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arxiv: 2508.16766 · v1 · pith:TXLVPO7Onew · submitted 2025-08-22 · 🧮 math.DS

A Koopman Operator Framework for Nonlinear Epidemic Dynamics: Application to an SIRSD Model

Achraf Zinihi , Matthias Ehrhardt , Moulay Rchid Sidi Ammi This is my paper

Pith reviewed 2026-05-21 22:40 UTC · model grok-4.3

classification 🧮 math.DS
keywords Koopman operatorextended dynamic mode decompositionSIRSD epidemic modelwaning immunitynonlinear dynamicsdata-driven modelingsynthetic epidemic datapeak infection prediction
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The pith

A Koopman operator with EDMD approximates nonlinear SIRSD dynamics and predicts peak infections from synthetic data across multiple diseases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first builds an SIRSD model that adds waning immunity and disease-induced mortality to the standard SIR setup, then proves solutions exist, are unique, positive, and bounded. It normalizes the variables and applies the Koopman operator via extended dynamic mode decomposition with two dictionaries of observables, one minimal and one enriched with nonlinear terms. Synthetic trajectories are generated with a nonstandard finite difference scheme for four epidemics including SARS-CoV-2, influenza, Ebola, and measles. Numerical tests show the resulting linear operator identifies dominant modes and forecasts key quantities such as peak infection timing. A sympathetic reader cares because the method converts hard nonlinear epidemic equations into a data-driven linear representation that could simplify both analysis and short-term prediction.

Core claim

The central claim is that the Koopman operator framework, realized through extended dynamic mode decomposition with an epidemiologically informed dictionary on the normalized SIRSD system, identifies dominant epidemic modes and accurately predicts outbreak features including peak infection dynamics when tested on synthetic data generated by a nonstandard finite difference scheme for SARS-CoV-2, seasonal influenza, Ebola, and measles.

What carries the argument

The extended dynamic mode decomposition (EDMD) approximation of the Koopman operator applied to a dictionary of epidemiological observables on the normalized SIRSD state variables, with comparison between a minimal dictionary and one augmented by nonlinear and cross terms.

If this is right

  • Dominant modes extracted by the operator can be used to analyze long-term epidemic behavior without repeated full nonlinear integration.
  • Peak infection timing and other outbreak metrics become directly computable from the linear Koopman representation for the tested diseases.
  • Enriching the dictionary with nonlinear and cross terms measurably improves fidelity to the original SIRSD trajectories.
  • The normalized formulation supports both rigorous existence proofs and the subsequent data-driven Koopman construction in a single consistent variable set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pipeline could be tested on streaming surveillance data to assess real-time forecasting performance beyond synthetic cases.
  • The framework may transfer to other compartmental models that include waning immunity or mortality terms once an appropriate dictionary is chosen.
  • Error bounds on the EDMD approximation could be derived to quantify prediction uncertainty for different epidemic parameter regimes.

Load-bearing premise

The synthetic trajectories produced by the nonstandard finite difference scheme are representative enough that the chosen dictionary and EDMD procedure recover the essential nonlinear dynamics without large approximation errors.

What would settle it

If the predicted peak infection times from the Koopman model deviate substantially from observed historical peaks when the same procedure is applied to real reported case data for measles or Ebola, the accuracy claim would be refuted.

Figures

Figures reproduced from arXiv: 2508.16766 by Achraf Zinihi, Matthias Ehrhardt, Moulay Rchid Sidi Ammi.

Figure 1
Figure 1. Figure 1: presents numerical simulations of the SIRSD model (3)–(4) for the four represen￾tative infectious diseases. Each subplot shows how the proportion of susceptible s(t), infected i(t), recovered r(t), and deceased d(t) individuals in the population changes over time. For example, for the case of the top-left subplot representing the spread of the SARS-CoV-2 virus (COVID-19)with parameters set to β = 0.5, ω = … view at source ↗
Figure 2
Figure 2. Figure 2: Koopman-based reconstruction of SIRSD dynamics from synthetic data. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Overlay comparison between synthetic SIRSD trajectories and Koopman reconstructions. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Extended-time validation of Koopman convergence for the Measles outbreak using [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Workflow of the SIRSD epidemic model within the Koopman operator framework. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

We develop and analyze an SIRSD epidemic model, which extends the classical SIR framework by incorporating waning immunity and disease-induced mortality. A rigorous well-posedness analysis ensures the existence, uniqueness, positivity, and boundedness of solutions, guaranteeing the model's epidemiological feasibility. To facilitate theoretical investigations and data-driven modeling, we reformulated the system in normalized variables. To capture and predict complex nonlinear epidemic dynamics, we use the Koopman operator framework with extended dynamic mode decomposition (EDMD) and an epidemiologically informed dictionary of observables. We compare two Koopman approximations: one based on a minimal epidemiological dictionary and another enriched with nonlinear and cross terms. We generate synthetic data using a nonstandard finite difference (NSFD) scheme for four representative epidemics: SARS-CoV-2, seasonal influenza, Ebola, and measles. Numerical experiments demonstrate that the Koopman-based approach effectively identifies dominant epidemic modes and accurately predicts key outbreak characteristics, including peak infection dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops an SIRSD epidemic model extending the classical SIR framework with waning immunity and disease-induced mortality. It provides a rigorous well-posedness analysis ensuring existence, uniqueness, positivity, and boundedness of solutions, reformulates the system in normalized variables, and applies the Koopman operator with extended dynamic mode decomposition (EDMD) using an epidemiologically informed dictionary. Synthetic trajectories are generated via a nonstandard finite difference (NSFD) scheme for four representative epidemics (SARS-CoV-2, seasonal influenza, Ebola, measles). Numerical experiments compare a minimal epidemiological dictionary against an enriched version with nonlinear and cross terms, claiming that the approach identifies dominant epidemic modes and accurately predicts key outbreak characteristics including peak infection dynamics.

Significance. If the numerical validation is strengthened, the work provides a data-driven Koopman framework for capturing nonlinear dynamics in extended epidemic models, potentially useful for mode identification and peak prediction. The combination of theoretical well-posedness with EDMD on structure-preserving discretizations is a positive aspect, though the accuracy claims currently lack quantitative anchoring.

major comments (2)
  1. [Numerical Experiments] Numerical experiments section: The central claim that the Koopman-EDMD approach 'accurately predicts key outbreak characteristics, including peak infection dynamics' depends on the enriched dictionary applied to NSFD-generated data faithfully approximating the true continuous SIRSD flow. No a-posteriori residual norms, no comparison against a standard ODE integrator on identical initial conditions, and no sensitivity tests to NSFD step size are reported, leaving the prediction accuracy without an independent verification anchor.
  2. [Well-posedness Analysis and EDMD Approximation] Well-posedness and EDMD sections: The abstract asserts well-posedness results and prediction accuracy, yet the manuscript supplies no explicit derivations for the well-posedness theorems, no quantitative error metrics (e.g., peak timing or amplitude errors), and no verification details for the EDMD approximation quality on the synthetic trajectories.
minor comments (2)
  1. [Dictionary Construction] Clarify the precise composition of the 'enriched' dictionary observables (e.g., which nonlinear and cross terms are included) and ensure consistent notation for the normalized variables across sections.
  2. [Numerical Experiments] Consider adding a brief comparison table of prediction errors across the four epidemics and the two dictionaries to make the accuracy claims more transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have carefully considered the comments and will revise the manuscript to address the concerns regarding numerical validation and the presentation of theoretical results. Below, we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [Numerical Experiments] Numerical experiments section: The central claim that the Koopman-EDMD approach 'accurately predicts key outbreak characteristics, including peak infection dynamics' depends on the enriched dictionary applied to NSFD-generated data faithfully approximating the true continuous SIRSD flow. No a-posteriori residual norms, no comparison against a standard ODE integrator on identical initial conditions, and no sensitivity tests to NSFD step size are reported, leaving the prediction accuracy without an independent verification anchor.

    Authors: We agree that additional verification of the NSFD scheme would strengthen the numerical claims. The NSFD discretization was deliberately chosen to preserve positivity and boundedness, which are essential for the epidemiological validity of the SIRSD model. In the revised manuscript we will add direct comparisons of NSFD trajectories against a standard ODE integrator (e.g., MATLAB ode45) on identical initial conditions, report a-posteriori residual norms, and include sensitivity tests with respect to step size, quantifying the resulting errors in peak timing and amplitude. These additions will supply the requested independent verification anchor. revision: yes

  2. Referee: [Well-posedness Analysis and EDMD Approximation] Well-posedness and EDMD sections: The abstract asserts well-posedness results and prediction accuracy, yet the manuscript supplies no explicit derivations for the well-posedness theorems, no quantitative error metrics (e.g., peak timing or amplitude errors), and no verification details for the EDMD approximation quality on the synthetic trajectories.

    Authors: The well-posedness analysis appears in Section 3, where existence and uniqueness follow from the Picard-Lindelöf theorem and positivity/boundedness are established via invariant-region arguments. We acknowledge that the derivations can be made more explicit. In the revision we will expand the proofs with additional intermediate steps. We will also insert quantitative tables reporting relative errors in peak timing and amplitude for both dictionary choices across the four diseases, together with EDMD verification metrics such as Koopman residual norms and reconstruction errors on held-out synthetic trajectories. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper first defines the SIRSD model and performs a standard well-posedness analysis (existence, uniqueness, positivity, boundedness) in normalized variables. It then generates synthetic trajectories via an NSFD discretization of that same model and applies EDMD with an epidemiologically informed dictionary to obtain a linear Koopman approximation. The numerical claims concern the quality of this approximation on the generated data (dominant modes, peak prediction). This workflow is a conventional data-driven validation exercise: the EDMD step is a least-squares fit to observed trajectories rather than a redefinition or tautological renaming of the original vector field. No load-bearing step reduces by construction to a fitted parameter, a self-citation chain, or an imported uniqueness theorem; the central results rest on explicit numerical comparison rather than on any circular equivalence between inputs and outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard well-posedness results for ODE epidemic models and the assumption that the chosen numerical scheme produces faithful trajectories; no new entities are postulated.

free parameters (1)
  • Dictionary observables
    Selection of minimal versus enriched nonlinear and cross terms is chosen to improve approximation quality but is not derived from first principles.
axioms (2)
  • domain assumption Existence, uniqueness, positivity, and boundedness of solutions for the SIRSD system.
    Invoked to guarantee epidemiological feasibility before applying the Koopman framework.
  • domain assumption The NSFD scheme generates accurate synthetic trajectories for the continuous SIRSD model.
    Used to produce training data for EDMD without further validation shown in abstract.

pith-pipeline@v0.9.0 · 5702 in / 1568 out tokens · 53086 ms · 2026-05-21T22:40:40.749041+00:00 · methodology

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Works this paper leans on

42 extracted references · 42 canonical work pages · 1 internal anchor

  1. [1]

    Bitsouni, N

    V. Bitsouni, N. Gialelis, and V. Tsilidis. An age-structured SVEAIR epidemiological model. Mathematical Methods in the Applied Sciences , 47(16):12460–12486, 2024

    work page 2024

  2. [2]

    Sekiguchi and E

    M. Sekiguchi and E. Ishiwata. Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay. Journal of Computational and Applied Mathematics , 236(6):997–1008, 2011

    work page 2011

  3. [3]

    Zinihi, M

    A. Zinihi, M. R. Sidi Ammi, and D. F. M. Torres. Fractional differential equations of a reaction-diffusion SIR model involving the Caputo-fractional time-derivative and a nonlinear diffusion operator. Evolution Equations and Control Theory , 14(5):944–967, 2025

    work page 2025

  4. [4]

    Lopez and N

    S. Lopez and N. L. Komarova. An optimal network that promotes the spread of an advantageous variant in an SIR epidemic. Journal of Theoretical Biology, 605:112095, 2025

    work page 2025

  5. [5]

    Guerrero-Flores, O

    S. Guerrero-Flores, O. Osuna, and C. Vargas-De-Le´ on. Periodic solutions of seasonal epidemiological models with information-dependent vaccination. Mathematical Methods in the Applied Sciences , 47(4):1961–1972, 2023

    work page 1961

  6. [6]

    Zinihi, M

    A. Zinihi, M. Ehrhardt, and M. R. Sidi Ammi. Spatiotemporal SEIQR epidemic mdeling with optimal control for vaccination, treatment, and social measures, 2025. arXiv preprint 2507.09328

    work page arXiv 2025

  7. [7]

    L. B. Puglisi, G. S.P. Malloy, T. D. Harvey, M. L. Brandeau, and E. A. Wang. Estimation of COVID-19 basic reproduction ratio in a large urban jail in the United States. Annals of Epidemiology , 53:103–105, 2021

    work page 2021

  8. [8]

    M. V. I. Soto-Rocha, O. Walle-Garc´ ıa, F. Salda˜ na-Jim´ enez, F. Hern´ andez-Cabrera, and F.J. Almaguer- Mart´ ınez. COVID-19 infection and recovery rates in Mexico: An ARMA time series model. Journal of Computational and Applied Mathematics , 472:116781, 2026

    work page 2026

  9. [9]

    Simeonov and C

    O. Simeonov and C. D. Eaton. Modeling the drivers of oscillations in COVID-19 data on college campuses. Annals of Epidemiology, 82:40–44, 2023

    work page 2023

  10. [10]

    A Nonstandard Finite Difference Scheme for an SEIQR Epidemiological PDE Model

    A. Zinihi, M. Ehrhardt, and M. R. Sidi Ammi. A nonstandard finite difference scheme for an SEIQR epidemiological PDE model, 2025. arXiv preprint 2508.02928

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  11. [11]

    Chang, S

    L. Chang, S. Gao, and Z. Wang. Optimal control of pattern formations for an SIR reaction–diffusion epidemic model. Journal of Theoretical Biology, 536:111003, 2022

    work page 2022

  12. [12]

    She and F

    G. She and F. Yi. Stability and bifurcation analysis of a reaction–diffusion SIRS epidemic model with the general saturated incidence rate. Journal of Nonlinear Science , 34(6), 2024

    work page 2024

  13. [13]

    Zinihi, M

    A. Zinihi, M. R. Sidi Ammi, and M. Ehrhardt. Mathematical modeling and Hyers-Ulam stability for a non- linear epidemiological model with Φp operator and Mittag-Leffler kernel. Advances in Applied Mathematics and Mechanics, 17(5):1481–1508, 2025

    work page 2025

  14. [14]

    M. T. Hoang and M. Ehrhardt. A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model. Journal of Difference Equations and Applications , 30(4):409–434, 2023. 16

    work page 2023

  15. [15]

    Suo and Y

    J. Suo and Y. Ge. Resonance and bifurcations of the discrete SIR model with unfixed incidence rate. Journal of Difference Equations and Applications , page 1–27, 2025

    work page 2025

  16. [16]

    Zinihi, M

    A. Zinihi, M. Ehrhardt, and M. R. Sidi Ammi. Actuarial analysis of an infectious disease insurance based on an SEIARD epidemiological model, 2025. arXiv preprint 2508.06580

    work page arXiv 2025

  17. [17]

    S. L. Brunton, M. Budiˇ si´ c, E. Kaiser, and J. N. Kutz. Modern Koopman theory for dynamical systems. SIAM Review, 64(2):229–340, 2022

    work page 2022

  18. [18]

    Berry and S

    T. Berry and S. Das. Limits of learning dynamical systems. SIAM Review, 67(1):107–137, 2025

    work page 2025

  19. [19]

    W. A. Manzoor, S. Rawashdeh, and A. Mohammadi. Vehicular applications of Koopman operator theory – A survey. IEEE Access, 11:25917–25931, 2023

    work page 2023

  20. [20]

    Mezi´ c, Z

    I. Mezi´ c, Z. Drmaˇ c, N.ˇCrnjari´ c, S. Ma´ ceˇ si´ c, M. Fonoberova, R. Mohr, A. M. Avila, I. Manojlovi´ c, and A. Andrejˇ cuk. A Koopman operator-based prediction algorithm and its application to COVID-19 pandemic and influenza cases. Scientific Reports, 14(1), 2024

    work page 2024

  21. [21]

    J. L. Proctor and P. A. Eckhoff. Discovering dynamic patterns from infectious disease data using dynamic mode decomposition. International Health, 7(2):139–145, 2015

    work page 2015

  22. [22]

    Mustavee, S

    S. Mustavee, S. Agarwal, C. Enyioha, and S. Das. A linear dynamical perspective on epidemiology: Interplay between early COVID-19 outbreak and human mobility. Nonlinear Dynamics, 109(2):1233–1252, 2022

    work page 2022

  23. [23]

    J. L. Proctor, S. L. Brunton, and J. N. Kutz. Generalizing Koopman theory to allow for inputs and control. SIAM Journal on Applied Dynamical Systems , 17(1):909–930, 2018

    work page 2018

  24. [24]

    Takeishi, Y

    N. Takeishi, Y. Kawahara, and T. Yairi. Subspace dynamic mode decomposition for stochastic Koopman analysis. Physical Review E, 96(3), 2017

    work page 2017

  25. [25]

    Z. Xu, R. Dai, and K. C. Howell. Tensor-based Koopman operator and its application to optimal control problems. Journal of Guidance, Control, and Dynamics , page 1–13, 2025

    work page 2025

  26. [26]

    Han and X

    M. Han and X. Yin. Deep neural Koopman operator-based economic model predictive control of shipboard carbon capture system. IEEE Transactions on Control Systems Technology , page 1–16, 2025

    work page 2025

  27. [27]

    Basit, J

    A. Basit, J. M. Zain, H. Z. Mojahid, A. K. Jumaat, N. Hamdan, and D. K. Bagal. Calculating SIRD model parameters and re-susceptibility in Malaysia, Pakistan, India and United Arab Emirates. Communications in Nonlinear Science and Numerical Simulation , 2025. 108902

    work page 2025

  28. [28]

    Verma, A

    M. Verma, A. K. Verma, and A. K. Misra. Mathematical modeling and optimal control of carbon dioxide emissions from energy sector. Environment, Development and Sustainability , 23(9):13919–13944, 2021

    work page 2021

  29. [29]

    M Korda and I. Mezi´ c. On convergence of extended dynamic mode decomposition to the Koopman operator. Journal of Nonlinear Science , 28(2):687–710, 2017

    work page 2017

  30. [30]

    Y. Jin, L. Hou, and S. Zhong. Extended dynamic mode decomposition with invertible dictionary learning. Neural Networks, 173:106177, 2024

    work page 2024

  31. [31]

    R. E. Mickens. Nonstandard finite difference models of differential equations . World Scientific, 1993

    work page 1993

  32. [32]

    Ehrhardt and R

    M. Ehrhardt and R. E. Mickens. A nonstandard finite difference scheme for convection–diffusion equations having constant coefficients. Applied Mathematics and Computation , 219(12):6591–6604, 2013

    work page 2013

  33. [33]

    Costa, M

    G. Costa, M. Lobosco, M. Ehrhardt, and R. Reis. Mathematical analysis and a nonstandard scheme for a model of the immune response against COVID-19. Mathematical and Computational Modeling of Phenomena Arising in Population Biology and Nonlinear Oscillations , page 251–270, 2024

    work page 2024

  34. [34]

    M. H. Maamar, M. Ehrhardt, and L. Tabharit. A nonstandard finite difference scheme for a time-fractional model of Zika virus transmission. Mathematical Biosciences and Engineering , 21(1):924–962, 2023

    work page 2023

  35. [35]

    COVID-19 dashboard; https://coronavirus.jhu.edu/data, Accessed on: August 18, 2025 (2020)

    Johns Hopkins University. COVID-19 dashboard; https://coronavirus.jhu.edu/data, Accessed on: August 18, 2025 (2020)

    work page 2025

  36. [36]

    CDC COVID data tracker; https://www.cdc.gov/ covid-data-tracker/index.html, Accessed on: August 18, 2025

    Centers for Disease Control and Prevention. CDC COVID data tracker; https://www.cdc.gov/ covid-data-tracker/index.html, Accessed on: August 18, 2025

    work page 2025

  37. [37]

    Seasonal influenza (Flu), https://www.cdc.gov/flu/index.htm, Accessed on: August 18, 2025

    Centers for Disease Control and Prevention. Seasonal influenza (Flu), https://www.cdc.gov/flu/index.htm, Accessed on: August 18, 2025

    work page 2025

  38. [38]

    Seasonal influenza data; https://www.ecdc.europa

    European Centre for Disease Prevention and Control. Seasonal influenza data; https://www.ecdc.europa. eu/en/seasonal-influenza, Accessed on: August 18, 2025

    work page 2025

  39. [39]

    Ebola virus disease outbreak, West Africa, october 2014; https://www.cdc.gov/mmwr/preview/mmwrhtml/mm6343a3.htm, Accessed on: August 18, 2025

    Centers for Disease Control and Prevention. Ebola virus disease outbreak, West Africa, october 2014; https://www.cdc.gov/mmwr/preview/mmwrhtml/mm6343a3.htm, Accessed on: August 18, 2025

    work page 2014

  40. [40]

    Ebola outbreak 2014–2016, West Africa; https://www.who.int/emergencies/ situations/ebola-outbreak-2014-2016-west-africa, Accessed on: August 18, 2025

    World Health Organization. Ebola outbreak 2014–2016, West Africa; https://www.who.int/emergencies/ situations/ebola-outbreak-2014-2016-west-africa, Accessed on: August 18, 2025

    work page 2014

  41. [41]

    Measles Surveillance Manual; https://www.cdc.gov/ surv-manual/php/table-of-contents/chapter-7-measles.html, Accessed on: August 18, 2025

    Centers for Disease Control and Prevention. Measles Surveillance Manual; https://www.cdc.gov/ surv-manual/php/table-of-contents/chapter-7-measles.html, Accessed on: August 18, 2025

    work page 2025

  42. [42]

    Pink Book – Measles; https://www.cdc.gov/pinkbook/hcp/ table-of-contents/chapter-13-measles.html, Accessed on: August 18, 2025

    Centers for Disease Control and Prevention. Pink Book – Measles; https://www.cdc.gov/pinkbook/hcp/ table-of-contents/chapter-13-measles.html, Accessed on: August 18, 2025. 17

    work page 2025