We study the spread of an SIRS-type epidemic with vaccination on network. Starting from an exact Markov description of the model, we investigate the mean epidemic lifetime by providing a sufficient condition for fast extinction that depends on the model parameters and the topology of the network. Then, we pass to consider a firstorder mean-field approximation of the exact model and its stability properties, by relying on the graph-theoretical notion of equitable partition. In the case of graphs possessing this kind of partition, we prove that the endemic equilibrium can be computed by using a lower-dimensional dynamical system. Finally, in the special case of regular graphs, we investigate the domain of attraction of the endemic equilibrium.
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