How fast does an epidemic die out on a network? Sharp decay envelopes for mean-field SIS epidemics and a proof of the Persoons–Sensi–Prasse–Van Mieghem conjectures

Unrefereed research candidate. For the N-intertwined mean-field (NIMFA) model of susceptible–infectious–susceptible (SIS) epidemics on networks, Persoons, Sensi, Prasse and Van Mieghem (Phys. Rev. E 109, 034308, 2024) conjectured from simulations two universal decay bounds. At or below the epidemic threshold the average infection level obeys y(t) ≤ 1/(1+t), with time in mean recovery times, whatever the network and starting state; above it, the excess over the endemic level, starting from above, obeys the same bound. This paper proves both (the second with the sharper prefactor 1 − p̄) by applying Jensen's inequality to the classical Volterra-type function h(s) = −s − log(1 − s). The argument covers time-varying networks, node-dependent recovery and, with Perron weights, directed networks, and via a known correlation inequality the expected prevalence of the exact stochastic SIS process with a common recovery rate. Regular networks attain the envelope. Exactly at the threshold the average decays like κ(G)/t, with an explicit eigenvector constant κ(G) ≤ 1 equal to 1 only for regular graphs; on standard synthetic networks of 1000 nodes κ ranges from about 0.9 (small-world, random) to about 0.04 (spatial, scale-free), so the same R0 = 1 epidemic takes about 920 recovery times to fall below 0.1% on a small-world network but about 40 on a random geometric one, while hubs and cores stay infected longer. Individual nodes can exceed the population envelope by a factor of order n^(1/3). For the timescale-separation question that motivated the conjectures the paper gives unconditional transition-time bounds, and two-solver numerics show that the growth formula of Persoons et al. (their eq. 20) is not a general bound away from R0 → 1 (numerical, not interval-certified). The archive contains the manuscript (PDF, Markdown source and generated LaTeX), the prior-art search record, a rating-free review summary, the bibliography, four checking scripts with retained outputs (exact SymPy identities, numerical tests of every theorem, synthetic-network illustrations and a two-solver check), a replay comparer, CI and SHA-256 sums. Assurance: written proofs and computational checks; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text, data and diagrams CC0-1.0; code MIT.