Site-bond percolation model of epidemic spreading with vaccination in complex networks

To study disease transmission with vaccination based on the network, we map an SIR model to a site-bond percolation model. In the case where the vaccination probability is zero, this model degenerates into a bond percolation model without the immunization. Using the method of generation functions, we obtain exact theoretical results for the epidemic threshold and the average outbreak size. From these exact solutions, we find that the epidemic threshold increases while the average outbreak size decreases with vaccination probability. Numerical simulations show that the size of giant component S increases with transmissibility T but decreases with the probability of vaccination. In addition, we compare the epidemic threshold and the size of the giant component for a Poisson network, an exponential network and a power-law network using numerical simulations. When the probability of vaccination is fixed, the epidemic threshold is the smallest for heterogeneous networks and the size of giant component S in homogeneous networks becomes largest for large transmissibility T(T close to 1).