An age structured $s$-$i$-$s$ epidemic model with random diffusion
is studied. The model is described by the system of nonlinear and
nonlocal integro-differential equations. Finite differences along
the characteristics in age-time domain combined with Galerkin
finite elements in spatial domain are used in the approximation.
It is shown that a positive periodic solution to the discrete
system resulting from the approximation can be generated, if the
initial condition is fertile. It is proved that the endemic
periodic solution is globally stable once it exists.