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

Transition from time-variant to static networks: Timescale separation in N-intertwined mean-field approximation of susceptible-infectious-susceptible epidemics

We extend the N-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times [under T]̲(r) and T[over ¯](r) as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance r. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for T[over ¯](r). Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time T[over ¯](r) is almost entirely determined by the basic reproduction number R_{0} of the network. The value of the upper-transition time T[over ¯](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.

Analysis of continuous-time Markovian ɛ-SIS epidemics on networks

We analyze continuous-time Markovian ɛ-SIS epidemics with self-infections on the complete graph. The majority of the graphs are analytically intractable, but many physical features of the ɛ-SIS process observed in the complete graph can occur in any other graph. In this work, we illustrate that the timescales of the ɛ-SIS process are related to the eigenvalues of the tridiagonal matrix of the SIS Markov chain. We provide a detailed analysis of all eigenvalues and illustrate that the eigenvalues show staircases, which are caused by the nearly degenerate (but strictly distinct) pairs of eigenvalues. We also illustrate that the ratio between the second-largest and third-largest eigenvalue is a good indicator of metastability in the ɛ-SIS process. Additionally, we show that the epidemic threshold of the Markovian ɛ-SIS process can be accurately approximated by the effective infection rate for which the third-largest eigenvalue of the transition matrix is the smallest. Finally, we derive the exact mean-field solution for the ɛ-SIS process on the complete graph, and we show that the mean-field approximation does not correctly represent the metastable behavior of Markovian ɛ-SIS epidemics.

Time dependence of susceptible-infected-susceptible epidemics on networks with nodal self-infections

The average fraction of infected nodes, in short the prevalence, of the Markovian ɛ-SIS (susceptible-infected-susceptible) process with small self-infection rate ɛ>0 exhibits, as a function of time, a typical "two-plateau" behavior, which was first discovered in the complete graph K_{N}. Although the complete graph is often dismissed as an unacceptably simplistic approximation, its analytic tractability allows to unravel deeper details, that are surprisingly also observed in other graphs as demonstrated by simulations. The time-dependent mean-field approximation for K_{N} performs only reasonably well for relatively large self-infection rates, but completely fails to mimic the typical Markovian ɛ-SIS process with small self-infection rates. While self-infections, particularly when their rate is small, are usually ignored, the interplay of nodal self-infection and spread over links may explain why absorbing processes are hardly observed in reality, even over long time intervals.