Reconstruction and continuation in network SIS epidemics

Summary

The practical question. Suppose two deterministic epidemic models match a recorded outbreak closely. Must their predictions remain close after the observation window ends? This is a continuation question. It differs from reconstruction: whether an exact, complete trajectory uniquely determines the contact matrix.

The answer has three parts. At every fixed number of communities, bounded SIS models do admit a finite linear continuation factor. No single linear factor works across all network sizes: the paper constructs exactly reconstructible models whose least factor grows at least exponentially. Uniform control is nevertheless recovered by an explicit dimension-free Hölder estimate. A new corollary also handles finitely sampled, community-resolved trajectories through the sample fill distance and a bounded measurement-error term.

Observation scope. The theorem observes the full vector of infected fractions—one trajectory for every community—not total prevalence alone. Its finite-sample extension is deterministic; it is not a stochastic data or calibration model.

Status. Anonymous, AI-assisted and unrefereed. The universal results are written analytic proofs. Exact code checks replay selected finite constructions and hostile controls; there has been no external peer review, unaffiliated rerun or proof-assistant formalisation.

Central theorem and exact scope

Let $z_i(t)$ be the infected fraction in community $i$. The deterministic SIS model is

The paper compares the size of an observable model-fitting residual on an initial interval $[0,T_0]$ with its size on $[0,T]$.

The negative examples are especially informative because they are exactly reconstructible. Nothing is perfectly hidden: instead, one explicit direction is extremely hard to see during the observation interval and becomes relatively more visible later. The binomial theorem supplies that direction.

Technical mechanism: from histories to samples

The direct trajectory theorem does not require differentiating the observations. If sample times have fill distance $\eta$, and the largest community-resolved discrepancy at a sample is $\delta$, then the continuous observed-interval discrepancy is controlled by

If each recorded sample has error at most $\sigma$, replace $\delta$ by the recorded between-model discrepancy plus $2\sigma$. This is the tractable extension added during review. It makes explicit what dense sampling contributes while retaining the theorem's actual scope: deterministic bounded error and complete community-resolved observations.

Reconstruction is a separate problem

If the trajectory span has dimension $r$, the symmetric contact-matrix ambiguity space has dimension

The paper also gives nonstationary failures of reconstruction without a nontrivial block-regular partition and classifies positive invariant rays. One complete-network reduction is explicitly credited to an earlier 2021 theorem; applying it to the later reconstruction conjecture is the new use, not the underlying reduction.

Evidence, assurance and limitations

The synthetic audio briefing is explanatory, not additional evidence. Its complete transcript is available for reading.

• Analytic proof: fixed-dimensional continuation, the growing-dimensional obstruction, the Hölder estimates, reconstruction-fibre dimension and invariant-ray classification.
• Exact supplementary replay: rational finite constructions, Taylor determinants and cancellation coefficients; ordinary and optimized Python agree.
• Hostile controls: a source mutation is rejected in both ordinary and optimized execution, so acceptance does not disappear under python -O.
• Novelty boundary: polynomial-ideal closure, complete-network reduction and bounded analytic continuation have prior literature. The claimed contribution is the model-specific resolution, admissible obstruction, explicit SIS modulus and sampled-observation extension.
• Not established: optimal constants, practical forecasting sharpness, aggregate-prevalence inference, a stochastic observation theorem, empirical epidemic calibration, independent reproduction, external specialist review or formal verification.

Who should read this

The result is aimed at researchers working on epidemic inverse problems, analytic continuation, network dynamics and identifiability. Applied readers should begin with the three-way split above and the observation scope; specialists can then follow the proof architecture and exact finite checks in the paper and archive.

Paper, archive and evidence package: how to inspect and replay

Start with the paper, then use CLAIM_MAP.md to match every theorem to its proof route and limits. The archive also contains CITATION_AUDIT.md, ASSURANCE.md, AI_INDEX.md and replay receipts.

From the extracted archive:

python3 code/verify_sis.py
python3 -O code/verify_sis.py
python3 code/test_negative_control.py
shasum -a 256 -c MANIFEST.sha256

The most useful next checks

1. Audit the analytic-continuation constants independently and seek a sharper exponent.
2. Reimplement the exponential family and Gram-ratio calculation in separate software.
3. Determine whether comparable dimension-free bounds can be proved for partial or aggregate observation schemes.
4. Test whether the deterministic finite-sample guarantee can be embedded in a justified stochastic observation model.

@misc{sisreconstructioncontinuation2026,
  title        = {Reconstruction and continuation in network SIS epidemics},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23525167},
  url          = {https://doi.org/10.5281/zenodo.23525167},
  version      = {1.0.1-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/sis-reconstruction-continuation/}
}