Global Stability of a Nonlocal Epidemic Model with Delay

In this work, we investigate the nonlocal time-delayed and reaction-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 latent period and non-local infections, J. Math. Biol. 65 (2012) 1387-410]. In the case that the coefficients are independent of the spatial variable, we obtain the global stability of the disease-free equilibrium and the unique endemic equilibrium, which partially answers the open problem proposed by Guo et al.