Model validation for a noninvasive arterial stenosis detection problem
H. Thomas Banks Shuhua Hu Zackary R. Kenz Carola Kruse Simon Shaw John Whiteman Mark P. Brewin Stephen E. Greenwald Malcolm J. Birch
Mathematical Biosciences & Engineering 2014, 11(3): 427-448 doi: 10.3934/mbe.2014.11.427
A current thrust in medical research is the development of a non-invasive method for detection, localization, and characterization of an arterial stenosis (a blockage or partial blockage in an artery). A method has been proposed to detect shear waves in the chest cavity which have been generated by disturbances in the blood flow resulting from a stenosis. In order to develop this methodology further, we use one-dimensional shear wave experimental data from novel acoustic phantoms to validate a corresponding viscoelastic mathematical model. We estimate model parameters which give a good fit (in a sense to be precisely defined) to the experimental data, and use asymptotic error theory to provide confidence intervals for parameter estimates. Finally, since a robust error model is necessary for accurate parameter estimates and confidence analysis, we include a comparison of absolute and relative models for measurement error.
keywords: asymptotic theory. inverse problem Viscoelastic model sensitivity analysis
Modeling shrimp biomass and viral infection for production of biological countermeasures
H. Thomas Banks V. A. Bokil Shuhua Hu A. K. Dhar R. A. Bullis C. L. Browdy F.C.T. Allnutt
Mathematical Biosciences & Engineering 2006, 3(4): 635-660 doi: 10.3934/mbe.2006.3.635
In this paper we develop a mathematical model for the rapid production of large quantities of therapeutic and preventive countermeasures. We couple equations for biomass production with those for vaccine production in shrimp that have been infected with a recombinant viral vector expressing a foreign antigen. The model system entails both size and class-age structure.
keywords: latently-acutely infected. size/class-age structured population models shrimp growth/viral pro-gression dynamics
A comparison of computational efficiencies of stochastic algorithms in terms of two infection models
H. Thomas Banks Shuhua Hu Michele Joyner Anna Broido Brandi Canter Kaitlyn Gayvert Kathryn Link
Mathematical Biosciences & Engineering 2012, 9(3): 487-526 doi: 10.3934/mbe.2012.9.487
In this paper, we investigate three particular algorithms: a stochastic simulation algorithm (SSA), and explicit and implicit tau-leaping algorithms. To compare these methods, we used them to analyze two infection models: a Vancomycin-resistant enterococcus (VRE) infection model at the population level, and a Human Immunodeficiency Virus (HIV) within host infection model. While the first has a low species count and few transitions, the second is more complex with a comparable number of species involved. The relative efficiency of each algorithm is determined based on computational time and degree of precision required. The numerical results suggest that all three algorithms have the similar computational efficiency for the simpler VRE model, and the SSA is the best choice due to its simplicity and accuracy. In addition, we have found that with the larger and more complex HIV model, implementation and modification of tau-Leaping methods are preferred.
keywords: continuous time Markov chain models Gillespie tau-leaping Dynamical models stochastic simulation algorithms bacterial and viral infection models.
Nonlinear stochastic Markov processes and modeling uncertainty in populations
H.Thomas Banks Shuhua Hu
Mathematical Biosciences & Engineering 2012, 9(1): 1-25 doi: 10.3934/mbe.2012.9.1
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.
keywords: forward Kolmogorov uncertainty Fokker-Planck probabilistic structures on deterministic systems pointwise equivalence. Nonlinear Markov processes
Comparison between stochastic and deterministic selection-mutation models
Azmy S. Ackleh Shuhua Hu
Mathematical Biosciences & Engineering 2007, 4(2): 133-157 doi: 10.3934/mbe.2007.4.133
We present a deterministic selection-mutation model with a discrete trait variable. We show that for an irreducible selection-mutation matrix in the birth term the deterministic model has a unique interior equilibrium which is globally stable. Thus all subpopulations coexist. In the pure selection case, the outcome is known to be that of competitive exclusion, where the subpopulation with the largest growth-to-mortality ratio will survive and the remaining subpopulations will go extinct. We show that if the selection-mutation matrix is reducible, then competitive exclusion or coexistence are possible outcomes. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, unlike the deterministic one, if the differences in the growth-to-mortality ratios are small in the pure selection case, it cannot be determined a priori which subpopulation will have the highest probability of surviving and winning the competition.
keywords: coexistence stochastic differential equations. selection-mutation models competitive exclusion
Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations
Azmy S. Ackleh H.T. Banks Keng Deng Shuhua Hu
Mathematical Biosciences & Engineering 2005, 2(2): 289-315 doi: 10.3934/mbe.2005.2.289
A least squares technique is developed for identifying unknown parameters in a coupled system of nonlinear size-structured populations. Convergence results for the parameter estimation technique are established. Ample numerical simulations and statistical evidence are provided to demonstrate the feasibility of this approach.
keywords: finite difference approximation coupled system of nonlinear size-structured populations parameter estimation standard deviation. numerical simulation
A comparison of nonlinear filtering approaches in the context of an HIV model
H. Thomas Banks Shuhua Hu Zackary R. Kenz Hien T. Tran
Mathematical Biosciences & Engineering 2010, 7(2): 213-236 doi: 10.3934/mbe.2010.7.213
In this paper three different filtering methods, the Extended Kalman Filter (EKF), the Gauss-Hermite Filter (GHF), and the Unscented Kalman Filter (UKF), are compared for state-only and coupled state and parameter estimation when used with log state variables of a model of the immunologic response to the human immunodeficiency virus (HIV) in individuals. The filters are implemented to estimate model states as well as model parameters from simulated noisy data, and are compared in terms of estimation accuracy and computational time. Numerical experiments reveal that the GHF is the most computationally expensive algorithm, while the EKF is the least expensive one. In addition, computational experiments suggest that there is little difference in the estimation accuracy between the UKF and GHF. When measurements are taken as frequently as every week to two weeks, the EKF is the superior filter. When measurements are further apart, the UKF is the best choice in the problem under investigation.
keywords: HIV. Gauss-Hermite filter unscented Kalman filter extended Kalman filter
Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective
Wei Feng Shuhua Hu Xin Lu
Conference Publications 2003, 2003(Special): 544-553 doi: 10.3934/proc.2003.2003.544
keywords: Optimal control cancer chemotherapy. three compartment model field of extremals

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