Units of time in simple differential equation 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
 A: 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. 
A: It is possible to solve analytically the system of three ODEs. This could be useful for numerical applications, in order to obtain more accurate numerical results than with the the usual methods of numerical solving of the ODEs.
The solution is presented on a parametric form : For given $R$, compute the corresponding $S , I, t$ with the formulas.
This is not a direct answer of the question raised, but might help anyways.

A: In the differential equation
  dR/dt =  α·I

the left member has the dimension size of population
   over time, and the right side has the same dimension
   if the dimension of α is the inverse of time. This makes
   the equation unit-independent: one can choose any unit
   for the common dimension -- the corresponding numeric
   equations are all equivalent since all the units of a
   dimension differ by a non-zero numerical factor.
On the other hand, the two members of
    dS/dt = -R·I·S

have different dimensions: the dimension of the
   right member is (size of population)³ while that
   of the left member is size of population over
   time. The meaning of that equation depends on a
   factor (unit) of dimension size of population
   squared times time; the equation is not meaningful
   without the specification of that factor.
Only if both sides of an equation have the same
   dimension is the equation independent of the
   choice of units. It is customary in physics to
   check equations for the same dimension of each
   member ("dimension calculus") -- other equations
   cannot represent a law of nature unless the units
   are spelled out.
Michael Deckers.
   2020-03-30
