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.

Epistasis may play an important role in evolution and speciation. Under multiplicative interactions between different loci, an analytical model is proposed to estimate genetic parameters at the individual locus level that contribute to interspecific differences in outcrossing species. The multiplicative epistasis model, inferred from a number of animal and plant experiments, suggests that genotypes at a pair of loci have genotypic values equal to the product of genotypic values at the two different loci. By considering the genetic property of outcrossing species (i.e., high polymorphisms) in the multilevel family structure analysis for an intra- and interspecific factorial mating design, a method is developed to provide estimates for allele frequencies and additive and dominant effects at individual loci in each of the two parental populations, the genotypic values of newly formed heterozygotes through species combination each with one allele from a parental population and the second from the other parental population, and the numbers of genetic factors that lead to species differentiation. Use of clones offers a tremendous power to test the adequacy of the model. However, the utilization of the model with species that cannot be cloned is also discussed. An example with interspecific hybrids of two forest tree species is used to demonstrate the model.

Seasonal and pandemic influenza A virus (IAV) continues to be a public health threat. However, we lack a detailed and quantitative understanding of the immune response kinetics to IAV infection and which biological parameters most strongly influence infection outcomes. To address these issues, we use modeling approaches combined with experimental data to quantitatively investigate the innate and adaptive immune responses to primary IAV infection. Mathematical models were developed to describe the dynamic interactions between target (epithelial) cells, influenza virus, cytotoxic T lymphocytes (CTLs), and virus-specific IgG and IgM. IAV and immune kinetic parameters were estimated by fitting models to a large data set obtained from primary H3N2 IAV infection of 340 mice. Prior to a detectable virus-specific immune response (before day 5), the estimated half-life of infected epithelial cells is ϳ1.2 days, and the half-life of free infectious IAV is ϳ4 h. During the adaptive immune response (after day 5), the average half-life of infected epithelial cells is ϳ0.5 days, and the average half-life of free infectious virus is ϳ1.8 min. During the adaptive phase, model fitting confirms that CD8 ؉ CTLs are crucial for limiting infected cells, while virus-specific IgM regulates free IAV levels. This may imply that CD4 T cells and class-switched IgG antibodies are more relevant for generating IAV-specific memory and preventing future infection via a more rapid secondary immune response. Also, simulation studies were performed to understand the relative contributions of biological parameters to IAV clearance. This study provides a basis to better understand and predict influenza virus immunity.

Transcriptional and post-transcriptional regulation of gene expression is of fundamental importance to numerous biological processes. Nowadays, an increasing amount of gene regulatory relationships have been documented in various databases and literature. However, to more efficiently exploit such knowledge for biomedical research and applications, it is necessary to construct a genome-wide regulatory network database to integrate the information on gene regulatory relationships that are widely scattered in many different places. Therefore, in this work, we build a knowledge-based database, named ‘RegNetwork’, of gene regulatory networks for human and mouse by collecting and integrating the documented regulatory interactions among transcription factors (TFs), microRNAs (miRNAs) and target genes from 25 selected databases. Moreover, we also inferred and incorporated potential regulatory relationships based on transcription factor binding site (TFBS) motifs into RegNetwork. As a result, RegNetwork contains a comprehensive set of experimentally observed or predicted transcriptional and post-transcriptional regulatory relationships, and the database framework is flexibly designed for potential extensions to include gene regulatory networks for other organisms in the future. Based on RegNetwork, we characterized the statistical and topological properties of genome-wide regulatory networks for human and mouse, we also extracted and interpreted simple yet important network motifs that involve the interplays between TF-miRNA and their targets. In summary, RegNetwork provides an integrated resource on the prior information for gene regulatory relationships, and it enables us to further investigate context-specific transcriptional and post-transcriptional regulatory interactions based on domain-specific experimental data.Database URL: http://www.regnetworkweb.org

Differential equation (DE) models are widely used in many scientific fields that include engineering, physics and biomedical sciences. The so-called "forward problem", the problem of simulations and predictions of state variables for given parameter values in the DE models, has been extensively studied by mathematicians, physicists, engineers and other scientists. However, the "inverse problem", the problem of parameter estimation based on the measurements of output variables, has not been well explored using modern statistical methods, although some least squares-based approaches have been proposed and studied. In this paper, we propose parameter estimation methods for ordinary differential equation models (ODE) based on the local smoothing approach and a pseudoleast squares (PsLS) principle under a framework of measurement error in regression models. The asymptotic properties of the proposed PsLS estimator are established. We also compare the PsLS method to the corresponding SIMEX method and evaluate their finite sample performances via simulation studies. We illustrate the proposed approach using an application example from an HIV dynamic study.

HIV dynamics studies have significantly contributed to the understanding of HIV infection and antiviral treatment strategies. But most studies are limited to short-term viral dynamics due to the difficulty of establishing a relationship of antiviral response with multiple treatment factors such as drug exposure and drug susceptibility during long-term treatment. In this article, a mechanism-based dynamic model is proposed for characterizing long-term viral dynamics with antiretroviral therapy, described by a set of nonlinear differential equations without closed-form solutions. In this model we directly incorporate drug concentration, adherence, and drug susceptibility into a function of treatment efficacy, defined as an inhibition rate of virus replication. We investigate a Bayesian approach under the framework of hierarchical Bayesian (mixed-effects) models for estimating unknown dynamic parameters. In particular, interest focuses on estimating individual dynamic parameters. The proposed methods not only help to alleviate the difficulty in parameter identifiability, but also flexibly deal with sparse and unbalanced longitudinal data from individual subjects. For illustration purposes, we present one simulation example to implement the proposed approach and apply the methodology to a data set from an AIDS clinical trial. The basic concept of the longitudinal HIV dynamic systems and the proposed methodologies are generally applicable to any other biomedical dynamic systems.

The cellular immune response to primary influenza virus infection is complex, involving multiple cell types and anatomical compartments, and is difficult to measure directly. Here we develop a two-compartment model that quantifies the interplay between viral replication and adaptive immunity. The fidelity of the model is demonstrated by accurately confirming the role of CD4 help for antibody persistence and the consequences of immune depletion experiments. The model predicts that drugs to limit viral infection and/or production must be administered within 2 days of infection, with a benefit of combination therapy when administered early, and cytotoxic CD8 T cells in the lung are as effective for viral clearance as neutralizing antibodies when present at the time of challenge. The model can be used to investigate explicit biological scenarios and generate experimentally testable hypotheses. For example, when the adaptive response depends on cellular immune cell priming, regulation of antigen presentation has greater influence on the kinetics of viral clearance than the efficiency of virus neutralization or cellular cytotoxicity. These findings suggest that the modulation of antigen presentation or the number of lung resident cytotoxic cells and the combination drug intervention are strategies to combat highly virulent influenza viruses. We further compared alternative model structures, for example, B-cell activation directly by the virus versus that through professional antigen-presenting cells or dendritic cell licensing of CD8 T cells.

We consider a nonparametric mixed-effects model y i 4t ij 5 D ‡4t ij 5 Cv i 4t ij 5 C … i 4t ij 5, j D 11 21 : : : 1 n i ; i D 11 21 : : : 1 n for longitudinal data. We propose combining local polynomial kernel regression and linear mixed-effects (LME) model techniques to estimate both xedeffects (population) curve ‡4t5 and random-effects curves v i 4t5. The resulting estimator, called the local polynomial LME (LLME) estimator, takes the local correlation structure of the longitudinal data into account naturally. We also propose new bandwidth selection strategies for estimating ‡4t5 and v i 4t5. Simulation studies show that our estimator for ‡4t5 is superior to the existing estimators in the sense of mean squared errors. The asymptotic bias, variance, mean squared errors, and asymptotic normality are established for the LLME estimators of ‡4t5. When n i is bounded and n tends to in nity, our LLME estimator converges in a standard nonparametric rate, and the asymptotic bias and variance are essentially the same as those of the kernel generalized estimating equation estimator proposed by Lin and Carroll. But when both n i and n tend to in nity, the LLME estimator is consistent with a slower rate of n 1=2 compared to the standard nonparametric rate, due to the existence of within-subject correlations of longitudinal data. We illustrate our methods with an application to a longitudinal dataset.