A spatial and delayed epidemic model with media coverage is studied, and the global asymptotically stability of the equilibria is proved. Furthermore, a discreted edition of this model is established by applying a nonstandard finite difference scheme which has the same dynamics as the original system. Moveover, it is shown that the global stability of the disease-free equilibrium, and the globally asymptotically stability of the endemic equilibrium is studied by constructing a suitable Lyapunov function. Finally, the NSFD scheme can be well simulated to support the theoretical results.