In this work, we analyze the stability properties of a recently proposed dynamical system that describes the evolution of the probability of infection in a network. We show that this model can be viewed as a concave game among the nodes. This characterization allows us to provide a simple condition, that can be checked in a distributed fashion, for stabilizing the origin. When the curing rates at the nodes are low, a residual infection stays within the network. Using properties of Hurwitz Mertzel matrices, we show that the residual epidemic state is locally exponentially stable. We also demonstrate that this state is globally asymptotically stable. Furthermore, we investigate the problem of stabilizing the network when the curing rates of a limited number of nodes can be controlled. In particular, we characterize the number of controllers required for a class of undirected graphs. Several simulations demonstrate our results.