  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 immunising a small positive
proportion of vertices. So, we instead immunise those with largest degree:
above a threshold for the maximum permitted degree, we discover the epidemic
with immunisation 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.
