Targeted immunisation thresholds for the contact process on power-law trees

Scale-free configuration models are intimately connected to power law Galton–Watson trees. It is known that contact process epidemics can propagate on these trees and therefore these networks with arbitrarily small infection rate, and this continues to be true after uniformly immunizing a small positive proportion of vertices. So, we instead immunize those with largest degree: above a threshold for the maximum permitted degree, we discover the epidemic with immunization has survival probability similar to without, by duality corresponding to comparable metastable density. With maximal degree below a threshold on the same order, this survival probability is severely reduced or zero.