  Following \cite{ipel1}, we consider a nonlinear SIS-type nonlocal system
describing the spread of epidemics on networks, assuming nonlimited
transmission, We prove local existence of a unique solution for any diffusion
coefficients and global existence in the case of equal diffusion coefficients.
Next we study the asymptotic behaviour of the solution and show that the
disease-free equilibrium (DFE) is linearly and globally asymptotically stable
when the total mean population is small. Finally, we prove that the solution of
the system converge to the $DFE$.
