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  "slug": "acyclic-spreading-algebraic-extinction",
  "title": "Algebraic extinction in acyclic spreading models: explicit counterexamples, genericity and support-checkable repairs",
  "shortTitle": "When an outbreak ends on the tipping point",
  "url": "https://evidencepress.org/releases/acyclic-spreading-algebraic-extinction/",
  "oneLine": "A successor to an earlier Evidence Press refutation of the Chan–Crommelin–Mandjes exponential-extinction conjecture: a four-compartment spreading model with both graphs acyclic fades like t^(-3/2) with an exact solution, nothing slower is possible with four compartments, and such slow tails need finely tuned starts and can be ruled out by checkable conditions.",
  "abstract": "Chan, Crommelin and Mandjes conjectured that, in their class of interaction-driven spreading models, every vanishing compartment decays exponentially whenever two graphs built from the model are acyclic or nearly so. An earlier Evidence Press release (doi:10.5281/zenodo.23501636) refuted this conjecture with a four-compartment model decaying like 18/t², proved four compartments necessary and gave a spectral repair. This successor explains the mechanism: an endgame in which the effective reproduction number of the active compartment tends to exactly one. It gives a four-compartment staged-rumour model in which both graphs are acyclic and the trajectory is explicit, with activity x(t) ~ √(3/8) t^(−3/2), and a second example decaying like 4t^(−2) that needs less tuning; within the source's encoding of recovery, a no-direct-infection SWIR-type submodel also admits a t^(−3/2) counterexample. With four compartments no tail is slower than t^(−3/2), and chains give t^(−1−1/m) for every even m. Algebraic extinction requires non-generic initial data whenever the fluxes share one active catalyst, a general law gives how fast exponential decay returns after a perturbation, the slow tails never follow an outbreak peak in the chain, SWIR and hierarchy families, and support- and rate-checkable conditions guarantee exponential decay. An exploratory simulation suggests finite populations die out much earlier. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-10-21",
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    "spreading models",
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    "algebraic extinction",
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    "Successor result: the conjecture was first refuted by the earlier release (18/t², cyclic dependency graph); here both graphs are acyclic and the trajectory is explicit, x(t) ~ √(3/8) t^(−3/2), with R_eff → 1 and 1 − R_eff ~ 9/(2t) (Theorem 3.1, proof; exact).",
    "A no-direct-infection SWIR-type submodel of the Janssen–Müller–Stenull generalised epidemic process, with recovery encoded as in the source, is in the class and gives a t^(−3/2) counterexample at a tricritical-type triple zero (§3.4, proof; exact).",
    "Limits: three compartments never suffice (as in the earlier release); any four-compartment model decays exponentially, like t^(−2) or like t^(−3/2), so nothing is slower (Theorem 5.7, proof); chains give t^(−1−1/m) for every even m (Theorem 5.1).",
    "Fragility: algebraic extinction lies on a null set of starting points in single-catalyst models (codimensions 1 and 2 in the chain; Theorem 6.2), exponential decay returns after perturbation with rate ∝ |ε|^((k−1)/k) (Theorem 6.3), and no outbreak peak precedes a slow tail in the chain, SWIR and hierarchy families (Proposition 6.5).",
    "Repairs: the log-rate equals the spectral abscissa of the limiting Metzler block (same statement as the earlier release), and support- and rate-checkable conditions guarantee exponential decay (Corollaries 7.3–7.4, proof). Finite populations die out at about 1.5 N^0.42 (exploratory)."
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    {
      "citation": "Anonymous (2026), Algebraic extinction in acyclic compartment models. Evidence Press, version 1.0.0-candidate (parent release; first refutation of the conjecture).",
      "url": "https://doi.org/10.5281/zenodo.23501636",
      "doi": "10.5281/zenodo.23501636"
    },
    {
      "citation": "Chan, Crommelin and Mandjes (2025), Asymptotic behavior for a general class of spreading models (arXiv:2511.02437v1).",
      "url": "https://doi.org/10.48550/arXiv.2511.02437",
      "doi": "10.48550/arXiv.2511.02437"
    },
    {
      "citation": "Janssen, Müller and Stenull (2004), Generalized epidemic process and tricritical dynamic percolation. Physical Review E 70, 026114.",
      "url": "https://doi.org/10.1103/PhysRevE.70.026114",
      "doi": "10.1103/PhysRevE.70.026114"
    },
    {
      "citation": "Choi, Lee and Kahng (2017), Mixed-order phase transition in a two-step contagion model with a single infectious seed. Physical Review E 95, 022304.",
      "url": "https://doi.org/10.1103/PhysRevE.95.022304",
      "doi": "10.1103/PhysRevE.95.022304"
    },
    {
      "citation": "Kermack and McKendrick (1927), A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A 115, 700–721.",
      "url": "https://doi.org/10.1098/rspa.1927.0118",
      "doi": "10.1098/rspa.1927.0118"
    },
    {
      "citation": "de Arruda, Lebensztayn, Rodrigues and Rodríguez (2015), A process of rumour scotching on finite populations. Royal Society Open Science 2, 150240.",
      "url": "https://doi.org/10.1098/rsos.150240",
      "doi": "10.1098/rsos.150240"
    },
    {
      "citation": "Hethcote (2000), The mathematics of infectious diseases. SIAM Review 42, 599–653.",
      "url": "https://doi.org/10.1137/S0036144500371907",
      "doi": "10.1137/S0036144500371907"
    },
    {
      "citation": "Angeli, De Leenheer and Sontag (2007), A Petri net approach to the study of persistence in chemical reaction networks. Mathematical Biosciences 210, 598–618.",
      "url": "https://doi.org/10.1016/j.mbs.2007.07.003",
      "doi": "10.1016/j.mbs.2007.07.003"
    },
    {
      "citation": "Dodds and Watts (2004), Universal behavior in a generalized model of contagion. Physical Review Letters 92, 218701.",
      "url": "https://doi.org/10.1103/PhysRevLett.92.218701",
      "doi": "10.1103/PhysRevLett.92.218701"
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