  Thanks to the successful performance of the James Webb Space Telescope, our
understanding of the epoch of reionization of the Universe has been advanced.
The ultraviolet luminosity functions (UV LFs) of galaxies span a wide range of
redshift, not only revealing the connection between galaxies and dark matter
(DM) halos but also providing the information during reionization. In this
work, we develop a model connecting galaxy counts and apparent magnitude based
on UV LFs, which incorporates redshift-dependent star formation efficiency
(SFE) and corrections for dust attenuation. By synthesizing some observations
across the redshift range $4\le z \le 10$ from various galaxy surveys, we
discern the evolving SFE with increasing redshift and DM halo mass through
model fitting. Subsequent analyses indicate that the Thomson scattering optical
depth was $\tau_{\rm e} = 0.052^{+0.003}_{-0.002}$ and the epoch of
reionization started (ended) at $z=20.58^{+6.25}_{-6.75}$
($z=5.38^{+0.65}_{-0.70}$) which is insensitive to the choice of the truncated
magnitude of the UV LFs. Incorporating additional dataset and some reasonable
constraints, the amplitude of matter perturbation is found to be
$\sigma_8=0.79\pm0.05$, which is consistent with the standard $\Lambda$CDM
model. Future galaxy surveys and the dynamical simulations of galaxy evolution
will break the degeneracy between SFE and cosmological parameters, improving
the accuracy and the precision of the UV LF model further.
