In this paper, an operator splitting method based on meshless and finite difference procedures, is being considered for numerical solution of compartmental epidemiological population models with and without diffusion. A one step explicit meshless procedure is also applied for the numerical solution of the nonlinear model. The compartmental model contains susceptible, vaccinated, exposed, infected, and recovered (SVEIR) classes of the population. Effects of the diffusion on the simulation results of the model are being studied. Stability of endemic equilibrium point along with bifurcation analysis has also been investigated. Due to non-availability of the exact solution, the numerical results obtained are mutually compared and their correctness is being verified by the theoretical results as well.