A quasi-particle model has been employed to describe the $(2+1)$-flavor lattice QCD equation of state with physical quark masses. The interaction part of the equation of state has been mapped to the effective fugacities of otherwise non-interacting quasi-gluons and quasi-quarks. The mapping is found to be exact for the equation of state. The model leads to non-trivial dispersion relations for quasi-partons. The dispersion relations, effective quasi-particle number densities, and trace anomaly have been investigated employing the model.A Virial expansion for the EOS has further been obtained to investigate the role of interactions in quark-gluon plasma (QGP). Finally, Debye screening in QGP has been studied employing the model.