Hulin Wu

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.

Modelling the distribution of plant species using the autologistic regression model

For modeling the distribution of plant species in terms of climate covariates, we consider an autologistic regression model for spatial binary data on a regularly spaced lattice. This model belongs to the class of autologistic models introduced by Besag (1974). Three estimation methods, the coding method, maximum pseudolikelihood method and Markov chain Monte Carlo method are studied and comparedvia simulation and real data examples. As examples, we use the proposed methodology to model the distributions of two plant species in the state of Florida.

MARKOV CHAIN MONTE CARLO FOR AUTOLOGISTIC REGRESSION MODELS WITH APPLICATION TO THE DISTRIBUTION OF PLANT SPECIES

SUMMARY In this paper, we explore using autologistic regression models for spatial binary data with covariates. Autologistic regression models can handle binary responses exhibiting both spatial correlation and dependence on covariates. We use Markov chain Monte Carlo (MCMC) to estimate the parameters in these models. The distributional behavior of the MCMC maximum likelihood estimates (MCMC MLEs) is studied via simulation. We find that the MCMC MLEs are approximately normally distributed and that the MCMC estimates of Fisher information may be used to estimate the variance of the MCMC MLEs and to construct confidence intervals. Finally, we illustrate by example how our studies may be applied to model the distribution of plant species.

Quantifying the Early Immune Response and Adaptive Immune Response Kinetics in Mice Infected with Influenza A Virus

ABSTRACT 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 ∼1.2 days, and the half-life of free infectious IAV is ∼4 h. During the adaptive immune response (after day 5), the average half-life of infected epithelial cells is ∼0.5 days, and the average half-life of free infectious virus is ∼1.8 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.

Estimation of HIV dynamic parameters

Investigation of HIV viral dynamics is important for understanding the HIV pathogenesis and for development of treatment strategies. Perelson et al. demonstrated that simple viral dynamic models fit to data on viral load as measured by plasma HIV-RNA could produce estimates of rates of clearance of virus and of infected CD4+ T-lymphocytes. In this paper we extend the work of Perelson et al. by proposing models with less restrictive assumptions about drug activity. Our models take into account the fact that infectious and non-infectious virions are produced by infected T-cells both before and after the treatment. We also show that direct measurement of infectious virus load provides sufficient information for estimation of antiretroviral drug efficacy parameter. For characterizing viral dynamics of populations and estimation of dynamic parameters, we propose a hierarchical non-linear model. Compared to other methods such as the non-linear least square method used by Perelson et al., we show that the proposed approach has the following advantages: (i) it is more appropriate for modelling within-patient and between-patient variation and to characterize the population dynamics; (ii) it is flexible enough to deal with both rich and sparse individual data; (iii) it has more power to detect model misspecification; (iv) it allows incorporation of covariates for viral dynamic parameters; (v) it makes more efficient use of between-subject information to get better parameter estimates. We give two simulation examples to illustrate the proposed approach and its advantages. Finally, we discuss practical issues regarding the clinical trial design for viral dynamic studies.

Local Polynomial Mixed-Effects Models for Longitudinal Data

We consider a nonparametric mixed-effects model yi(tij)=η(tij)+vi(tij)+ϵi(tij),j=1,2,…,ni;i=1,2,…,n for longitudinal data. We propose combining local polynomial kernel regression and linear mixed-effects (LME) model techniques to estimate both fixedeffects (population) curve η(t) and random-effects curves vi(t). 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 η(t) and vi(t). Simulation studies show that our estimator for η(t) 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 η(t). When ni is bounded and n tends to infinity, 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 ni and n tend to infinity, the LLME estimator is consistent with a slower rate of n1/2 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.

Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models

Differential equation (DE) models are widely used in many scientific fields, including 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 article we propose parameter estimation methods for ordinary differential equation (ODE) models based on the local smoothing approach and a pseudo–least 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 simulation-extrapolation (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.

Simulation and Prediction of the Adaptive Immune Response to Influenza A Virus Infection

ABSTRACT 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.

Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity.

Highly active antiretroviral therapies consisting of reverse transcriptase inhibitor drugs and protease inhibitor drugs, which can rapidly suppress HIV below the limit of detection, are currently the most effective treatment for HIV infected patients. In spite of this, many patients fail to achieve viral suppression, probably due to existing or developing drug resistance, poor adherence, pharmacokinetic problems and other clinical factors. In this paper, we develop a viral dynamic model to evaluate how time-varying drug exposure and drug susceptibility affect antiviral response. Plasma concentrations, in turn, are modeled using a standard pharmacokinetic (PK) one-compartment open model with first order absorption and elimination as a function of fixed individual PK parameters and dose times. Imperfect adherence is considered as missed doses in PK models. We discuss the analytic properties of the viral dynamic model and study how time-varying treatment efficacies affect antiviral responses, specifically viral load and T cell counts. The relationship between actual failure time (the time at which the viral growth rate changes from negative to positive) and detectable failure time (the time at which viral load rebounds to above the limit of detection) is investigated. We find that an approximately linear relationship can be used to estimate the actual rebound failure time from the detectable rebound failure time. In addition, the effect of adherence on antiviral response is studied. In particular, we examine how different patterns of adherence affect antiviral response. Results suggest that longer sequences of missed doses increase the chance of treatment failure and accelerate the failure. Simulation experiments are presented to illustrate the relationship between antiviral response and pharmacokinetics, time-varying adherence and drug resistance. The proposed models and methods may be useful in AIDS clinical trial simulations.

Parameter identifiability and estimation of HIV/AIDS dynamic models.

Abstract We use a technique from engineering (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005) to investigate the algebraic identifiability of a popular three-dimensional HIV/AIDS dynamic model containing six unknown parameters. We find that not all six parameters in the model can be identified if only the viral load is measured, instead only four parameters and the product of two parameters (N and λ) are identifiable. We introduce the concepts of an identification function and an identification equation and propose the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005). We show that the newly proposed MTP method has advantages over the HOD method in the practical implementation. We also discuss the effect of the initial values of state variables on the identifiability of unknown parameters. We conclude that the initial values of output (observable) variables are part of the data that can be used to estimate the unknown parameters, but the identifiability of unknown parameters is not affected by these initial values if the exact initial values are measured with error. These noisy initial values only increase the estimation error of the unknown parameters. However, having the initial values of the latent (unobservable) state variables exactly known may help to identify more parameters. In order to validate the identifiability results, simulation studies are performed to estimate the unknown parameters and initial values from simulated noisy data. We also apply the proposed methods to a clinical data set to estimate HIV dynamic parameters. Although we have developed the identifiability methods based on an HIV dynamic model, the proposed methodologies are generally applicable to any ordinary differential equation systems.