  $Background:$ The determination of the nuclear neutron properties suffers
from uncontrolled uncertainties, which attracted considerable attention
recently, such as in the context of the PREX experiment.
  $Purpose:$ Our aim is to analyze the sensitivity of charge-exchange ($p, n$)
reactions to the neutron density distribution $\rho_{n}$ and constrain the
neutron characteristics in the nuclear structure models.
  $Method:$ By combing the folding and the mean-field models, the
nucleon-nucleus ($NA$) potential can be obtained from the nuclear density
distribution. Further, the ($p, p$) and ($p, n$) cross sections for $^{48}$Ca
and $^{208}$Pb are calculated following the distorted-wave Born approximation
(DWBA) method.
  $Results:$ Compared with the ($p, p$) cross section, the effects of
$\rho_{n}$ variation on the ($p, n$) cross section are significant, which is
due to the impact of isovector properties. Based on the global folding model
analyses of data, it is found that $^{48}$Ca and $^{208}$Pb have relatively
large neutron skin thickness $\Delta R_{n p}$.
  $Conclusions:$ Results illustrate that the charge-exchange ($p, n$) reaction
is a sensitive probe of $\rho_{n}$. The results in this paper can offer useful
guides for future experiments of neutron characteristics.
