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We study evolution of single subhaloes with their masses of $\sim10^9M_\odot$ in a Milky-Way-sized host halo for self-interacting dark matter (SIDM)models. We perform dark-matter-only N-body simulations of dynamical evolutionof individual subhaloes orbiting its host by varying self-scattering crosssections (including a velocity-dependent scenario), subhalo orbits, andinternal properties of the subhalo. We calibrate a gravothermal fluid model topredict time evolution in spherical mass density profiles of isolated SIDMhaloes with the simulations. We find that tidal effects of SIDM subhaloes canbe described with a framework developed for the case of collision-less colddark matter (CDM), but a shorter typical time scale for the mass loss due totidal stripping is required to explain our SIDM simulation results. As long asthe cross section is less than $\sim10\, \mathrm{cm}^2/\mathrm{g}$ and initialstates of subhaloes are set within a $2\sigma$-level scatter at redshifts of$\sim2$ predicted by the standard $\Lambda$CDM cosmology, our simulations donot exhibit a prominent feature of gravothermal collapse in the subhalo centraldensity for 10 Gyr. We develop a semi-analytic model of SIDM subhaloes in atime-evolving density core of the host with tidal stripping and self-scatteringram pressure effects. Our semi-analytic approach provides a simple, efficientand physically-intuitive prediction of SIDM subhaloes, but further improvementsare needed to account for baryonic effects in the host and the gravothermalinstability accelerated by tidal stripping effects.

Modelling self-interacting dark matter substructures I: Calibration with N-body simulations of a Milky-Way-sized halo and its satellite