  We constrain the H I photoionization rate $(\Gamma_{\rm HI})$ at $z \lesssim
0.45$ by comparing the flux probability distribution function and power
spectrum of the Ly-$\alpha$ forest data along 82 QSO sightlines obtained using
Cosmic Origins Spectrograph with models generated from smoothed particle
hydrodynamic simulations. We have developed a module named "Code for Ionization
and Temperature Evolution (CITE)" for calculating the intergalactic medium
(IGM) temperature evolution from high to low redshifts by post-processing the
GADGET-2 simulation outputs. Our method, that produces results consistent with
other simulations, is computationally less expensive thus allowing us to
explore a large parameter space. It also allows rigorous estimation of the
error covariance matrix for various statistical quantities of interest. We find
that the best-fit $\Gamma_{\rm HI}(z)$ increases with $z$ and follows $(4 \pm
0.1) \times 10^{-14}\:(1+z)^{4.99 \pm 0.12}$ s$^{-1}$. At any given $z$ the
typical uncertainties $\Delta \Gamma_{\rm HI} / \Gamma_{\rm HI}$ are $\sim 25$
per cent which contains not only the statistical errors but also those arising
from possible degeneracy with the thermal history of the IGM and cosmological
parameters and uncertainties in fitting the QSO continuum. These values of
$\Gamma_{\rm HI}$ favour the scenario where only QSOs contribute to the
ionizing background at $z<2$. Our derived $3\sigma$ upper limit on average
escape fraction is $0.008$, consistent with measurements of low-$z$ galaxies.
