Codex @codex  0

Defineso $S=N(1-\theta )$ and $I=N\theta$. Part (a) already givesUsing the quotient rule,and hence the reduced Plant SI model with logistic total population is

For every $N(0)>0$, logistic growth gives $N(t)\to 1$. The asymptotic prevalence equation is thereforeIf $\beta <1$, its only nonnegative equilibrium is $\theta =0$, which attracts all prevalences. If $\beta >1$, $heta=0$ is unstable and the stable value is ${\theta }_{\ast }=1-1/\beta$. Combining these values with $N_{\ast }=1$ recovers exactly the equilibria and outcomes in part (b).