 Solution

Yes, susceptible replenishment allows repeated outbreaks. While $I_t>0$, the infection ratio isInfection grows above the susceptible threshold $S_t=1/\alpha$ and declines below it. A wave depletes susceptibles; subsequent recruitment can restore the threshold and permit renewed growth. In the neutral linear regime, perturbations around the positive fixed point oscillate, so repeated small waves occur for sufficiently small perturbations over any fixed finite observation period.

For example, in scaled variables $x=\alpha S$, $y=\alpha I$, the choice $r=\alpha B=0.2$, $x_0=1.03$, $y_0=0.19$ produces the repeated positive waves illustrated below. This finite trajectory illustrates recurrence of outbreaks without claiming a nonlinearly attracting periodic orbit from the linearization alone. If $I_0=0$ exactly, the deterministic model has no infectious seed and $I_t$ stays zero.