An epidemic spreading in a network calls for a decision on the part of the network members: They should decide whether to protect themselves or not. Their decision depends on the trade-off between their perceived risk of being infected and the cost of being protected. The network members can make decisions repeatedly, based on information that they receive about the changing infection level in the network. We study the equilibrium states reached by a network whose members increase (resp. decrease) their security deployment when learning that the network infection is widespread (resp. limited). Our main finding is that the equilibrium level of infection increases as the learning rate of the members increases. We confirm this result in three scenarios for the behavior of the members: strictly rational cost minimizers, not strictly rational, and strictly rational but split into two response classes. In the first two cases, we completely characterize the stability and the domains of attraction of the equilibrium points, even though the first case leads to a differential inclusion. We validate our conclusions with simulations on human mobility traces.