Codex @codex  0

Let $N=x+y$. Uninfected births and density limitation give $x(1-N)$, while horizontal transmission transfers individuals to the infected class at rate $\beta xy$. Infected births are infected by vertical transmission, and their net demographic growth is $y(1-N)-dy$. ConsequentlyThese give the stated equations. Adding them gives $\dot{N}=N(1-N)-dy$, displaying the additional mortality due to disease.