Shuhua Hu

Experimental design and estimation of growth rate distributions in size-structured shrimp populations

We discuss inverse problem results for problems involving the estimation of probability distributions using aggregate data for growth in populations. We begin with a mathematical model describing variability in the early growth process of size-structured shrimp populations and discuss a computational methodology for the design of experiments to validate the model and estimate the growth-rate distributions in shrimp populations. Parameter-estimation findings using experimental data from experiments so designed for shrimp populations cultivated at Advanced BioNutrition Corporation are presented, illustrating the usefulness of mathematical and statistical modeling in understanding the uncertainty in the growth dynamics of such populations.

Comparison of Probabilistic and Stochastic Formulations in Modeling Growth Uncertainty and Variability

We compare two approaches for inclusion of uncertainty/variability in modelling growth in size-structured population models. One entails imposing a probabilistic structure on growth rates in the population while the other involves formulating growth as a stochastic Markov diffusion process. We present a theoretical analysis that allows one to include comparable levels of uncertainty in the two distinct formulations in making comparisons of the two approaches.

A hierarchical Bayesian approach for parameter estimation in HIV models

A hierarchical Bayesian approach is developed to estimate parameters at both the individual and the population levels in a HIV model, with the implementation carried out by Markov Chain Monte Carlo (MCMC) techniques. Sample numerical simulations and statistical results are provided to demonstrate the feasibility of this approach.

A Computational Comparison of Alternatives to Including Uncertainty in Structured Population Models

Two conceptually different approaches to incorporate growth uncertainty into size-structured population models have recently been investigated. One entails imposing a probabilistic structure on all the possible growth rates across the entire population, which results in a growth rate distribution model. The other involves formulating growth as a Markov stochastic diffusion process, which leads to a Fokker-Planck model. Numerical computations verify that a Fokker-Planck model and a growth rate distribution model can, with properly chosen parameters, yield quite similar time dependent population densities. The relationship between the two models is based on the theoretical analysis in [7].

Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems

Abstract Normalized differences of several adjacent observations, referred to as pseudo-measurement errors in this paper, are used in so-called difference-based estimation methods as building blocks for the variance estimate of measurement errors. Numerical results demonstrate that pseudo-measurement errors can be used to serve the role of measurement errors. Based on this information, we propose the use of pseudo-measurement errors to determine an appropriate statistical model and then to subsequently investigate whether there is a mathematical model misspecification or error. We also propose to use the information provided by pseudo-measurement errors to quantify uncertainty in parameter estimation by bootstrapping methods. A number of numerical examples are given to illustrate the effectiveness of these proposed methods.

Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

We consider probability measure estimation in a nonparametric model using a least-squares approach under the Prohorov metric framework. We summarize the computational methods and related convergence results that were recently developed by our group. New results are presented on the bias and the variance due to the approximation and new pointwise asymptotic normality of the approximated probability measure estimator. We propose the use of a model selection criterion to balance the bias and the variance, and compare the new pointwise confidence bands constructed using the asymptotic normality results with those obtained by Monte Carlo simulations.

Modelling and optimal control of immune response of renal transplant recipients

We consider the increasingly important and highly complex immunological control problem: control of the dynamics of immunosuppression for organ transplant recipients. The goal in this problem is to maintain the delicate balance between over-suppression (where opportunistic latent viruses threaten the patient) and under-suppression (where rejection of the transplanted organ is probable). First, a mathematical model is formulated to describe the immune response to both viral infection and introduction of a donor kidney in a renal transplant recipient. Some numerical results are given to qualitatively validate and demonstrate that this initial model exhibits appropriate characteristics of primary infection and reactivation for immunosuppressed transplant recipients. In addition, we develop a computational framework for designing adaptive optimal treatment regimes with partial observations and low-frequency sampling, where the state estimates are obtained by solving a second deterministic optimal tracking problem. Numerical results are given to illustrate the feasibility of this method in obtaining optimal treatment regimes with a balance between under-suppression and over-suppression of the immune system.

Host Immune Responses That Promote Initial HIV Spread

The host inflammatory response to HIV invasion is a necessary component of the innate antiviral activity that vaccines and early interventions seek to exploit/enhance. However, the response is dependent on CD4+ T-helper cell 1 (Th1) recruitment and activation. It is this very recruitment of HIV-susceptible target cells that is associated with the initial viral proliferation. Hence, global enhancement of the inflammatory response by T-cells and dendritic cells will likely feed viral propagation. Mucosal entry sites contain inherent pathways, in the form of natural regulatory T-cells (nTreg), that globally dampen the inflammatory response. We created a model of this inflammatory response to virus as well as inherent nTreg-mediated regulation of Th1 recruitment and activation. With simulations using this model we sought to address the net effect of nTreg activation and its specific functions as well as identify mechanisms of the natural inflammatory response that are best targeted to inhibit viral spread without compromising initial antiviral activity. Simulation results provide multiple insights that are relevant to developing intervention strategies that seek to exploit natural immune processes: (i) induction of the regulatory response through nTreg activation expedites viral proliferation due to viral production by nTreg itself and not to reduced Natural Killer (NK) cell activity; (ii) at the same time, induction of the inflammation response through either DC activation or Th1 activation expedites viral proliferation; (iii) within the inflammatory pathway, the NK response is an effective controller of viral proliferation while DC-mediated stimulation of T-cells is a significant driver of viral proliferation; and (iv) nTreg-mediated DC deactivation plays a significant role in slowing viral proliferation by inhibiting T-cell stimulation, making this function an aide to the antiviral immune response.

Modeling Immune Response to BK Virus Infection and Donor Kidney in Renal Transplant Recipients.

In this paper, we develop and validate with bootstrapping techniques a mechanistic mathematical model of immune response to both BK virus infection and a donor kidney based on known and hypothesized mechanisms in the literature. The model presented does not capture all the details of the immune response but possesses key features that describe a very complex immunological process. We then estimate model parameters using a least squares approach with a typical set of available clinical data. Sensitivity analysis combined with asymptotic theory is used to determine the number of parameters that can be reliably estimated given the limited number of observations.

Nonlinear Stochastic Markov Processes and Modeling Uncertainty in Populations

We consider an alternative approach to the use of nonlinear stochastic Markov processes (which have a Fokker-Planck or Forward Kolmogorov representation for density) in modeling uncertainty in populations. These alternate formulations, which involve imposing probabilistic structures on a family of deterministic dynamical systems, are shown to yield pointwise equivalent population densities. Moreover, these alternate formulations lead to fast efficient calculations in inverse problems as well as in forward simulations. Here we derive a class of stochastic formulations for which such an alternate representation is readily found.