A two-dimensional spatially independent rate equation model of vertical-cavity surface-emitting lasers (VCSELs) is derived and then used to analysis the multi-mode behaviour of VCSELs. The transverse mode characteristics of VCSELs, the carrier distribution in both the radial and the azimuthal directions, and the effects of the azimuthal non-uniformity of the injection current on the transverse mode behaviours are investigated in detail. By using both Bessel and Fourier expansion of carrier density, the 2D spatially independent rate equations for transverse mode are formulated, which take into account carrier diffusion both in the radial and in the azimuthal direction as well as gain non-uniformity in the lateral direction. The equations are numerical solved self-consistently using the Runge-Kutta method for different spatial periodic injection current. Results show that a proper current injection profile can separate the sine mode and cosine mode of the same order transverse modes observably. It is found that an injection current with periodic change in the azimuthal direction is favourable for the excitation of the modes whose mode profile match the current profile best. The results are useful to the design and control of transverse mode characteristics of a VCSEL.