  JHKs photometry is presented from a three-year survey of the central regions
of the Local Group dwarf irregular galaxy IC1613. The morphologies of the
colour-magnitude and colour-colour diagrams are discussed with particular
reference to the supergiants and M- and C-type asymptotic giant branch (AGB)
stars. Mean JHKs magnitudes, amplitudes and periods are given for five O-rich
and nine C-rich Mira variables for which bolometric magnitudes are also
estimated. A distance of 750 kpc ($(m-M)_0=24.37\pm 0.08$ mag) is derived for
IC1613 by fitting a period-luminosity relation to the C-rich Miras. This is in
agreement with values from the literature. The AGB stars exhibit a range of
ages. A comparison with theoretical isochrones suggests that four luminous
O-rich Miras are as young as $2\times 10^8$ yrs. One of these has a lithium
absorption line in its spectrum, demonstrating that it is undergoing hot bottom
burning (HBB). This supports the idea that HBB is the cause of the high
luminosity of these AGB stars, which puts them above the fundamental
period-luminosity (PL) relation. Further studies of similar stars, selected
from their positions in the PL diagram, could provide insight into HBB. A much
fainter, presumed O-rich, Mira is similar to those found in Galactic globular
clusters. The C Miras are of intermediate age.
  The O-rich variables are not all recognized as O-rich, or even as AGB stars,
on the basis of their J-Ks colour. It is important to appreciate this when
using near-infrared surveys to classify AGB stars in more distant galaxies.
