# DATA…GOOD OR BAD? — Part One: A MATHEMATICAL MODEL EXAMPLE

Extrapolation:  The riskiest form of prediction.

It has been said that when you are afraid that people will think you are a fool, don’t open your mouth and remove all doubt.  That is precisely the risk involved in making a prediction based upon someone else’s theory, especially if you do not fully understand the theory.

To that end I will tell you a true story.  I have left out some of the names as my intent is not to embarrass.

The propagation of disease by the inhaling of airborne particles of the disease is of no small interest.  And I will tell you up front that it is not my area of expertise…but when has that ever stopped me.

A colleague of mine was studying airborne disease propagation, and I kept hearing about the “Wells-Riley” equation.  It is an equation that looks at propagation as a process of inhaling disease carrying particles in quantities called, oddly enough, “quanta”.  The theory is that if you inhale a single quanta of diseased particles, you will get the disease.  I’m purposely leaving out two important issues for the moment, but don’t worry about them for now.

OK, so along comes the equation.  It’s relatively simple, it’s elegant, and unfortunately for a particular group of researchers, it was beguiling.

Do you remember the anthrax scare?  Sure you do.  There were stories of letters and packages being delivered by mail.  In them was a powder which usually was harmless, but was thought initially to be anthrax, a deadly source of disease.  One such incident involved what I think was the main Washington D.C. Post Office.  This then became the focus of a now published study.  The study was done for good reason.  There were unanswered questions about how difficult it was to provide sufficient protection against the threat of anthrax, and since that post office had actually been attacked, it appeared to be an ideal case to study using the Wells-Riley equation.  Sounds perfect, doesn’t it.

When my collegue first showed me the equation, it looked strangely familiar.  So, I looked at a statistics book and confirmed that the equation was actually a form of the Poisson Distribution equation for the probability of having one or more encounters with a single item that is randomly distributed.  So, here is the first thing I neglected to tell you.  In their original paper, to their credit, Wells and Riley mentioned that very fact.  We’ll get back to this.

The next thing I didn’t tell you is that each type of disease is thought to require a different number of separate particles (identical particles in theory) in order to form a single “quanta”.  That is, not all diseases have a single particle quanta.  Most require several particles to form a quanta.  The only disease with a single particle quanta as far as I know is tuberculosis, and as I said. this is not my field of expertise.  I have heard that the common cold requires a few hundred particles before a quanta is reached.  Anthrax supposedly requires thousands of particles to make up one quanta.  This is important to know because the Wells-Riley equation only works for a single particle quanta.  If it takes more than a single particle to make up one quanta, the equation has to be modified to include more terms.

And there lies the problem.  The Wells-Riley theory freely admits that it only covers the one particle quanta problem.  If you are dealing with a disease that requires thousands of particles to form one quanta, you need to add thousands of separate terms of the Poisson Distribution equation to run the proper calculation.  As it turns out, the conclusion reached by the anthrax researchers was off by a very substantial amount for just this reason.  As mentioned above, anthrax requires thousands of separate particles to constitute one quanta.

A foot note for this is that no one seemed to be sure at that time how many particles are in a quanta for any given disease or any type of particle that carries the disease.  I would also guess that a quanta for one person is not necessarily a quanta for some other person.  So at best, you can only look at this from the standpoint of the “average” person, whoever that is.

As I said up front, this is not my area of expertise.  I presume that the study of disease propagation has moved on since this incident.  This is just an example of one of the pitfalls of dealing with data where adequate theories have yet to be promulgated.  “All that glitters is not gold.”

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