Transition from time-variant to static networks: timescale separation in NIMFA SIS

We extend the N-intertwined mean-field approximation (NIMFA) for the Susceptible-Infectious-Susceptible (SIS) epidemiological process to time-varying networks. We investigate the timescale separation between disease spreading and topology updates of the network. We introduce the transition times $\mathrm{\underline{T}}(r)$ and $\mathrm{\overline{T}}(r)$ as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively. By analysing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for $\mathrm{\overline{T}}(r)$. We then illustrate these bounds numerically and we compare our simulations with a heuristic alternative for $\mathrm{\overline{T}}(r)$. We show that, under our assumptions, the upper transition time $\mathrm{\overline{T}}(r)$ is almost entirely determined by the basic reproduction number $R_0$. The value of the upper transition time $\mathrm{\overline{T}}(r)$ around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.