  In recent years, discrete fractional epidemic models with reaction-diffusion
have become increasingly popular in the literature, not only for its necessity
of numerical simulation, but also for its defined physical processes. In this
paper, by second order central difference scheme and L1 nonstandard finite
difference scheme, a discrete counterpart of time-fractional reaction-diffusion
epidemic model with generalized incidence rate is considered. More importantly,
the main idea in choosing an nonstandard finite difference scheme is to obtain
unconditionally positivity in the proposed system, which leads to the proposal
of the discrete epidemic model with time delay. Furthermore, the global
properties of the proposed discrete system are studied, including the global
boundedness of positive solutions, the existence and the global stability of
equilibrium points, which are consistent with the corresponding continuous
systems. Meanwhile, it shows that L1 nonstandard finite difference scheme and
second order central difference scheme can keep the properties of the
corresponding continuous system well. It is worth noting that, different from
discrete epidemic model with integer-order, the memory Lyapunov function is
constructed in this paper, which depends on the previous historical information
of the proposed system. This is consistent with the non-local property of
Caputo fractional derivatives. Finally, numerical results are given to verify
the theoretical results.
