli’s proof of the Law of Large Numbers is one of the most important theorems in probability and statistics. It demonstrates for a large enough system, the mean value of a random variable is stable over long periods of time or large numbers of samples.

A useful example of this theorem is the outcome of repeated flipping of an unbiased or “fair” coin. An unbiased coin will land heads or tails with a probability of 50%. After a single flip, the number of heads will be either zero or one. This is not a good statistical sample as one would not want to conclude from a single trial that one expects to get a heads 0% or 100% of the time. After a larger and larger number of trials, the average result will get closer and closer to 50%. After 100 flips, there might be 60 heads (10 excess) and after 1000 flips perhaps 520 heads (20 excess). The number of excess heads would have doubled while the mean would have decreased from 60% to 52%, converging on the long-term stable value of 50%.

One of the earliest attempts to use the principles of statistics and probability to understand infectious disease was made by yet another member of the Bernoulli family. In a paper published in 1766, Daniel Bernoulli used census data and statistical methods to study the advantages of variolation or variolization as inoculation against smallpox. In a process originally developed in Asia, smallpox scabs were dried, ground, and inhaled by people, who then contracted, hopefully, a mild form of the disease. Approximately 2% of individuals who contracted smallpox by variolation died compared to 30% of those normally exposed to the disease.

The ability to study large numbers of cases and large populations makes possible the application of statistical methods to the study of infectious disease. In particular, we can make accurate predictions using models based on continuous variables. Consider again the coin flip example. At any given point of time, by definition, a coin will be found in a particular state, heads or tails. Imagine a collection of many thousands of coins being repeatedly flipped from one state to another. For a large population, P, the number of coins likely to be found in the heads state is simply: Nheads = pheads x P

Section 1

For now please see a great wikipedia article on Compartment models in Epidemiology.