  The role of waning immunity in basic epidemic models on networks has been
undervalued while being noticeable fundamental for real epidemic outbreaks. One
central question is which mean-field approach is more accurate in describing
the epidemic dynamics. We tackled this problem considering the
susceptible-infected-recovered-susceptible (SIRS) epidemic model on networks.
Two pairwise mean-field theories, one based on recurrent dynamical
message-passing (rDMP) and the other on the pair quenched mean-field theory
(PQMF), are compared with extensive stochastic simulations on large networks of
different levels of heterogeneity. For waning immunity times longer than or
comparable with the recovering time, rDMP outperforms PQMF theory on power-law
networks with degree distribution $P(k) \sim k^{-\gamma}$. In particular, for
$\gamma > 3$, the epidemic threshold observed in simulations is finite, in
qualitative agreement with rDMP, while PQMF leads to an asymptotically null
threshold. The critical epidemic prevalence for $\gamma > 3$ is localized in a
finite set of vertices in the case of the PQMF theory. In contrast, the
localization happens in a subextensive fraction of the network in rDMP theory.
Simulations, however, indicate that localization patterns of the actual
epidemic lay between the two mean-field theories, and improved theoretical
approaches are necessary to understanding the SIRS dynamics.
