In this work, we investigate the nonlocal time-delayed andreaction-diffusion epidemic model studied by Guo et al. [Z. Guo, F. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latentperiod and non-local infections, J. Math. Biol. 65 (2012) 1387–410]. In thecase that the coefficients are independent of the spatial variable, we obtainthe global stability of the disease-free equilibrium and the unique endemicequilibrium, which partially answers the open problem proposed by Guo et al.