--- title: "Reconstruction and continuation in network SIS epidemics" date: 2026-10-21 version: "1.0.1-candidate" doi: 10.5281/zenodo.23525167 pdf: https://github.com/ipitchford/sis-reconstruction-continuation/releases/download/v1.0.1-candidate/sis-reconstruction-continuation-v1.0.1-candidate.pdf repository: https://github.com/ipitchford/sis-reconstruction-continuation archive: https://zenodo.org/records/23525167 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Reconstruction and continuation in network SIS epidemics ## Summary **The practical question.**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 $$ \dot z_i=(1-z_i)(W\Pi z)_i-\gamma_i z_i. $$ The paper compares the size of an observable model-fitting residual on an initial interval $[0,T_0]$ with its size on $[0,T]$. | Question | Answer | Boundary | |---|---|---| | Is the least linear continuation factor uniformly finite when network dimension is fixed? | **Yes.** | Existence is proved; a useful numerical value is not computed. | | Is there one linear factor independent of the number of communities? | **No.** An exactly reconstructible family has exponential lower growth. | The obstruction is for the stated residual norm; it is not automatically a lower bound for nonlinear trajectory error in another norm. | | Is any dimension-free stability statement left? | **Yes.** An explicit Hölder estimate. | The exponent is conservative and not claimed optimal or practically sharp. | 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 $$ \min\{1,\delta+4M\eta\}. $$ 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 $$ \frac{(n-r)(n-r+1)}2. $$ 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](/assets/audio/sis-reconstruction-continuation.txt) 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: ```sh 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. ## Verification status Unrefereed AI-assisted mathematical candidate. Universal statements rest on written proofs; exact code checks have a finite supplementary scope. External specialist review, independent reproduction and formal verification have not been established. ## References 1. Keliger and Horváth (2026), Reconstruction of the network in the N-intertwined mean-field approximation of the SIS epidemic model. 2. Trefethen (2020), Quantifying the ill-conditioning of analytic continuation. 3. Prasse, Devriendt and Van Mieghem (2021), Clustering for epidemics on networks: A geometric approach.