Estimation for dynamical systems using a population-based Kalman filter – Applications in computational biology
MathematicS In Action, Volume 11 (2022) no. 1, pp. 213-242.

Estimation of dynamical systems (in particular, identification of their parameters) is fundamental in computational biology, e.g., pharmacology, virology, or epidemiology, to reconcile model runs with available measurements. Unfortunately, the mean and variance priors of the parameters must be chosen very appropriately to balance our distrust of the measurements when the data are sparse or corrupted by noise. Otherwise, the identification procedure fails. One option is to use repeated measurements collected in configurations with common priors (for example, with multiple subjects in a clinical trial or clusters in an epidemiological investigation). This shared information is beneficial and is typically modeled in statistics using nonlinear mixed-effects models. In this paper, we present a data assimilation method that is compatible with such a mixed-effects strategy without being compromised by the potential curse of dimensionality. We define population-based estimators through maximum likelihood estimation. We then develop an equivalent robust sequential estimator for large populations based on filtering theory that sequentially integrates data. Finally, we limit the computational complexity by defining a reduced-order version of this population-based Kalman filter that clusters subpopulations with common observational backgrounds. The performance of the resulting algorithm is evaluated against classical pharmacokinetics benchmarks. Finally, the versatility of the proposed method is tested in an epidemiological study using real data on the hospitalisation of COVID-19 patients in the regions and departments of France.

Published online:
DOI: 10.5802/msia.25
Classification: 62L12,  93B53,  92-08,  62P10
Keywords: Data Assimilation, Non linear mixed-effect models, Kalman Filters, Epidemiology, COVID-19, Pharmacokinetics
Annabelle Collin 1; Mélanie Prague 2; Philippe Moireau 3

1 IMB UMR 5251, Université Bordeaux – Inria, Université Bordeaux, Talence, France
2 Univ. Bordeaux, Department of Public Health, Inserm Bordeaux Population Health Research Center, Inria SISTM, UMR 1219, Bordeaux, France; Vaccine Research institute, Créteil, France
3 Inria – LMS, CNRS UMR 7649, Ecole Polytechnique, Institut Polytechnique de Paris, 1 Rue Honoré d’Estienne d’Orves, Palaiseau, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Annabelle Collin; Mélanie Prague; Philippe Moireau. Estimation for dynamical systems using a population-based Kalman filter – Applications in computational biology. MathematicS In Action, Volume 11 (2022) no. 1, pp. 213-242. doi : 10.5802/msia.25. https://msia.centre-mersenne.org/articles/10.5802/msia.25/

[1] F. J. Angulo; L. Finelli; D. L. Swerdlow Estimation of US SARS-CoV-2 Infections, Symptomatic Infections, Hospitalizations, and Deaths Using Seroprevalence Surveys, JAMA Network Open, Volume 4 (2021) no. 1, p. e2033706 | DOI

[2] M. Asch; M. Bocquet; M. Nodet Data assimilation: methods, algorithms, and applications, Fundamentals of Algorithms, Society for Industrial and Applied Mathematics, 2016, xviii+306 pages | DOI | HAL

[3] A. Barrau; S. Bonnabel The invariant extended Kalman filter as a stable observer, IEEE Trans. Autom. Control, Volume 62 (2017) no. 4, pp. 1797-1812 | DOI | MR | Zbl

[4] A. Bensoussan Estimation and Control of Dynamical Systems, Interdisciplinary Applied Mathematics, Springer, 2018 | DOI

[5] J. Blum; F.-X. Le Dimet; I. M. Navon Data assimilation for geophysical fluids, Handbook of Numerical Analysis: Computational Methods for the Atmosphere and the Oceans (R. Temam; J. Tribbia, eds.), Elsevier, 2008, pp. 385-441

[6] A. Caiazzo; F. Caforio; G. Montecinos; L. O. Muller; P. J. Blanco; E. F. Toro Assessment of reduced-order unscented Kalman filter for parameter identification in 1-dimensional blood flow models using experimental data., Int. J. Numer. Methods Biomed. Eng., Volume 33 (2017) no. 8, p. e2843 | DOI | MR

[7] B. Carpenter; A. Gelman; M. D. Hoffman; D. Lee; B. Goodrich; M. Betancourt; M. Brubaker; J. Guo; P. Li; A. Riddell Stan: A probabilistic programming language, J. Stat. Softw., Volume 76 (2017) no. 1 | DOI

[8] M. Cevik; M. Tate; O. Lloyd; A. E. Maraolo; J. Schafers; A. Ho SARS-CoV-2, SARS-CoV, and MERS-CoV viral load dynamics, duration of viral shedding, and infectiousness: a systematic review and meta-analysis, The Lancet Microbe (2020)

[9] D. Chapelle; M. Fragu; V. Mallet; P. Moireau Fundamental principles of data assimilation underlying the Verdandi library: applications to biophysical model personalization within euHeart, Med. Biol. Eng. Comput., Volume 51 (2013), pp. 1221-1233 | DOI

[10] D. Chapelle; A. Gariah; P. Moireau; J. Sainte-Marie A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems – Analysis, assessments and applications to parameter estimation, ESAIM, Math. Model. Numer. Anal., Volume 47 (2013) no. 6, pp. 1821-1843 | DOI | Numdam | MR | Zbl

[11] E. Comets; A. Lavenu; M. Lavielle Parameter estimation in nonlinear mixed effect models using saemix, an R implementation of the SAEM algorithm, J. Stat. Softw., Volume 80 (2017), pp. 1-41 | DOI

[12] M. Delattre; M. Lavielle Coupling the SAEM algorithm and the extended Kalman filter for maximum likelihood estimation in mixed-effects diffusion models, Stat. Interface, Volume 6 (2013) no. 4, pp. 519-532 | DOI | MR | Zbl

[13] J.-F. Delfraissy; L. Atlani Duault; D. Benamouzig; L. Bouadma; S. Cauchemez; F. Chauvin; A. Fontanet; A. Hoang; D. Malvy; Y. Yazdanpanah Une deuxième vague entrainant une situation sanitaire critique, 2020 (Note du Conseil scientifique COVID-19)

[14] M. J. Denwood runjags: An R package providing interface utilities, model templates, parallel computing methods and additional distributions for MCMC models in JAGS, J. Stat. Softw., Volume 71 (2016) no. 9, pp. 1-25 | DOI

[15] L. Di Domenico; G. Pullano; C. E. Sabbatini; P.-Y. Boëlle; V. Colizza Impact of lockdown on COVID-19 epidemic in Île-de-France and possible exit strategies, BMC Medicine, Volume 18 (2020) no. 1, pp. 1-13 | DOI

[16] S. B. Duffull; C. M. J. Kirkpatrick; B. Green; N. H. G. Holford Analysis of population pharmacokinetic data using NONMEM and WinBUGS, J. Biopharm. Stat., Volume 15 (2004) no. 1, pp. 53-73 | DOI | MR

[17] L. C. Evans An introduction to stochastic differential equations, American Mathematical Society, 2012

[18] G. Evensen Data Assimilation: The Ensemble Kalman Filter, Springer, 2009 | DOI

[19] A. Gelb; J. F. Kasper; R. A. Nash; C. F. Price; A. A. Sutherland Applied Optimal Estimation, MIT Press, 1974

[20] J. He; Y. Guo; R. Mao; J. Zhang Proportion of asymptomatic coronavirus disease 2019: A systematic review and meta-analysis, J. Med. Virol., Volume 93 (2021) no. 2, pp. 820-830 | DOI

[21] J. M. Heffernan; R. J. Smith; L. M. Wahl Perspectives on the basic reproductive ratio, J. R. Soc. Interface, Volume 2 (2005) no. 4, pp. 281-293 | DOI

[22] S. J. Julier; J. K. Uhlmann A new extension of the Kalman filter to nonlinear systems, Proc. of AeroSense: The 11th Int. Symp. on Aerospace/Defence Sensing, Simulation and Controls (1997)

[23] R. Kalman; R. Bucy New results in linear filtering and prediction theory, Trans. ASME J. Basic. Eng., Volume 83 (1961), p. 95--108 | DOI | MR

[24] S. Klim; S. B. Mortensen; N. R. Kristensen; R. V. Overgaard; H. Madsen Population stochastic modelling (PSM)—an R package for mixed-effects models based on stochastic differential equations, Comput. Methods Programs Biomed., Volume 94 (2009) no. 3, pp. 279-289 | DOI

[25] E. Kuhn; M. Lavielle Maximum likelihood estimation in nonlinear mixed effects models, Comput. Stat. Data Anal., Volume 49 (2005) no. 4, pp. 1020-1038 | DOI | MR | Zbl

[26] N. M. Laird; J. H. Ware Random-effects models for longitudinal data, Biometrics, Volume 38 (1982) no. 4, pp. 963-974 | DOI | Zbl

[27] S. Lauer; K. Grantz; Q. Bi; F. Jones; Q. Zheng; H. Meredith; A. Azman; N. Reich; J. Lessler The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. Intern. Med., Volume 172 (2020) no. 9, pp. 577-582 | DOI

[28] M. Lavielle Mixed effects models for the population approach: models, tasks, methods and tools, CRC Press, 2014 | DOI

[29] M. Lavielle; M. Faron; J. Lefevre; J.-D. Zeitoun Extension of a SIR model for modelling the propagation of Covid-19 in several countries. (2020) (https://www.medrxiv.org/content/early/2020/05/21/2020.05.17.20104885) | DOI

[30] K. Law; A. Stuart; K. Zygalakis Data assimilation: A mathematical introduction, Texts in Applied Mathematics, 62, Springer, 2015 | DOI

[31] R. Li; S. Pei; B. Chen; Y. Song; T. Zhang; W. Yang; J. Shaman Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2), Science, Volume 368 (2020) no. 6490, pp. 489-493 | DOI

[32] X. Liu; Y. Wang Comparing the performance of [FOCE] and different expectation-maximization methods in handling complex population physiologically-based pharmacokinetic models, J. Pharmacokinet. Pharmacodyn., Volume 43 (2016) no. 4, pp. 359-370 | DOI

[33] P. Moireau; D. Chapelle Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems, ESAIM, Control Optim. Calc. Var., Volume 17 (2011) no. 2, pp. 380-405 | DOI | Numdam | MR | Zbl

[34] P. Moireau; D. Chapelle Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems, ESAIM, Control Optim. Calc. Var., Volume 17 (2011) no. 2, pp. 380-405 | DOI | Numdam | MR | Zbl

[35] A. Oberg; M. Davidian Estimating Data Transformations in Nonlinear Mixed Effects Models, Biometrics, Volume 56 (2000) no. 1, pp. 65-72 | DOI | Zbl

[36] R. V. Overgaard; N. Jonsson; C. W. Tornøe; H. Madsen Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, J. Pharmacokinet. Pharmacodyn., Volume 32 (2005) no. 1, pp. 85-107 | DOI

[37] A. Pan; L. Liu; C. Wang; H. Guo; X. Hao; Q. Wang; J. Huang; N. He; H. Yu; X. Lin et al. Association of public health interventions with the epidemiology of the COVID-19 outbreak in Wuhan, China, J. Am. Med. Soc., Volume 323 (2020) no. 19, pp. 1915-1923

[38] A. Perasso An introduction to the basic reproduction number in mathematical epidemiology, ESAIM, Proc. Surv., Volume 62 (2018), pp. 123-138 | DOI | MR | Zbl

[39] A. S. Perelson; A. U. Neumann; M. Markowitz; J. M. Leonard; D. D. Ho HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, Volume 271 (1996) no. 5255, pp. 1582-1586 | DOI

[40] D. T. Pham Stochastic methods for sequential data assimilation in strongly nonlinear systems, Monthly Weather Review, Volume 129 (2001) no. 5, pp. 1194-1207 | DOI

[41] D. T. Pham; J. Verron; L. Gourdeau Filtres de Kalman singuliers évolutifs pour l’assimilation de données en océanographie, C. R. Acad. Sci., Sér. IIA Earth Planet. Sci., Volume 326 (1998) no. 4, pp. 255-260

[42] D. T. Pham; J. Verron; C. M. Roubaud A singular evolutive extended Kalman filter for data assimilation in oceanography, J. Marine Syst., Volume 16 (1998) no. 3-4, pp. 323-340 | DOI

[43] J. C. Pinheiro; D. M. Bates Approximations to the log-likelihood function in the nonlinear mixed-effects model, J. Comput. Graph. Stat., Volume 4 (1995) no. 1, pp. 12-35

[44] E. L. Plan; A. Maloney; F. Mentré; M. O. Karlsson; J. Bertrand Performance comparison of various maximum likelihood nonlinear mixed-effects estimation methods for dose–response models, AAPS J., Volume 14 (2012) no. 3, pp. 420-432 | DOI

[45] M. Prague Use of dynamical models for treatment optimization in HIV infected patients: a sequential Bayesian analysis approach., J. Soc. Fr. Stat., Volume 157 (2016) no. 2, p. 20 | Numdam | MR | Zbl

[46] M. Prague; D. Commenges; J. Guedj; J. Drylewicz; R. Thiébaut NIMROD : A program for inference via a normal approximation of the posterior in models with random effects based on ordinary differential equations, Comput. Methods Programs Biomed., Volume 111 (2013) no. 2, pp. 447-458 | DOI

[47] H. Salje; C. T. Kiem; N. Lefrancq; N. Courtejoie; P. Bosetti; J. Paireau; A. Andronico; N. Hozé; J. Richet; C.-L. Dubost et al. Estimating the burden of SARS-CoV-2 in France, Science, Volume 369 (2020) no. 6500, pp. 208-211 | DOI

[48] F. L. Schumacher; C. S. Ferreira; M. O. Prates; A. Lachos; V. H. Lachos A robust nonlinear mixed-effects model for COVID-19 deaths data, Int. J. Numer. Methods Biomed. Eng., Volume 14 (2021) no. 1, pp. 39-57 | MR

[49] D. Simon Optimal State Estimation: Kalman, H , and Nonlinear Approaches, Wiley-Interscience, 2006 | DOI

[50] C. W. Tornøe; R. V. Overgaard; H. Agersø; H. A . Nielsen; H. Madsen; E. N. Jonsson Stochastic differential equations in NONMEM®: implementation, application, and comparison with ordinary differential equations, Pharm. Res., Volume 22 (2005) no. 8, pp. 1247-1258 | DOI

[51] R. A. Upton Pharmacokinetic interactions between theophylline and other medication (Part I), Clin. Pharmacokin., Volume 20 (1991) no. 1, pp. 66-80 | DOI

[52] G. Verbeke Linear mixed models for longitudinal data, Linear mixed models in practice, Springer, 1997, pp. 63-153 | DOI

[53] J. Wakefield; A. Racine-Poon An application of Bayesian population pharmacokinetic/pharmacodynamic models to dose recommendation, Stat. Med., Volume 14 (1995) no. 9, pp. 971-986 | DOI

[54] H. Wu Statistical methods for HIV dynamic studies in AIDS clinical trials, Stat. Methods Med. Res., Volume 14 (2005) no. 2, pp. 171-192 | DOI | MR | Zbl

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