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Mathematical Biosciences & Engineering

2005 , Volume 2 , Issue 4

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Spatial Buffering Mechanism: Mathematical Model and Computer Simulations
Benjamin Steinberg, Yuqing Wang, Huaxiong Huang and Robert M. Miura
2005, 2(4): 675-702 doi: 10.3934/mbe.2005.2.675 +[Abstract](311) +[PDF](1133.5KB)
Abstract:
It is generally accepted that the spatial buffering mechanism is important to buffer extracellular-space potassium in the brain-cell microenvironment. In the past, this phenomenon, generally associated with glial cells, has been treated analytically and numerically using a simplified one-dimensional description. The present study extends the previous research by using a novel numerical scheme for the analysis of potassium buffering mechanisms in the extracellular brain-cell microenvironment. In particular, a lattice-cellular automaton was employed to simulate a detailed two-compartment model of a two-dimensional brain-cell system. With this numerical approach, the present study elaborates upon previous theoretical work on spatial buffering (SB) by incorporating a more realistic structure of the brain-cell microenvironment, which was not feasible earlier. We use the experimental paradigm consisting of iontophoretic injection of KCl to study the SB mechanism. Our simulations confirmed the results reported in the literature obtained by an averaged model. The results also show that the additional effects captured by a simplified two-dimensional geometry do not alter significantly the conclusions obtained from the averaged model. The details of applying such a numerical method to the study of ion movements in cellular environments, as well as its potential for future study, are discussed.
Predator-Prey Dynamics with Disease in the Prey
Peter A. Braza
2005, 2(4): 703-717 doi: 10.3934/mbe.2005.2.703 +[Abstract](376) +[PDF](557.7KB)
Abstract:
The Holling-Tanner model for predator-prey systems is adapted to incorporate the spread of disease in the prey. The analysis of the dynamics centers on bifurcation diagrams in which the disease transmission rate is the primary parameter. The ecologically reasonable assumption that the diseased prey are easier to catch enables tractable analytic results to be obtained for the stability of the steady states and the locations of Hopf bifurcation points as a function of the ecological parameters. Two parameters of particular relevance are the ratio of the predator's intrinsic growth rate to the prey's growth rate and the maximum number of infected prey that can be eaten per time. The dynamics are shown to be qualitatively different depending on the comparative size of these parameters. Numerical results obtained with AUTO are used to extend the local analysis and further illustrate the rich dynamics.
An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery
Blaise Faugeras and Olivier Maury
2005, 2(4): 719-741 doi: 10.3934/mbe.2005.2.719 +[Abstract](424) +[PDF](390.3KB)
Abstract:
We develop an advection-diffusion size-structured fish population dynamics model and apply it to simulate the skipjack tuna population in the Indian Ocean. The model is fully spatialized, and movements are parameterized with oceanographical and biological data; thus it naturally reacts to environment changes. We first formulate an initial-boundary value problem and prove existence of a unique positive solution. We then discuss the numerical scheme chosen for the integration of the simulation model. In a second step we address the parameter estimation problem for such a model. With the help of automatic differentiation, we derive the adjoint code which is used to compute the exact gradient of a Bayesian cost function measuring the distance between the outputs of the model and catch and length frequency data. A sensitivity analysis shows that not all parameters can be estimated from the data. Finally twin experiments in which pertubated parameters are recovered from simulated data are successfully conducted.
Host Extinction Dynamics in a Simple Parasite-Host Interaction Model
Tzy-Wei Hwang and Yang Kuang
2005, 2(4): 743-751 doi: 10.3934/mbe.2005.2.743 +[Abstract](282) +[PDF](210.4KB)
Abstract:
This short article carefully formulate a simple SI model for a parasite-host interaction through the basic birth and death processes analysis. This model reveals and corrects an error in similar models studied recently by various authors. Complete mathematical investigation of this simple model shows that the host extinction dynamics can happen and the outcomes may depend on the initial conditions. We also present biological implications of our findings.
The Role of Vaccination in the Control of SARS
Julijana Gjorgjieva, Kelly Smith, Gerardo Chowell, Fabio Sánchez, Jessica Snyder and Carlos Castillo-Chavez
2005, 2(4): 753-769 doi: 10.3934/mbe.2005.2.753 +[Abstract](535) +[PDF](2607.7KB)
Abstract:
We assess pre-outbreak and during-outbreak vaccination as control strategies for SARS epidemics using a mathematical model that includes susceptible, latent (traced and untraced), infectious, isolated and recovered individuals. Scenarios focusing on policies that include contact tracing and levels of self-isolation among untraced infected individuals are explored. Bounds on the proportion of pre-outbreak successfully vaccinated individuals are provided using the the basic reproductive number. Uncertainty and sensitivity analyses on the reproductive number are carried out. The final epidemic size under different vaccination scenarios is computed.
The Effect of the HIV/AIDS Epidemic on Africa's Truck Drivers
Christopher M. Kribs-Zaleta, Melanie Lee, Christine Román, Shari Wiley and Carlos M. Hernández-Suárez
2005, 2(4): 771-788 doi: 10.3934/mbe.2005.2.771 +[Abstract](344) +[PDF](260.7KB)
Abstract:
The AIDS epidemic is having a growing impact on the transport sector of the economy of sub-Saharan Africa, where long-distance truck drivers are at an increased risk of infection due to their frequent contacts with commercial sex workers. The spread of AIDS in the transport industry is especially significant to the economy, as truck drivers are largely responsible for transporting crops and supplies needed for daily subsistence. In this paper we analyze these effects via two models, one employing a switch and the other a Verhulst saturation function, to describe the rate at which new drivers are recruited in terms of the supply and demand for them in the general population. Results provide an estimate of the epidemic's economic impact on the transportation sector through the loss of truck drivers (an estimated 10% per year, with endemic levels near 90%).
A Mathematical Model for Fibroblast Growth Factor Competition Based on Enzyme
Justin P. Peters, Khalid Boushaba and Marit Nilsen-Hamilton
2005, 2(4): 789-810 doi: 10.3934/mbe.2005.2.789 +[Abstract](318) +[PDF](294.3KB)
Abstract:
In this paper, we develop a mathematical model for the competition of two species of fibroblast growth factor, FGF-1 and FGF-2, for the same cell surface receptor. We provide pathways for this interaction using experimental data obtained by Neufeld and Gospodarowicz reported in 1986 [9]. These pathways demonstrate how the interaction of two fibroblast growth factors affects cell proliferation. Upon development of these pathways, we use simulations in MATLAB and optimization to extrapolate the values of a variety of biochemical parameters imbedded within the model. Furthermore, it should be possible to use the model as the basis for a testable hypothesis. We explore this predictive ability with further simulations in MATLAB.
A two-strain HIV-1 mathematical model to assess the effects of chemotherapy on disease parameters
Tinevimbo Shiri, Winston Garira and Senelani D. Musekwa
2005, 2(4): 811-832 doi: 10.3934/mbe.2005.2.811 +[Abstract](428) +[PDF](316.9KB)
Abstract:
Treatment of human immunodeficiency virus type 1 (HIV-1) infection during the symptomatic phase has significantly improved patient survival. We present a two-strain HIV mathematical model that captures the dynamics of the immune system and two HIV-1 variants under antiretroviral therapy. We explore the effects of chemotherapy on the dynamics of two viral strains and T lymphocytes with one mutant strain phenotypically resistant to drug effects. Model calculations show that there is a common pattern for CD4+ T cell count increase. There is a drastic increase of CD4+ T cells during the first few weeks of treatment, followed by a gradual increase, and these increases are strictly by clonal expansion of preexisting CD4+ T cells. Plasma HIV RNA dramatically decline to zero levels during the first week of drug administration. If drug efficacy is equal to or above a threshold efficacy, viral load remains at zero levels and if drug efficacy is less than the threshold efficacy, viral load gradually increases until it stabilizes. Viral rebound during treatment is entirely due to the recovery of CD4+ T cells. The results also reveal that there is a dynamic equilibrium between viral load and cytotoxic T lymphocyte (CTL) response in an infected individual during drug administration.
Edge-linked dynamics and the scale-dependence of competitive
Robert Stephen Cantrell, Chris Cosner and William F. Fagan
2005, 2(4): 833-868 doi: 10.3934/mbe.2005.2.833 +[Abstract](295) +[PDF](288.8KB)
Abstract:
Empirical data for several ecological systems suggest that how resource availability scales with patch geometry may influence the outcome of species interactions. To study this process, we assume a pseudoequilibrium to reduce the dimensionality of a two-consumer-two-resource model in which different resources are available in the interior of a patch versus at the edge. We analyze the resulting two species competition model to understand how the outcome of competition between consumers changes as the size of the patch changes, paying particular attention to the differential scaling of interior and edge-linked allochthonous resources as a function of patch size. We characterize conditions on patch size and parameters under which competitive exclusion, coexistence, and a reversal in competitive dominance occur. We find that the degree of exclusivity in the use of edge versus interior habitats influences the potential for transitions in competitive outcomes, but that differences in resource quality between interior and edge habitats can, depending on the scenario, have either qualitative or quantitative influences on the transitions. The work highlights the importance of patch size to understanding species interactions and demonstrates that competitive dominance can be a scale- dependent trait.

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