Dimensional analysis
Dimensional analysis checks whether the quantities in a mathematical model are combined in a physically and biologically meaningful way. It is a simple but powerful method for detecting mistakes and for determining the units that model parameters must have.
Dimensions and units are different
A dimension describes the kind of quantity being measured. A unit is the particular scale used to measure it.
| Quantity | Dimension | Possible unit |
|---|---|---|
| Time \(t\) | time | day or hour |
| Population \(N\) | population | individuals |
| Per-capita growth rate \(r\) | inverse time | day\(^{-1}\) |
For example, changing from days to hours changes the unit of time, but time is still the same dimension.
Checking a population-growth model
Consider the model
\[ \frac{dN}{dt}=rN. \]If \(N\) is measured in individuals and \(t\) in days, then \(dN/dt\) represents population change per unit time, so its units are
\[ \frac{\text{individuals}}{\text{day}}. \]The right-hand side must have exactly the same units. Since \(N\) has units individuals, \(r\) must have units
\[ \text{day}^{-1}. \]Therefore,
\[ (rN)=\frac{1}{\text{day}}\times\text{individuals} =\frac{\text{individuals}}{\text{day}}. \]Both sides have the same units, so the equation is dimensionally consistent.
Units can reveal what a parameter means
Dimensional analysis is not only a final check. It can tell us what units an unknown parameter must have.
Using square brackets to mean “units or dimensions of”, the equation
\[ \frac{dN}{dt}=rN \]requires
\[ \left[\frac{dN}{dt}\right]=[r][N]. \]Hence
\[ [r]=\frac{[dN/dt]}{[N]} =\frac{\text{individuals/day}}{\text{individuals}} =\text{day}^{-1}. \]Detecting an impossible expression
Suppose someone wrote
\[ \frac{dN}{dt}=r+N. \]If \(r\) has units day\(^{-1}\) and \(N\) has units individuals, the right-hand side tries to add
\[ \text{day}^{-1}+\text{individuals}. \]These are different kinds of quantities and cannot meaningfully be added. The equation is therefore dimensionally inconsistent.
An epidemic example
Consider the infection term in a frequency-dependent SIR model:
\[ \frac{dS}{dt}=-\beta\frac{SI}{N}. \]Suppose \(S\), \(I\) and \(N\) are all measured in individuals. Then
\[ \left[\frac{SI}{N}\right] =\frac{\text{individuals}\times\text{individuals}}{\text{individuals}} =\text{individuals}. \]Because \(dS/dt\) has units individuals/day, \(\beta\) must have units
\[ [\beta]=\text{day}^{-1}. \]Thus dimensional analysis tells us the units required for the transmission parameter from the structure of the equation itself.
What dimensional analysis can and cannot tell us
If the dimensions do not match, the model contains an error or the quantities have not been defined consistently. If the dimensions do match, the equation has passed an important check, but this does not prove that the biological assumptions are correct.
For example, two different biological models can both be dimensionally consistent while making very different assumptions about how organisms interact.