  In this paper, the in-medium $NN\rightarrow N\Delta$ cross section is
calculated in the framework of the one-boson exchange model by including the
isovector mesons, i.e. $\delta$ and $\rho$ mesons. Due to the isospin exchange
in the $NN\rightarrow N\Delta$ process, the vector self-energies of the
outgoing particles are modified relative to the incoming particles in isospin
asymmetric nuclear matter, and it leads to the effective energies of the
incoming $NN$ pair being different from the outgoing $N\Delta$ pair. This
effect is investigated in the calculation of the in-medium $NN\rightarrow
N\Delta$ cross section. With the corrected energy conservation, the cross
sections of the $\Delta^{++}$ and $\Delta^+$ channels are suppressed, and the
cross sections of the $\Delta^0$ and $\Delta^-$ channels are enhanced relative
to the results obtained without properly considering the potential energy
changes. Our results further confirm the dependence of medium correction
factor, $R=\sigma_{ NN\rightarrow N\Delta}^*/\sigma_{NN\rightarrow
N\Delta}^{\text{free}}$, on the charge state of $NN\rightarrow N\Delta$
especially around the threshold energy, but the isospin splitting of medium
correction factor $R$ becomes weak at high beam energies.
