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An epidemic that spreads in a network calls for a decision on the part of the network users. They have to decide whether to protect themselves or not. Their decision depends on the tradeoff between the perceived infection and the protection cost. Aiming to help users reach an informed decision, security advisories provide periodic information about the infection level in the network. We study the best-response dynamic in a network whose users repeatedly activate or de-activate security, depending on what they learn about the infection level. Our main result is the counterintuitive fact that the equilibrium level of infection increases as the users' learning rate increases. The same is true when the users follow smooth best-response dynamics, or any other continuous response function that implies higher probability of protection when learning a higher level of infection. In both cases, we characterize the stability and the domains of attraction of the equilibrium points. Our finding also holds when the epidemic propagation is simulated on human contact traces, both when all users are of the same best-response behavior type and when they are of two distinct behavior types.