  Classical Cepheids are primary distance indicators playing a fundamental role
in the calibration of the extragalactic distance scale. The possible dependence
of their characteristic Period-Luminosity (PL) relation on chemical composition
is still debated in the literature, and the behaviour of these pulsators at
very low metallicity regimes is almost unexplored. In order to derive
constraints on the application of the Period-Luminosity relation at low metal
abundances, we investigate the properties of the few ultra-low metallicity (Z ~
0.0004) Cepheids recently discovered in the Blue Compact Dwarf galaxy IZw18. To
this purpose we have computed an updated and extended set of nonlinear
convective models for Z=0.0004 and Y=0.24, spanning a wide range of stellar
masses, and taking into account the evolutionary constraints for selected
luminosity levels. As a result we are able to predict the topology of the
instability strip, the variations of all the relevant quantities along the
pulsation cycle, including the morphology of the light curves, the theoretical
Period-Luminosity-Color, Period-Wesenheit and Period-Luminosity relations at
such a low metallicity. For each of these relations we provide the appropriate
coefficients for fundamental mode pulsators with Z=0.0004. By comparing these
results with the properties of more metal rich Cepheids we find that the
synthetic PL relations for Z=0.0004 are steeper than at higher Z, but similar
to the Z=0.004 ones, thus suggesting a leveling off of the metallicity effect
towards the lowest Zs.
