This article studies epidemic processes over discrete-time periodic time-varying networks. We focus on the susceptible-infected-susceptible (SIS) model that accounts for a (possibly) mutating virus. We say that an agent is in the disease-free state if it is not infected by the virus. Our objective is to devise a control strategy which ensures that all agents in a network exponentially (respectively asymptotically) converge to the disease-free equilibrium (DFE). Toward this end, we first provide 1) sufficient conditions for exponential (respectively, asymptotic) convergence to the DFE and 2) a necessary and sufficient condition for asymptotic convergence to the DFE. The sufficient condition for global exponential stability (GES) [respectively global asymptotic stability (GAS)] of the DFE is in terms of the joint spectral radius of a set of suitably defined matrices, whereas the necessary and sufficient condition for GAS of the DFE involves the spectral radius of an appropriately defined product of matrices. Subsequently, we leverage the stability results in order to design a distributed control strategy for eradicating the epidemic.