Difference between revisions of "Introduction to Compartment Models"

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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%.

File:Figure 1. For a large number of files.zip

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:

\begin{math}N_heads_= p_heads_xP\end{math}

where P is the total number of population of coins and pheads is the probability of a coin landing heads (\begin{math}p_heads_= 0.5\end{math}). Compartment models are mathematical models designed to describe the transport of information, material, energy, heat, etc., between and among the various compartments of a dynamic system. Formally, a dynamic (or dynamical) system is any system wherein objects interact by a fixed set of rules or equations that describe the time dependence of the state variables of the system. For example, a system of planets orbiting a star is a dynamic system where the position of each planet varies with time in accordance with Newton’s Laws of Motion. Compartment models were first introduced by Jay Wright Forrester who developed them for the field of Operations Research. Forrester is credited as the father of System Dynamics.

Compartment models have been applied to the study of dynamical systems in fields as distinct as manufacturing (workflows), systems biology, environmental studies, economics, politics, engineering, and medicine. First published in the Harvard Business Review in 1958, Forrester’s work establishes a paradigm that can be applied wherever one wants to describe how a system changes in time.

A compartment model provides a framework for the study of transport between different compartments of a system. Think of each compartment as a room in a house. A person living in that house travels from room to room over time. At any given time the person may be found in the living room, the kitchen, the bedroom, etc. At no time will the same person be found simultaneously to be in two different rooms. Entities within a compartment model will exist in well defined states. “John is now in the living room.” There may be more than one person living in the house. “John and Jane are in the living room, Julia is in the kitchen.” If the house is a closed system so nobody ever enters or leaves, then summing the number of people in every compartment or room of the house will always yield the total occupancy of the house. In a closed system the number of people is conserved. If the system is open, the occupancy of the house might vary from time to time, but the number of people in a house, plus the number entering, minus the number leaving, is still conserved.

A house may have many rooms. In general, there can be a large number, N, of components and they may connected in different ways. It may be possible to enter the kitchen from the living room or from the dining room. Our coin flip example is a very simple model with two compartments. After any given flip of a coin, it is found to be in one of two states (either head or tails).

In medicine compartment models have been used to study the dynamic flow of chemicals (nutrients, hormones, drugs, radio-isotopes, etc) between different organs of the human body. The flow of material in a compartment model follows certain rule. (For a blood alcohol concentration of x%, alcohol a will pass through the blood/brain barrier at rate y mg/hour.”) These rules, in turn, may be expressed as mathematical equations.

Section 1

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