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Mathematical Biosciences & Engineering

2006 , Volume 3 , Issue 2

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Competition and Dispersal Delays in Patchy Environments
Nancy Azer and  P. van den Driessche
2006, 3(2): 283-296 doi: 10.3934/mbe.2006.3.283 +[Abstract](28) +[PDF](251.2KB)
Dispersal delays are introduced into a competition model for two species that disperse among $n$ identical patches. The model is formulated as a system of integro-differential equations with an arbitrary distribution of dispersal times between patches. By identifying steady states and analyzing local stability, conditions for competitive exclusion, coexistence or extinction are determined in terms of the system parameters. These are confirmed by numerical simulations with a delta function distribution, showing that all solutions approach a steady state and that high dispersal is generally a disadvantage to a species. However, if the two species have identical local dynamics, then small dispersal rates (with certain parameter restrictions) can be an advantage to the dispersing species. If the number of species is increased to three, then oscillatory coexistence with dispersal delay is possible.
The Effects of Vertical Transmission on the Spread of HIV/AIDS in the Presence of Treatment
Moatlhodi Kgosimore and  Edward M. Lungu
2006, 3(2): 297-312 doi: 10.3934/mbe.2006.3.297 +[Abstract](30) +[PDF](231.2KB)
In this study, we develop a model that incorporates treatment of both juveniles who were infected with HIV/AIDS through vertical transmission and HIV/AIDS-infected adults. We derive conditions under which the burden of HIV/AIDS can be reduced in the population both in the absence of and in the presence of vertical transmission. We have determined the critical threshold parameter ($R_v^*$), which represents the demographic replacement of infectives through vertical transmission, below which treated infected juveniles can reach adulthood without causing an epidemic. Five countries in sub-Saharan Africa are used to illustrate our results. We have concluded that $R_v^*$ is dependent on the current prevalence rate but that a significant proportion of infected juveniles receiving treatment can reach adulthood without causing an epidemic.
Modeling Cancer in HIV-1 Infected Individuals: Equilibria, Cycles and Chaotic Behavior
Jie Lou , Tommaso Ruggeri and  Claudio Tebaldi
2006, 3(2): 313-324 doi: 10.3934/mbe.2006.3.313 +[Abstract](24) +[PDF](2830.3KB)
For HIV-infected individuals, cancer remains a significant burden. Gaining insight into the epidemiology and mechanisms that underlie AIDS-related cancers can provide us with a better understanding of cancer immunity and viral oncogenesis. In this paper, an HIV-1 dynamical model incorporating the AIDS-related cancer cells was studied. The model consists of three components, cancer cells, healthy CD4+ T lymphocytes and infected CD4+ T lymphocytes, and can have six steady states. We discuss the existence, the stability properties and the biological meanings of these steady states, in particular for the positive one: cancer-HIV-healthy cells steady state. We find conditions for Hopf bifurcation of the positive steady state, leading to periodic solutions, sequences of period doubling bifurcations and appearance of chaos. Further, chaos and periodic behavior alternate. Our results are consistent with some clinical and experimental observations.
Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model
Fabien Crauste
2006, 3(2): 325-346 doi: 10.3934/mbe.2006.3.325 +[Abstract](37) +[PDF](392.2KB)
We analyze the asymptotic stability of a nonlinear system of two differential equations with delay, describing the dynamics of blood cell production. This process takes place in the bone marrow, where stem cells differentiate throughout division in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells in the introduction of cells in cycle, we are led to study a characteristic equation with delay-dependent coefficients. We determine a necessary and sufficient condition for the global stability of the first steady state of our model, which describes the population's dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial positive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature.
Energy Considerations in a Model of Nematode Sperm Crawling
Borys V. Bazaliy , Ya. B. Bazaliy , Avner Friedman and  Bei Hu
2006, 3(2): 347-370 doi: 10.3934/mbe.2006.3.347 +[Abstract](26) +[PDF](267.4KB)
In this paper we propose a mathematical model for nematode sperm cell crawling. The model takes into account both force and energy balance in the process of lamellipodium protrusion and cell nucleus drag. It is shown that by specifying the (possibly variable) efficiency of the major sperm protein biomotor one completely determines a self-consistent problem of the lamellipodium-nucleus motion. The model thus obtained properly accounts for the feedback of the load on the lamellipodium protrusion, which in general should not be neglected. We study and analyze the steady crawling state for a particular efficiency function and find that all nonzero modes, up to a large magnitude, are linearly asymptotically stable, thus reproducing the experimental observations of the long periods of steady crawling exhibited by the nematode sperm cells.
Simulation of Pulsatile Flow of Blood in Stenosed Coronary Artery Bypass with Graft
B. Wiwatanapataphee , D. Poltem , Yong Hong Wu and  Y. Lenbury
2006, 3(2): 371-383 doi: 10.3934/mbe.2006.3.371 +[Abstract](26) +[PDF](1265.2KB)
In this paper, we investigate the behavior of the pulsatile blood flow in a stenosed right coronary artery with a bypass graft. The human blood is assumed to be a non-Newtonian fluid and its viscous behavior is described by the Carreau model. The transient phenomena of blood flow though the stenosed region and the bypass grafts are simulated by solving the three dimensional unsteady Navier-Stokes equations and continuity equation. The influence of the bypass angle on the flow interaction between the jet flow from the native artery and the flow from the bypass graft is investigated. Distributions of velocity, pressure and wall shear stresses are determined under various conditions. The results show that blood pressure in the stenosed artery drops dramatically in the stenosis area and that high wall shear stresses occur around the stenosis site.
Remark on the Paper by Rao And Kakehashi (2005)
Saralees Nadarajah
2006, 3(2): 385-387 doi: 10.3934/mbe.2006.3.385 +[Abstract](112) +[PDF](122.0KB)
The paper by Rao and Kakehashi [1] uses the Weibull distribution (given by equation (3) in the paper) and a truncated version of it (given by equations (4) and (5)) to model HIV/AIDS data in India. After a careful reading, I have found that all of the results presented in the appendix are incorrect (starting with equation (11) itself). Instead of pointing out all of the specific errors, I have chosen to give the correct formulas (and a brief outline of their derivation) for the $r$th moment of the truncated Weibull distribution.
Multiscale Image Registration
Dana Paquin , Doron Levy , Eduard Schreibmann and  Lei Xing
2006, 3(2): 389-418 doi: 10.3934/mbe.2006.3.389 +[Abstract](36) +[PDF](1664.9KB)
A multiscale image registration technique is presented for the registration of medical images that contain significant levels of noise. An overview of the medical image registration problem is presented, and various registration techniques are discussed. Experiments using mean squares, normalized correlation, and mutual information optimal linear registration are presented that determine the noise levels at which registration using these techniques fails. Further experiments in which classical denoising algorithms are applied prior to registration are presented, and it is shown that registration fails in this case for significantly high levels of noise, as well. The hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and L. Vese [20] is presented, and accurate registration of noisy images is achieved by obtaining a hierarchical multiscale decomposition of the images and registering the resulting components. This approach enables successful registration of images that contain noise levels well beyond the level at which ordinary optimal linear registration fails. Image registration experiments demonstrate the accuracy and efficiency of the multiscale registration technique, and for all noise levels, the multiscale technique is as accurate as or more accurate than ordinary registration techniques.
Analysis of Blood Flow Velocity and Pressure Signals using the Multipulse Method
Derek H. Justice , H. Joel Trussell and  Mette S. Olufsen
2006, 3(2): 419-440 doi: 10.3934/mbe.2006.3.419 +[Abstract](30) +[PDF](1380.7KB)
This paper shows how the multipulse method from digital signal processing can be used to accurately synthesize signals obtained from blood pressure and blood flow velocity sensors during posture change from sitting to standing. The multipulse method can be used to analyze signals that are composed of pulses of varying amplitudes. One of the advantages of the multipulse method is that it is able to produce an accurate and efficient representation of the signals at high resolution. The signals are represented as a set of input impulses passed through an autoregressive (AR) filter. The parameters that define the AR filter can be used to distinguish different conditions. In addition, the AR coefficients can be transformed to tube radii associated with digital wave guides, as well as pole-zero representation. Analysis of the dynamics of the model parameters have potential to provide better insight and understanding of the underlying physiological control mechanisms. For example, our data indicate that the tube radii may be related to the diameter of the blood vessels.

2016  Impact Factor: 1.035




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