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Discrete & Continuous Dynamical Systems - B

2009 , Volume 12 , Issue 2

A Special Issue on Mathematical Biology and Medicine

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Preface
Yang Kuang, Jiaxu Li, Bingtuan Li, Urszula Ledzewicz and Ami Radunskaya
2009, 12(2): i-ii doi: 10.3934/dcdsb.2009.12.2i +[Abstract](350) +[PDF](37.6KB)
Abstract:
This special issue of Discrete and Continuous Dynamical Systems, Series B (DCDS-B), is based on the timely special session on dynamical systems in biology and medicine in the 7th AIMS Conference on Dynamical Systems and Differential Equations, which took place at University of Texas at Arlington, Texas, USA, in the period of May 18 - 21, 2008. All papers are carefully refereed and selected based on the mathematical originality and biological relevance of the presented research work.
   The bi-annual AIMS international Conference on Dynamical Systems and Differential Equations has grown steadily in size, quality and scope. Indeed, in a short period of 12 years, it has become the largest, most popular and well organized international meeting of its kind, featuring many impressive keynote speakers, dynamic and engaging special sessions, and most importantly, effective and economical conference management. A total of 867 researchers participated in this well organized meeting at Arlington.

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Predicting the drug release kinetics of matrix tablets
Boris Baeumer, Lipika Chatterjee, Peter Hinow, Thomas Rades, Ami Radunskaya and Ian Tucker
2009, 12(2): 261-277 doi: 10.3934/dcdsb.2009.12.261 +[Abstract](465) +[PDF](1247.9KB)
Abstract:
In this paper we develop two mathematical models to predict the release kinetics of a water soluble drug from a polymer/excipient matrix tablet. The first of our models consists of a random walk on a weighted graph, where the vertices of the graph represent particles of drug, excipient and polymer, respectively. The graph itself is the contact graph of a multidisperse random sphere packing. The second model describes the dissolution and the subsequent diffusion of the active drug out of a porous matrix using a system of partial differential equations. The predictions of both models show good qualitative agreement with experimental release curves. The models will provide tools for designing better controlled release devices.
The impact of vaccination and coinfection on HPV and cervical cancer
Britnee Crawford and Christopher M. Kribs-Zaleta
2009, 12(2): 279-304 doi: 10.3934/dcdsb.2009.12.279 +[Abstract](483) +[PDF](482.8KB)
Abstract:
Understanding the relationship between coinfection with multiple strains of human papillomavirus and cervical cancer may play a key role in vaccination strategies for the virus. In this article we formulate a model with two strains of infection and vaccination for one of the strains (strain 1, oncogenic) in order to investigate how multiple strains of HPV and vaccination may affect the number of cervical cancer cases and deaths due to infections with both types of HPV. We calculate the basic reproductive number $R_i$ for both strains independently as well as the basic reproductive number for the system based on $R_1$ and $R_2$. We also compute the invasion reproductive number Ř i for strain i when strain j is at endemic equilibrium ($i\ne j$). We show that the disease-free equilibrium is locally stable when $R_0=max\{R_1,R_2\}<1$ and each single strain endemic equilibrium $E_i$ exists when $R_i>1$. We determine stability of the single strain equilibria using the invasion reproductive numbers. The $R_1,R_2$ parameter space is partitioned into 4 regions by the curves $R_1=1, R_2=1,$ Ř 1 = 1, and Ř 2 = 1. In each region a different equilibrium is dominant. The presence of strain 2 can increase strain 1 related cancer deaths by more than 100 percent, but strain 2 prevalence can be reduced by more than 90 percent with 50 percent vaccination coverage. Under certain conditions, we show that vaccination against strain 1 can actually eradicate strain 2.
Mathematical modelling of internal HIV dynamics
Nirav Dalal, David Greenhalgh and Xuerong Mao
2009, 12(2): 305-321 doi: 10.3934/dcdsb.2009.12.305 +[Abstract](423) +[PDF](317.3KB)
Abstract:
We study a mathematical model for the viral dynamics of HIV in an infected individual in the presence of HAART. The paper starts with a literature review and then formulates the basic mathematical model. An expression for $R_0$, the basic reproduction number of the virus under steady state application of HAART, is derived followed by an equilibrium and stability analysis. There is always a disease-free equilibrium (DFE) which is globally asymptotically stable for $R_0 < 1$. Deterministic simulations with realistic parameter values give additional insight into the model behaviour.
A preliminary mathematical model of skin dendritic cell trafficking and induction of T cell immunity
Amy H. Lin Erickson, Alison Wise, Stephen Fleming, Margaret Baird, Zabeen Lateef, Annette Molinaro, Miranda Teboh-Ewungkem and Lisette dePillis
2009, 12(2): 323-336 doi: 10.3934/dcdsb.2009.12.323 +[Abstract](389) +[PDF](544.8KB)
Abstract:
Chronic inflammation is a process where dendritic cells (DCs) are constantly sampling antigen in the skin and migrating to lymph nodes where they induce the activation and proliferation of T cells. The T cells then travel back to the skin where they release cytokines that induce/maintain the inflammatory condition. This process is cyclic and ongoing. We created a differential equations model to reflect the initial stages of the inflammatory process. In particular, we modeled antigen stimulation of DCs in the skin, movement of DCs from the skin to a lymph node, and the subsequent activation of T cells in the lymph node. The model was able to simulate DC and T cell responses to antigen introduction taking place within realistic time scales. The goal of such a preliminary model is simply to be able to capture biologically realistic dynamics. Future models can then build on this preliminary model in directions that can potentially allow not only for model validation, but for predictions and hypothesis testing.
Analysis of a model of two parallel food chains
Sze-Bi Hsu, Christopher A. Klausmeier and Chiu-Ju Lin
2009, 12(2): 337-359 doi: 10.3934/dcdsb.2009.12.337 +[Abstract](347) +[PDF](1264.3KB)
Abstract:
In this paper we study a mathematical model of two parallel food chains in a chemostat. Each food chain consists of a prey species $x$ and a predator species $y$. Two food chains are symmetric in the sense that the prey species are identical and so are the specialized predator species. We assume that both of the prey species in the parallel food chains share the same nutrient $R$. In this paper we show that as the input concentration $R^{(0)}$ of the nutrient varies, there are several possible outcomes: (1) all species go extinct; (2) only the two prey species survive; (3) all species coexist at equilibrium; (4) all species coexist in the form of oscillations. We analyze cases (1)-(3) rigorously; for case (4) we do extensive numerical studies to present all possible phenomena, which include limit cycles, heteroclinic cycles, and chaos.
Modelling the dynamic response of oxygen uptake to exercise
Alex James, Simon Green and Mike Plank
2009, 12(2): 361-370 doi: 10.3934/dcdsb.2009.12.361 +[Abstract](452) +[PDF](148.3KB)
Abstract:
The response of oxygen uptake ($\textrVO_2$) to exercise is multiphasic, each phase being exponential and often achieving a plateau before the next phase begins. Although the physiological processes underlying this multiphasic response are unclear, we assume that to some extent they reflect processes within contracting skeletal myocytes. To explore this further, a simple and novel dynamical model of motor unit behaviour during exercise is presented that captures essential features of the exercise $\textrVO_2$ response.
A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions
Hassan Khassehkhan, Messoud A. Efendiev and Hermann J. Eberl
2009, 12(2): 371-388 doi: 10.3934/dcdsb.2009.12.371 +[Abstract](365) +[PDF](1146.0KB)
Abstract:
We study a mathematical model that describes how a "good" bacterial biofilm controls the growth of a harmful pathogenic bacterial biofilm. The underlying mechanism is a modification of the local protonated acid concentration, which in turn decreases the local pH and, thus, makes growth conditions for the pathogens less favorable, while the control-agent itself is more tolerant to these changes. This system is described by a system of 5 density-dependent diffusion-reaction equations that show two nonlinear diffusion effects: porous medium degeneracy and fast diffusion. This is a multi-species expansion of a previously studied single species biofilm model. In this paper we prove the existence of solutions to this model and show in numerical simulations the effectiveness of the control mechanism.
Some remarks on traveling wave solutions in competition models
Bingtuan Li
2009, 12(2): 389-399 doi: 10.3934/dcdsb.2009.12.389 +[Abstract](437) +[PDF](172.1KB)
Abstract:
We study the existence of traveling wave solutions for competition models in the form of integro-difference equations. We show that for a two-species competition model it is possible that two species spread at different speeds, and there exists a traveling wave solution. For an $m$-species competition model, under the assumption that species have the same dispersal and growth properties but have different competition abilities, we establish the existence of traveling wave solutions.
Mathematical models of subcutaneous injection of insulin analogues: A mini-review
Jiaxu Li and James D. Johnson
2009, 12(2): 401-414 doi: 10.3934/dcdsb.2009.12.401 +[Abstract](288) +[PDF](252.8KB)
Abstract:
In the last three decades, several models relevant to the subcutaneous injection of insulin analogues have appeared in the literature. Most of them model the absorption of insulin analogues in the injection depot and then compute the plasma insulin concentration. The most recent systemic models directly simulate the plasma insulin dynamics. These models have been and/or can be applied to the technology of the insulin pump or to the coming closed-loop systems, also known as the artificial pancreas. In this paper, we selectively review these models in detail and at point out that these models provide key building blocks for some important endeavors into physiological questions of insulin secretion and action. For example, it is not clear at this time whether or not picomolar doses of insulin are found near the islets and there is no experimental method to assess this in vivo. This is of interest because picomolar concentrations of insulin have been found to be effective at blocking beta-cell death and increasing beta-cell growth in recent cell culture experiments.
Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models
Urszula Ledzewicz, James Munden and Heinz Schättler
2009, 12(2): 415-438 doi: 10.3934/dcdsb.2009.12.415 +[Abstract](434) +[PDF](413.4KB)
Abstract:
The problem of scheduling a given amount of angiogenic inhibitors is considered as an optimal control problem with the objective of maximizing the achievable tumor reduction. For a dynamical model for the evolution of the carrying capacity of the vasculature formulated in [15] optimal controls are computed for both a Gompertzian and logistic model of tumor growth. While optimal controls for the Gompertzian model typically contain a segment along which the control is singular, for the logistic model optimal controls are bang-bang with at most two switchings.
Response of yeast mutants to extracellular calcium variations
Pamela A. Marshall, Eden E. Tanzosh, Francisco J. Solis and Haiyan Wang
2009, 12(2): 439-453 doi: 10.3934/dcdsb.2009.12.439 +[Abstract](407) +[PDF](317.9KB)
Abstract:
We study, both experimentally and through mathematical modeling, the response of wild type and mutant yeast strains to systematic variations of extracellular calcium abundance. We extend a previously developed mathematical model (Cui and Kaandorp, Cell Calcium, 39, 337 (2006))[3], that explicitly considers the population and activity of proteins with key roles in calcium homeostasis. Modifications of the model can directly address the responses of mutants lacking these proteins. We present experimental results for the response of yeast cells to sharp, step-like variations in external $Ca^{++}$ concentrations. We analyze the properties of the model and use it to simulate the experimental conditions investigated. The model and experiments diverge more markedly in the case of mutants laking the Pmc1 protein. We discuss possible extensions of the model to address these findings.
Robust closed-loop control of plasma glycemia: A discrete-delay model approach
Pasquale Palumbo, Pierdomenico Pepe, Simona Panunzi and Andrea De Gaetano
2009, 12(2): 455-468 doi: 10.3934/dcdsb.2009.12.455 +[Abstract](347) +[PDF](247.0KB)
Abstract:
The paper investigates the problem of tracking a desired plasma glucose evolution by means of intra-venous insulin administration. A model-based approach is followed. A recent model of the glucose/insulin regulatory system which consists of discrete-delay nonlinear differential equations is used. A disturbance is added to the insulin kinetics in order to model uncertainties concerning both the insulin delivery rate and the mechanism actuating the insulin pump. A feedback control law which yields input-to-state stability of the closed loop error system with respect to the disturbance is provided. Such control law depends on the glucose and insulin measurements at the present and at a delayed time. In silico simulations validate the theoretical results.
Finite-time perturbations of dynamical systems and applications to tumor therapy
Jianjun Paul Tian
2009, 12(2): 469-479 doi: 10.3934/dcdsb.2009.12.469 +[Abstract](391) +[PDF](160.0KB)
Abstract:
We study finite-time perturbations of dynamical systems. We prove that finite-time perturbed dynamical systems are asymptotically equivalent to unperturbed dynamical systems. And so the asymptotical behavior of finite-time perturbed systems can be studied by unperturbed systems. As an example, we study a system perturbed by drug treatments.
Daphnia species invasion, competitive exclusion, and chaotic coexistence
Hao Wang, Katherine Dunning, James J. Elser and Yang Kuang
2009, 12(2): 481-493 doi: 10.3934/dcdsb.2009.12.481 +[Abstract](517) +[PDF](1064.7KB)
Abstract:
The cladoceran Daphnia lumholtzi has invaded many US rivers and lakes. To better understand the ecological factors and consequences associated with D. lumholtzi invasion, we carried out a microcosm experiment evaluating competition of D. lumholtzi with a widespread native daphnid, D. pulex. We applied two light treatments to these two different microcosms and found strong context-dependent competitive exclusion in both treatments. We observed that D. lumholtzi out-competed D. pulex in the high light treatment, while D. pulex out-competed D. lumholtzi in the low light treatment. To better understand these results we developed and tested a mechanistically formulated stoichiometric population interaction model. This model exhibits chaotic coexistence of the competing species of Daphnia. The rich dynamics of this model allow us to suggest some plausible strategies to control the invasive species D. lumholtzi.
A metapopulation model with local competitions
Dashun Xu and Z. Feng
2009, 12(2): 495-510 doi: 10.3934/dcdsb.2009.12.495 +[Abstract](428) +[PDF](414.6KB)
Abstract:
A metapopulation model with explicit local dynamics is studied. Unlike many patch-based metapopulation models which assume that the local population within each patch is at its equilibrium, our model incorporates population changes in local patches that interact with metapopulation dynamics. The model keeps track of the fractions of patches that have species 1 only, species 2 only, or both species. For patches with both species, the Lotka-Volterra type of competition is assumed. It is shown that when the local dynamics is coupled with the metapopulation dynamics the model outcomes can be very different comparing with metapopulation models that do not explicitly include local population dynamics. The analysis of the coupled system is carried out by using techniques in singular perturbation theory.
Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay
Huiyan Zhu and Xingfu Zou
2009, 12(2): 511-524 doi: 10.3934/dcdsb.2009.12.511 +[Abstract](822) +[PDF](264.3KB)
Abstract:
In this paper, we consider a mathematical model for HIV-1 infection with intracellular delay and cell-mediated immune response. A novel feature is that both cytotoxic T lymphocytes (CTLs) and the intracellular delay are incorporated into the model. We obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the two infection equilibria: one without CTLs being activated and the other with. We also perform some numerical simulations which support the obtained theoretical results. These results show that larger intracellular delay may help eradicate the virus, while the activation of CTLs can only help reduce the virus load and increase the healthy CD$_4^+$ cells population in the long term sense.

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