  Starting from a stochastic individual-based description of an SIS epidemic
spreading on a random network, we study the dynamics when the size n of the
network tends to infinity. We recover in the limit an infinite-dimensional
integro-differential equation studied by Delmas, Dronnier and Zitt (2022) for
an SIS epidemic propagating on a graphon. Our work covers the case of dense and
sparse graphs, provided that the number of edges grows faster than n, but not
the case of very sparse graphs with O(n) edges. In order to establish our limit
theorem, we have to deal with both the convergence of the random graphs to the
graphon and the convergence of the stochastic process spreading on top of these
random structures: in particular, we propose a coupling between the process of
interest and an epidemic that spreads on the complete graph but with a modified
infection rate.
  Keywords: random graph, mathematical models of epidemics, measure-valued
process, large network limit, limit theorem, graphon.
