  Despite observational evidences, InfraRed (IR) excess of classical Cepheids
are seldom studied and poorly understood, but probably induces systematics on
the Period-Luminosity (PL) relation used in the calibration of the
extragalactic distance scale. This study aims to understand the physical origin
of the IR excess found in the spectral energy distribution (SED) of 5 Cepheids
: RS Pup (P=41.46d), zeta Gem (P=10.15d), eta Aql (P=7.18d), V Cen (P=5.49d)
and SU Cyg (P=3.85d). A time series of atmospheric models along the pulsation
cycle are fitted to a compilation of data, including optical and near-IR
photometry, Spitzer spectra (secured at a specific phase), interferometric
angular diameters, effective temperature and radial velocity measurements.
Herschel images in two bands are also analyzed qualitatively. In this fitting
process, based on the SPIPS algorithm, a residual is found in the SED, whatever
the pulsation phase, and for wavelengths larger than about $1.2\mu$m, which
corresponds to the so-determined infrared excess of Cepheids. This IR excess is
then corrected from interstellar medium absorption in order to infer or not the
presence of dust shells, and is finally used in order to fit a model of a shell
of ionized gas. For all Cepheids, we find a continuum IR excess increasing up
to about -0.1 magnitudes at 30$\mu$m, which cannot be explained by a hot or
cold dust model of CircumStellar Environment (CSE). We show, for the first
time, that the IR excess of Cepheids can be explained by free-free emission
from a thin shell of ionized gas, with a thickness of about 15% of the star
radius, a mass of $10^{-9}-10^{-7}$ Msol and a temperature ranging from 3500 to
4500K. This result has to be tested with interferometers operating in visible,
in the mid-IR or in the radio domain. The impact of such CSEs of ionized gas on
the PL relation of Cepheids needs also more investigations.
