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Rubio FJ, Espindola JA, Montoya JA. On near-redundancy and identifiability of parametric hazard regression models under censoring. Biom J 2023; 65:e2300006. [PMID: 37394716 DOI: 10.1002/bimj.202300006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2023] [Revised: 04/13/2023] [Accepted: 05/09/2023] [Indexed: 07/04/2023]
Abstract
We study parametric inference on a rich class of hazard regression models in the presence of right-censoring. Previous literature has reported some inferential challenges, such as multimodal or flat likelihood surfaces, in this class of models for some particular data sets. We formalize the study of these inferential problems by linking them to the concepts of near-redundancy and practical nonidentifiability of parameters. We show that the maximum likelihood estimators of the parameters in this class of models are consistent and asymptotically normal. Thus, the inferential problems in this class of models are related to the finite-sample scenario, where it is difficult to distinguish between the fitted model and a nested nonidentifiable (i.e., parameter-redundant) model. We propose a method for detecting near-redundancy, based on distances between probability distributions. We also employ methods used in other areas for detecting practical nonidentifiability and near-redundancy, including the inspection of the profile likelihood function and the Hessian method. For cases where inferential problems are detected, we discuss alternatives such as using model selection tools to identify simpler models that do not exhibit these inferential problems, increasing the sample size, or extending the follow-up time. We illustrate the performance of the proposed methods through a simulation study. Our simulation study reveals a link between the presence of near-redundancy and practical nonidentifiability. Two illustrative applications using real data, with and without inferential problems, are presented.
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Affiliation(s)
- Francisco J Rubio
- Department of Statistical Science, University College London, London, UK
| | | | - José A Montoya
- Department of Mathematics, University of Sonora, Hermosillo, México
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Tönsing C, Steiert B, Timmer J, Kreutz C. Likelihood-ratio test statistic for the finite-sample case in nonlinear ordinary differential equation models. PLoS Comput Biol 2023; 19:e1011417. [PMID: 37738254 PMCID: PMC10550180 DOI: 10.1371/journal.pcbi.1011417] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2023] [Revised: 10/04/2023] [Accepted: 08/08/2023] [Indexed: 09/24/2023] Open
Abstract
Likelihood ratios are frequently utilized as basis for statistical tests, for model selection criteria and for assessing parameter and prediction uncertainties, e.g. using the profile likelihood. However, translating these likelihood ratios into p-values or confidence intervals requires the exact form of the test statistic's distribution. The lack of knowledge about this distribution for nonlinear ordinary differential equation (ODE) models requires an approximation which assumes the so-called asymptotic setting, i.e. a sufficiently large amount of data. Since the amount of data from quantitative molecular biology is typically limited in applications, this finite-sample case regularly occurs for mechanistic models of dynamical systems, e.g. biochemical reaction networks or infectious disease models. Thus, it is unclear whether the standard approach of using statistical thresholds derived for the asymptotic large-sample setting in realistic applications results in valid conclusions. In this study, empirical likelihood ratios for parameters from 19 published nonlinear ODE benchmark models are investigated using a resampling approach for the original data designs. Their distributions are compared to the asymptotic approximation and statistical thresholds are checked for conservativeness. It turns out, that corrections of the likelihood ratios in such finite-sample applications are required in order to avoid anti-conservative results.
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Affiliation(s)
- Christian Tönsing
- Institute of Physics, University of Freiburg, Germany
- CIBSS Centre for Integrative Biological Signalling Studies, University of Freiburg, Germany
- FDM Freiburg Center for Data Analysis and Modeling, University of Freiburg, Germany
| | | | - Jens Timmer
- Institute of Physics, University of Freiburg, Germany
- CIBSS Centre for Integrative Biological Signalling Studies, University of Freiburg, Germany
- FDM Freiburg Center for Data Analysis and Modeling, University of Freiburg, Germany
| | - Clemens Kreutz
- CIBSS Centre for Integrative Biological Signalling Studies, University of Freiburg, Germany
- FDM Freiburg Center for Data Analysis and Modeling, University of Freiburg, Germany
- Faculty of Medicine and Medical Center, Institute of Medical Biometry and Statistics, University of Freiburg, Germany
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Grodd M, Refisch L, Lorenz F, Fischer M, Lottes M, Hackenberg M, Kreutz C, Grabenhenrich L, Binder H, Wolkewitz M. [Forecasting models to guide intensive care COVID-19 capacities in Germany]. Med Klin Intensivmed Notfmed 2023; 118:125-131. [PMID: 35267045 PMCID: PMC8907553 DOI: 10.1007/s00063-022-00903-x] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2021] [Revised: 12/14/2021] [Accepted: 01/31/2022] [Indexed: 11/02/2022]
Abstract
BACKGROUND Time-series forecasting models play a central role in guiding intensive care coronavirus disease 2019 (COVID-19) bed capacity in a pandemic. A key predictor of future intensive care unit (ICU) COVID-19 bed occupancy is the number of new severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infections in the general population, which in turn is highly associated with week-to-week variability, reporting delays, regional differences, number of unknown cases, time-dependent infection rates, vaccinations, SARS-CoV‑2 virus variants, and nonpharmaceutical containment measures. Furthermore, current and also future COVID ICU occupancy is significantly influenced by ICU discharge and mortality rates. METHODS Both the number of new SARS-CoV‑2 infections in the general population and intensive care COVID-19 bed occupancy rates are recorded in Germany. These data are statistically analyzed on a daily basis using epidemic SEIR (susceptible, exposed, infection, recovered) models using ordinary differential equations and multiple regression models. RESULTS Forecast results of the immediate trend (20-day forecast) of ICU occupancy by COVID-19 patients are made available to decision makers at various levels throughout the country. CONCLUSION The forecasts are compared with the development of available ICU bed capacities in order to identify capacity limitations at an early stage and to enable short-term solutions to be made, such as supraregional transfers.
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Affiliation(s)
- Marlon Grodd
- Institut für Medizinische Biometrie und Statistik, Medizinische Fakultät und Universitätsklinikum, Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland
| | - Lukas Refisch
- Institut für Medizinische Biometrie und Statistik, Medizinische Fakultät und Universitätsklinikum, Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland
| | - Fabian Lorenz
- Institut für Medizinische Biometrie und Statistik, Medizinische Fakultät und Universitätsklinikum, Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland
| | | | | | - Maren Hackenberg
- Institut für Medizinische Biometrie und Statistik, Medizinische Fakultät und Universitätsklinikum, Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland
| | - Clemens Kreutz
- Institut für Medizinische Biometrie und Statistik, Medizinische Fakultät und Universitätsklinikum, Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland
| | | | - Harald Binder
- Institut für Medizinische Biometrie und Statistik, Medizinische Fakultät und Universitätsklinikum, Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland
| | - Martin Wolkewitz
- Institut für Medizinische Biometrie und Statistik, Medizinische Fakultät und Universitätsklinikum, Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland.
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Refisch L, Lorenz F, Riedlinger T, Taubenböck H, Fischer M, Grabenhenrich L, Wolkewitz M, Binder H, Kreutz C. Data-driven prediction of COVID-19 cases in Germany for decision making. BMC Med Res Methodol 2022; 22:116. [PMID: 35443607 PMCID: PMC9019290 DOI: 10.1186/s12874-022-01579-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2021] [Accepted: 03/15/2022] [Indexed: 11/28/2022] Open
Abstract
Background The COVID-19 pandemic has led to a high interest in mathematical models describing and predicting the diverse aspects and implications of the virus outbreak. Model results represent an important part of the information base for the decision process on different administrative levels. The Robert-Koch-Institute (RKI) initiated a project whose main goal is to predict COVID-19-specific occupation of beds in intensive care units: Steuerungs-Prognose von Intensivmedizinischen COVID-19 Kapazitäten (SPoCK). The incidence of COVID-19 cases is a crucial predictor for this occupation. Methods We developed a model based on ordinary differential equations for the COVID-19 spread with a time-dependent infection rate described by a spline. Furthermore, the model explicitly accounts for weekday-specific reporting and adjusts for reporting delay. The model is calibrated in a purely data-driven manner by a maximum likelihood approach. Uncertainties are evaluated using the profile likelihood method. The uncertainty about the appropriate modeling assumptions can be accounted for by including and merging results of different modelling approaches. The analysis uses data from Germany describing the COVID-19 spread from early 2020 until March 31st, 2021. Results The model is calibrated based on incident cases on a daily basis and provides daily predictions of incident COVID-19 cases for the upcoming three weeks including uncertainty estimates for Germany and its subregions. Derived quantities such as cumulative counts and 7-day incidences with corresponding uncertainties can be computed. The estimation of the time-dependent infection rate leads to an estimated reproduction factor that is oscillating around one. Data-driven estimation of the dark figure purely from incident cases is not feasible. Conclusions We successfully implemented a procedure to forecast near future COVID-19 incidences for diverse subregions in Germany which are made available to various decision makers via an interactive web application. Results of the incidence modeling are also used as a predictor for forecasting the need of intensive care units. Supplementary Information The online version contains supplementary material available at (10.1186/s12874-022-01579-9).
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Affiliation(s)
- Lukas Refisch
- Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center, University of Freiburg, Stefan Meier Str. 26, Freiburg, 79104, Germany.,Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, Freiburg, 79104, Germany
| | - Fabian Lorenz
- Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center, University of Freiburg, Stefan Meier Str. 26, Freiburg, 79104, Germany.,Centre for Integrative Biological Signalling Studies (CIBSS), Schänzlestr. 18, Freiburg, 79104, Germany
| | - Torsten Riedlinger
- German Aerospace Center, Earth Observation Center, Münchener Str. 20, Weßling, 82234, Germany
| | - Hannes Taubenböck
- German Aerospace Center, Earth Observation Center, Münchener Str. 20, Weßling, 82234, Germany.,Institute for Geography and Geology, Julius-Maximilians-Universität Würzburg, Am Hubland, Würzburg, 97074, Germany
| | - Martina Fischer
- Robert-Koch-Institute, Department for Methodology and Research Infrastructure, Nordufer 20, Berlin, 13353, Germany
| | - Linus Grabenhenrich
- Robert-Koch-Institute, Department for Methodology and Research Infrastructure, Nordufer 20, Berlin, 13353, Germany.,Charité - Universitätsmedizin Berlin, Department of Dermatology, Venerology and Allergology, Luisenstraße 2, Berlin, 10117, Germany
| | - Martin Wolkewitz
- Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center, University of Freiburg, Stefan Meier Str. 26, Freiburg, 79104, Germany
| | - Harald Binder
- Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center, University of Freiburg, Stefan Meier Str. 26, Freiburg, 79104, Germany.,Freiburg Center for Data Analysis and Modelling (FDM), University of Freiburg, Ernst-Zermelo-Str. 1, Freiburg, 79104, Germany
| | - Clemens Kreutz
- Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center, University of Freiburg, Stefan Meier Str. 26, Freiburg, 79104, Germany. .,Centre for Integrative Biological Signalling Studies (CIBSS), Schänzlestr. 18, Freiburg, 79104, Germany. .,Freiburg Center for Data Analysis and Modelling (FDM), University of Freiburg, Ernst-Zermelo-Str. 1, Freiburg, 79104, Germany.
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Wang Y, Lu G, Du J. Calibration and prediction for the inexact SIR model. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:2800-2818. [PMID: 35240807 DOI: 10.3934/mbe.2022128] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
A Susceptible Infective Recovered (SIR) model is usually unable to mimic the actual epidemiological system exactly. The reasons for this inaccuracy include observation errors and model discrepancies due to assumptions and simplifications made by the SIR model. Hence, this work proposes calibration and prediction methods for the SIR model with a one-time reported number of infected cases. Given that the observation errors of the reported data are assumed to be heteroscedastic, we propose two predictors to predict the actual epidemiological system by modeling the model discrepancy through a Gaussian Process model. One is the calibrated SIR model, and the other one is the discrepancy-corrected predictor, which integrates the calibrated SIR model with the Gaussian Process predictor to solve the model discrepancy. A wild bootstrap method quantifies the two predictors' uncertainty, while two numerical studies assess the performance of the proposed method. The numerical results show that, the proposed predictors outperform the existing ones and the prediction accuracy of the discrepancy-corrected predictor is improved by at least 49.95%.
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Affiliation(s)
- Yan Wang
- School of Statistics and Data Science, Beijing University of Technology, Beijing 100124, China
| | - Guichen Lu
- School of Science, Chongqing University of Technology, Chongqing 400054, China
| | - Jiang Du
- School of Science, Chongqing University of Technology, Chongqing 400054, China
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Adlung L, Stapor P, Tönsing C, Schmiester L, Schwarzmüller LE, Postawa L, Wang D, Timmer J, Klingmüller U, Hasenauer J, Schilling M. Cell-to-cell variability in JAK2/STAT5 pathway components and cytoplasmic volumes defines survival threshold in erythroid progenitor cells. Cell Rep 2021; 36:109507. [PMID: 34380040 DOI: 10.1016/j.celrep.2021.109507] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2020] [Revised: 04/01/2021] [Accepted: 07/19/2021] [Indexed: 12/25/2022] Open
Abstract
Survival or apoptosis is a binary decision in individual cells. However, at the cell-population level, a graded increase in survival of colony-forming unit-erythroid (CFU-E) cells is observed upon stimulation with erythropoietin (Epo). To identify components of Janus kinase 2/signal transducer and activator of transcription 5 (JAK2/STAT5) signal transduction that contribute to the graded population response, we extended a cell-population-level model calibrated with experimental data to study the behavior in single cells. The single-cell model shows that the high cell-to-cell variability in nuclear phosphorylated STAT5 is caused by variability in the amount of Epo receptor (EpoR):JAK2 complexes and of SHP1, as well as the extent of nuclear import because of the large variance in the cytoplasmic volume of CFU-E cells. 24-118 pSTAT5 molecules in the nucleus for 120 min are sufficient to ensure cell survival. Thus, variability in membrane-associated processes is sufficient to convert a switch-like behavior at the single-cell level to a graded population-level response.
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Affiliation(s)
- Lorenz Adlung
- Division Systems Biology of Signal Transduction, German Cancer Research Center (DKFZ), 69120 Heidelberg, Germany; Department of Medicine, University Medical Center Hamburg-Eppendorf, 20246 Hamburg, Germany; Hamburg Center for Translational Immunology (HCTI), University Medical Center Hamburg-Eppendorf, 20246 Hamburg, Germany
| | - Paul Stapor
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, 85764 Neuherberg, Germany; Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, 85748 Garching, Germany
| | - Christian Tönsing
- Institute of Physics and Freiburg Center for Data Analysis and Modelling (FDM), University of Freiburg, 79104 Freiburg, Germany; CIBSS-Centre for Integrative Biological Signalling Studies, University of Freiburg, 79104 Freiburg, Germany
| | - Leonard Schmiester
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, 85764 Neuherberg, Germany; Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, 85748 Garching, Germany
| | - Luisa E Schwarzmüller
- Division Systems Biology of Signal Transduction, German Cancer Research Center (DKFZ), 69120 Heidelberg, Germany
| | - Lena Postawa
- Division Systems Biology of Signal Transduction, German Cancer Research Center (DKFZ), 69120 Heidelberg, Germany
| | - Dantong Wang
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, 85764 Neuherberg, Germany; Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, 85748 Garching, Germany
| | - Jens Timmer
- Institute of Physics and Freiburg Center for Data Analysis and Modelling (FDM), University of Freiburg, 79104 Freiburg, Germany; CIBSS-Centre for Integrative Biological Signalling Studies, University of Freiburg, 79104 Freiburg, Germany.
| | - Ursula Klingmüller
- Division Systems Biology of Signal Transduction, German Cancer Research Center (DKFZ), 69120 Heidelberg, Germany; Translational Lung Research Center (TLRC), Member of the German Center for Lung Research (DZL), 69120 Heidelberg, Germany.
| | - Jan Hasenauer
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, 85764 Neuherberg, Germany; Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, 85748 Garching, Germany; Faculty of Mathematics and Natural Sciences, University of Bonn, 53113 Bonn, Germany.
| | - Marcel Schilling
- Division Systems Biology of Signal Transduction, German Cancer Research Center (DKFZ), 69120 Heidelberg, Germany.
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Piazzola C, Tamellini L, Tempone R. A note on tools for prediction under uncertainty and identifiability of SIR-like dynamical systems for epidemiology. Math Biosci 2020; 332:108514. [PMID: 33217409 DOI: 10.1016/j.mbs.2020.108514] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Revised: 11/09/2020] [Accepted: 11/09/2020] [Indexed: 12/18/2022]
Abstract
We provide an overview of the methods that can be used for prediction under uncertainty and data fitting of dynamical systems, and of the fundamental challenges that arise in this context. The focus is on SIR-like models, that are being commonly used when attempting to predict the trend of the COVID-19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIR-like models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it non-trivial to use these models for meaningful predictions. Most of the points that we touch upon are actually generally valid for inverse problems in more general setups.
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Affiliation(s)
- Chiara Piazzola
- Consiglio Nazionale delle Ricerche - Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" (CNR-IMATI), Via Ferrata 5/A, 27100 Pavia, Italy.
| | - Lorenzo Tamellini
- Consiglio Nazionale delle Ricerche - Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" (CNR-IMATI), Via Ferrata 5/A, 27100 Pavia, Italy.
| | - Raúl Tempone
- Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, Pontdriesch 14-16, 52062, Aachen, Germany; King Abdullah University of Science and Technology (KAUST) - Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE), Thuwal, 23955-6900, Saudi Arabia.
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Antonini C, Calandrini S, Stracci F, Dario C, Bianconi F. Mathematical Modeling and Robustness Analysis to Unravel COVID-19 Transmission Dynamics: The Italy Case. BIOLOGY 2020; 9:E394. [PMID: 33187109 PMCID: PMC7697740 DOI: 10.3390/biology9110394] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/05/2020] [Revised: 11/02/2020] [Accepted: 11/09/2020] [Indexed: 01/12/2023]
Abstract
This study started from the request of providing predictions on hospitalization and Intensive Care Unit (ICU) rates that are caused by COVID-19 for the Umbria region in Italy. To this purpose, we propose the application of a computational framework to a SEIR-type (Susceptible, Exposed, Infected, Removed) epidemiological model describing the different stages of COVID-19 infection. The model discriminates between asymptomatic and symptomatic cases and it takes into account possible intervention measures in order to reduce the probability of transmission. As case studies, we analyze not only the epidemic situation in Umbria but also in Italy, in order to capture the evolution of the pandemic at a national level. First of all, we estimate model parameters through a Bayesian calibration method, called Conditional Robust Calibration (CRC), while using the official COVID-19 data of the Italian Civil Protection. Subsequently, Conditional Robustness Analysis (CRA) on the calibrated model is carried out in order to quantify the influence of epidemiological and intervention parameters on the hospitalization rates. The proposed pipeline properly describes the COVID-19 spread during the lock-down phase. It also reveals the underestimation of new positive cases and the need of promptly isolating asymptomatic and presymptomatic cases. The results emphasize the importance of the lock-down timeliness and provide accurate predictions on the current evolution of the pandemic.
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Affiliation(s)
- Chiara Antonini
- ICT4life srl, Via Mario Donati Guerrieri, 16, 06132 Perugia, Italy;
| | - Sara Calandrini
- ICT4life srl, Via Mario Donati Guerrieri, 16, 06132 Perugia, Italy;
- Department of Engineering, University of Perugia, Via Goffredo Duranti, 93, 06125 Perugia, Italy
| | - Fabrizio Stracci
- Department of Experimental Medicine, University of Perugia, Piazzale Settimio Gambuli, 06132 Perugia, Italy;
| | - Claudio Dario
- Regional Government of Umbria, Corso Vannucci, 96, 06121 Perugia, Italy;
| | - Fortunato Bianconi
- COVID-19 Epidemiological Unit, Regional Government of Umbria, Corso Vannucci, 96, 06121 Perugia, Italy;
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Testing structural identifiability by a simple scaling method. PLoS Comput Biol 2020; 16:e1008248. [PMID: 33141821 PMCID: PMC7665633 DOI: 10.1371/journal.pcbi.1008248] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2020] [Revised: 11/13/2020] [Accepted: 08/14/2020] [Indexed: 12/12/2022] Open
Abstract
Successful mathematical modeling of biological processes relies on the expertise of the modeler to capture the essential mechanisms in the process at hand and on the ability to extract useful information from empirical data. A model is said to be structurally unidentifiable, if different quantitative sets of parameters provide the same observable outcome. This is typical (but not exclusive) of partially observed problems in which only a few variables can be experimentally measured. Most of the available methods to test the structural identifiability of a model are either too complex mathematically for the general practitioner to be applied, or require involved calculations or numerical computation for complex non-linear models. In this work, we present a new analytical method to test structural identifiability of models based on ordinary differential equations, based on the invariance of the equations under the scaling transformation of its parameters. The method is based on rigorous mathematical results but it is easy and quick to apply, even to test the identifiability of sophisticated highly non-linear models. We illustrate our method by example and compare its performance with other existing methods in the literature.
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