Very simple question:
There is a well-known model in epidemiology called SIR model. It describes the changes in the number of susceptible, infectious and recovered individuals in a population. It is described simply as:
$$\frac{dS}{dt} = -IRS$$
$$\frac{dI}{dt} = IRS-\alpha I$$
$$\frac{dR}{dt} = \alpha I$$
where $S$, $I$, $R$ are the proportions of susceptible, infectious and recovered people, respectively. $\alpha$ is the recovery rate, also described as the inverse of the average infectious period.
My question is the following: Does it make sense to have an equation such as $\displaystyle \frac{dR}{dt} = \alpha I$ where $dR/dt$ is measured in $1/\text{time}$ and consequently, $\alpha I$ is also measured in $1/\text{time}$, but the right-hand side uses days, for example, $\displaystyle \frac{1}{3 \text{ days}}$ and the left-hand side uses other measure of time?. Mathematically, I think, everything looks correct, since the units match both sides. However, the use of this equation looks suspicious to me, because the left-hand side is telling me the variation per unit time and I expect this variation to be measured in a very short time interval. Therefore, when I see that, in a very small time interval, the change of $R$ (of recovered people) is equal to the proportion of infectious people that recovered from disease in a matter of days, it just seems a bit incongruous.
Maybe, the left-hand side also uses days, but then how should I justify that $dR/dt$ would have units of 1/time measured in days?
Thanks in advance
Best Answer
Mathematics is agnostic to units.
Mathematical models, however, do care very much about units. (The numeral 1 attached to the units 'seconds' and 'hour' give two very different meanings.) So it is extremely important that when you do write down a model and do actual analysis and computations on it, you include conversion factors which makes your unit agree.
To take your model: $R$ is the number of people recovered, and $I$ is the number of people infected; both are either regarded as "unit-less" or "ones". $\alpha$ is the rate of recovery, which has unit "inverse-time". The value of $\alpha$, however, depends on the unit chosen for time: since
$$ 1~\mathrm{s}^{-1} = 3600~\mathrm{hr}^{-1} = 86400~\mathrm{day}^{-1} $$
The fact that the left hand side is written as a differential does not mean that its unit must be "small"! Go back to something simple, like physics. The speed of an object is the "instantaneous displacement" divided by the "instantaneous time change", but on the highways we usually measure speed by kilometers per hour (or miles per hour, if you are in certain English speaking countries).
So a recovery rate of $dR/dt = 1 / \mathrm{day}$ can be analogously interpreted as $dR/dt = \frac1{12} \frac{\text{persons}}{\text{hour}}$: yes, it makes no sense to say that 1/12 of a patient recovered in the last hour, but on average over the course of the day that is the rate at which patients recover.