  We combine the semi-analytical structure formation model, SASHIMI, which
predicts subhalo populations in collisionless, cold dark matter (CDM), with a
parametric model that maps CDM halos to self-interacting dark matter (SIDM)
halos. The resulting model, SASHIMI-SIDM, generates SIDM subhalo populations
down to sub-galactic mass scales, for an arbitrary input cross section, in
minutes. We show that SASHIMI-SIDM agrees with SIDM subhalo populations from
high-resolution cosmological zoom-in simulations in resolved regimes.
Crucially, we predict that the fraction of core-collapsed subhalos peaks at a
mass scale determined by the input SIDM cross section and decreases toward
higher halo masses, consistent with the predictions of gravothermal models and
cosmological simulations. For the first time, we also show that the
core-collapsed fraction decreases toward lower halo masses; this result is
uniquely enabled by our semi-analytical approach. As a proof of principle, we
apply SASHIMI-SIDM to predict the boost to the local dark matter density and
annihilation rate from core-collapsed SIDM subhalos, which can be enhanced
relative to CDM by an order of magnitude for viable SIDM models. Thus,
SASHIMI-SIDM provides an efficient and reliable tool for scanning SIDM
parameter space and testing it with astrophysical observations. The code is
publicly available at https://github.com/shinichiroando/sashimi-si.
