The spreading time in SIS epidemics on networks

In a Susceptible-Infected-Susceptible (SIS) process, we investigate the spreading time Tm, which is the time when the number of infected nodes in the metastable state is first reached, starting from the outbreak of the epidemics. We observe that the spreading time Tm resembles a lognormal-like distribution, though with different deep tails, both for the Markovian and the non-Markovian infection process, which implies that the spreading time can be very long with a relatively high probability. In addition, we show that a stronger virus, with a higher effective infection rate τ or an earlier timing of the infection attempts, does not always lead to a shorter average spreading time E[Tm]. We numerically demonstrate that the average spreading time E[Tm] in the complete graph and the star graph scales logarithmically as a function of the network size N for a fixed fraction of infected nodes in the metastable state.