  We consider a SIR model with vaccination strategy on a sparse configuration
model random graph. We show the convergence of the system when the number of
nodes grows and characterize the scaling limits. Then, we prove the existence
of optimal controls for the limiting equations formulated in the framework of
game theory, both in the centralized and decentralized setting.
  We show how the characteristics of the graph (degree distribution) influence
the vaccination efficiency for optimal strategies, and we compute the limiting
final size of the epidemic depending on the degree distribution of the graph
and the parameters of infection, recovery and vaccination. We also present
several simulations for two types of vaccination, showing how the optimal
controls allow to decrease the number of infections and underlining the crucial
role of the network characteristics in the propagation of the disease and the
vaccination program.
