We consider the problem of decontaminating an infected network using as few mobile cleaning agents as possible and avoiding recontamination. After a cleaning agent has left a vertex v, this vertex will become recontaminated if m or more of its neighbours are infected, where m ≥ 1 is a threshold parameter of the system indicating the local immunity level of the network. This network decontamination problem, also called monotone connected graph search and intruder capture, has been extensively studied in the literature when m = 1 (no immunity). In this paper, we extend these investigations and consider for the first time the network decontamination problem when the parameter m is an arbitrary in