The stories units tell

Grammage, fuel efficiency, and recovery time

One of the ways to describe paper is its grammage, which is most commonly expressed in gsm, i.e. grams per squared meters. Your usual notebook will be somewhere around 60 gsm, a really heavy paper can be 120, a business card is up to 400, and the standard copy paper is about 30. Higher grammage does not mean that the paper will be “better” - this is simply an indication of how thick, rigid, etc, the paper will be. But we are not here to talk about paper quality.

Expressing paper “density” as grams (a weight) per squared meters (a surface) is an interesting choice, as it assumes that a sheet of paper is a two-dimensional object. And this is an acceptable approximation, as your average sheet of paper is thinner than a tenth of a millimeter. We know that this object exists in three dimensions, but we are willing to think of it as being two-dimensional. In short, our expression of grammage, in terms of the units we picked, is technically incorrect, but it tells the correct story.

Another interesting unit is mpg, or miles per gallon, as a measure of fuel efficiency; this is, quite literally, how many miles you will drive on a single gallon of fuel. Because this is not necessarily intuitive, let’s express this instead as litres per kilometers, which amounts to a volume (a distance cubed) divided by a distance - the resulting unit is therefore… a surface? How would it make sense to measure the efficiency of a vehicle as a surface? Well, it doesn’t. We are perfectly justified in bringing all of these units to the same basis, and the simplifying, but this misses the point. The point is that a litre of fuel contains some quantity of energy, and the correct way to think about miles per gallon is as an answer to the question: how much energy do I need to expend to advance one unit of distance. Realizing that a volume is a distance cubed is correct, but tells the wrong story.

Now, what does it have to do with anything?

One of the first exercises I assign in my modeling class is to think about ways to come up with approximate values for the parameters of an epidemic model in which individuals are initially susceptible, can become infectious, and finally recover. The specific question I ask is, “can you think of a way to come up with a good enough guess for the rate of recovery”. This problem becomes really simple if you think about units. In this model, we are describing individuals, so the units of all variables are “individual”. Because this is a temporal model, the derivatives are expressed as changes in individuals, or individuals per time unit. If we assume that the only way to recover is to start as an infectious individual, then we know that we are multiplying something which is expressed in individuals (the number of infectious), and we must get something expressed as individuals per time.

Or to put it in not-words:

$$\text{individuals} \times u = \frac{\text{individuals}}{\text{time}}$$

From this, it is easy to see that the units of the quantity $u$ (i.e. the unit of the recovery rate) is $\text{time}^{-1}$. This is a rate, and one way to estimate a rate is to guess the time spent in the state, and then invert it – in this case, by guessing how long an individual remains infectious for, we can express the rate of recovery as the inverse of this quantity.

Neat, right? This is an example where units can serve as a check that the model parameters are correct (everything must balance at the end), and also provide some guidance about what should be measured. Discussing units (of variables and parameters) is one of my favorite ways to go through the process of debugging a model, and it helps decide on the “correct” way to represent a mechanism.