Sieve Estimation of Constant and Time-Varying Coefficients in Nonlinear Ordinary Differential Equation Models by Considering Both Numerical Error and Measurement Error

Specification testing for ordinary differential equation models with fixed design and applications to COVID-19 epidemic models

Checking the models about the ongoing Coronavirus Disease 2019 (COVID-19) pandemic is an important issue. Some famous ordinary differential equation (ODE) models, such as the SIR and SEIR models have been used to describe and predict the epidemic trend. Still, in many cases, only part of the equations can be observed. A test is suggested to check possibly partially observed ODE models with a fixed design sampling scheme. The asymptotic properties of the test under the null, global and local alternative hypotheses are presented. Two new propositions about U-statistics with varying kernels based on independent but non-identical data are derived as essential tools. Some simulation studies are conducted to examine the performances of the test. Based on the available public data, it is found that the SEIR model, for modeling the data of COVID-19 infective cases in certain periods in Japan and Algeria, respectively, maybe not be appropriate by applying the proposed test.

Combining data assimilation and machine learning to build data-driven models for unknown long time dynamics-Applications in cardiovascular modeling

We propose a method to discover differential equations describing the long-term dynamics of phenomena featuring a multiscale behavior in time, starting from measurements taken at the fast-scale. Our methodology is based on a synergetic combination of data assimilation (DA), used to estimate the parameters associated with the known fast-scale dynamics, and machine learning (ML), used to infer the laws underlying the slow-scale dynamics. Specifically, by exploiting the scale separation between the fast and the slow dynamics, we propose a decoupling of time scales that allows to drastically lower the computational burden. Then, we propose a ML algorithm that learns a parametric mathematical model from a collection of time series coming from the phenomenon to be modeled. Moreover, we study the interpretability of the data-driven models obtained within the black-box learning framework proposed in this paper. In particular, we show that every model can be rewritten in infinitely many different equivalent ways, thus making intrinsically ill-posed the problem of learning a parametric differential equation starting from time series. Hence, we propose a strategy that allows to select a unique representative model in each equivalence class, thus enhancing the interpretability of the results. We demonstrate the effectiveness and noise-robustness of the proposed methods through several test cases, in which we reconstruct several differential models starting from time series generated through the models themselves. Finally, we show the results obtained for a test case in the cardiovascular modeling context, which sheds light on a promising field of application of the proposed methods.

Integration of single-cell multi-omics for gene regulatory network inference

The advancement of single-cell sequencing technology in recent years has provided an opportunity to reconstruct gene regulatory networks (GRNs) with the data from thousands of single cells in one sample. This uncovers regulatory interactions in cells and speeds up the discoveries of regulatory mechanisms in diseases and biological processes. Therefore, more methods have been proposed to reconstruct GRNs using single-cell sequencing data. In this review, we introduce technologies for sequencing single-cell genome, transcriptome, and epigenome. At the same time, we present an overview of current GRN reconstruction strategies utilizing different single-cell sequencing data. Bioinformatics tools were grouped by their input data type and mathematical principles for reader's convenience, and the fundamental mathematics inherent in each group will be discussed. Furthermore, the adaptabilities and limitations of these different methods will also be summarized and compared, with the hope to facilitate researchers recognizing the most suitable tools for them.

Mapping epileptic directional brain networks using intracranial EEG data

The human brain is a directional network system, in which brain regions are network nodes and the influence exerted by one region on another is a network edge. We refer to this directional information flow from one region to another as directional connectivity. Seizures arise from an epileptic directional network; abnormal neuronal activities start from a seizure onset zone and propagate via a network to otherwise healthy brain regions. As such, effective epilepsy diagnosis and treatment require accurate identification of directional connections among regions, i.e., mapping of epileptic patients' brain networks. This article aims to understand the epileptic brain network using intracranial electroencephalographic data-recordings of epileptic patients' brain activities in many regions. The most popular models for directional connectivity use ordinary differential equations (ODE). However, ODE models are sensitive to data noise and computationally costly. To address these issues, we propose a high-dimensional state-space multivariate autoregression (SSMAR) model for the brain's directional connectivity. Different from standard multivariate autoregression and SSMAR models, the proposed SSMAR features a cluster structure, where the brain network consists of several clusters of densely connected brain regions. We develop an expectation-maximization algorithm to estimate the proposed model and use it to map the interregional networks of epileptic patients in different seizure stages. Our method reveals the evolution of brain networks during seizure development.

Modeling Quorum Sensing Dynamics and Interference on Escherichia coli

Bacteria control the expression of specific genes by Quorum Sensing (QS). This works using small signaling molecules called Autoinducers (AIs), for example, the Autoinducer-2 (AI-2). In this work, we present a mathematical model that represents the AI-2 dynamics on Escherichia coli, which is linked to the cell growth and the lsr operon expression. The model is adjusted using experimental data. Our results suggest that the extracellular AI-2 activity level depends on the cell growth rate, and this activity depends on the cell exponential growth phase. The model was adapted to simulate the interference of QS mechanisms in a co-culture of two E. coli strains: a wild type strain and a knock out strain that detects AI-2 but does not produce it. Co-culture simulations unveiled two conditions to avoid the QS on the wild strain: when the knock out takes control of the growth medium and overcomes the wild strain, or when is pre-cultured to its mid-exponential phase and then added to the wild strain culture. Model simulations unveiled new insights about the interference of bacterial communication and offer new tools for QS control.

LatinPSO: An algorithm for simultaneously inferring structure and parameters of ordinary differential equations models

Simultaneously inferring both the structure and parameters of Ordinary Differential Equations (ODEs) for a complex dynamic system is more practical in many systems identification problems, but it remains challenging due to the complexity of the underlying search space. In this research, we propose a novel algorithm based on Particle Swarm Optimization (PSO) and Latin Hypercube Sampling (LHS) to address the above problem. The proposed algorithm is termed LatinPSO, and it can be effectively used for inferring the structure and parameters of ODE models through time course data. To start with, the real Human Immunodeficiency Virus (HIV) model and several synthetic models are used for evaluating the performance of LatinPSO. Experimental results demonstrated that LatinPSO could find satisfactory candidate ODE models with appropriate structure and parameters.

The influence of numerical error on parameter estimation and uncertainty quantification for advective PDE models

Advective partial differential equations can be used to describe many scientific processes. Two significant sources of error that can cause difficulties in inferring parameters from experimental data on these processes include (i) noise from the measurement and collection of experimental data and (ii) numerical error in approximating the forward solution to the advection equation. How this second source of error alters parameter estimation and uncertainty quantification during an inverse problem methodology is not well understood. As a step towards a better understanding of this problem, we present both analytical and computational results concerning how a least squares cost function and parameter estimator behave in the presence of numerical error in approximating solutions to the underlying advection equation. We investigate residual patterns to derive an autocorrelative statistical model that can improve parameter estimation and confidence interval computation for first order methods. Building on our results and their general nature, we provide guidelines for practitioners to determine when numerical or experimental error is the main source of error in their inference, along with suggestions of how to efficiently improve their results.

A Functional Data Method for Causal Dynamic Network Modeling of Task-Related fMRI

Functional MRI (fMRI) is a popular approach to investigate brain connections and activations when human subjects perform tasks. Because fMRI measures the indirect and convoluted signals of brain activities at a lower temporal resolution, complex differential equation modeling methods (e.g., Dynamic Causal Modeling) are usually employed to infer the neuronal processes and to fit the resulting fMRI signals. However, this modeling strategy is computationally expensive and remains to be mostly a confirmatory or hypothesis-driven approach. One major statistical challenge here is to infer, in a data-driven fashion, the underlying differential equation models from fMRI data. In this paper, we propose a causal dynamic network (CDN) method to estimate brain activations and connections simultaneously. Our method links the observed fMRI data with the latent neuronal states modeled by an ordinary differential equation (ODE) model. Using the basis function expansion approach in functional data analysis, we develop an optimization-based criterion that combines data-fitting errors and ODE fitting errors. We also develop and implement a block coordinate-descent algorithm to compute the ODE parameters efficiently. We illustrate the numerical advantages of our approach using data from realistic simulations and two task-related fMRI experiments. Compared with various effective connectivity methods, our method achieves higher estimation accuracy while improving the computational speed by from tens to thousands of times. Though our method is developed for task-related fMRI, we also demonstrate the potential applicability of our method (with a simple modification) to resting-state fMRI, by analyzing both simulated and real data from medium-sized networks.

Parameter Estimation and Variable Selection for Big Systems of Linear Ordinary Differential Equations: A Matrix-Based Approach

Ordinary differential equations (ODEs) are widely used to model the dynamic behavior of a complex system. Parameter estimation and variable selection for a "Big System" with linear ODEs are very challenging due to the need of nonlinear optimization in an ultra-high dimensional parameter space. In this article, we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares (SLS). Simulation studies demonstrate that the proposed matrix-based SLS method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ODE system with thousands of dimensions and millions of parameters much better than the direct least squares (LS) method and the vector-based two-stage method that are currently available. We applied this new method to two real data sets: a yeast cell cycle gene expression data set with 30 dimensions and 930 unknown parameters and the Standard & Poor 1500 index stock price data with 1250 dimensions and 1,563,750 unknown parameters, to illustrate the utility and numerical performance of the proposed parameter estimation and variable selection method for big systems in practice.

Parameter Estimation for Semiparametric Ordinary Differential Equation Models

We propose a new class of two-stage parameter estimation methods for semiparametric ordinary differential equation (ODE) models. In the first stage, state variables are estimated using a penalized spline approach; In the second stage, form of numerical discretization algorithms for an ODE solver is used to formulate estimating equations. Estimated state variables from the first stage are used to obtain more data points for the second stage. Asymptotic properties for the proposed estimators are established. Simulation studies show that the method performs well, especially for small sample. Real life use of the method is illustrated using Influenza specific cell-trafficking study.

Independence screening for high dimensional nonlinear additive ODE models with applications to dynamic gene regulatory networks

Mechanism-driven low-dimensional ordinary differential equation (ODE) models are often used to model viral dynamics at cellular levels and epidemics of infectious diseases. However, low-dimensional mechanism-based ODE models are limited for modeling infectious diseases at molecular levels such as transcriptomic or proteomic levels, which is critical to understand pathogenesis of diseases. Although linear ODE models have been proposed for gene regulatory networks (GRNs), nonlinear regulations are common in GRNs. The reconstruction of large-scale nonlinear networks from time-course gene expression data remains an unresolved issue. Here, we use high-dimensional nonlinear additive ODEs to model GRNs and propose a 4-step procedure to efficiently perform variable selection for nonlinear ODEs. To tackle the challenge of high dimensionality, we couple the 2-stage smoothing-based estimation method for ODEs and a nonlinear independence screening method to perform variable selection for the nonlinear ODE models. We have shown that our method possesses the sure screening property and it can handle problems with non-polynomial dimensionality. Numerical performance of the proposed method is illustrated with simulated data and a real data example for identifying the dynamic GRN of Saccharomyces cerevisiae.

Statistical inference in mechanistic models: time warping for improved gradient matching

Inference in mechanistic models of non-linear differential equations is a challenging problem in current computational statistics. Due to the high computational costs of numerically solving the differential equations in every step of an iterative parameter adaptation scheme, approximate methods based on gradient matching have become popular. However, these methods critically depend on the smoothing scheme for function interpolation. The present article adapts an idea from manifold learning and demonstrates that a time warping approach aiming to homogenize intrinsic length scales can lead to a significant improvement in parameter estimation accuracy. We demonstrate the effectiveness of this scheme on noisy data from two dynamical systems with periodic limit cycle, a biopathway, and an application from soft-tissue mechanics. Our study also provides a comparative evaluation on a wide range of signal-to-noise ratios.

Estimating varying coefficients for partial differential equation models

Partial differential equations (PDEs) are used to model complex dynamical systems in multiple dimensions, and their parameters often have important scientific interpretations. In some applications, PDE parameters are not constant but can change depending on the values of covariates, a feature that we call varying coefficients. We propose a parameter cascading method to estimate varying coefficients in PDE models from noisy data. Our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed. The performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a PDE model arising from LIDAR data.

Network Reconstruction From High-Dimensional Ordinary Differential Equations

We consider the task of learning a dynamical system from high-dimensional time-course data. For instance, we might wish to estimate a gene regulatory network from gene expression data measured at discrete time points. We model the dynamical system nonparametrically as a system of additive ordinary differential equations. Most existing methods for parameter estimation in ordinary differential equations estimate the derivatives from noisy observations. This is known to be challenging and inefficient. We propose a novel approach that does not involve derivative estimation. We show that the proposed method can consistently recover the true network structure even in high dimensions, and we demonstrate empirical improvement over competing approaches. Supplementary materials for this article are available online.

Hierarchical effects of pro-inflammatory cytokines on the post-influenza susceptibility to pneumococcal coinfection

In the course of influenza A virus (IAV) infections, a secondary bacterial infection frequently leads to serious respiratory conditions provoking high hospitalization and death tolls. Although abundant pro-inflammatory responses have been reported as key contributing factors for these severe dual infections, the relative contributions of cytokines remain largely unclear. In the current study, mathematical modelling based on murine experimental data dissects IFN-γ as a cytokine candidate responsible for impaired bacterial clearance, thereby promoting bacterial growth and systemic dissemination during acute IAV infection. We also found a time-dependent detrimental role of IL-6 in curtailing bacterial outgrowth which was not as distinct as for IFN-γ. Our numerical simulations suggested a detrimental effect of IFN-γ alone and in synergism with IL-6 but no conclusive pathogenic effect of IL-6 and TNF-α alone. This work provides a rationale to understand the potential impact of how to manipulate temporal immune components, facilitating the formulation of hypotheses about potential therapeutic strategies to treat coinfections.

Controllability and stability analysis of large transcriptomic dynamic systems for host response to influenza infection in human

Background

Gene regulatory networks are complex dynamic systems and the reverse-engineering of such networks from high-dimensional time course transcriptomic data have attracted researchers from various fields. It is also interesting and important to study the behavior of the reconstructed networks on the basis of dynamic models and the biological mechanisms. We focus on the gene regulatory networks reconstructed using the ordinary differential equation (ODE) modelling approach and investigate the properties of these networks.

Results

Controllability and stability analyses are conducted for the reconstructed gene response networks of 17 influenza infected subjects based on ODE models. Symptomatic subjects tend to have larger numbers of driver nodes, higher proportions of critical links and lower proportions of redundant links than asymptomatic subjects. We also show that the degree distribution, rather than the structure of networks, plays an important role in controlling the network in response to influenza infection. In addition, we find that the stability of high-dimensional networks is very sensitive to randomness in the reconstructed systems brought by errors in measurements and parameter estimation.

Conclusions

The gene response networks of asymptomatic subjects are easier to be controlled than those of symptomatic subjects. This may indicate that the regulatory systems of asymptomatic subjects are easier to recover from disease stimulations, so these subjects are less likely to develop symptoms. Our results also suggest that stability constraint should be considered in the modelling of high-dimensional networks and the estimation of network parameters.

Statistical analysis of differential equations: introducing probability measures on numerical solutions

In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncertainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.

Inference in dynamic systems using B-splines and quasilinearized ODE penalties

Nonlinear (systems of) ordinary differential equations (ODEs) are common tools in the analysis of complex one-dimensional dynamic systems. We propose a smoothing approach regularized by a quasilinearized ODE-based penalty. Within the quasilinearized spline-based framework, the estimation reduces to a conditionally linear problem for the optimization of the spline coefficients. Furthermore, standard ODE compliance parameter(s) selection criteria are applicable. We evaluate the performances of the proposed strategy through simulated and real data examples. Simulation studies suggest that the proposed procedure ensures more accurate estimates than standard nonlinear least squares approaches when the state (initial and/or boundary) conditions are not known.

Modeling Influenza Virus Infection: A Roadmap for Influenza Research

Influenza A virus (IAV) infection represents a global threat causing seasonal outbreaks and pandemics. Additionally, secondary bacterial infections, caused mainly by Streptococcus pneumoniae, are one of the main complications and responsible for the enhanced morbidity and mortality associated with IAV infections. In spite of the significant advances in our knowledge of IAV infections, holistic comprehension of the interplay between IAV and the host immune response (IR) remains largely fragmented. During the last decade, mathematical modeling has been instrumental to explain and quantify IAV dynamics. In this paper, we review not only the state of the art of mathematical models of IAV infection but also the methodologies exploited for parameter estimation. We focus on the adaptive IR control of IAV infection and the possible mechanisms that could promote a secondary bacterial coinfection. To exemplify IAV dynamics and identifiability issues, a mathematical model to explain the interactions between adaptive IR and IAV infection is considered. Furthermore, in this paper we propose a roadmap for future influenza research. The development of a mathematical modeling framework with a secondary bacterial coinfection, immunosenescence, host genetic factors and responsiveness to vaccination will be pivotal to advance IAV infection understanding and treatment optimization.

Diversity in Compartmental Dynamics of Gene Regulatory Networks: The Immune Response in Primary Influenza A Infection in Mice

Current approaches to study transcriptional profiles post influenza infection typically rely on tissue sampling from one or two sites at a few time points, such as spleen and lung in murine models. In this study, we infected female C57/BL6 mice intranasally with mouse-adapted H3N2/Hong Kong/X31 avian influenza A virus, and then analyzed the gene expression profiles in four different compartments (blood, lung, mediastinal lymph nodes, and spleen) over 11 consecutive days post infection. These data were analyzed by an advanced statistical procedure based on ordinary differential equation (ODE) modeling. Vastly different lists of significant genes were identified by the same statistical procedure in each compartment. Only 11 of them are significant in all four compartments. We classified significant genes in each compartment into co-expressed modules based on temporal expression patterns. We then performed functional enrichment analysis on these co-expression modules and identified significant pathway and functional motifs. Finally, we used an ODE based model to reconstruct gene regulatory network (GRN) for each compartment and studied their network properties.

Dynamic transcriptional signatures and network responses for clinical symptoms in influenza-infected human subjects using systems biology approaches

Recent studies demonstrate that human blood transcriptional signatures may be used to support diagnosis and clinical decisions for acute respiratory viral infections such as influenza. In this article, we propose to use a newly developed systems biology approach for time course gene expression data to identify significant dynamically response genes and dynamic gene network responses to viral infection. We illustrate the methodological pipeline by reanalyzing the time course gene expression data from a study with healthy human subjects challenged by live influenza virus. We observed clear differences in the number of significant dynamic response genes (DRGs) between the symptomatic and asymptomatic subjects and also identified DRG signatures for symptomatic subjects with influenza infection. The 505 common DRGs shared by the symptomatic subjects have high consistency with the signature genes for predicting viral infection identified in previous works. The temporal response patterns and network response features were carefully analyzed and investigated.

A model-based initial guess for estimating parameters in systems of ordinary differential equations

The inverse problem of parameter estimation from noisy observations is a major challenge in statistical inference for dynamical systems. Parameter estimation is usually carried out by optimizing some criterion function over the parameter space. Unless the optimization process starts with a good initial guess, the estimation may take an unreasonable amount of time, and may converge to local solutions, if at all. In this article, we introduce a novel technique for generating good initial guesses that can be used by any estimation method. We focus on the fairly general and often applied class of systems linear in the parameters. The new methodology bypasses numerical integration and can handle partially observed systems. We illustrate the performance of the method using simulations and apply it to real data.

Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters

Influenza A virus (IAV) infection continues to be a global health threat, as evidenced by the outbreak of the novel A/California/7/2009 IAV strain. Previous flu vaccines have proven less effective than hoped for emerging IAV strains, indicating a more thorough understanding of immune responses to primary infection is needed. One issue is the difficulty in directly measuring many key parameters and variables of the immune response. To address these issues, we considered a comprehensive workflow for statistical inference for ordinary differential question (ODE) models with partially observed variables and time-varying parameters, including identifiability analysis, two-stage and NLS estimation, and model selection etc‥ In particular, we proposed a novel one-step method to verify parameter identifiability and formulate estimating equations simultaneously. Thus, the pseudo-LS method can now deal with general ODE models with partially observed state variables for the first time. Using this workflow, we verified the relative significance of various immune factors to virus control, including target epithelial cells, cytotoxic T-lymphocyte (CD8+) cells and IAV specific antibodies (IgG and IgM). Factors other than cytotoxic T-lymphocyte (CTL) killing contributed the most to the loss of infected epithelial cells, though the effects of CTL are still significant. IgM antibody was found to be the major contributor to neutralization of free infectious viral particles. Also, the maximum viral load, which correlates well with mortality, was found to depend more on viral replication rates than infectivity. In contrast to current hypotheses, the results obtained via our methods suggest that IgM antibody and viral replication rates may be worth of further explorations in vaccine development.

Ebola virus infection modeling and identifiability problems

The recent outbreaks of Ebola virus (EBOV) infections have underlined the impact of the virus as a major threat for human health. Due to the high biosafety classification of EBOV (level 4), basic research is very limited. Therefore, the development of new avenues of thinking to advance quantitative comprehension of the virus and its interaction with the host cells is urgently needed to tackle this lethal disease. Mathematical modeling of the EBOV dynamics can be instrumental to interpret Ebola infection kinetics on quantitative grounds. To the best of our knowledge, a mathematical modeling approach to unravel the interaction between EBOV and the host cells is still missing. In this paper, a mathematical model based on differential equations is used to represent the basic interactions between EBOV and wild-type Vero cells in vitro. Parameter sets that represent infectivity of pathogens are estimated for EBOV infection and compared with influenza virus infection kinetics. The average infecting time of wild-type Vero cells by EBOV is slower than in influenza infection. Simulation results suggest that the slow infecting time of EBOV could be compensated by its efficient replication. This study reveals several identifiability problems and what kind of experiments are necessary to advance the quantification of EBOV infection. A first mathematical approach of EBOV dynamics and the estimation of standard parameters in viral infections kinetics is the key contribution of this work, paving the way for future modeling works on EBOV infection.

Generalized Ordinary Differential Equation Models

Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method.

Progressive contraction of the latent HIV reservoir around a core of less-differentiated CD4⁺ memory T Cells

In patients who are receiving prolonged antiretroviral treatment (ART), HIV can persist within a small pool of long-lived resting memory CD4⁺ T cells infected with integrated latent virus. This latent reservoir involves distinct memory subsets. Here we provide results that suggest a progressive reduction of the size of the blood latent reservoir around a core of less-differentiated memory subsets (central memory and stem cell-like memory (T_SCM) CD4⁺ T cells). This process appears to be driven by the differences in initial sizes and decay rates between latently infected memory subsets. Our results also suggest an extreme stability of the T_SCM sub-reservoir, the size of which is directly related to cumulative plasma virus exposure before the onset of ART, stressing the importance of early initiation of effective ART. The presence of these intrinsic dynamics within the latent reservoir may have implications for the design of optimal HIV therapeutic purging strategies.

HIV can persist in CD4⁺ T cells of patients receiving long-term antiretroviral therapy. Here the authors show the presence of intrinsic dynamics that progressively contract the latent HIV reservoir around a core of less-differentiated CD4 T-cell memory subsets.

Variable Selection for Sparse High-Dimensional Nonlinear Regression Models by Combining Nonnegative Garrote and Sure Independence Screening

In many regression problems, the relations between the covariates and the response may be nonlinear. Motivated by the application of reconstructing a gene regulatory network, we consider a sparse high-dimensional additive model with the additive components being some known nonlinear functions with unknown parameters. To identify the subset of important covariates, we propose a new method for simultaneous variable selection and parameter estimation by iteratively combining a large-scale variable screening (the nonlinear independence screening, NLIS) and a moderate-scale model selection (the nonnegative garrote, NNG) for the nonlinear additive regressions. We have shown that the NLIS procedure possesses the sure screening property and it is able to handle problems with non-polynomial dimensionality; and for finite dimension problems, the NNG for the nonlinear additive regressions has selection consistency for the unimportant covariates and also estimation consistency for the parameter estimates of the important covariates. The proposed method is applied to simulated data and a real data example for identifying gene regulations to illustrate its numerical performance.

Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling

The gene regulation network (GRN) is a high-dimensional complex system, which can be represented by various mathematical or statistical models. The ordinary differential equation (ODE) model is one of the popular dynamic GRN models. High-dimensional linear ODE models have been proposed to identify GRNs, but with a limitation of the linear regulation effect assumption. In this article, we propose a sparse additive ODE (SA-ODE) model, coupled with ODE estimation methods and adaptive group LASSO techniques, to model dynamic GRNs that could flexibly deal with nonlinear regulation effects. The asymptotic properties of the proposed method are established and simulation studies are performed to validate the proposed approach. An application example for identifying the nonlinear dynamic GRN of T-cell activation is used to illustrate the usefulness of the proposed method.

Estimation of nonlinear differential equation model for glucose-insulin dynamics in type I diabetic patients using generalized smoothing

In this work, we develop an ordinary differential equations (ODE) model of physiological regulation of glycemia in type 1 diabetes mellitus (T1DM) patients in response to meals and intravenous insulin infusion. Unlike for majority of existing mathematical models of glucose-insulin dynamics, parameters in our model are estimable from a relatively small number of noisy observations of plasma glucose and insulin concentrations. For estimation, we adopt the generalized smoothing estimation of nonlinear dynamic systems of Ramsay et al. (2007). In this framework, the ODE solution is approximated with a penalized spline, where the ODE model is incorporated in the penalty. We propose to optimize the generalized smoothing by using penalty weights that minimize the covariance penalties criterion (Efron, 2004). The covariance penalties criterion provides an estimate of the prediction error for nonlinear estimation rules resulting from nonlinear and/or non-homogeneous ODE models, such as our model of glucose-insulin dynamics. We also propose to select the optimal number and location of knots for B-spline bases used to represent the ODE solution. The results of the small simulation study demonstrate advantages of optimized generalized smoothing in terms of smaller estimation errors for ODE parameters and smaller prediction errors for solutions of differential equations. Using the proposed approach to analyze the glucose and insulin concentration data in T1DM patients we obtained good approximation of global glucose-insulin dynamics and physiologically meaningful parameter estimates.

Modeling genome-wide dynamic regulatory network in mouse lungs with influenza infection using high-dimensional ordinary differential equations

The immune response to viral infection is regulated by an intricate network of many genes and their products. The reverse engineering of gene regulatory networks (GRNs) using mathematical models from time course gene expression data collected after influenza infection is key to our understanding of the mechanisms involved in controlling influenza infection within a host. A five-step pipeline: detection of temporally differentially expressed genes, clustering genes into co-expressed modules, identification of network structure, parameter estimate refinement, and functional enrichment analysis, is developed for reconstructing high-dimensional dynamic GRNs from genome-wide time course gene expression data. Applying the pipeline to the time course gene expression data from influenza-infected mouse lungs, we have identified 20 distinct temporal expression patterns in the differentially expressed genes and constructed a module-based dynamic network using a linear ODE model. Both intra-module and inter-module annotations and regulatory relationships of our inferred network show some interesting findings and are highly consistent with existing knowledge about the immune response in mice after influenza infection. The proposed method is a computationally efficient, data-driven pipeline bridging experimental data, mathematical modeling, and statistical analysis. The application to the influenza infection data elucidates the potentials of our pipeline in providing valuable insights into systematic modeling of complicated biological processes.