We study stationary epidemic processes in scale-free networks with local awareness behavior adopted by only susceptible, only infected, or all nodes. We find that while the epidemic size in the susceptible-aware and the all-aware models scales linearly with the network size, the scaling becomes sublinear in the infected-aware model. Hence, fewer aware nodes may reduce the epidemic size more effectively; a phenomenon reminiscent of Braess's paradox. We present numerical and theoretical analysis, and highlight the role of influential nodes and their disassortativity to raise epidemic awareness.