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The isospin splitting of the in-medium $NN\rightarrow N\Delta$ cross sectionsin asymmetric nuclear medium are investigated in the framework of the one-bosonexchange model by including $\delta$ and $\rho$ mesons. Our results show thatthe medium correction factors $R=\sigma_{ NN\rightarrowN\Delta}^*/\sigma_{NN\rightarrow N\Delta}^{\text{free}}$ have $R_{pp \ton\Delta ^{++}} < R_{nn \to p\Delta ^{-}}$ and $R_{NN \to N\Delta ^{+}} <R_{NN\to N\Delta ^{0}}$ by using the without-$\delta$ sets. By including the$\delta$ meson, it appears the totally opposite results in the $R$ fordifferent channels, i.e., $R_{pp \to n\Delta ^{++}} > R_{nn \to p\Delta ^{-}}$and $R_{NN \to N\Delta ^{+}} >R_{NN \to N\Delta ^{0}}$.

The influence of $\delta$ meson on the isospin splitting of in-medium $NN\to N\Delta$ cross sections