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Mathematical Biosciences & Engineering

2010 , Volume 7 , Issue 2

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A comparison of nonlinear filtering approaches in the context of an HIV model
H. Thomas Banks , Shuhua Hu , Zackary R. Kenz and  Hien T. Tran
2010, 7(2): 213-236 doi: 10.3934/mbe.2010.7.213 +[Abstract](34) +[PDF](598.3KB)
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.
Morphological spatial patterns in a reaction diffusion model for metal growth
Benedetto Bozzini , Deborah Lacitignola and  Ivonne Sgura
2010, 7(2): 237-258 doi: 10.3934/mbe.2010.7.237 +[Abstract](52) +[PDF](1752.7KB)
In this paper a reaction-diffusion system modelling metal growth processes is considered, to investigate - within the electrodeposition context- the formation of morphological patterns in a finite two-dimensional spatial domain. Nonlinear dynamics of the system is studied from both the analytical and numerical points of view. Phase-space analysis is provided and initiation of spatial patterns induced by diffusion is shown to occur in a suitable region of the parameter space. Investigations aimed at establishing the role of some relevant chemical parameters on stability and selection of solutions are also provided. By the numerical approximation of the equations, simulations are presented which turn out to be in good agreement with experiments for the electrodeposition of Au-Cu and Au-Cu-Cd alloys.
Surface tension and modeling of cellular intercalation during zebrafish gastrulation
Colette Calmelet and  Diane Sepich
2010, 7(2): 259-275 doi: 10.3934/mbe.2010.7.259 +[Abstract](47) +[PDF](2630.7KB)
In this paper we discuss a model of zebrafish embryo notochord development based on the effect of surface tension of cells at the boundaries. We study the process of interaction of mesodermal cells at the boundaries due to adhesion and cortical tension, resulting in cellular intercalation. From in vivo experiments, we obtain cell outlines of time-lapse images of cell movements during zebrafish embryo development. Using Cellular Potts Model, we calculate the total surface energy of the system of cells at different time intervals at cell contacts. We analyze the variations of total energy depending on nature of cell contacts. We demonstrate that our model can be viable by calculating the total surface energy value for experimentally observed configurations of cells and showing that in our model these configurations correspond to a decrease in total energy values in both two and three dimensions.
A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions
Fabrizio Clarelli and  Roberto Natalini
2010, 7(2): 277-300 doi: 10.3934/mbe.2010.7.277 +[Abstract](31) +[PDF](1942.9KB)
Mycobacterium tuberculosis (Mtb) is a widely diffused infection. However, in general, the human immune system is able to contain it. In this work, we propose a mathematical model which describes the early immune response to the Mtb infection in the lungs, also including the possible evolution of the infection in the formation of a granuloma. The model is based on coupled reaction-diffusion-transport equations with chemotaxis, which take into account the interactions among bacteria, macrophages and chemoattractant. The novelty of this approach is in the modeling of the velocity field, proportional to the gradient of the pressure developed between the cells, which makes possible to deal with a full multidimensional description and efficient numerical simulations. We perform a linear stability analysis of the model and propose a robust implicit-explicit scheme to deal with long time simulations. Both in one and two-dimensions, we find that there are threshold values in the parameters space, between a contained infection and the uncontrolled bacteria growth, and the generation of granuloma-like patterns can be observed numerically.
A kinetic mechanism inducing oscillations in simple chemical reactions networks
Julien Coatléven and  Claudio Altafini
2010, 7(2): 301-312 doi: 10.3934/mbe.2010.7.301 +[Abstract](37) +[PDF](206.4KB)
It is known that a kinetic reaction network in which one or more secondary substrates are acting as cofactors may exhibit an oscillatory behavior. The aim of this work is to provide a description of the functional form of such a cofactor action guaranteeing the onset of oscillations in sufficiently simple reaction networks.
Vector control for the Chikungunya disease
Yves Dumont and  Frederic Chiroleu
2010, 7(2): 313-345 doi: 10.3934/mbe.2010.7.313 +[Abstract](75) +[PDF](742.8KB)
We previously proposed a compartmental model to explain the outbreak of Chikungunya disease in Réunion Island, a French territory in Indian Ocean, and other countries in 2005 and possible links with the explosive epidemic of 2006. In the present paper, we asked whether it would have been possible to contain or stop the epidemic of 2006 through appropriate mosquito control tools. Based on new results on the Chikungunya virus, its impact on mosquito life-span, and several experiments done by health authorities, we studied several types of control tools used in 2006 to contain the epidemic. We present an analysis of the model, and we develop a new nonstandard finite difference scheme to provide several simulations with and without mosquito control. Our preliminary study shows that an early use of a combination of massive spraying and mechanical control (like the destruction of breeding sites) can be efficient, to stop or contain the propagation of Chikungunya infection, with a low impact on the environment.
Global stability for a class of discrete SIR epidemic models
Yoichi Enatsu , Yukihiko Nakata and  Yoshiaki Muroya
2010, 7(2): 347-361 doi: 10.3934/mbe.2010.7.347 +[Abstract](59) +[PDF](197.7KB)
In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distributed delays by using a variation of the backward Euler method. Applying a Lyapunov functional technique, it is shown that the global dynamics of each discrete SIR epidemic model are fully determined by a single threshold parameter and the effect of discrete time delays are harmless for the global stability of the endemic equilibrium of the model.
Mathematically modeling PCR: An asymptotic approximation with potential for optimization
Martha Garlick , James Powell , David Eyre and  Thomas Robbins
2010, 7(2): 363-384 doi: 10.3934/mbe.2010.7.363 +[Abstract](39) +[PDF](225.7KB)
A mathematical model for PCR (Polymerase Chain Reaction) is developed using the law of mass action and simplifying assumptions regarding the structure of the reactions. Differential equations are written from the chemical equations, preserving the detail of the complementary DNA single strand being extended one base pair at a time. The equations for the annealing stage are solved analytically. The method of multiple scales is used to approximate solutions for the extension stage, and a map is developed from the solutions to simulate PCR. The map recreates observed PCR well, and gives us the ability to optimize the PCR process. Our results suggest that dynamically optimizing the extension and annealing stages of individual samples may significantly reduce the total time for a PCR run. Moreover, we present a nearly optimal design that functions almost as well and does not depend on the specifics of a single reaction, and so would work for multi sample and multiplex applications.
Diffusion-limited tumour growth: Simulations and analysis
Philip Gerlee and  Alexander R. A. Anderson
2010, 7(2): 385-400 doi: 10.3934/mbe.2010.7.385 +[Abstract](34) +[PDF](498.8KB)
The morphology of solid tumours is known to be affected by the background oxygen concentration of the tissue in which the tumour grows, and both computational and experimental studies have suggested that branched tumour morphology in low oxygen concentration is caused by diffusion-limited growth. In this paper we present a simple hybrid cellular automaton model of solid tumour growth aimed at investigating this phenomenon. Simulation results show that for high consumption rates (or equivalently low oxygen concentrations) the tumours exhibit branched morphologies, but more importantly the simplicity of the model allows for an analytic approach to the problem. By applying a steady-state assumption we derive an approximate solution of the oxygen equation, which closely matches the simulation results. Further, we derive a dispersion relation which reveals that the average branch width in the tumour depends on the width of the active rim, and that a smaller active rim gives rise to thinner branches. Comparison between the prediction of the stability analysis and the results from the simulations shows good agreement between theory and simulation.
Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors
Nikolaos Kazantzis and  Vasiliki Kazantzi
2010, 7(2): 401-419 doi: 10.3934/mbe.2010.7.401 +[Abstract](30) +[PDF](381.6KB)
A new approach to the problem of characterizing the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using the notion of slow invariant manifold is proposed. The problem of interest is addressed within the context of singular partial differential equations (PDE) theory, and in particular, through a system of singular quasi-linear invariance PDEs for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system's slow invariant manifold exponentially attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original biosystem dynamics on the slow manifold. The analyticity property of the solution to the invariance PDEs enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, examples of certain immobilized enzyme bioreactors are considered to elucidate aspects of the proposed context of analysis.
Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections
Felicia Maria G. Magpantay and  Xingfu Zou
2010, 7(2): 421-442 doi: 10.3934/mbe.2010.7.421 +[Abstract](50) +[PDF](310.1KB)
We consider a neuronal network model with both axonal connections (in the form of synaptic coupling) and delayed non-local feedback connections. The kernel in the feedback channel is assumed to be a standard non-local one, while for the kernel in the synaptic coupling, four types are considered. The main concern is the existence of travelling wave front. By employing the speed index function, we are able to obtain the existence of a travelling wave front for each of these four types within certain range of model parameters. We are also able to describe how the feedback coupling strength and the magnitude of the delay affect the wave speed. Some particular kernel functions for these four cases are chosen to numerically demonstrate the theoretical results.
Mathematical observations on the relation between eclosion periods and the copulation rate of cicadas
Yasumasa Saisho
2010, 7(2): 443-453 doi: 10.3934/mbe.2010.7.443 +[Abstract](40) +[PDF](363.2KB)
In many species of cicadas the peak of eclosion of males precedes that of females. In this paper, we construct a stochastic model and consider whether this sexual difference of eclosion periods works against mating or not. We also discuss the relation between the peak period of copulations and the development of population number by using this model.
Pulse vaccination strategies in a metapopulation SIR model
Alan J. Terry
2010, 7(2): 455-477 doi: 10.3934/mbe.2010.7.455 +[Abstract](42) +[PDF](1048.8KB)
We examine a model for a disease with SIR-type dynamics circulating in a population living on two or more patches between any pair of which migration is allowed. We suppose that a pulse vaccination strategy (PVS) is carried out on each patch. Conditions are derived on each PVS such that the disease will be eradicated on all patches. The PVS on one patch is assumed to be essentially independent of the PVS on the other patches except in so far as they are all performed simultaneously. This independence is of practical value when we bear in mind that the patches may represent regions or countries with autonomous public health authorities, which may make individual decisions about the days appropriate for a vaccination pulse to occur in their own region or country. Simulations corroborate our theoretical results.

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