  We study the interaction between epidemic spreading and a vaccination
process. We assume that, similar to the disease spreading, also the vaccination
process occurs through direct contact, i.e., it follows the standard
susceptible-infected-susceptible (SIS) dynamics. The two competing processes
are asymmetrically coupled as vaccinated nodes can directly become infected at
a reduced rate with respect to susceptible ones. We study analytically the
model in the framework of mean-field theory finding a rich phase-diagram. When
vaccination provides little protection toward infection, two continuous
transitions separate a disease-free immunized state from vaccinated-free
epidemic state, with an intermediate mixed state where susceptible, infected
and vaccinated individuals coexist. As vaccine efficiency increases, a
tricritical point leads to a bistable regime and discontinuous phase
transitions emerge. Numerical simulations for homogeneous random networks agree
very well with analytical predictions.
