  Based on the quasi-particle description of the QCD medium at finite
temperature and density we formulate the phenomenological model for the
equation of state that exhibits crossover or the first order deconfinement
phase transition. The models are constructed in such a way to be
thermodynamically consistent and to satisfy the properties of the ground state
nuclear matter comply with constraints from intermediate heavy--ion collision
data. Our equations of states show quite reasonable agreement with the recent
lattice findings on temperature and baryon chemical potential dependence of
relevant thermodynamical quantities in the parameter range covering both the
hadronic and quark--gluon sectors. The model predictions on the isentropic
trajectories in the phase diagram are shown to be consistent with the recent
lattice results. Our nuclear equations of states are to be considered as an
input to the dynamical models describing the production and the time evolution
of a thermalized medium created in heavy ion collisions in a broad energy range
from SIS up to LHC.
