## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
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- Mathematical Biosciences & Engineering
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- Mathematical Foundations of Computing
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### Open Access Journals

MBE

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.

MBE

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.

MBE

A comparison of computational efficiencies of stochastic algorithms in terms of two infection models

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.

MBE

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.

MBE

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.

MBE

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.

MBE

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.

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