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  "title": "Algebraic extinction in acyclic compartment models",
  "shortTitle": "Dying out need not mean exponential decay",
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  "oneLine": "A minimal four-state counterexample separates finite total activity from exponential extinction, then identifies the missing trajectory-level spectral condition.",
  "abstract": "This mathematical candidate disproves a stated exponential-extinction conjecture for conservative quadratic compartment models under acyclic transitions and the specified dependency-cycle condition. A four-state model with positive integer rates and strictly positive rational initial fractions has vanishing state I(t) asymptotic to 18/t² and integrated activity log 2. Four states are necessary. A family on m+2 states has exponents m/(m−1), approaching the integrability threshold one. The paper classifies the four-state boundary and proves a trajectory-dependent spectral repair, including the aggregate vanishing mass logarithmic rate. These are existence and structural results, not an observed epidemic or a genericity claim. Finite exact checks support but do not replace the written universal proofs.",
  "datePublished": "2026-10-20",
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    "compartment models",
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  "relatedWorks": [
    {
      "citation": "Chan, Crommelin and Mandjes (2025), Asymptotic behavior for a general class of spreading models.",
      "url": "https://doi.org/10.48550/arXiv.2511.02437",
      "doi": "10.48550/arXiv.2511.02437"
    },
    {
      "citation": "Luhar et al. (2022), Accounting for super-spreader events and algebraic decay in SIR models.",
      "url": "https://doi.org/10.1016/j.cma.2022.115286",
      "doi": "10.1016/j.cma.2022.115286"
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