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

Abstract

The average fraction of infected nodes, in short the prevalence, of the Markovian ε-SIS (susceptible-infectedsusceptible) process with small self-infection rate ε > 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 ε-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.

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

“…[33], Lemma 7.7.1). The same conclusion follows by computing the probability generating function of the ε-SIS process and concluding that the resulting differential equation is of Sturm-Liouville type [21], which is known to have simple eigenvalues. The transition matrix P of the ε-SIS Markov chain has a unique, largest eigenvalue ξ 1 = 0, which corresponds to the steady state π .…”

“…Our primary motivation for researching the eigenvalues of the transition matrix P in Eq. ( 2) is the observation of plateau-behavior in the ε-SIS process [21], which is illustrated FIG. 2.…”

“…Unfortunately, solving the partial differential equation seems infeasible for ε-SIS dynamics (see Ref. [21], Appendix A). Alternatively, one may compute the Rayleigh-Ritz coefficient of the partial differential equations to derive bounds for the eigenvalues.…”