The Isaac Newton Telescope monitoring survey of Local Group dwarf
  galaxies. II. The star formation history of Andromeda I derived from long
  period variables
  An optical monitoring survey in the nearby dwarf galaxies was carried out
with the 2.5-m Isaac Newton Telescope (INT). 55 dwarf galaxies and four
isolated globular clusters in the Local Group (LG) were observed with the Wide
Field Camera (WFC). The main aims of this survey are to identify the most
evolved asymptotic giant branch (AGB) stars and red supergiants at the
end-point of their evolution based on their pulsational instability, use their
distribution over luminosity to reconstruct the star formation history (SFH),
quantify the dust production and mass loss from modelling the multi-wavelength
spectral energy distributions, and relate this to luminosity and radius
variations. In this second of a series of papers, we present the methodology
used to estimate SFH based on long-period variable (LPV) stars and then derive
it for Andromeda\,I (And\,I) dwarf galaxy as an example of the survey. Using
our identified 59 LPV candidates within two half-light radii of And\,I and
Padova stellar evolution models, we estimated the SFH of this galaxy. A major
epoch of star formation occurred in And\,I peaking around 6.6 Gyr ago, reaching
$0.0035\pm0.0016$ M$_\odot$ yr$^{-1}$ and only slowly declining until 1--2 Gyr
ago. The presence of some dusty LPVs in this galaxy corresponds to a slight
increase in recent star formation peaking around 800 Myr ago. We evaluate a
quenching time around 4 Gyr ago ($z<0.5$), which makes And\,I a late-quenching
dSph. A total stellar mass $(16\pm7)\times10^6$ M$_\odot$ is calculated within
two half-light radii of And\,I for a constant metallicity $Z=0.0007$.
