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The ɛ-susceptible-infected-susceptible (SIS) epidemic model on a graph adds an independent, Poisson self-infection process with rate ɛ to the "classical" Markovian SIS process. The steady state in the classical SIS process (with ɛ=0) on any finite graph is the absorbing or overall-healthy state, in which the virus is eradicated from the network. We report that there always exists a phase transition around τ_{c}^{ɛ}=O(ɛ^{-1/N-1}) in the ɛ-SIS process on the complete graph K_{N} with N nodes, above which the effective infection rate τ>τ_{c}^{ɛ} causes the average steady-state fraction of infected nodes to approach that of the mean-field approximation, no matter how small, but not zero, the self-infection rate ɛ is. For τ<τ_{c}^{ɛ} and small ɛ, the network is almost overall healthy. The observation was found by mathematical analysis on the complete graph K_{N}, but we claim that the phase transition of explosive type may also occur in any other finite graph. We thus conclude that the overall-healthy state of the classical Markovian SIS model is unstable in the ɛ-SIS process and, hence, unlikely to exist in reality, where "background" infection ɛ>0 is imminent.

Explosive phase transition in susceptible-infected-susceptible epidemics with arbitrary small but nonzero self-infection rate