  The inclusive reactions $pp \rightarrow e^+ e^- X$ and $np \rightarrow e^+
e^- X$ at the laboratory kinetic energy of 1.25 GeV are investigated in a model
of dominance of nucleon and $\Delta$ resonances. Experimental data for these
reactions have recently been reported by the HADES Collaboration. In the
original model, the dileptons are produced either from the decays of nucleon
and $\Delta$ resonances $R \rightarrow N e^+ e^-$ or from the Dalitz decays of
$\pi^0$- and $\eta$-mesons created in the $R \to N\pi^0$ and $R \to N\eta$
decays. We found that the distribution of dilepton invariant masses in the $pp
\rightarrow e^+ e^- X$ reaction is well reproduced by the contributions of $R
\rightarrow N e^+ e^-$ decays and $R \rightarrow N \pi^0$, $\pi^0 \to \gamma
e^+e^-$ decays. Among the resonances, the predominant contribution comes from
the $\Delta(1232)$, which determines both the direct decay channel $R
\rightarrow N e^+ e^-$ and the pion decay channel. In the collisions $np
\rightarrow e^+ e^- X$, additional significant contributions arise from the
$\eta$-meson Dalitz decays, produced in the $np \rightarrow np\eta$ and $np
\rightarrow d\eta$ reactions, the radiative capture $np \rightarrow d e^+ e^-$,
and the $np \rightarrow np e^+ e^-$ bremsstrahlung. These mechanisms may partly
explain the strong excess of dileptons in the cross section for collisions of
$np$ versus $pp$, which ranges from 7 to 100 times for the dilepton invariant
masses of 0.2 to 0.5 GeV.
