Competition between vaccination and disease spreading.

We study the interaction between epidemic spreading and a vaccination process. We assume that, similar to the disease spreading, the vaccination process also 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.