Global Identifiability of Differential Models

Many real-world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if it can be uniquely recovered from input and output data. The main contribution of this paper is to provide theory, an algorithm, and software for deciding global identifiability. First, we rigorously derive an algebraic criterion for global identifiability (this is an analytic property), which yields a deterministic algorithm. Second, we improve the efficiency by randomizing the algorithm while guaranteeing the probability of correctness. With our new algorithm, we can tackle problems that could not be tackled before. A software based on the algorithm (called SIAN) is available at https://github.com/pogudingleb/SIAN

Observability and Structural Identifiability of Nonlinear Biological Systems

Observability is a modelling property that describes the possibility of inferring the internal state of a system from observations of its output. A related property, structural identifiability, refers to the theoretical possibility of determining the parameter values from the output. In fact, structural identifiability becomes a particular case of observability if the parameters are considered as constant state variables. It is possible to simultaneously analyse the observability and structural identifiability of a model using the conceptual tools of differential geometry. Many complex biological processes can be described by systems of nonlinear ordinary differential equations, and can therefore be analysed with this approach. The purpose of this review article is threefold: (I) to serve as a tutorial on observability and structural identifiability of nonlinear systems, using the differential geometry approach for their analysis; (II) to review recent advances in the field; and (III) to identify open problems and suggest new avenues for research in this area.Comment: Accepted for publication in the special issue "Computational Methods for Identification and Modelling of Complex Biological Systems" of Complexit

Benchmarking tools for a priori identifiability analysis

Motivation: The theoretical possibility of determining the state and parameters of a dynamic model by measuring its outputs is given by its structural identifiability and observability. These properties should be analysed before attempting to calibrate a model, but their a priori analysis can be challenging, requiring symbolic calculations that often have a high computational cost. In recent years a number of software tools have been developed for this task, mostly in the systems biology community. These tools have vastly different features and capabilities, and a critical assessment of their performance is still lacking. Results: Here we present a comprehensive study of the computational resources available for analysing structural identifiability. We consider 13 software tools developed in 7 programming languages and evaluate their performance using a set of 25 case studies created from 21 models. Our results reveal their strengths and weaknesses, provide guidelines for choosing the most appropriate tool for a given problem, and highlight opportunities for future developments. Availability: https://github.com/Xabo-RB/Benchmarking_files.Xunta de Galicia | Ref. ED431F 2021/003Agencia Estatal de Investigación | Ref. PID2020-113992RA-I00MCIN/AEI/10.13039/501100011033 | Ref. RYC-2019-027537-IFinanciado para publicación en acceso aberto: Universidade de Vigo/CISU

Web-based Structural Identifiability Analyzer

Parameter identifiability describes whether, for a given differential model, one can determine parameter values from model equations. Knowing global or local identifiability properties allows construction of better practical experiments to identify parameters from experimental data. In this work, we present a web-based software tool that allows to answer specific identifiability queries. Concretely, our toolbox can determine identifiability of individual parameters of the model and also provide all functions of parameters that are identifiable (also called identifiable combinations) from single or multiple experiments. The program is freely available at https://maple.cloud/app/6509768948056064

Review: to be or not to be an identifiable model. Is this a relevant question in animal science modelling?

International audienceWhat is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODE) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and highly informative experiments. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design

Input-dependent structural identifiability of nonlinear systems

A dynamic model is structurally identifiable if it is possible to infer its unknown parameters by observing its output. Structural identifiability depends on the system dynamics, output, and input, as well as on the specific values of initial conditions and parameters. Here we present a symbolic method that characterizes the input that a model requires to be structurally identifiable. It determines which derivatives must be non-zero in order to have a sufficiently exciting input. Our approach considers structural identifiability as a generalization of nonlinear observability and incorporates extended Lie derivatives. The methodology assesses structural identifiability for time-varying inputs and, additionally, it can be used to determine the input profile that is required to make the parameters structurally locally identifiable. Furthermore, it is sometimes possible to replace an experiment with time-varying input with multiple experiments with constant inputs. We implement the resulting method as a MATLAB toolbox named STRIKE-GOLDD2. This tool can assist in the design of new experiments for the purpose of parameter estimation

Structural identifiability analysis of epidemic models based on differential equations: A Primer

The successful application of epidemic models hinges on our ability to estimate model parameters from limited observations reliably. An often-overlooked step before estimating model parameters consists of ensuring that the model parameters are structurally identifiable from a given dataset. Structural identifiability analysis uncovers any existing parameter correlations that preclude their estimation from the observed variables. Here we review and illustrate methods for structural identifiability analysis based on a differential algebra approach using DAISY and Mathematica (Wolfram Research). We demonstrate this approach through examples of compartmental epidemic models previously employed to study transmission dynamics and control. We show that lack of structural identifiability may be remedied by incorporating additional observations from different model states or fixing some parameters based on existing parameter correlations, or by reducing the number of parameters or state variables involved in the system dynamics. We also underscore how structural identifiability analysis can help improve compartmental diagrams of differential-equation models by indicating the observed variables and the results of the structural identifiability analysis