  We have worked out predictions for the radio counts of star-forming galaxies
down to nJy levels, along with redshift distributions down to the detection
limits of the phase 1 Square Kilometer Array MID telescope (SKA1-MID) and of
its precursors. Such predictions were obtained by coupling epoch dependent star
formation rate (SFR) functions with relations between SFR and radio
(synchrotron and free-free) emission. The SFR functions were derived taking
into account both the dust obscured and the unobscured star-formation, by
combining far-infrared (FIR), ultra-violet (UV) and H_alpha luminosity
functions up to high redshifts. We have also revisited the South Pole Telescope
(SPT) counts of dusty galaxies at 95\,GHz performing a detailed analysis of the
Spectral Energy Distributions (SEDs). Our results show that the deepest
SKA1-MID surveys will detect high-z galaxies with SFRs two orders of magnitude
lower compared to Herschel surveys. The highest redshift tails of the
distributions at the detection limits of planned SKA1-MID surveys comprise a
substantial fraction of strongly lensed galaxies. We predict that a survey down
to 0.25 microJy at 1.4 GHz will detect about 1200 strongly lensed galaxies per
square degree, at redshifts of up to 10. For about 30% of them the SKA1-MID
will detect at least 2 images. The SKA1-MID will thus provide a comprehensive
view of the star formation history throughout the re-ionization epoch,
unaffected by dust extinction. We have also provided specific predictions for
the EMU/ASKAP and MIGHTEE/MeerKAT surveys.
