= Solution
Define
$$
N=S+I,
\qquad
\theta=\frac IN,
$$
so $S=N(1-\theta)$ and $I=N\theta$. Part (a) already gives
$$
\boxed{\dot N=N(1-N).}
$$
Using the <quotient rule>,
$$
\dot\theta
=\frac{\dot I N-I\dot N}{N^2}
=\theta\{\beta N(1-\theta)-1\},
$$
and hence the reduced <Plant SI model with logistic total population> is
$$
\boxed{
\dot N=N(1-N),
\qquad
\dot\theta=\theta\{\beta N(1-\theta)-1\}.}
$$
For every $N(0)>0$, logistic growth gives $N(t)\to1$. The asymptotic prevalence equation is therefore
$$
\dot\theta=\theta\{(\beta-1)-\beta\theta\}.
$$
If $\beta<1$, its only nonnegative equilibrium is $\theta=0$, which attracts all prevalences. If $\beta>1$, $ heta=0$ is unstable and the stable value is $\theta_*=1-1/\beta$. Combining these values with $N_*=1$ recovers exactly the equilibria and outcomes in part (b).
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