  A significant number of active galactic nuclei (AGNs) are observed to be
hidden behind dust and gas. The distribution of material around AGNs plays an
important role in modeling the cosmic X-ray background (CXB), especially the
fraction of Type-2 AGNs ($f_2$). One of the possible explanations for the
obscuration in Seyfert galaxies at intermediate redshift is dusty starburst
discs. We compute the 2D hydrostatic structure of 768 nuclear starburst discs
(NSDs) under various physical conditions and also the distribution of column
density along the line of sight ($N_{\text{H}}$) associated with these discs.
Then, the $N_{\text{H}}$ distribution is evolved with redshift by using the
redshift dependent distribution function of input parameters. The $f_2$ shows a
strong positive evolution up to $z=2$, but only a weak level of enhancement at
higher $z$. The Compton-thin and Compton-thick AGN fractions associated with
these starburst regions increase as $\propto (1+z)^{\delta}$ where the $\delta$
is estimated to be 1.12 and 1.45, respectively. The reflection parameter $R_f$
associated with column density $N_{\text{H}} \geq 10^{23.5}$ cm$^{-2}$ extends
from 0.13 at $z=0$ to 0.58 at $z=4$. A CXB model employing this evolving
$N_{\text{H}}$ distribution indicates more compact ($R_{\text{out}}<120$ pc)
NSDs provide a better fit to the CXB. In addition to "Seyfert-like" AGNs
obscured by nuclear starbursts, we predict that 40 to 60 per cent of quasars
must be Compton-thick to produce the peak of the CXB spectrum within
observational uncertainty. The predicted total number counts of AGNs in 8-24
keV band are in fair agreement with observations from NuSTAR.
