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A susceptible-infectious-susceptible reaction-diffusion equation is used to model the spatial spread of a disease. It is shown that the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when the basic reproduction number is less than or equal to 1 and unstable when it is greater than 1. In the latter case, there exists an endemic equilibrium (EE) which is GAS. We construct nonstandard finite difference (NSFD) schemes which theoretically and computationally replicate the stability properties of the equilibria.Key words: Disease-free equilibrium, endemic equilibrium, nonstandard finite differencemethod, fixed point, locally asymptotically stable, globally asymptotically stable.