= Solution
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 $y(1-N)-dy$. Consequently
$$
\dot x=x(1-N)-\beta xy=x[1-x-(1+\beta)y],\qquad \dot y=y(1-N)-dy+\beta xy.
$$
These give the stated equations. Adding them gives $\dot N=N(1-N)-dy$, displaying the additional mortality due to disease.
Back to article page
