Practical identifiability of HIV dynamics models

Design, optimization, and inference of biphasic decay of infectious virus particles

Virus population dynamics are driven by counter-balancing forces of production and loss. Whereas viral production arises from complex interactions with susceptible hosts, the loss of infectious virus particles is often approximated as a first-order kinetic process. As such, experimental protocols to measure infectious virus loss are not typically designed to identify non-exponential decay processes. Here, we propose methods to evaluate if an experimental design is adequate to identify multiphasic virus particle decay and to optimize the sampling times of decay experiments, accounting for uncertainties in viral kinetics. First, we evaluate synthetic scenarios of biphasic decays, with varying decay rates and initial proportions of subpopulations. We show that robust inference of multiphasic decay is more likely when the faster decaying subpopulation predominates insofar as early samples are taken to resolve the faster decay rate. Moreover, design optimization involving non-equal spacing between observations increases the precision of estimation while reducing the number of samples. We then apply these methods to infer multiple decay rates associated with the decay of bacteriophage ('phage') ΦD9, an evolved isolate derived from phage Φ21. A pilot experiment confirmed that ΦD9 decay is multiphasic, but was unable to resolve the rate or proportion of the fast decaying subpopulation(s). We then applied a Fisher information matrix-based design optimization method to propose nonequally spaced sampling times. Using this strategy, we were able to robustly estimate multiple decay rates and the size of the respective subpopulations. Notably, we conclude that the vast majority (94%) of the phage ΦD9 population decays at a rate 16-fold higher than the slow decaying population. Altogether, these results provide both a rationale and a practical approach to quantitatively estimate heterogeneity in viral decay.

Practical Identifiability of Plant Growth Models: A Unifying Framework and Its Specification for Three Local Indices

Amid the rise of machine learning models, a substantial portion of plant growth models remains mechanistic, seeking to capture an in-depth understanding of the underlying phenomena governing the system's dynamics. The development of these models typically involves parameter estimation from experimental data. Ensuring that the estimated parameters align closely with their respective "true" values is crucial since they hold biological interpretation, leading to the challenge of uniqueness in the solutions. Structural identifiability analysis addresses this issue under the assumption of perfect observations of system dynamics, whereas practical identifiability considers limited measurements and the accompanying noise. In the literature, definitions for structural identifiability vary only slightly among authors, whereas the concept and quantification of practical identifiability lack consensus, with several indices coexisting. In this work, we provide a unified framework for studying identifiability, accommodating different definitions that need to be instantiated depending on each application case. In a more applicative second step, we focus on three widely used methods for quantifying practical identifiability: collinearity indices, profile likelihood, and average relative error. We show the limitations of their local versions, and we propose a new risk index built on the profile likelihood-based confidence intervals. We illustrate the usefulness of these concepts for plant growth modeling using a discrete-time individual plant growth model, LNAS, and a continuous-time plant population epidemics model. Through this work, we aim to underline the significance of identifiability analysis as a complement to any parameter estimation study and offer guidance to the modeler.

Introducing Parameter Clustering to the OED Procedure for Model Calibration of a Synthetic Inducible Promoter in S. cerevisiae

In recent years, synthetic gene circuits for adding new cell features have become one of the most powerful tools in biological and pharmaceutical research and development. However, because of the inherent non-linearity and noisy experimental data, the experiment-based model calibration of these synthetic parts is perceived as a laborious and time-consuming procedure. Although the optimal experimental design (OED) based on the Fisher information matrix (FIM) has been proved to be an effective means to improve the calibration efficiency, the required calculation increases dramatically with the model size (parameter number). To reduce the OED complexity without losing the calibration accuracy, this paper proposes two OED approaches with different parameter clustering methods and validates the accuracy of calibrated models with in-silico experiments. A model of an inducible synthetic promoter in S. cerevisiae is adopted for bench-marking. The comparison with the traditional off-line OED approach suggests that the OED approaches with both of the clustering methods significantly reduce the complexity of OED problems (for at least 49.0%), while slightly improving the calibration accuracy (11.8% and 19.6% lower estimation error in average for FIM-based and sensitivity-based approaches). This study implicates that for calibrating non-linear models of biological pathways, cluster-based OED could be a beneficial approach to improve the efficiency of optimal experimental design.

Zika virus dynamics: Effects of inoculum dose, the innate immune response and viral interference

Experimental Zika virus infection in non-human primates results in acute viral load dynamics that can be well-described by mathematical models. The inoculum dose that would be received in a natural infection setting is likely lower than the experimental infections and how this difference affects the viral dynamics and immune response is unclear. Here we study a dataset of experimental infection of non-human primates with a range of doses of Zika virus. We develop new models of infection incorporating both an innate immune response and viral interference with that response. We find that such a model explains the data better than models with no interaction between virus and the immune response. We also find that larger inoculum doses lead to faster dynamics of infection, but approximately the same total amount of viral production.

The relationship between the infecting dose of a pathogen and the subsequent viral dynamics is unclear in many disease settings, and this relationship has implications for both the timing and the required efficacy of antiviral therapy. Since experimental challenge studies often employ higher doses of virus than would generally be present in natural infection assessment of this relationship is particularly important for translation of findings. In this study we used mathematical modelling of viral load data from a multi-dose study of Zika virus infection in a macaque model to describe the impact of varying the dose of Zika virus on model parameters, and developed a novel mathematical model incorporating viral interference with the innate immune response.

Computational solutions of the HIV-1 infection of CD4 + T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy

This paper investigates the exact traveling wave solutions of the fractional model of the human immunodeficiency virus (HIV-1) infection for CD4 + T-cells. This model also treats with the effect of antiviral drug therapy. These solutions calculate both the boundary and initial conditions that allow employing the septic-B-spline scheme which is one of the most recent schemes in the numerical field. We use the obtained computational solutions via the modified Khater, the extended simplest equation, and sech-tanh methods through Atangana-Baleanu derivative operator. The comparison between the exact and numerical evaluated solutions is illustrated by some distinct sketches. The functioning of our numerical method is tested under three computational obtained solutions.

Model Averaging in Viral Dynamic Models

The paucity of experimental data makes both inference and prediction particularly challenging in viral dynamic models. In the presence of several candidate models, a common strategy is model selection (MS), in which models are fitted to the data but only results obtained with the "best model" are presented. However, this approach ignores model uncertainty, which may lead to inaccurate predictions. When several models provide a good fit to the data, another approach is model averaging (MA) that weights the predictions of each model according to its consistency to the data. Here, we evaluated by simulations in a nonlinear mixed-effect model framework the performances of MS and MA in two realistic cases of acute viral infection, i.e., (1) inference in the presence of poorly identifiable parameters, namely, initial viral inoculum and eclipse phase duration, (2) uncertainty on the mechanisms of action of the immune response. MS was associated in some scenarios with a large rate of false selection. This led to a coverage rate lower than the nominal coverage rate of 0.95 in the majority of cases and below 0.50 in some scenarios. In contrast, MA provided better estimation of parameter uncertainty, with coverage rates ranging from 0.72 to 0.98 and mostly comprised within the nominal coverage rate. Finally, MA provided similar predictions than those obtained with MS. In conclusion, parameter estimates obtained with MS should be taken with caution, especially when several models well describe the data. In this situation, MA has better performances and could be performed to account for model uncertainty.

Ebola viral dynamics in nonhuman primates provides insights into virus immuno-pathogenesis and antiviral strategies

Despite several clinical trials implemented, no antiviral drug could demonstrate efficacy against Ebola virus. In non-human primates, early initiation of polymerase inhibitors favipiravir and remdesivir improves survival, but whether they could be effective in patients is unknown. Here we analyze the impact of antiviral therapy by using a mathematical model that integrates virological and immunological data of 44 cynomolgus macaques, left untreated or treated with favipiravir. We estimate that favipiravir has a ~50% efficacy in blocking viral production, which results in reducing virus growth and cytokine storm while IFNα reduces cell susceptibility to infection. Simulating the effect of delayed initiations of treatment, our model predicts survival rates of 60% for favipiravir and 100% for remdesivir when treatment is initiated within 3 and 4 days post infection, respectively. These results improve the understanding of Ebola immuno-pathogenesis and can help optimize antiviral evaluation in future outbreaks.

Mathematical modeling of within-host Zika virus dynamics

Recent Zika virus outbreaks have been associated with severe outcomes, especially during pregnancy. A great deal of effort has been put toward understanding this virus, particularly the immune mechanisms responsible for rapid viral control in the majority of infections. Identifying and understanding the key mechanisms of immune control will provide the foundation for the development of effective vaccines and antiviral therapy. Here, we outline a mathematical modeling approach for analyzing the within-host dynamics of Zika virus, and we describe how these models can be used to understand key aspects of the viral life cycle and to predict antiviral efficacy.

An evaluation of the operational model when applied to quantify functional selectivity

BACKGROUND AND PURPOSE

Functional selectivity describes the ability of ligands to differentially regulate multiple signalling pathways when coupled to a single receptor, and the operational model is commonly used to analyse these data. Here, we assess the mathematical properties of the operational model and evaluate the outcomes of fixing parameters on model performance.

EXPERIMENTAL APPROACH

The operational model was evaluated using both a mathematical identifiability analysis and simulation.

KEY RESULTS

Mathematical analysis revealed that the parameters R0 and KE were not independently identifiable which can be solved by considering their ratio, τ. The ratio parameter, τ, was often imprecisely estimated when only functional assay data were available and generally only the transduction coefficient R ( τKA) could be estimated precisely. The general operational model (that includes baseline and the Hill coefficient) required either the parameters Em or KA to be fixed. The normalization process largely cancelled out the mean error of the calculated Δlog (R) caused by fixing these parameters. From this analysis, it was determined that we can avoid the need for a full agonist ligand to be included in an experiment to determine Δlog (R).

CONCLUSION AND IMPLICATIONS

This analysis has provided a ready-to-use understanding of current methods for quantifying functional selectivity. It showed that current methods are generally tolerant to fixing parameters. A new method was proposed that removes the need for including a high efficacy ligand in any given experiment, which allows application to large-scale screening to identify compounds with desirable features of functional selectivity.

Mathematical analysis of an HIV infection model including quiescent cells and periodic antiviral therapy

In this paper, we revisit the model by Guedj et al. [J. Guedj, R. Thibaut and D. Commenges, Maximum likelihood estimation in dynamical models of HIV, Biometrics 63 (2007) 198–206; J. Guedj, R. Thibaut and D. Commenges, Practical identifiability of HIV dynamics models, Bull. Math. Biol. 69 (2007) 2493–2513] which describes the effect of treatment with reverse transcriptase (RT) inhibitors and incorporates the class of quiescent cells. We prove that there is a threshold value [Formula: see text] of drug efficiency [Formula: see text] such that if [Formula: see text], the basic reproduction number [Formula: see text] and the infection is cleared and if [Formula: see text], the infectious equilibrium is globally asymptotically stable. When the drug efficiency function [Formula: see text] is periodic and of the bang–bang type we establish a threshold, in terms of spectral radius of some matrix, between the clearance and the persistence of the disease. As stated in related works [L. Rong, Z. Feng and A. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol. 69 (2007) 2027–2060; P. De Leenheer, Within-host virus models with periodic antiviral therapy, Bull. Math. Biol. 71 (2009) 189–210.], we prove that the increase of the drug efficiency or the active duration of drug must clear the infection more quickly. We illustrate our results by some numerical simulations.

What do we mean by identifiability in mixed effects models?

We discuss the question of model identifiability within the context of nonlinear mixed effects models. Although there has been extensive research in the area of fixed effects models, much less attention has been paid to random effects models. In this context we distinguish between theoretical identifiability, in which different parameter values lead to non-identical probability distributions, structural identifiability which concerns the algebraic properties of the structural model, and practical identifiability, whereby the model may be theoretically identifiable but the design of the experiment may make parameter estimation difficult and imprecise. We explore a number of pharmacokinetic models which are known to be non-identifiable at an individual level but can become identifiable at the population level if a number of specific assumptions on the probabilistic model hold. Essentially if the probabilistic models are different, even though the structural models are non-identifiable, then they will lead to different likelihoods. The findings are supported through simulations.

A unified framework for estimating parameters of kinetic biological models

Background

Utilizing kinetic models of biological systems commonly require computational approaches to estimate parameters, posing a variety of challenges due to their highly non-linear and dynamic nature, which is further complicated by the issue of non-identifiability. We propose a novel parameter estimation framework by combining approaches for solving identifiability with a recently introduced filtering technique that can uniquely estimate parameters where conventional methods fail. This framework first conducts a thorough analysis to identify and classify the non-identifiable parameters and provides a guideline for solving them. If no feasible solution can be found, the framework instead initializes the filtering technique with informed prior to yield a unique solution.

Results

This framework has been applied to uniquely estimate parameter values for the sucrose accumulation model in sugarcane culm tissue and a gene regulatory network. In the first experiment the results show the progression of improvement in reliable and unique parameter estimation through the use of each tool to reduce and remove non-identifiability. The latter experiment illustrates the common situation where no further measurement data is available to solve the non-identifiability. These results show the successful application of the informed prior as well as the ease with which parallel data sources may be utilized without increasing the model complexity.

Conclusion

The proposed unified framework is distinct from other approaches by providing a robust and complete solution which yields reliable and unique parameter estimation even in the face of non-identifiability.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-015-0500-9) contains supplementary material, which is available to authorized users.

Influence of covariance between random effects in design for nonlinear mixed-effect models with an illustration in pediatric pharmacokinetics

Nonlinear mixed-effect models are used increasingly during drug development. For design, an alternative to simulations is based on the Fisher information matrix. Its expression was derived using a first-order approach, was then extended to include covariance and implemented into the R function PFIM. The impact of covariance on standard errors, amount of information, and optimal designs was studied. It was also shown how standard errors can be predicted analytically within the framework of rich individual data without the model. The results were illustrated by applying this extension to the design of a pharmacokinetic study of a drug in pediatric development.

Quantifying the turnover of transcriptional subclasses of HIV-1-infected cells

HIV-1-infected cells in peripheral blood can be grouped into different transcriptional subclasses. Quantifying the turnover of these cellular subclasses can provide important insights into the viral life cycle and the generation and maintenance of latently infected cells. We used previously published data from five patients chronically infected with HIV-1 that initiated combination antiretroviral therapy (cART). Patient-matched PCR for unspliced and multiply spliced viral RNAs combined with limiting dilution analysis provided measurements of transcriptional profiles at the single cell level. Furthermore, measurement of intracellular transcripts and extracellular virion-enclosed HIV-1 RNA allowed us to distinguish productive from non-productive cells. We developed a mathematical model describing the dynamics of plasma virus and the transcriptional subclasses of HIV-1-infected cells. Fitting the model to the data allowed us to better understand the phenotype of different transcriptional subclasses and their contribution to the overall turnover of HIV-1 before and during cART. The average number of virus-producing cells in peripheral blood is small during chronic infection. We find that a substantial fraction of cells can become defectively infected. Assuming that the infection is homogenous throughout the body, we estimate an average in vivo viral burst size on the order of 10⁴ virions per cell. Our study provides novel quantitative insights into the turnover and development of different subclasses of HIV-1-infected cells, and indicates that cells containing solely unspliced viral RNA are a good marker for viral latency. The model illustrates how the pool of latently infected cells becomes rapidly established during the first months of acute infection and continues to increase slowly during the first years of chronic infection. Having a detailed understanding of this process will be useful for the evaluation of viral eradication strategies that aim to deplete the latent reservoir of HIV-1.

Gaining a quantitative understanding of the development and turnover of different HIV-1-infected subpopulations of cells is crucial to improve the outcome of patients on combination antiretroviral therapy (cART). The population of latently infected cells is of particular interest as they represent the major barrier to a cure of HIV-1 infection. We developed a mathematical model that describes the dynamics of different transcriptionally active subclasses of HIV-1-infected cells and the viral load in peripheral blood. The model was fitted to previously published data from five chronically HIV-1-infected patients starting cART. This allowed us to estimate critical parameters of the within-host dynamics of HIV-1, such as the the number of virions produced by a single infected cell. The model further allowed investigation of HIV-1 dynamics during the acute phase. Computer simulations illustrate that latently infected cells become rapidly established during the first months of acute infection and continue to increase slowly during the first years of chronic infection. This illustrates the opportunity for strategies that aim to eradicate the virus during early cART as the pool of HIV-1 infected cells is substantially smaller during acute infection than during chronic infection.

Trend and fluctuations: analysis and design of population dynamics measurements in replicate ecosystems

The dynamical evolution of complex systems is often intrinsically stochastic and subject to external random forces. In order to study the resulting variability in dynamics, it is essential to make measurements on replicate systems and to separate arbitrary variation of the average dynamics of these replicates from fluctuations around the average dynamics. Here we do so for population time-series data from replicate ecosystems sharing a common average dynamics or common trend. We explain how model parameters, including the effective interactions between species and dynamical noise, can be estimated from the data and how replication reduces errors in these estimates. For this, it is essential that the model can fit a variety of average dynamics. We then show how one can judge the quality of a model, compare alternate models, and determine which combinations of parameters are poorly determined by the data. In addition we show how replicate population dynamics experiments could be designed to optimize the acquired information of interest about the systems. Our approach is illustrated on a set of time series gathered from replicate microbial closed ecosystems.

Identification and experimental validation of an HIV model for HAART treated patients

The objective of this paper is to identify the parameters of a human immunodeficiency virus (HIV) evolution model from a clinical data set of patients treated with two different highly active antiretroviral therapy (HAART) protocols. After introducing a model with six state variables, a preliminary step considers the reduction of the number of parameters to be identified by means of sensitivity analysis, and then identifiability items are discussed. A nonlinear optimization-based procedure for identification is developed, which divides the unknown parameters into two families: the group dependent and the patient dependent parameters. Numerical results show that the identified model can be individually adapted to each patient and this result is promising for predicting the effects (e.g., failures or successes) of therapeutic actions.

NIMROD: a program for inference via a normal approximation of the posterior in models with random effects based on ordinary differential equations

Models based on ordinary differential equations (ODE) are widespread tools for describing dynamical systems. In biomedical sciences, data from each subject can be sparse making difficult to precisely estimate individual parameters by standard non-linear regression but information can often be gained from between-subjects variability. This makes natural the use of mixed-effects models to estimate population parameters. Although the maximum likelihood approach is a valuable option, identifiability issues favour Bayesian approaches which can incorporate prior knowledge in a flexible way. However, the combination of difficulties coming from the ODE system and from the presence of random effects raises a major numerical challenge. Computations can be simplified by making a normal approximation of the posterior to find the maximum of the posterior distribution (MAP). Here we present the NIMROD program (normal approximation inference in models with random effects based on ordinary differential equations) devoted to the MAP estimation in ODE models. We describe the specific implemented features such as convergence criteria and an approximation of the leave-one-out cross-validation to assess the model quality of fit. In pharmacokinetics models, first, we evaluate the properties of this algorithm and compare it with FOCE and MCMC algorithms in simulations. Then, we illustrate NIMROD use on Amprenavir pharmacokinetics data from the PUZZLE clinical trial in HIV infected patients.

Dynamical models of biomarkers and clinical progression for personalized medicine: the HIV context

Mechanistic models, based on ordinary differential equation systems, can exhibit very good predictive abilities that will be useful to build treatment monitoring strategies. In this review, we present the potential and the limitations of such models for guiding treatment (monitoring and optimizing) in HIV-infected patients. In the context of antiretroviral therapy, several biological processes should be considered in addition to the interaction between viruses and the host immune system: the mechanisms of action of the drugs, their pharmacokinetics and pharmacodynamics, as well as the viral and host characteristics. Another important aspect to take into account is clinical progression, although its implementation in such modelling approaches is not easy. Finally, the control theory and the use of intrinsic properties of mechanistic models make them very relevant for dynamic treatment adaptation. Their implementation would nevertheless require their evaluation through clinical trials.

An approach for identifiability of population pharmacokinetic-pharmacodynamic models

Mathematical models are routinely used in clinical pharmacology to study the pharmacokinetic and pharmacodynamic properties of a drug in the body. Identifiability of these models is an important requirement for the success of these clinical studies. Identifiability is classified into two types, structural identifiability related to the structure of the mathematical model and deterministic identifiability which is related to the study design. There are existing approaches for assessment of structural identifiability of fixed-effects models, although their use appears uncommon in the literature. In this study, we develop an informal unified approach for simultaneous assessment of structural and deterministic identifiability for fixed and mixed-effects pharmacokinetic or pharmacokinetic-pharmacodynamic models. This approach uses an information theoretic framework. The method is applied both to simple examples to explore known identifiability properties and to a more complex example to illustrate its utility.CPT: Pharmacometrics & Systems Pharmacology (2013) 2, e49; doi:10.1038/psp.2013.25; advance online publication 19 June 2013.

STUDYING THE IDENTIFIABILITY OF EPIDEMIOLOGICAL MODELS USING MCMC

Studying different theoretical properties of epidemiological models has been widely addressed, while numerical studies and especially the calibration of models, which are often complicated and loaded with a high number of unknown parameters, against measured data have received less attention. In this paper, we describe how a combination of simulated data and Markov Chain Monte Carlo (MCMC) methods can be used to study the identifiability of model parameters with different type of measurements. Three known models are used as case studies to illustrate the importance of parameter identifiability: a basic SIR model, an influenza model with vaccination and treatment and a HIV–Malaria co-infection model. The analysis reveals that calibration of complex models commonly studied in mathematical epidemiology, such as the HIV–Malaria co-dynamics model, can be difficult or impossible, even if the system would be fully observed. The presented approach provides a tool for design and optimization of real-life field campaigns of collecting data, as well as for model selection.

Joint modeling of the clinical progression and of the biomarkers' dynamics using a mechanistic model

Joint models are used to rigorously explore the relationship between the dynamics of biomarkers and clinical events. In the context of HIV infection, where the multivariate dynamics of HIV-RNA and CD4 are complex, a mechanistic approach based on a system of nonlinear differential equations naturally takes into account the correlation between the biomarkers. Using data from a randomized clinical trial comparing dual antiretroviral therapy to a single drug regimen, a full maximum likelihood approach is proposed to explore the relationship between the evolution of the biomarkers and the time to a clinical event. The role of each marker as an independent predictor of disease progression is assessed. We show that the joint dynamics of HIV-RNA and CD4 captures the effect of antiretroviral treatment; the CD4 dynamics alone is found to capture most but not all of the treatment effect.

Maximum likelihood estimation of long-term HIV dynamic models and antiviral response

HIV dynamics studies, based on differential equations, have significantly improved the knowledge on HIV infection. While first studies used simplified short-term dynamic models, recent works considered more complex long-term models combined with a global analysis of whole patient data based on nonlinear mixed models, increasing the accuracy of the HIV dynamic analysis. However statistical issues remain, given the complexity of the problem. We proposed to use the SAEM (stochastic approximation expectation-maximization) algorithm, a powerful maximum likelihood estimation algorithm, to analyze simultaneously the HIV viral load decrease and the CD4 increase in patients using a long-term HIV dynamic system. We applied the proposed methodology to the prospective COPHAR2-ANRS 111 trial. Very satisfactory results were obtained with a model with latent CD4 cells defined with five differential equations. One parameter was fixed, the 10 remaining parameters (eight with between-patient variability) of this model were well estimated. We showed that the efficacy of nelfinavir was reduced compared to indinavir and lopinavir.

Design evaluation and optimization for models of hepatitis C viral dynamics

Mathematical modeling of hepatitis C viral (HCV) kinetics is widely used for understanding viral pathogenesis and predicting treatment outcome. The standard model is based on a system of five non-linear ordinary differential equations (ODE) that describe both viral kinetics and changes in drug concentration after treatment initiation. In such complex models parameter estimation is challenging and requires frequent sampling measurements on each individual. By borrowing information between study subjects, non-linear mixed effect models can deal with sparser sampling from each individual. However, the search for optimal designs in this context has been limited by the numerical difficulty of evaluating the Fisher information matrix (FIM). Using the software PFIM, we show that a linearization of the statistical model avoids most of the computational burden, while providing a good approximation to the FIM. We then compare the precision of the parameters that can be expected using five study designs from the literature. We illustrate the usefulness of rationalizing data sampling by showing that, for a given level of precision, optimal design could reduce the total number of measurements by up 50 per cent. Our approach can be used by a statistician or a clinician aiming at designing an HCV viral kinetics study.

ON IDENTIFIABILITY OF NONLINEAR ODE MODELS AND APPLICATIONS IN VIRAL DYNAMICS

Ordinary differential equations (ODE) are a powerful tool for modeling dynamic processes with wide applications in a variety of scientific fields. Over the last 2 decades, ODEs have also emerged as a prevailing tool in various biomedical research fields, especially in infectious disease modeling. In practice, it is important and necessary to determine unknown parameters in ODE models based on experimental data. Identifiability analysis is the first step in determing unknown parameters in ODE models and such analysis techniques for nonlinear ODE models are still under development. In this article, we review identifiability analysis methodologies for nonlinear ODE models developed in the past one to two decades, including structural identifiability analysis, practical identifiability analysis and sensitivity-based identifiability analysis. Some advanced topics and ongoing research are also briefly reviewed. Finally, some examples from modeling viral dynamics of HIV, influenza and hepatitis viruses are given to illustrate how to apply these identifiability analysis methods in practice.

A perspective on modelling hepatitis C virus infection

By mathematically describing early hepatitis C virus (HCV) RNA decay after initiation of interferon (IFN)-based antiviral therapy, crucial parameters of the in vivo viral kinetics have been estimated, such as the rate of production and clearance of free virus, and the rate of loss of infected cells. Furthermore, by suggesting mechanisms of action for IFN and ribavirin mathematical modelling has provided a means for evaluating and optimizing treatment strategies. Here, we review recent modelling developments for understanding complex viral kinetics patterns, such as triphasic HCV RNA declines and viral rebounds observed in patients treated with pegylated interferon and ribavirin. Moreover, we discuss new modelling approaches developed to interpret the viral kinetics observed in clinical trials with direct-acting antiviral agents, which induce a rapid decline of wild-type virus but also engender a higher risk for emergence of drug-resistant variants. Lastly, as in vitro systems have allowed a better characterization of the virus lifecycle, we discuss new modelling approaches that combine the intracellular and the extracellular viral dynamics.

Score tests for exploring complex models: application to HIV dynamics models

In biostatistics, more and more complex models are being developed. This is particularly the case in system biology. Fitting complex models can be very time-consuming, since many models often have to be explored. Among the possibilities are the introduction of explanatory variables and the determination of random effects. The particularity of this use of the score test is that the null hypothesis is not itself very simple; typically, some random effects may be present under the null hypothesis. Moreover, the information matrix cannot be computed, but only an approximation based on the score. This article examines this situation with the specific example of HIV dynamics models. We examine the score test statistics for testing the effect of explanatory variables and the variance of random effect in this complex situation. We study type I errors and the statistical powers of this score test statistics and we apply the score test approach to a real data set of HIV-infected patients.