  The in-medium $NN\rightarrow N\Delta$ cross section and its differential
cross section in isospin asymmetric nuclear medium are investigated in the
framework of the one-boson exchange model by including the isovector mesons,
i.e., $\delta$ and $\rho$ mesons. Our results show that the in-medium
$NN\rightarrow N\Delta$ cross sections are suppressed with density increasing,
and the differential cross sections become isotropic with the density
increasing at the beam energy around the $\Delta$ threshold energy. The isospin
splitting on the medium correction factor, $R=\sigma_{ NN\rightarrow
N\Delta}^*/\sigma_{NN\rightarrow N\Delta}^{\text{free}}$ is observed for
different channels of $NN\to N\Delta$, especially around the threshold energy
for all the effective Lagrangian parameters. By analyzing the selected
effective Lagrangian parameters, our results show that the larger effective
mass is, the weaker medium correction $R$ is.
