  In the last decade a Quasi-Particle Model ($QPM$) has supplied the basis for
the study of HQ production in ultra-relativistic AA collisions, allowing for a
phenomenological estimate of the HQ diffusion coefficient $D_s(T)$. Taking
advantage of the new lattice QCD results for the Equation of State (EoS) with
2+1+1 dynamical flavors, we extend our $QPM$ approach from $N_f=2+1$ to
$N_f=2+1+1$, in which the charm quark is included. Given an effective coupling
$g(T)$ fixed by a fit to the lQCD energy density $\epsilon(T)$, we evaluate the
impact of different temperature parametrizations of charm quark mass on EoS and
susceptibilities $\chi_q(T)$ of light, $\chi_s(T)$ of strange and $\chi_c(T)$
of charm quarks, the last favouring a charm quark mass increasing toward $T_c$.
We also explore the extension of the $QPM$ approach to a more realistic
approach, that we label $QPM_p$, in which quark and gluon masses explicitly
depend on their momentum converging to the current quark mass at high momenta,
as expected from asymptotic free dynamics. The $QPM_p$ is seen to allow for a
simultaneous quantitative description not only of the EoS but also of the quark
susceptibilities ($\chi_q(T)$, $\chi_s(T)$), which instead are underestimated
in the simple $QPM$ model. Furthermore, evaluating the spatial diffusion
coefficient $2\pi T D_s(T)$ in the $QPM_p$, we find it is also significantly
closer than $QPM$ to the recent lQCD data performed including dynamical
fermions. Finally, in a 1+1D expanding system, we evaluate the $R_{AA}(p_T)$ in
the $QPM$ and $QPM_p$, finding a significant reduction at low momenta for
$QPM_p$ which could lead in a realistic scenario to a better agreement to
experimental data.
