Identifiability analysis and parameter estimation of a single Hodgkin-Huxley type voltage dependent ion channel under voltage step measurement conditions

Identifiability analysis of a single Hodgkin-Huxley (HH) type voltage dependent ion channel model under voltage clamp circumstances is performed in order to decide if one can uniquely determine the model parameters from measured data in this simple case. It is shown that the two steady-state parameters (m_∞, h_∞) and the conductance (g) are not globally identifiable together using a single step voltage input. Moreover, no pair from these three parameters is identifiable. Based on the results of the identifiability analysis, a novel optimization-based identification method is proposed and demonstrated on in silico data. The proposed method is based on the decomposition of the parameter estimation problem into two parts using multiple voltage step traces. The results of the article are used to formulate explicit criteria for the design of voltage clamp protocols.     

Dynamical behavior of a vector-host epidemic model with demographic structure

In the paper, we propose a model that tracks the dynamics of many diseases spread by vectors, such as malaria, dengue, or West Nile virus (all spread by mosquitoes). Our model incorporates demographic structure with variable population size which is described by nonlinear birth rate and linear death rate. The stability of the system is analyzed for the existence of the disease-free and endemic equilibria points. We find the basic reproduction number R₀ in terms of measurable epidemiological and demographic parameters is the threshold condition that determines the dynamics of disease infection: if R₀<1 the disease fades out, and for R₀>1 the disease remains endemic. The threshold condition provides important guidelines for accessing control of the vector diseases, and implies that it is an efficient way to halt the spread of vector epidemic by reducing the carrying capacity of the environment for the vector and the host. Moreover, sufficient conditions are also obtained for the global stability of the unique endemic equilibrium E^*.     

The combined impact of external computers and network topology on the spread of computer viruses

This paper aims to study the combined impact of external computers and network topology on the spread of computer viruses over the Internet. By assuming that the network underlying a recently proposed model capturing virus spreading behaviour under the influence of external computers follows a power-law degree distribution, a new virus epidemic model is proposed. A comprehensive study of the model shows the global stability of the virus-free equilibrium or the global attractivity of the viral equilibrium, depending on the basic reproduction number R₀. Next, the impacts of different model parameters on R₀ are analysed. In particular, it is found that (a) higher network heterogeneity benefits virus spreading, (b) higher-degree nodes are more susceptible to infections than lower-degree nodes, and (c) a lower rate at which external computers enter the Internet could restrain virus spreading. On this basis, some practical measures of inhibiting virus diffusion are suggested.     

Identification and Identifiability of Unknown Parameters in Multibody Dynamic Systems

Assessing design changes in mechanical systems from simulationresults requires both accurate dynamic models and accurate values forparameters in the models. Model parameters are often unavailable ordifficult to measure. This study details an identification procedure fordetermining optimal values for unknown or estimated model parametersfrom experimental test data. The resulting optimization problem issolved by Levenberg–Marquardt methods. Partial derivative matricesneeded for the optimization are computed through sensitivity analysis.The sensitivity equations to be solved are generated analytically.Unfortunately, not all parameters can be uniquely determined using anidentification procedure. An issue of parameter identifiability remains.Since a global identifiability test is impractical for even the simplestmodels, a local identifiability test is developed. Two examples areprovided. The first example highlights the test for parameteridentifiability, while the second shows the usefulness of parameteridentification by determining vehicle suspension parameters fromexperimentally measured data.     

Kinetic modelling of the role of the aldehyde dehydrogenase enzyme and the breast cancer resistance protein in drug resistance and transport

A compartmental model for the in vitro uptake kinetics of the anti-cancer agent topotecan (TPT) has been extended from a previously published model. The extended model describes the drug activity and delivery of the pharmacologically active form to the DNA target as well as the catalysis of the aldehyde dehydrogenase (ALDH) enzyme and the elimination of drug from the cytoplasm via the efflux pump. Verification of the proposed model is achieved using scanning-laser microscopy data from live human breast cancer cells. Before estimating the unknown model parameters from the experimental in vitro data it is essential to determine parameter uniqueness (or otherwise) from this imposed output structure. This is formally performed as a structural identifiability analysis, which demonstrates that all of the unknown model parameters are uniquely determined by the output structure corresponding to the experiment.     

统计机器学习中参数可辨识性研究及其关键问题

参数可辨识性研究在统计机器学习中具有重要的理论意义和应用价值.参数可辨识性是关于模型参数能否被惟一确定的性质.在包含物理参数的学习模型中,可辨识性不仅是物理参数获得正确估计的前提条件,更重要的是,它反映了学习机器中由参数决定的物理特征.为扩展到未来类人智能机器研究的考察视角,我们将学习模型纳入"知识与数据共同驱动模型"的框架中讨论.在此框架下,我们提出两个关键问题.第一是参数可辨识性准则问题.该问题考察与可辨识性密切相关的各种判断准则,其中知识驱动子模型与数据驱动子模型的耦合方式为参数可辨识性问题提供了新的研究空间.第二是参数可辨识性与机器学习理论和应用相关联的研究.该研究包括可辨识性对参数估计、模型选择、学习算法、学习动态过程、奇异学习理论、贝叶斯推断等内容的深刻影响.     

DAISY: a new software tool to test global identifiability of biological and physiological systems

A priori global identifiability is a structural property of biological and physiological models. It is considered a prerequisite for well-posed estimation, since it concerns the possibility of recovering uniquely the unknown model parameters from measured input-output data, under ideal conditions (noise-free observations and error-free model structure). Of course, determining if the parameters can be uniquely recovered from observed data is essential before investing resources, time and effort in performing actual biomedical experiments. Many interesting biological models are nonlinear but identifiability analysis for nonlinear system turns out to be a difficult mathematical problem. Different methods have been proposed in the literature to test identifiability of nonlinear models but, to the best of our knowledge, so far no software tools have been proposed for automatically checking identifiability of nonlinear models. In this paper, we describe a software tool implementing a differential algebra algorithm to perform parameter identifiability analysis for (linear and) nonlinear dynamic models described by polynomial or rational equations. Our goal is to provide the biological investigator a completely automatized software, requiring minimum prior knowledge of mathematical modelling and no in-depth understanding of the mathematical tools. The DAISY (Differential Algebra for Identifiability of SYstems) software will potentially be useful in biological modelling studies, especially in physiology and clinical medicine, where research experiments are particularly expensive and/or difficult to perform. Practical examples of use of the software tool DAISY are presented. DAISY is available at the web site http://www.dei.unipd.it/~pia/.     

A priori identifiability: An overview on definitions and approaches

For a system, a priori identifiability is a theoretical property depending only on the model and guarantees that its parameters can be uniquely determined from observations. This paper provides a survey of the various and numerous definitions of a priori identifiability given in the literature, for both deterministic continuous and discrete-time models. A classification is done by distinguishing analytical and algebraic definitions as well as local and global ones. Moreover, this paper provides an overview on the distinct methods to test the parameter identifiability. They are classified into the so-called output equality approaches, local state isomorphism approaches and differential algebra approaches. A few examples are detailed to illustrate the methods and complete this survey.     

Identifiability study of an aquatic ecosystem model

An example of ecosystem modelization is presented and used to underline the problems in this area of system analysis. The model construction is analysed. The parameters identification requires the test of a priori identifiability of a complex non-linear model (structural identifiability) ; it then calls for the choice of a good identification method ns the input signals of an ecosystem cannot be manipulated, to guarantee a posteriori identifiability. This method is applied to data collected on an alpine lake     

Stability of an equilibrium state in a multiinfectious-type SIS model on a truncated network

In a homogeneous constant population, the basic SIS model potentially has an epidemic equilibrium state with global asymptotic stability since it can be reduced to the logistic equation. On the basic SIS model with a nonhomogeneous constant population, viewed as a multitype SIS model, the global or local asymptotic stability of an epidemic equilibrium state has also been studied.^1–4 However, this kind of analysis in other models with nonhomogeneous populations has rarely been developed, even though the corresponding models with homogeneous populations are well known. In addition, recent studies of complex networks have revealed that heterogeneity of the link number of vertices drastically changes the epidemic thresholds.^5–9 For these reasons, figuring out the roles of heterogeneity is a major topic in epidemic modeling. Here, we consider a multiinfectious-type SIS model on a network, and show the (local or global) asymptotic stability of an epidemic equilibrium state whenever it exists. This work was presented in part at the 11th International Symposium of Artificial Life and Robotics, Oita, Japan, January 23–25, 2006     

Kinetic modelling of haemodialysis removal of myoglobin in rhabdomyolysis patients

An extended two compartment model is proposed to describe the dynamics of myoglobin in rhabdomyolysis patients undergoing dialysis. Before using clinical data to estimate the model's unknown parameters, structural identifiability analysis was performed to determine the parameters uniqueness given certain clinical observations. A Taylor series expansion method was implemented which found that the model was structurally globally/uniquely identifiable for both on- and off-dialysis phases. The fitted model was then used in a predictive capacity showing that the use of Theralite high cut-off (HCO) or HCO 1100 dialyser gave a significant reduction in myoglobin renal exposure compared to standard haemodialysis (HD).     

Identifiability and identification of a class of parabolic distributed systems under approximation

The identifiability and identification problem of linear distributed parameter systems is studied. A parabolic distributed parameter system is approximated into a finite-dimensional model using orthogonal polynomials. The identifiability problem of the lumped model is studied. Then the problem of estimating the parameters of the finite-dimensional model is investigated, and the local uniqueness of the identified values is discussed. An illustrative example is given and good results are obtained.     

Bounded control and discrete-time controllability†

The discrete-time linear system x_k+1 = Ax_k + u_k (k = 0, 1,[tdot]), for which the input u_k belongs to an arbitrary bounded and convex set Ω, is considered. The error in the sufficiency proof of the controllability result when 0 ? ri (Ω) owing to Wing and Desoer is avoided by using convexity arguments, and the result is extended to encompass the case 0 ? ri (Ω)     

挖掘回归类的混合模型的可识别性

复杂海量数据往往表现为多种结构特征的混合体，回归类混合模型就是对这种混合体的一个描述．该文基于统计学的有限混合分布理论和可识别性的相关结果，针对回归变量的三种情形：(1)解释变量固定，(2)解释变量随机，(3)解释变量固定且类别参数指定，分别讨论挖掘一般回归类的混合模型的可识别性问题，并给出同族回归类混合模型可识别的相应充分条件．这些条件的一个共同特点是它们都与一类特别的解释变量集合有关，而该类集合是由同族的回归函数与回归参数唯一确定的，其元素使不同的回归参数对应回归函数的相同值．特别地，当回归函数线性时，这类集合就是解释变量空间中的超平面．     

Structural identifiability and observability of compartmental models of the COVID-19 pandemic

The recent coronavirus disease (COVID-19) outbreak has dramatically increased the public awareness and appreciation of the utility of dynamic models. At the same time, the dissemination of contradictory model predictions has highlighted their limitations. If some parameters and/or state variables of a model cannot be determined from output measurements, its ability to yield correct insights – as well as the possibility of controlling the system – may be compromised. Epidemic dynamics are commonly analysed using compartmental models, and many variations of such models have been used for analysing and predicting the evolution of the COVID-19 pandemic. In this paper we survey the different models proposed in the literature, assembling a list of 36 model structures and assessing their ability to provide reliable information. We address the problem using the control theoretic concepts of structural identifiability and observability. Since some parameters can vary during the course of an epidemic, we consider both the constant and time-varying parameter assumptions. We analyse the structural identifiability and observability of all of the models, considering all plausible choices of outputs and time-varying parameters, which leads us to analyse 255 different model versions. We classify the models according to their structural identifiability and observability under the different assumptions and discuss the implications of the results. We also illustrate with an example several alternative ways of remedying the lack of observability of a model. Our analyses provide guidelines for choosing the most informative model for each purpose, taking into account the available knowledge and measurements.     

Fault detection and identification relying on set-membership identifiability

Identifiability is the property that a mathematical model must satisfy to guarantee an unambiguous mapping between its parameters and the output trajectories. It is of prime importance when parameters must be estimated from experimental data representing input–output behavior and clearly when parameter estimation is used for fault detection and identification. Definitions of identifiability and methods for checking this property for linear and nonlinear systems are now well established and, interestingly, some scarce works (Braems et al., 2001, Jauberthie et al., 2011) have provided identifiability definitions and numerical tests in a bounded-error context. This paper resumes and better formalizes the two complementary definitions of set-membership identifiability and μ-set-membership identifiability of Jauberthie et al. (2011) and presents a method applicable to nonlinear systems for checking them. This method is based on differential algebra and makes use of relations linking the observations, the inputs and the unknown parameters of the system. Using these results, a method for fault detection and identification is proposed. The relations mentioned above are used to estimate the uncertain parameters of the model. By building the parameter estimation scheme on the analysis of identifiability, the solution set is guaranteed to reduce to one connected set, avoiding this way the pessimism of classical set-membership estimation methods. Fault detection and identification are performed at once by checking the estimated values against the parameter nominal ranges. The method is illustrated with an example describing the capacity of a macrophage mannose receptor to endocytose a specific soluble macromolecule.     

A parameter estimation and identifiability analysis methodology applied to a street canyon air pollution model

Mathematical models are increasingly used in environmental science thus increasing the importance of uncertainty and sensitivity analyses. In the present study, an iterative parameter estimation and identifiability analysis methodology is applied to an atmospheric model – the Operational Street Pollution Model (OSPM^®). To assess the predictive validity of the model, the data is split into an estimation and a prediction data set using two data splitting approaches and data preparation techniques (clustering and outlier detection) are analysed. The sensitivity analysis, being part of the identifiability analysis, showed that some model parameters were significantly more sensitive than others. The application of the determined optimal parameter values was shown to successfully equilibrate the model biases among the individual streets and species. It was as well shown that the frequentist approach applied for the uncertainty calculations underestimated the parameter uncertainties. The model parameter uncertainty was qualitatively assessed to be significant, and reduction strategies were identified.     

Factors affecting the identifiability of compartmental models

The identifiability of compartmental models is analysed through a series of examples which have been used to describe physiological or pharmacokinetic processes. Emphasis is placed on aspects of experimental identifiability which have hitherto received little attention in identifiability analysis. It is shown that where a single-input, single-output experiment results in non-identifiability or local identifiability, it is often possible to improve the situation by measuring more responses or simultaneously perturbing more inputs. Identifiability is then shown usually to depend on whether the observation gains are known and on the shape of the inputs, when more than one is applied. The relative merits of the Laplace transform and normal mode methods of analysing identifiability are discussed and illustrated with a substantial example. The identifiability analysis of a nonlinear compartmental model, with state-dependent rate coefficients, is presented. It is shown that inclusion of a neglected (nonlinear) relationship can make a previously non-identifiable model uniquely identifiable.     

Indistinguishability and identifiability of kinetic models for the MurC reaction in peptidoglycan biosynthesis

An important question in Systems Biology is the design of experiments that enable discrimination between two (or more) competing chemical pathway models or biological mechanisms. In this paper analysis is performed between two different models describing the kinetic mechanism of a three-substrate three-product reaction, namely the MurC reaction in the cytoplasmic phase of peptidoglycan biosynthesis. One model involves ordered substrate binding and ordered release of the three products; the competing model also assumes ordered substrate binding, but with fast release of the three products. The two versions are shown to be distinguishable; however, if standard quasi-steady-state assumptions are made distinguishability cannot be determined. Once model structure uniqueness is ensured the experimenter must determine if it is possible to successfully recover rate constant values given the experiment observations, a process known as structural identifiability. Structural identifiability analysis is carried out for both models to determine which of the unknown reaction parameters can be determined uniquely, or otherwise, from the ideal system outputs. This structural analysis forms an integrated step towards the modelling of the full pathway of the cytoplasmic phase of peptidoglycan biosynthesis.