In this paper, we study the dynamics of epidemic processes taking place in adaptive networks of arbitrary topology. We focus our study on the adaptive susceptible-infected-susceptible (ASIS) model, where healthy individuals are allowed to temporarily cut edges connecting them to infected nodes in order to prevent the spread of the infection. In this paper, we derive a closed-form expression for a lower bound on the epidemic threshold of the ASIS model in arbitrary networks with heterogeneous node and edge dynamics. For networks with homogeneous node and edge dynamics, we show that the resulting \blue{lower bound} is proportional to the epidemic threshold of the standard SIS model over static networks, with a proportionality constant that depends on the adaptation rates. Furthermore, based on our results, we propose an efficient algorithm to optimally tune the adaptation rates in order to eradicate epidemic outbreaks in arbitrary networks. We confirm the tightness of the proposed lower bounds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures.