SIR model differential equations

Question

I'm trying to solve the SIR model differential equations by separation of variables to get $S$,$I$,$R$ as functions of time , for example $I$ solved the Infected differential equation as follows: $$ dI/dt= BIS-YI, \\ dI/dt = I(BS-Y), \\ dI/I= BS-Y dt, \\ \int dI/I = \int (BS-Y) \, dt = \ln{I} ⁡= BSt-Yt+C $$ Is that integration by separation of variables possible ? (Please answer)

Answer 1

No, you can't do that. The integral $\int (BS-Y) \, dt$ is not equal to $(BS-Y)t+C$, since $S$ depends on $t$.