SIR epidemic model with vital dynamics

I am reading the Wikipedia article on the SIR model with vital dynamics.
I am wondering about the birth and death rate. The birth rate seems to be constant, ie, it seems like the population in all 3 compartments reproduce, and overall, newborns are added to the S compartment at a constant rate. It sounds reasonable.

Based on the definition of R compartment, it seems like dead people are also added in this section. I quote

R for the number of removed (and immune) or deceased individuals.

What I am wondering is shouldn’t the last equation with R be?
$$ frac{dR}{dt} = gamma I + mu S + mu I $$
The dead people from S and I should get added to R compartment. This further raises another doubt, if compartment R is indeed filled by dead people as well, then my justification of constant birth rate fails. My logic was that the birth rate would be proportional to $ propto (S + I + R)=N $
so it is a constant. However, now dead people can’t reproduce and hence R compartment can’t reproduce at a rate proportional to $R$. It would be much lower due to dead people.