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Discrete-time compartmental models with partially observed data: a comparison among frequentist and Bayesian approaches for addressing likelihood intractability

  • Cecilia Viscardi , Alessio Lachi ORCID logo EMAIL logo and Michela Baccini
Published/Copyright: September 26, 2025
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Epidemiologic Methods
From the journal Epidemiologic Methods Volume 14 Issue 1

Abstract

Compartmental models have emerged as useful tools in epidemiology due to their mechanistic nature. They provide insights into complex dynamic systems and allow predictions under different scenarios. However, despite their widespread use, there is still a gap in the literature, concerning their statistical formalization and a systematic discussion of the statistical methods suitable for both tasks of inference and forecasting. In this work, starting from the fundamental distinction between deterministic and stochastic compartmental models, we focus on how the formulation of the likelihood function becomes a necessary and challenging step in the transition from a deterministic to a stochastic framework. We then analyse the various difficulties encountered in evaluating the likelihood function associated with discrete-time stochastic models. We distinguish two reasons for the intractability of the likelihood function, the high dimension of missing data and the complexity of the model structure, and discuss suitable methods for addressing the problem both from a frequentist and Bayesian perspective. We overview likelihood-based methods and explore the use of likelihood-free approaches in this framework, namely approximate Bayesian computation algorithms and a method that combines model calibration with a parametric bootstrap procedure. We emphasize their ability to make inferences from data that are partially observed, or only observed in some aggregated form. To showcase their feasibility and reliability, we compare the likelihood-free and likelihood-based methods at work with a toy example of the Susceptible-Infected-Removed. Finally, we explore the relevance of likelihood-free methods in a real-world framework through an example of a complex compartmental model developed to study smoking dynamics in Tuscany (Italy).

Keywords: approximate Bayesian computation; calibration; compartmental models; incomplete data; intractable likelihood
Corresponding author: Alessio Lachi, Department of Medicine, Saint Camillus International University of Health and Medical Sciences, Rome, Italy, E-mail: 

Funding source: Regione Toscana

Funding source: Dipartimenti di Eccellenza

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: C.V. and M.B. conceptualized the project; A.L. wrote the codes and conducted the statistical analysis under the statistical supervision of C.V. and the epidemiological supervision of M.B.; C.V. and A.L. wrote the first version of the paper; M.B. provided critical feedback. All the authors read and approved the final version of the paper.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: The authors acknowledge the financial support provided by the Attributable Cancer Burden in Tuscany (ACAB) project funded by Regione Toscana and the “Dipartimenti Eccellenti 2018–2022” ministerial funds.

  7. Data availability: Not applicable.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/em-2024-0032).

Received: 2024-12-30
Accepted: 2025-08-06
Published Online: 2025-09-26

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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  3. Discrete-time compartmental models with partially observed data: a comparison among frequentist and Bayesian approaches for addressing likelihood intractability
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https://doi.org/10.1515/em-2024-0032
Keywords for this article
approximate Bayesian computation; calibration; compartmental models; incomplete data; intractable likelihood

Articles in the same Issue

  1. Causal mediation analysis for difference-in-difference design and panel data
  2. Research Articles
  3. Discrete-time compartmental models with partially observed data: a comparison among frequentist and Bayesian approaches for addressing likelihood intractability
  4. Sensitivity analysis for unmeasured confounding for a joint effect with an application to survey data
  5. Investigating the association between school substance programs and student substance use: accounting for informative cluster size
  6. The quantiles of extreme differences matrix for evaluating discriminant validity
  7. Finite-sample improved confidence intervals based on the estimating equation theory for the modified Poisson and least-squares regressions
  8. What if dependent causes of death were independent?
  9. Bot invasion: protecting the integrity of online surveys against spamming
  10. A study of a stochastic model and extinction phenomenon of meningitis epidemic
  11. Understanding the impact of media and latency in information response on the disease propagation: a mathematical model and analysis
  12. Time-varying reproductive number estimation for practical application in structured populations
  13. Perspective
  14. Should we still use pointwise confidence intervals for the Kaplan–Meier estimator?
  15. Leveraging data from multiple sources in epidemiologic research: transportability, dynamic borrowing, external controls, and beyond
  16. Regression calibration for time-to-event outcomes: mitigating bias due to measurement error in real-world endpoints
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