The paper studies the spread of a virus over a (possibly) time-varying graph, with the spread being (possibly) worsened by the presence of a shared resource. We propose a time-varying susceptible-infected-water-susceptible (SIWS) model, with the water compartment representing the contamination level in the shared resource. We say that the system is in the disease-free equilibrium (DFE) if none of the nodes (representative of sub-populations, such as cities, districts, etc.) are infected, and the shared resource is contaminant-free. We identify multiple sufficient conditions for exponential convergence to the DFE. Based on one of the aforementioned sufficient conditions, an on/off lockdown strategy that eradicates the infection spread is proposed. More specifically, we design a switching rule between lockdown and free (i.e., no lockdown) modes to guarantee exponential convergence to the DFE.