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Observations of the Lyman-$α$ (Ly$α$) forest in spectra of distant quasars enable us to probe the matter power spectrum at relatively small scales. With several upcoming surveys, it is expected that there will be a many-fold increase in the quantity and quality of data, and hence it is important to develop efficient simulations to forward model these data sets. One such semi-numerical method is based on the assumption that the baryonic densities in the intergalactic medium (IGM) follow a lognormal distribution. In this work, we test the robustness of the lognormal model of the Ly$α$ forest in recovering a set of IGM parameters by comparing with high-resolution Sherwood SPH simulations. We study the recovery of the parameters $T_0$ (temperature of the mean-density IGM), $γ$ (slope of the temperature-density relation) and $Γ_{12}$ (hydrogen photoionization rate) at $z \sim 2.5$ using a Markov Chain Monte Carlo (MCMC) technique for parameter estimation. Using three flux statistics, the probability distribution, the mean flux and the power spectrum, values of all three parameters, $T_0$, $γ$ and $Γ_{12}$ implied in the SPH simulations are recovered within $1 - σ$ ($\sim$ 9, 4 and 1% respectively) of the median (best-fit) values. We verify the validity of our results at different baryon smoothing filter, SNR, box size & resolution, and data seed and confirm that the lognormal model can be used as an efficient tool for modelling the Ly$α$ transmitted flux at $z \sim 2.5$.