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arxiv: 2301.06385 · v2 · submitted 2023-01-16 · 📊 stat.ME · stat.AP· stat.CO

Dynamic SIR/SEIR-like models comprising a time-dependent transmission rate: Hamiltonian Monte Carlo approach with applications to COVID-19

Hristo Inouzhe , Mar\'ia Xos\'e Rodr\'iguez-\'Alvarez , Lorenzo Nagar , Elena Akhmatskaya This is my paper

Pith reviewed 2026-05-24 10:02 UTC · model grok-4.3

classification 📊 stat.ME stat.APstat.CO
keywords time-dependent transmission rateBayesian P-splinesHamiltonian Monte CarloSIKR modelSEMIKR modelCOVID-19under-reportingprobabilistic model
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The pith

SIKR and SEMIKR models with Bayesian P-spline time-dependent transmission rates recover plausible COVID-19 patterns via HMC sampling.

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

The paper develops SIKR and SEMIKR compartmental models that extend standard SIR/SEIR structures by adding multiple sub-compartments for infectious and exposed individuals and by replacing constant transmission rates with a flexible time-dependent rate. This rate is parameterised using Bayesian P-splines, which are embedded directly into a probabilistic model that also accounts for under-reporting of new cases. The resulting model is made differentiable so that Hamiltonian Monte Carlo sampling becomes feasible after targeted initialisation and tuning, even when posterior geometry is difficult. When applied to COVID-19 data from the Basque Country between February 2020 and January 2021, the models recover temporal patterns in transmission while making the dependence of those patterns on specific modelling choices explicit through convergence diagnostics.

Core claim

The SIKR and SEMIKR models, enriched with a time-dependent transmission rate parameterised using Bayesian P-splines, are embedded in a probabilistic model that relies on the solutions of the underlying ODEs; this construction can be differentiated efficiently, which makes Hamiltonian Monte Carlo sampling feasible after careful initialisation and tuning, and when applied to COVID-19 data it recovers plausible temporal patterns in transmission while making explicit the dependence of results on modelling choices and convergence diagnostics.

What carries the argument

Bayesian P-splines used to parameterise the time-dependent transmission rate inside the ODE-based SIKR and SEMIKR compartmental models, enabling both flexibility and differentiability for HMC sampling within the probabilistic model that incorporates under-reporting.

If this is right

  • The models capture effects of non-pharmacological measures, population behaviour changes and random events without resorting to a priori or ad-hoc specifications.
  • Information about under-reporting of new infected cases improves estimates for diseases with large asymptomatic fractions.
  • Hamiltonian Monte Carlo sampling becomes practical in settings that otherwise present weakly identified directions.
  • Results on transmission dynamics are presented together with explicit diagnostics of their dependence on modelling choices.

Where Pith is reading between the lines

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

  • The same differentiable embedding strategy could be tested on other infectious diseases that exhibit high under-reporting.
  • Persistent challenges with posterior geometry point toward possible gains from reparameterisation or additional regularisation not explored in the current work.
  • Ensemble comparisons across different spline knot numbers or compartment counts would quantify how sensitive recovered patterns are to those specific choices.
  • Real-time updating of the P-spline coefficients as new data arrive might allow ongoing tracking of transmission shifts.

Load-bearing premise

The solutions of the ODEs can be embedded directly into a differentiable probabilistic model that remains well-behaved under HMC sampling despite weakly identified directions and challenging posterior geometries.

What would settle it

If the HMC sampler, after the described initialisation and tuning, produces non-convergent chains or transmission-rate trajectories that fail to reflect documented changes in COVID-19 spread in the Basque Country data, the claim that the models recover plausible patterns would be falsified.

Figures

Figures reproduced from arXiv: 2301.06385 by Elena Akhmatskaya, Hristo Inouzhe, Lorenzo Nagar, Mar\'ia Xos\'e Rodr\'iguez-\'Alvarez.

Figure 1
Figure 1. Figure 1: Flow Diagrams for SIKR (top) and SEMIKR (bottom) models with time-dependent transmission rate β(t). The S compartment represents the number of individuals in the population susceptible to be infected. The E1, . . . , EM compartments reflect the number of individuals at the different stages of exposure, when individuals are infected but are not yet infectious. The I1, . . . , IK compartments represent the n… view at source ↗
Figure 2
Figure 2. Figure 2: For the synthetic data: posterior predictive checks on daily incidence for a spline-based [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: For the synthetic data: posterior medians (solid lines) and 95% credible intervals (shaded [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For the Basque Country data: estimated P-spline regression (posterior mean) on daily [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: For the Basque Country data: posterior predictive checks on daily incidence data, cor [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: For the Basque Country data: posterior medians (solid lines) and 95% credible intervals [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: For the synthetic data: posterior medians (solid lines) and 95% credible intervals (shaded [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: For the synthetic data: trace-plot of the epidemiologically relevant variables for 10 chains [PITH_FULL_IMAGE:figures/full_fig_p039_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: For the synthetic data: posterior predictive checks on daily incidence for a spline-based [PITH_FULL_IMAGE:figures/full_fig_p041_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: For the synthetic data: posterior medians (solid lines) and 95% credible intervals (shaded [PITH_FULL_IMAGE:figures/full_fig_p042_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: For the synthetic data: posterior medians (solid lines) and 95% credible intervals (shaded [PITH_FULL_IMAGE:figures/full_fig_p043_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p044_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p045_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p046_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: For the synthetic data: parameters’ posterior densities for GHMC (top row) and SMR [PITH_FULL_IMAGE:figures/full_fig_p047_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: For the synthetic data: trace-plot of the epidemiologically relevant variables for 10 chains [PITH_FULL_IMAGE:figures/full_fig_p048_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: For the Basque Country data: posterior medians (solid lines) and 95% credible intervals [PITH_FULL_IMAGE:figures/full_fig_p050_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: For the Basque Country data: parameters’ posterior densities corresponding to 10 chains [PITH_FULL_IMAGE:figures/full_fig_p051_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: For the Basque Country data: parameters’ posterior densities corresponding to 10 chains [PITH_FULL_IMAGE:figures/full_fig_p052_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: For the Basque Country data: trace-plot of the epidemiologically relevant variables for [PITH_FULL_IMAGE:figures/full_fig_p053_24.png] view at source ↗
read the original abstract

A study of changes in the transmission of a disease, in particular, a new disease like COVID-19, requires very flexible models which can capture, among others, the effects of non-pharmacological and pharmacological measures, changes in population behaviour and random events. We favour data-driven approaches over a priori and ad-hoc methods and introduce a generalised family of epidemiologically informed mechanistic models, guided by Ordinary Differential Equations and embedded in a probabilistic model. The mechanistic models SIKR and SEMIKR which divide the population into disjoint compartments for individuals Susceptible to infection, Infectious (K sub-compartments), Exposed (M sub-compartments), and Removed from the pool of susceptible are enriched with a time-dependent transmission rate, parameterised using Bayesian P-splines. Such a parameterisation enables an extensive flexibility in the transmission dynamics, without resorting to ad-hoc specifications. Our probabilistic model relies on the solutions of a mechanistic model and benefits from access to the information about under-reporting of new infected cases, a crucial property when studying diseases with a large fraction of asymptomatic infections. Such a model can be differentiated efficiently, which makes Hamiltonian-based Monte Carlo sampling feasible after a careful initialisation and tuning strategy. This is particularly important in the present setting with weakly identified directions and challenging posterior geometries. Furthermore, we apply our methodology to study the transmission dynamics of COVID-19 in the Basque Country (Spain) from mid February 2020 to the end of January 2021, showing how the framework can recover plausible temporal patterns in transmission while making explicit the dependence of the results on modelling choices and convergence diagnostics.

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 paper introduces the SIKR and SEMIKR compartmental models (extensions of SIR/SEIR with K infectious and M exposed sub-compartments) that incorporate a time-dependent transmission rate parameterized via Bayesian P-splines. These are embedded in a differentiable probabilistic model that uses ODE solutions and accounts for under-reporting, enabling Hamiltonian Monte Carlo sampling after careful initialization and tuning. The framework is applied to COVID-19 case data from the Basque Country (mid-February 2020 to end-January 2021), with the central claim being that it recovers plausible temporal transmission patterns while making the dependence on modeling choices explicit and providing convergence diagnostics.

Significance. If the HMC inference is reliable, the approach supplies a flexible, data-driven alternative to ad-hoc transmission specifications that quantifies uncertainty and under-reporting effects, which could aid in evaluating intervention impacts in emerging diseases.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (probabilistic model and sampling strategy): The central claim requires reliable HMC sampling of the P-spline coefficients despite 'weakly identified directions and challenging posterior geometries.' The manuscript describes a 'careful initialisation and tuning strategy' but reports no quantitative diagnostics (effective sample size, R-hat, or divergent transitions) for the COVID-19 application; without these, the recovered transmission trajectories cannot be trusted as more than artifacts of poor mixing.
  2. [Results (Basque Country application)] Results section on Basque Country application: The claim that results 'make explicit the dependence of the results on modelling choices' is load-bearing, yet the manuscript provides no sensitivity analysis (e.g., varying the P-spline order, knot placement, or smoothing-parameter prior) with quantitative metrics comparing inferred transmission rates across specifications; plausibility alone does not establish robustness.
minor comments (2)
  1. [Model definition] Notation for the transmission-rate function β(t) should be introduced with an explicit equation number in the model-definition section to improve traceability when discussing its P-spline expansion.
  2. [Figures] Figure captions for the transmission-rate trajectories should include the specific HMC settings (chains, iterations, adaptation phase) used to generate each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback, which highlights important aspects for strengthening the credibility of our results. We address each major comment below and commit to revisions that directly respond to the concerns raised.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (probabilistic model and sampling strategy): The central claim requires reliable HMC sampling of the P-spline coefficients despite 'weakly identified directions and challenging posterior geometries.' The manuscript describes a 'careful initialisation and tuning strategy' but reports no quantitative diagnostics (effective sample size, R-hat, or divergent transitions) for the COVID-19 application; without these, the recovered transmission trajectories cannot be trusted as more than artifacts of poor mixing.

    Authors: We agree that the absence of specific numerical convergence diagnostics in the Basque Country application limits the ability to fully substantiate the reliability of the HMC samples. Although the manuscript describes the initialization and tuning strategy and references convergence diagnostics in the abstract, quantitative metrics such as effective sample sizes, R-hat values, and divergent transition counts were not reported for the application. In the revised manuscript we will add these diagnostics for the reported chains to allow direct assessment of mixing and sampling quality. revision: yes

  2. Referee: [Results (Basque Country application)] Results section on Basque Country application: The claim that results 'make explicit the dependence of the results on modelling choices' is load-bearing, yet the manuscript provides no sensitivity analysis (e.g., varying the P-spline order, knot placement, or smoothing-parameter prior) with quantitative metrics comparing inferred transmission rates across specifications; plausibility alone does not establish robustness.

    Authors: We acknowledge that the manuscript discusses modeling choices but does not present a formal sensitivity analysis with quantitative metrics comparing transmission-rate inferences across specifications such as P-spline order, knot placement, or smoothing-parameter priors. To strengthen the claim that dependence on modeling choices is made explicit, we will perform and report such a sensitivity analysis in the revised version, including quantitative comparisons of the resulting transmission trajectories. revision: yes

Circularity Check

0 steps flagged

No circularity: transmission rate learned from external data via P-splines

full rationale

The paper parameterizes time-dependent transmission rates in the SIKR/SEMIKR ODE models using Bayesian P-splines, then embeds the ODE solutions into a differentiable probabilistic model for HMC sampling. This is a standard data-driven fitting procedure applied to external COVID-19 case counts; the recovered trajectories are outputs of inference on observed data rather than inputs redefined as predictions. No equation or section shows a self-definitional loop, a fitted parameter renamed as a prediction, or a load-bearing claim resting solely on self-citation. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on standard ODE compartment assumptions plus the differentiability of the spline-augmented transmission function; no new physical entities are postulated.

free parameters (1)
  • P-spline coefficients and smoothing parameter
    The time-dependent transmission rate is parameterised by Bayesian P-splines whose coefficients are estimated from data.
axioms (2)
  • domain assumption The population can be partitioned into the stated disjoint compartments whose flows obey the given ODEs.
    Invoked when the mechanistic models SIKR and SEMIKR are defined.
  • domain assumption The transmission rate function is sufficiently smooth for the spline representation and the resulting ODE system remains numerically stable.
    Required for the probabilistic embedding and HMC sampling to be feasible.

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