Hematologic disorders such as the myelodysplastic syndromes (MDS) are discussed. Thelingering controversies related to various diseases are highlighted. A simplebiomathematical model of bone marrow - peripheral blood dynamics in the normal state isproposed and used to investigate cell behavior in normal hematopoiesis from a mathematicalviewpoint. Analysis of the steady state and properties of the model are used to makepostulations about the phenomenon of massive apoptosis in MDS. Simulations of the modelshow situations in which homeostatic equilibrium can be achieved and maintained.Consequently, it is postulated that hematopoietic growth factors may possess thecapabilities of preventing oscillatory dynamics and enhancing faster evolution towardshomeostatic equilibrium.

Epidermal wound healing is a complex process that repairs injured tissue. The complexityof this process increases when bacteria are present in a wound; the bacteria interactiondetermines whether infection sets in. Because of underlying physiological problemsinfected wounds do not follow the normal healing pattern. In this paper we present amathematical model of the healing of both infected and uninfected wounds. At the core ofour model is an account of the initiation of angiogenesis by macrophage-derived growthfactors. We express the model as a system of reaction-diffusion equations, and we presentresults of computations for a version of the model with one spatial dimension.

In this paper, we review some of our recent results in the study of the dynamics ofinteracting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on thewell-posedness of the associated Cauchy problem for the infinite particle system describedby the GP hierarchy.

In this paper, we propose a computational model to investigate the coupling between cell’s adhesions and actin fibres and how this coupling affects cell shape and stability. To accomplish that, we take into account the successive stages of adhesion maturation from adhesion precursors to focal complexes and ultimately to focal adhesions, as well as the actin fibres evolution from growing filaments, to bundles and finally contractile stress fibres.

We use substrates with discrete patterns of adhesive patches, whose inter-patches distance can be modulated in order to control the location of the adhesions and the resulting fibres architecture. We then investigate the emergence of stable cell morphologies as a function of the inter-patches distance, for two different cell phenotypes generated from the model. Force generated by the stress fibres on the focal adhesions and specifically the influence of the cell contractility are also investigated.

Our results suggest that adhesion lifetime and fibre growing rate are the key parameters in the emergence of stable cell morphologies and the limiting factors for the magnitude of the mean tension force from the fibres on the focal adhesions.

In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDSepidemic model with an imperfect HIV vaccine, varying total population size anddistributed time delay to become infectious due to intracellular delay between initialinfection of a cell by HIV and the release of new virions. Here, we have established somesufficient conditions on the permanence and extinction of the disease by using inequalityanalytical technique. We have obtained the explicit formula of the eventual lower boundsof infected persons. We have introduced some new threshold valuesR 0 and R ∗ and further obtainedthat the disease will be permanent when R 0 > 1 and thedisease will be going to extinct when R ∗ < 1. ByLyapunov functional method, we have also obtained some sufficient conditions for globalasymptotic stability of this model. The aim of the analysis of this model is to trace theparameters of interest for further study, with a view to informing and assistingpolicy-maker in targeting prevention and treatment resources for maximumeffectiveness.

We study the coexistence of multiple periodic solutions for an analogue of theintegrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayedfeedback, which incorporates the firing process and absolute refractory period. Uponreceiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits aspike with a pattern-related delay, in addition to the synaptic delay. We present atheoretical framework to view the inhibitory signal from the inhibitory neuron as aself-feedback of the excitatory neuron with this additional delay. Our analysis shows thatthe inhibitory feedbacks with firing and the absolute refractory period can generate fourbasic types of oscillations, and the complicated interaction among these basicoscillations leads to a large class of periodic patterns and the occurrence ofmultistability in the recurrent inhibitory loop. We also introduce the average time ofconvergence to a periodic pattern to determine which periodic patterns have the potentialto be used for neural information transmission and cognition processing in the nervoussystem.

The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation oftruncated vectorial series, for path following problems [2]. In this paper, we present anddiscuss some techniques to define local parameterization [4, 6, 7] in the ANM. We givesome numerical comparisons of pseudo arc-length parameterization and localparameterization on non-linear elastic shells problems

A biophysical model describing long-range cell-to-cell communication by a diffusiblesignal mediated by autocrine loops in developing epithelia in the presence of amorphogenetic pre-pattern is introduced. Under a number of approximations, the modelreduces to a particular kind of bistable reaction-diffusion equation with strongheterogeneity. In the case of the heterogeneity in the form of a long strip a detailedanalysis of signal propagation is possible, using a variational approach. It is shown thatunder a number of assumptions which can be easily verified for particular sets of modelparameters, the equation admits a unique (up to translations) variational traveling wavesolution. A global bifurcation structure of these solutions is investigated in a number ofparticular cases. It is demonstrated that the considered setting may provide a robustdevelopmental regulatory mechanism for delivering chemical signals across large distancesin developing epithelia.

Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use asmodels of biological phenomena. This paper begins with a survey of applications toecology, cell biology and bacterial colony patterns. The author then reviews mathematicalresults on the existence of travelling wave front solutions of these equations, and theirgeneration from given initial data. A detailed study is then presented of the form ofsmooth-front waves with speeds close to that of the (unique) sharp-front solution, for theparticular equation ut =(uux)x + u(1− u). Using singular perturbation theory, the author derives anasymptotic approximation to the wave, which gives valuable information about the structureof smooth-front solutions. The approximation compares well with numerical results.

We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valuedDirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated withone-dimensional Schrödinger operators on a compact interval [0, R] withseparated boundary conditions at 0 and R. Most of our results areformulated in the non-self-adjoint context.

Our principal results include explicit representations of these boundary data maps interms of the resolvent of the underlying Schrödinger operator and the associated boundarytrace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding todifferent (separated) boundary conditions, and a derivation of the Herglotz property ofboundary data maps (up to right multiplication by an appropriate diagonal matrix) in thespecial self-adjoint case.

Solid tumors and hematological cancers contain small population of tumor cells that arebelieved to play a critical role in the development and progression of the disease. Thesecells, named Cancer Stem Cells (CSCs), have been found in leukemia, myeloma, breast,prostate, pancreas, colon, brain and lung cancers. It is also thought that CSCs drive themetastatic spread of cancer. The CSC compartment features a specific and phenotypicallydefined cell population characterized with self-renewal (through mutations), quiescence orslow cycling, overexpression of anti-apoptotic proteins, multidrug resistance and impaireddifferentiation. CSCs show resistance to a number of conventional therapies, and it isbelieved that this explains why it is difficult to completely eradicate the disease andwhy recurrence is an ever-present threat. A hierarchical phenomenological model isproposed based on eight compartments following the stem cell lineage at the normal andcancer cell levels. As an empirical test, the tumor grading and progression, typicallycollected in the pathologic lab, is used to correlate the outcome of this model with thetumor development stages. In addition, the model is able to quantitatively account for thetemporal development of the population of observed cell types. Two types of therapeutictreatment models are considered, with dose-density chemotherapy (a pulsatile scenario) aswell as continuous, metronomic delivery. The drug hit is considered at the stem cellprogenitor and early differentiated specialized cell levels for both normal and cancercells, while the quiescent stem cell and fully differentiated compartments are consideredfavorable outcome for cancer treatment. Circulating progenitors are neglected in thisanalysis. The model provides a number of experimentally testable predictions. The relativeimportance of the cell kill and survival is demonstrated through a deterministicparametric study. The significance of the stem cell compartment is underlined based onthis simulation study. This predictive mathematical model for cancer stem cell hypothesisis used to understand tumor responses to chemotherapeutic agents and judge theefficacy.

We present a simple mechanism of cell motility in a confined geometry, inspired by recentmotility assays in microfabricated channels. This mechanism relies mainly on the couplingof actin polymerisation at the cell membrane to geometric confinement. We first showanalytically using a minimal model of polymerising viscoelastic gel confined in a narrowchannel that spontaneous motion occurs due to polymerisation alone. Interestingly, thismechanism does not require specific adhesion with the channel walls, and yields velocitiespotentially larger than the polymerisation velocity of the gel. We then study the effectof the contractile activity of myosin motors, and show that whilst it is not necessary toinduce motion, it quantitatively increases the velocity of motion in the polymerisationmechanism we describe. Our model qualitatively accounts for recent experiments which showthat cells without specific adhesion proteins are motile only in confined environmentswhile they are unable to move on a flat surface. It also constitutes a first step in thestudy of cell migration in more complex confined geometries such as living tissues.

The CDC launched the National Plan to Eliminate Syphilis from the USA in October 1999[4]. In order to reach this goal, a goodunderstanding of the transmission dynamics of the disease is necessary. Based on a SIRSmodel Breban et al. [3] providedsome evidence that supports the feasibility of the plan proving that no recurringoutbreaks should occur for syphilis. We study in this work a syphilis model that includespartial immunity and vaccination. This model suggests that a backward bifurcation verylikely occurs for the real-life estimated epidemiological parameters for syphilis. Thismay explain the resurgence of syphilis after mass treatment [21]. Occurrence of backward bifurcation brings a new challenge for theplan of the CDC’s –striking a balance between treatment of early infection, vaccinationdevelopment and health education. Our models suggest that the development of an effectivevaccine, as well as health education that leads to enhanced biological and behavioralprotection against infection in high-risk populations, are among the best ways to achievethe goal of elimination of syphilis in the USA.

We present in this work a numerical study of a problem governed by Navier-Stokesequations and heat equation. The mathematical problem under consideration is obtained bymodelling the natural convection of an incompressible fluid, in laminar flow between twohorizontal concentric coaxial cylinders, the temperature of the inner cylinder is supposedto be greater than the outer one. The numerical simulation of the flow is carried out bycollocation-Legendre method. The influence of Prandtl and Rayleigh numbers isinvestigated

The interplay between intrinsic and network dynamics has been the focus of manyinvestigations. Here we use a combination of theoretical and numerical approaches to studythe effects of delayed global feedback on the information transmission properties ofneural networks. Specifically, we compare networks of neurons that display intrinsicinterspike interval correlations (nonrenewal) to networks that do not (renewal). We findthat excitatory and inhibitory delays can tune information transmission by single neuronsbut not by the entire network. Most surprisingly, addition of a delay can change thedependence of the information on the coupling strength for renewal neurons and not fornonrenewal neurons. Our results show that intrinsic ISI correlations can have nontrivialinteractions with network-induced phenomena.

Collective cell motility and its guidance via cell-cell contacts is instrumental in several morphogenetic and pathological processes such as vasculogenesis or tumor growth. Multicellular sprout elongation, one of the simplest cases of collective motility, depends on a continuous supply of cells streaming along the sprout towards its tip. The phenomenon is often explained as leader cells pulling the rest of the sprout forward via cell-cell adhesion. Building on an empirically demonstrated analogy between surface tension and cell-cell adhesion, we demonstrate that such a mechanism is unable to recruit cells to the sprout. Moreover, the expansion of such hypothetical sprouts is limited by a form of the Plateau-Taylor instability. In contrast, actively moving cells – guided by cell-cell contacts – can readily populate and expand linear sprouts. We argue that preferential attraction to the surfaces of elongated cells can provide a generic mechanism, shared by several cell types, for multicellular sprout formation.

This paper is devoted to the study of global existence of periodic solutions of a delayedtumor-immune competition model. Also some numerical simulations are given to illustratethe theoretical results

In this work we study a nonlocal reaction-diffusion equation arising in populationdynamics. The integral term in the nonlinearity describes nonlocal stimulation ofreproduction. We prove existence of travelling wave solutions by the Leray-Schauder methodusing topological degree for Fredholm and proper operators and special a priori estimatesof solutions in weighted Hölder spaces.

Computational models for human decision making are typically based on the properties of bistable dynamical systems where each attractor represents a different decision. A limitation of these models is that they do not readily account for the fragilities of human decision making, such as “choking under pressure”, indecisiveness and the role of past experiences on current decision making. Here we examine the dynamics of a model of two interacting neural populations with mutual time–delayed inhibition. When the input to each population is sufficiently high, there is bistability and the dynamics is determined by the relationship of the initial function to the separatrix (the stable manifold of a saddle point) that separates the basins of attraction of two co–existing attractors. The consequences for decision making include long periods of indecisiveness in which trajectories are confined in the neighborhood of the separatrix and wrong decision making, particularly when the effects of past history and irrelevant information (“noise”) are included. Since the effects of delay, past history and noise on bistable dynamical systems are generic, we anticipate that similar phenomena will arise in the setting of other physical, chemical and neural time–delayed systems which exhibit bistability.

In this paper, we look at a model depicting the relationship of cancer cells in differentdevelopment stages with immune cells and a cell cycle specific chemotherapy drug. Themodel includes a constant delay in the mitotic phase. By applying optimal control theory,we seek to minimize the cost associated with the chemotherapy drug and to minimize thenumber of tumor cells. Global existence of a solution has been shown for this model andexistence of an optimal control has also been proven. Optimality conditions andcharacterization of the control are discussed.