We consider an SIR model with nonlinear contact and vaccination rates influenced by media information. In the theoretical analysis, we include the well-posedness of the system, computation of the basic reproduction number, and local stability of disease-free and disease-persistent fixed points. Further, we construct a nonstandard finite difference scheme (NSFD) for numerically establishing the results. The NSFD system is shown to maintain positivity for all positive initial values and preserves the conservation law. We also find that the NSFD method shares the same fixed points and stability criteria as the corresponding continuous model. We also compare the method with existing finite difference schemes to demonstrate its superior performance.