Waning Immunity Fails to Restore a Positive Epidemic Threshold on Power-Law Networks

Abstract

In a seminal work, Chatterjee and Durrett (2009) established that for the SIS epidemic process on random graphs with power-law degree distributions, the infection survives for an exponentially long time (in the network size) for any fixed, positive infection rate. Equivalently, the critical infection rate separating polynomial and exponential survival regimes is zero. In contrast, a substantial body of work in the physics literature conjectures, based primarily on numerical evidence and heuristic mean-field arguments, that introducing waning immunity (as in the SIRS process) yields a strictly positive critical infection rate on random graphs with power-law degrees; see, e.g., Pastor-Satorras et al. (2015), Ferreira et al. (2016), Silva et al. (2022). In particular, below this threshold, the epidemic is expected to persist only for a polynomial duration. A recent work by Friedrich et al. (2024) reinforces this perspective by proving polynomial survival for the SIRS process on star graphs, which is in contrast to the exponential survival in the SIS case that underpins Chatterjee and Durrett's arguments. In this paper, we disprove this conjecture and show that the epidemic threshold is also zero for the SIRS process on the configuration model with power-law degree distribution with exponent $\tau>2$. Our proof uncovers a novel bottleneck structure for the SIRS dynamics, which we term a "hierarchical star" of order 2, and show that it sustains the infection for an exponentially long time with high probability.