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The aim of this article is to study the known susceptible-infectious (SI) epidemic model using fractional order reaction-diffusion fractional partial differential equations [FPDEs] in order to better describe the dynamics of a reaction-diffusion SI with a nonlinear incidence rate describing the infection dynamics of the HIV/AIDS virus. We initially examined the nonnegativity, global existence, and boundedness for solutions of the proposed system. After determining that the proposed model has two steady states, we derived sufficient conditions for the global and local asymptotic stability of the equilibrium of the proposed system and their relationship to basic reproduction in the case of fractional ordinary differential equations and FPDEs by analyzing the eigenvalues and using the appropriately chosen Lyapunov function. Finally, we used numerical examples to illustrate our theoretical results.