The structure of a network can significantly affect the course of infections on it. For example, a human-to-human contact network affects the epidemiology of infectious diseases, affecting both the rate of new infections and the sizes of outbreaks. Related results are also known for infrastructure systems like communication and power transmission systems that experience cascading breakdowns. Despite this dependence, some characteristics of outbreaks are predictable based only on the infection being transmitted. Here we consider SIR-like infections, and give an elementary proof that for any network, increasing the probability of transmission monotonically increases the mean outbreak size. We also introduce a simple model, termed 2FleeSIR, in which susceptibles protect themselves by avoiding contacts with infectees. The dynamics of 2FleeSIR are fundamentally different from SIR dynamics because 2FleeSIR seems to exhibit no outbreak transition in densely-connected networks. Moreover, 2FleeSIR exhibits non-monotonic phenomena: for some networks, increasing transmissibility actually decreases the final extent. 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 transmissibility. However, interventions that decrease transmissibility might actually cause more people to become infected.
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