Time to Extinction for the SIS Epidemic Model: New Bounds on the Tail Probabilities

In this paper, we present a novel estimation of the time to extinction of a Susceptible-Infected-Susceptible (SIS) epidemic model over a general network of interactions. Specifically, we prove that, for an effective infection rate above a threshold depending on the topology of the network, the time to extinction grows exponentially in the size of the population, with probability converging to 1 as the size of the population grows large. This provides new insights with respect to the results from the literature, which are focused on the expected time to extinction. In particular, our result yields a better characterization of the phase transition of the SIS model and a better understanding of the properties of the model in the regime above this threshold, depending on the topology of the network of interactions and on the initial condition. In view of the generality of the techniques used in this paper, we believe that similar arguments can be developed in order to analyze phase transitions of more complex epidemic models. Finally, explicit applications to popular families of networks (e.g., Erdős-Rényi random graphs, stars graphs and barbell graphs) show the effectiveness of the result.