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.