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

The -susceptible-infected-susceptible (SIS) epidemic model on a graph adds an independent, Poisson self-infection process with rate to the "classical" Markovian SIS process. The steady state in the classical SIS process (with =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 τc=O-1N-1 in the -SIS process on the complete graph KN with N nodes, above which the effective infection rate τ>τc 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 is. For τ<τc and small, the network is almost overall healthy. The observation was found by mathematical analysis on the complete graph KN, 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 -SIS process and, hence, unlikely to exist in reality, where "background" infection >0 is imminent.