  Contact networks can significantly change the course of epidemics, affecting
the rate of new infections and the mean size of an outbreak. Despite this
dependence, some characteristics of epidemics are not contingent on the contact
network and are probably predictable based only on the pathogen. Here we
consider SIR-like pathogens and give an elementary proof that for any network
increasing the probability of transmission increases the mean outbreak size. We
also introduce a simple model, termed 2FleeSIR, in which susceptibles protect
themselves by avoiding contacts with infectees. The 2FleeSIR model is
non-monotonic: for some networks, increasing transmissibility actually
decreases the final extent. The dynamics of 2FleeSIR are fundamentally
different from SIR because 2FleeSIR exhibits no outbreak transition in
densely-connected networks. We show that in non-monotonic epidemics, public
health officials might be able to intervene in a fundamentally new way to
change the network so as to control the effect of unexpectedly-high virulence.
However, interventions that decrease transmissibility might actually cause more
people to become infected.
