  It is the main purpose of this paper to introduce a graph-valued stochastic
process in order to model the spread of a communicable infectious disease. The
major novelty of the SIR model we promote lies in the fact that the social
network on which the epidemics is taking place is not specified in advance but
evolves through time, accounting for the temporal evolution of the interactions
involving infective individuals. Without assuming the existence of a fixed
underlying network model, the stochastic process introduced describes, in a
flexible and realistic manner, epidemic spread in non-uniformly mixing and
possibly heterogeneous populations. It is shown how to fit such a
(parametrised) model by means of Approximate Bayesian Computation methods based
on graph-valued statistics. The concepts and statistical methods described in
this paper are finally applied to a real epidemic dataset, related to the
spread of HIV in Cuba in presence of a contact tracing system, which permits
one to reconstruct partly the evolution of the graph of sexual partners
diagnosed HIV positive between 1986 and 2006.
