  We present new theoretical period-luminosity-metallicity (PLZ) relations for
RR Lyrae stars (RRL) at Spitzer and WISE wavelengths. The PLZ relations were
derived using nonlinear, time-dependent convective hydrodynamical models for a
broad range in metal abundances (Z=0.0001 to 0.0198). In deriving the light
curves, we tested two sets of atmospheric models (Brott & Hauschildt 2005,
Castelli & Kurucz 2003) and found no significant difference between the
resulting mean magnitudes. We also compare our theoretical relations to
empirical relations derived from RRL in both the field and in the globular
cluster M4. Our theoretical PLZ relations were combined with multi-wavelength
observations to simultaneously fit the distance modulus, mu_0, and extinction,
Av, of both the individual Galactic RRL and of the cluster M4. The results for
the Galactic RRL are consistent with trigonometric parallax measurements from
Gaia's first data release. For M4, we find a distance modulus of $\mu_0=11.257
\pm 0.035$ mag with $A_V = 1.45 \pm 0.12$ mag, which is consistent with
measurements from other distance indicators. This analysis has shown that when
considering a sample covering a range of iron abundances, the metallicity
spread introduces a dispersion in the PL relation on the order of 0.13 mag.
However, if this metallicity component is accounted for in a PLZ relation, the
dispersion is reduced to ~0.02 mag at MIR wavelengths.
