# How do the units of the SIR model cancel out?

I was having trouble trying to understand the parameters of the simplest SIR model.

If beta is the effective contact rate and s is the percentage of people who are susceptible, then how do the units cancel out such that ds/dt is measured in percentage per unit time? When I multiply -ßsi out, the units cancel out to people/time, which isn't how ds/dt is measured. Am I missing something here?

The variables $$s, i, n$$ are dimensionless. (You consider the dimension 'percentage' but that is not a dimension. A percentage is dimensionless.)
So the differential on the left $$\frac{di}{dt}$$ has units 'per unit time' and not 'percentage per unit time'.
On the right the term $$\beta si$$ has also units 'per unit time'. The $$\beta$$, a rate, has units 'per unit time', and the $$s$$ and $$i$$ are dimensionless.