  We aim to investigate the IR excess of 45 MW Cepheids combining different
observables to constrain the presence of CSE. We used the SPIPS algorithm, a
robust implementation of the parallax-of-pulsation method that combines
photometry, angular diameter, stellar effective temperature, and RV
measurements in a global modelling of the pulsation. We obtained new
photometric measurements at mid-IR with the VLT/VISIR complemented with
literature data. We then compared the mean magnitude from 0.5$\mu$m to 70$\mu$m
with stellar atmosphere models to infer the IR excess, which we attribute to
the presence of a CSE. We report that at least 29% of our sample have a
detected IR excess. We estimated a mean excess of 0.08mag in K and 0.13mag in
N. Other Cepheids possibly have IR excess, but they were rejected due to their
low detection level compared to a single-star model. We do not see any
correlation between the IR excess and the pulsation period as previously
suspected for MW Cepheids, but a rather constant trend. We also do not find any
correlation between the CO absorption and the presence of a CSE, but rather
with the stellar effective temperature, which confirms that the CO features
previously reported are mostly photospheric. No bias caused by the presence of
the CSE is detected on the average distance estimates from a SPIPS analysis
with a fitted colour excess. We also do not find correlation between the
presence of IR excess and the evolution stage. We report a fraction of 29% of
Cepheids with an IR excess likely produced by the CSE. Longer period Cepheids
do not exhibit greater excess than short periods as previously suspected from
observations and theoretical dusty-wind models. Other mechanisms such as
free-free emission, among others, may be at the origin of their formation. We
also show that not fitting the colour excess leads to a bias on the distance
estimates in our Galaxy.

Solar and Stellar Astrophysics