Spreading processes with a unique absorbing state and finite lifetime of populations

The presence of a unique absorbing state in the finite state space of an epidemic process always poses a challenge in the definition of its epidemiological threshold. Without the elimination of the absorbing state of the SIS process we show that an exponentially distributed finite lifetime of the population results in a threshold like beahivour in the long time limit. We illustrate our findings with the SIS dynamics on the complete graph and the star.