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Capturing the time-varying drivers of an epidemic using stochastic dynamical systems

Joseph Dureau, Konstantinos Kalogeropoulos, Marc Baguelin

arXiv:1203.5950v2stat.APstat.COstat.ME

TL;DR

Time-varying extrinsic influences complicate inference from epidemic data, motivating stochastic epidemic models and adaptive particle MCMC. In the England H1N1 application, the estimates supported a holiday-related decline in contact rates and evidence favoring a possible second wave after schools reopened.

  • Problem

    Epidemic trajectories reflect time-varying extrinsic influences, but relating climate and incidence alone does not disentangle them from intrinsic dynamics.

    Source: §1 · p. 2

  • Method

    The paper models time-varying epidemic parameters with diffusion processes and develops adaptive particle MCMC inference procedures.

    Source: §1 · p. 2, §2.2 · p. 4, §3.3 · p. 8

  • Results

    In the England application, βt changes aligned with school opening and closing, and real-time analysis provided evidence favoring holiday-related waning and a possible September second wave.

    Source: §5.1 · p. 10, §5.2 · p. 11

  • Takeaways & Limitations

    The framework can support developing richer epidemic models, testing public-health intervention scenarios, and exploring extrinsic influences such as climate.

    Source: §6 · p. 13

    As the diffusion-path partition is refined, the conditional posterior of σ concentrates and Gibbs MCMC convergence slows at O(m).

    Source: §3.2 · p. 6

Abstract

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Epidemics are often modelled using non-linear dynamical systems observed through partial and noisy data. In this paper, we consider stochastic extensions in order to capture unknown influences (changing behaviors, public interventions, seasonal effects etc). These models assign diffusion processes to the time-varying parameters, and our inferential procedure is based on a suitably adjusted adaptive particle MCMC algorithm. The performance of the proposed computational methods is validated on simulated data and the adopted model is applied to the 2009 H1N1 pandemic in England. In addition to estimating the effective contact rate trajectories, the methodology is applied in real time to provide evidence in related public health decisions. Diffusion driven SEIR-type models with age structure are also introduced.

1 Introduction

Epidemic models often focus on intrinsic dynamics, while time-varying extrinsic factors such as holidays, interventions, and climate also shape epidemic trajectories. This paper develops stochastic models and adaptive particle MCMC procedures to infer such changing influences.

1 Introduction

  • Time-varying extrinsic factors—including social cycles, interventions, and climate—can be crucial to epidemic trajectories.The introduction identifies these factors as distinct from intrinsic dynamics such as susceptible depletion and population immunity.Source: §1 · p. 2
  • Earlier studies either related climate and incidence without disentangling intrinsic from extrinsic effects or extrapolated experimental virus-resistance results to populations.Source: §1 · p. 2
  • The paper proposes stochastic differential equation models for time-varying epidemic aspects and robust inference using MCMC, particle MCMC, and adaptive methods.Source: §1 · p. 2
  • The paper evaluates its adaptive PMCMC schemes on simulated data and applies the methodology to the 2009 A/H1N1 pandemic.Source: §1 · p. 2

2 Modelling framework

The modelling framework uses SEIR dynamics while treating the effective contact rate as time-varying, then represents selected coefficients with diffusion processes. This allows flexible trajectories while making assumptions about their smoothness and using model uncertainty to accommodate possible structural misspecification.

2.1 Epidemic models with time-varying coefficients

  • The stochastic SEIR framework attributes model limitations mainly to time-varying effective contact rates, rather than individual or transmission-process randomness.It is intended for large-scale epidemics, where transmission random effects are treated as approximately deterministic.Source: §2.1 · p. 3
  • Estimating βt rather than Rt avoids combining evolving transmissibility with immunity, while trajectory flexibility trades lower model bias against higher variance.The framework notes regularisation as one way to manage this trade-off.Source: §2.1 · p. 3

2.2 Diffusion driven epidemic models

  • Diffusion processes represent selected epidemic coefficients, with drift and volatility choices encoding prior expectations about how the effective contact rate changes.An Ornstein–Uhlenbeck process is suggested when the rate is expected to converge; Brownian motion is an option when little prior restriction is desired.Source: §2.2 · pp. 3–4
  • Brownian motion gives a continuous, non-differentiable Markov path, while higher-order Brownian motions can represent smoother trajectories like spline-based approaches.The paper interprets βt as combining a smooth population-average transmission component and a rough component containing extrinsic factors.Source: §2.2 · p. 4
  • The Extended Kalman Filter offers fast computations but relies on Taylor and Gaussian approximations that may have non-negligible error.The paper also describes EKF as a possible proposal-building tool for MCMC schemes.Source: §2.2 · p. 4

3 Data augmentation via MCMC for diffusion driven epidemic models

The section sets up Bayesian inference for diffusion-driven epidemic models with noisy observations and latent state paths. It explains why standard Gibbs updates can mix poorly, then presents particle MCMC and adaptive parameter proposals to address those challenges.

3.1 Model and data augmentation setup

  • The model links a diffusion path to noisy prevalence or incidence observations through an ODE state trajectory, whose likelihood is intractable and requires path discretisation.Increasing the number of discretisation points can make the approximation error arbitrarily small.Source: §3.1 · pp. 5–6

3.2 Data augmentation via Gibbs schemes

  • Updating the path and parameters separately can produce high posterior correlations and reducible chains as m increases.The section identifies path-space sampling and path–parameter correlation as two difficulties with these schemes.Source: §3.2 · p. 6
  • Gibbs data augmentation can mix increasingly slowly as the grid is refined because quadratic variation identifies diffusion volatility and its conditional posterior concentrates.For constant volatility, the reported convergence rate is O(m); fixed-m schemes may have substantial approximation error.Source: §3.2 · p. 6

3.3 Adaptive Particle Markov Chain Monte Carlo algorithms

  • Particle filters provide an unbiased likelihood estimate and a posterior path sample, which particle MCMC uses to integrate over the discretised diffusion path.The framework targets posterior inference for the diffusion path and model parameters while accounting for parameter uncertainty and prior information.Source: §3 · p. 5, §3.3 · pp. 7–8
  • Particle MCMC with adaptive proposals updates the latent diffusion path and parameters jointly, while EKF-based covariance estimates improve the adaptive proposal.EK-Mode uses observed information at the EKF mode; EK-MCMC uses covariance from an approximate EKF-likelihood chain, with marginal added computational burden and substantial improvement reported in Section 4.Source: §3.2 · p. 7, §3.3 · pp. 8–9

4 Simulation Experiments

The simulations compare adaptive particle MCMC schemes and test trajectory estimation under noisy case observations. Adaptive covariance proposals achieve the strongest reported minimum effective sample sizes, and the estimated contact-rate path agrees with the simulated trajectories.

Simulation experiments

  • The simulations used a Brownian-motion model for log(β_t), with paths drawn either from that model or from a deterministic sigmoid curve.The experiments used 3,000 particles and 100,000 MCMC iterations after a long burn-in period.Source: §4 · p. 9
  • Adaptive PMCMC initialized with EK-MCMC estimated β_t in good agreement with simulated paths from partial, noisy observations.The observations represented 50 weeks of noisy new-case data; similar results were obtained at τ = 0.05.Source: §4 · p. 9

Comparison of adaptive PMCMC schemes

  • Minimum ESS was 0.008% (Id), 0.19% (EK-Mode), and 0.54% (EK-MCMC) for schemes adapting ε, versus 0.57%, 1.24%, and 1.38%, respectively, for schemes adapting Σ.The authors report that adapting Σ is crucial for reasonable performance unless EK-Mode or EK-MCMC matrices are used.Source: §4 · p. 9

5 The 2009 A/H1N1 pandemic

The England H1N1 analysis estimates time-varying contact rates and applies the model in real time to assess competing explanations for the first wave’s decline. A two-age-group SEIR extension is also fitted to separate child and adult incidence estimates.

5.1 Data, model and estimates

  • The England analysis used weekly ILI estimates corrected for over-reporting, asymptomatic infections, and consultation propensity, with c initially fixed at 10.The latent and infectious period priors came from additional data sources.Source: §5.1 · p. 10
  • The estimated βt changes aligned with school opening and closing, while inference was sensitive to priors for k and γ but not the other parameters.The paper notes that different βt values were observed during school and holiday periods.Source: §5.1 · p. 10

5.2 Application in real time. Was the first wave waning due to depletion of susceptibles?

  • The real-time analysis used partial data through three dates, fixed τ at 0.1, selected Brownian motion by DIC, and ran PMCMC in under two hours on a standard PC.Each run began from scratch; the model otherwise matched Section 5.1.Source: §5.2 · p. 11
  • For c = 10, the 97.5% posterior point for the difference in βt between July 13 and August 1 was −0.32, indicating a decrease and supporting scenario (i).The point reached 0 for a correction factor close to 70, which seemed unrealistic; a second wave in September actually occurred.Source: §5.2 · p. 11

5.3 A multiple age group diffusion driven SEIR model

  • The two-group model lets contact rates vary by age while treating cross-group rates as equal constants for parsimony.Brownian motion is assigned to log(βcc_t); βca and βac are set to b.Source: §5.3 · pp. 11–12
  • The age-group model fit well unlike earlier versions with one diffusion for all contact rates, and the children’s trajectory resembled Fig. 2.The authors say this stresses children’s role in epidemic evolution; EK-MCMC was essential for reasonable MCMC performance.Source: §5.3 · p. 12

6 Discussion

The paper develops diffusion-driven epidemic models and adaptive particle MCMC inference, with approximation error controllable by the user. It presents the approach as a route to richer epidemic models and notes that specific models or datasets need further study.

6 Discussion

  • The framework uses fine-partition data augmentation MCMC with user-controlled approximation error and jointly updates paths and parameters through particle MCMC.The continuous-time analysis identifies major issues with Gibbs data augmentation; EKF-based adaptive algorithms can improve performance, especially with many static parameters.Source: §6 · p. 12
  • The authors describe this as one of the first PMCMC applications to epidemic models and data, while noting prior standard PMCMC use by Rasmussen and others (2011).The paper’s application is therefore distinguished from the broader existence of PMCMC schemes in this area.Source: §6 · p. 13
  • The framework can support richer models, alternative intervention scenarios, and multiple data sources, while extending beyond influenza time series to other cases.The paper also proposes combining environmental and demographic stochasticity as a next step.Source: §6 · p. 13
  • The inference framework has a “plug and play” feature, but specific models or datasets require extra care and further study.The authors are working toward integration in a generic inference platform inspired by POMP.Source: §6 · p. 13

using stochastic dynamical systems SUPPLEMENTARY MATERIALS

The supplementary materials describe implementation choices, a particle-filter comparison, additional analyses, and model details.

Supplementary materials

Appendix A: details of the PMCMC implementation

The appendix gives practical guidance for PMCMC tuning, balancing numerical accuracy and computational cost when selecting the Euler time step and particle count.

Appendix A: details of the PMCMC implementation

  • The appendix identifies the Euler time step δ, particle-smoother count Nparts, and Metropolis parameter updates as key PMCMC implementation choices.Source: Appendix A · p. 22

Determining the Euler discretization time-step

  • As δ decreases, the Euler approximation converges toward the target posterior; monitoring posterior quantities across decreasing steps suggests δ = 0.1 day is reasonable in the influenza example.The computational cost is O(δ^-1).Source: Determining the Euler discretization… · p. 23

Determining an optimal number of particles

  • More particles reduce likelihood-estimate noise and improve MCMC acceptance until a plateau, but cost grows as O(Nparts).Lower observational noise requires more particles because they must be closer to observations.Source: Determining an optimal number of… · p. 23

Appendix B: Assessing the validity and limitations of the Extended Kalman approximation

The appendix compares particle-based estimates of the time-varying contact rate with Extended Kalman estimates. Particle methods reduce bias, and smoothing is associated with a substantial further reduction in MSE, while the approximation remains useful as a PMCMC proxy.

MSE Bias

  • The comparison used 100 simulated seven-month weekly influenza series, with biological and diffusion parameters assumed known to isolate estimation of β_t.The series were reverse-engineered from randomly selected Google FluTrend time series.Source: Appendix B · pp. 23–24
  • The EKF bias was 0.0285, while the particle-filter estimate reduced bias by about 78% (0.0063).The corresponding relative MSE reduction was 10%, from 0.0270 to 0.0242, indicating a bias-variance tradeoff.Source: Appendix B · p. 24
  • The smoothing-distribution estimator was associated with a further 87% (0.0032) reduction in MSE while keeping bias at the same low levels.The particle-filter estimates had a computational cost of 2 hours on a standard PC; the authors describe the EKF approximation as a robust proxy for initializing PMCMC.Source: Appendix B · p. 24
  • The authors conclude that EKF approximation bias is non-negligible relative to exact particle methods, although the approximation is robust enough to serve as a proxy.They suggest using it to initialize the particle MCMC algorithm.Source: Appendix B · p. 24

Appendix C: details on the simulations and results of sections 4 and 5

The supplementary analyses test particle MCMC on simulated epidemics and report additional results for the H1N1 and age-structured analyses. They also compare alternative real-time approaches, finding that the BM model performs consistently better by DIC, while noting uncertainty about that criterion.

Assessing the validity of the particle MCMC through simulations

  • With Brownian motion fitted to a smooth sigmoid contact-rate trajectory, the method captured β_t and other parameters reasonably well, but slightly underestimated τ.In experiment 2, the true parameter values fell within the 95% credible intervals; the authors suggest some variability may have been absorbed by Brownian-motion volatility.Source: Assessing the validity of the particle… · p. 25

A/H1N1 pandemic

  • Trace plots for the Section 5.2 parameter estimates indicated good PMCMC mixing.The authors attribute this to following the procedure presented in Appendix A.Source: A/H1N1 pandemic · p. 25

Age-structured model: children and adults

Illustration of alternative approaches on the real time example

  • The BM model was consistently better than the iBM model by DIC, but the authors note uncertainty about DIC performance in this setting.The alternative MIF approach fixes parameters at maximum-likelihood estimates before particle filtering, producing narrower 95% credible intervals—roughly 50% narrower on the six-month dataset and even more at early stages.Source: Illustration of alternative approaches… · pp. 25–26

Appendix D: Sensitivity analysis

The sensitivity analysis tilts prior means for the initial immune proportion and latent and infectious periods, then compares posterior summaries and β_t trajectories with the original-prior analysis.

Sensitivity analysis

  • Under prior-mean tilts of ±10% and ±20%, the β_t trajectories remained within the original 50% credible intervals.For all quantities except the parameters whose priors were tilted, posterior medians lay within both the original 50% and 95% credible intervals.Source: Appendix D · p. 26

Appendix E: details of Gibbs data augmentation scheme

The appendix describes diffusion-model data augmentation and its mixing challenges, motivating particle MCMC. It also presents supplementary figures and comparisons concerning algorithm behavior and contact-rate estimates.

Appendix E: details of Gibbs data augmentation scheme

  • The model represents the latent epidemic process with a diffusion, links observations to the ODE state vector through an observation model, and uses time-discretized paths with Euler approximation.The data-augmentation scheme alternates between updating the discretized trajectories given parameters and updating parameters given trajectories.Source: Appendix E · pp. 26–27
  • As the Euler time step δ tends to zero, the quadratic variation of the diffusion identifies its volatility parameter, causing Gibbs algorithms to degenerate and mix increasingly slowly.The stated practical mixing time is O(m).Source: Appendix E · p. 26
  • The Lamperti reparametrization transforms the diffusion path to a unit-diffusion process under the stated differentiability assumption on σ_x.The transformed process has drift ν and Brownian motion as its driving noise.Source: Appendix E · p. 26
  • The Gibbs schemes can mix poorly because proposed parameter values imply ODE-state trajectories that must remain close to noisy observations for acceptance.The problem intensifies as observation noise decreases and the epidemic time horizon increases; a simulation trace showed poor mixing for σ.Source: Appendix E · p. 28
  • The appendix concludes that both Gibbs data-augmentation formulations are likely to yield inaccurate and inefficient MCMC, making PMCMC essential in the paper’s implementation.It describes alternative reparametrizations for Gibbs schemes as a direction for future research.Source: Appendix E · p. 28
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