  In this work, we use the spectral properties of graphons to study stability
and sensitivity to noise of deterministic SIS epidemics over large networks. We
consider the presence of additive noise in a linearized SIS model and we derive
a noise index to quantify the deviation from the disease-free state due to
noise. For finite networks, we show that the index depends on the adjacency
eigenvalues of its graph. We then assume that the graph is a random sample from
a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of
the associated graphon operator, we find an approximation of the index that is
tight when the network size goes to infinity. A numerical example is included
to illustrate the results.
