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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 first‐order 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.

Some aspects of the Markovian SIRS epidemic on networks and its mean‐field approximation