The formulations and numerical algorithms of a three-level model for studying the Purcell effect produced by the scattering of an air/AlGaN interface and the surface plasmon (SP) coupling effect induced by a surface Al nanoparticle in a two-polarization emission system to simulate the transverse-electric- (TE-) and transverse-magnetic- (TM-) polarized emissions in an AlxGa1-xN/AlyGa1-yN (y > x) quantum well (QW) are built. In reasonably selected ranges of Al content for an AlGaN QW to emit deep-ultraviolet (UV) light, the enhancement (suppression) of TE- (TM-) polarized emission is mainly caused by the SP-coupling (interface-scattering) effect. Different from a two two-level model, in the three-level model the TE- and TM-polarized emissions compete for electron in the shared upper state, which is used for simulating the conduction band, such that either interface-scattering or SP-coupling effect becomes weaker. In a quite large range of emission wavelength, in which the intrinsic emission is dominated by TM polarization, with the interface-scattering and SP-coupling effects, the TE-polarized emission becomes dominant for enhancing the light extraction efficiency of a deep-UV light-emitting diode.