We prove the existence of a global bifurcation branch of 2π-periodic,smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset ofsolutions in the global branch contains a sequence which converges uniformly to somesolution of Hölder class Cα, α < 1/2. Bifurcation formulas are given, as well as some properties along theglobal bifurcation branch. In addition, a spectral scheme for computing approximations tothose waves is put forward, and several numerical results along the global bifurcationbranch are presented, including the presence of a turning point and a ‘highest’, cuspedwave. Both analytic and numerical results are compared to traveling-wave solutions of theKdV equation.

We formulate an immuno-epidemiological model of coupled “within-host” model of ODEs and“between-host” model of ODE and PDE, using the Human Immunodeficiency Virus (HIV) forillustration. Existence and uniqueness of solution to the “between-host” model isestablished, and an explicit expression for the basic reproduction number of the“between-host” model derived. Stability of disease-free and endemic equilibria isinvestigated. An optimal control problem with drug-treatment control on the within-hostsystem is formulated and analyzed; these results are novel for optimal control of ODEslinked with such first order PDEs. Numerical simulations based on the forward-backwardsweep method are obtained.

The oriented movement of biological cells or organisms in response to a chemical gradientis called chemotaxis. The most interesting situation related to self-organizationphenomenon takes place when the cells detect and response to a chemical which is secretedby themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) manyparticularized models have been proposed to describe the aggregation phase of thisprocess. Most of efforts were concentrated, so far, on mathematical models in which theformation of aggregate is interpreted as finite time blow-up of cell density. In recentlyproposed models cells are no more treated as point masses and their finite volume isaccounted for. Thus, arbitrary high cell densities are precluded in such description and athreshold value for cells density is a priori assumed. Different modelingapproaches based on this assumption lead to a class of quasilinear parabolic systems withstrong nonlinearities including degenerate or singular diffusion. We give a survey ofanalytical results on the existence and uniqueness of global-in-time solutions, theirconvergence to stationary states and on a possibility of reaching the density threshold bya solution. Unsolved problems are pointed as well.

The development of functional-structural plant models has opened interesting perspectivesfor a better understanding of plant growth as well as for potential applications inbreeding or decision aid in farm management. Parameterization of such models is however adifficult issue due to the complexity of the involved biological processes and theinteractions between these processes. The estimation of parameters from experimental databy inverse methods is thus a crucial step. This paper presents some results anddiscussions as first steps towards the construction of a general framework for theparametric estimation of functional-structural plant models. A general family of models ofCarbon allocation formalized as dynamic systems serves as the basis for our study. Anadaptation of the 2-stage Aitken estimator to this family of model is introduced as wellas its numerical implementation, and applied in two different situations: first amorphogenetic model of sugar beet growth with simple plant structure, multi-stage anddetailed observations, and second a tree growth model characterized by sparse observationsand strong interactions between functioning and organogenesis. The proposed estimationmethod appears robust, easy to adapt to a wide variety of models, and generally provides asatisfactory goodness-of-fit. However, it does not allow a proper evaluation of estimationuncertainty. Finally some perspectives opened by the theory of hidden models arediscussed.

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.

Properties of blood cells and their interaction determine their distribution in flow. Itis observed experimentally that erythrocytes migrate to the flow axis, platelets to thevessel wall, and leucocytes roll along the vessel wall. In this work, a three-dimensionalmodel based on Dissipative Particle Dynamics method and a new hybrid (discrete-continuous)model for blood cells is used to study the interaction of erythrocytes with platelets andleucocytes in flow. Erythrocytes are modelled as elastic highly deformable membranes,while platelets and leucocytes as elastic membranes with their shape close to a sphere.Separation of erythrocytes and platelets in flow is shown for different values ofhematocrit. Erythrocyte and platelet distributions are in a good qualitative agreementwith the existing experimental results. Migration of leucocyte to the vessel wall and itsrolling along the wall is observed.

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.

One of the most intriguing questions in life science is how living organisms develop and maintain their predominant form and shape via the cascade of the processes of differentiation starting from the single cell. Mathematical modeling of these developmental processes could be a very important tool to properly describe the complex processes of evolution and geometry of morphogenesis in time and space. Here, we summarize the most important biological knowledge on plant development, exploring the different layers of investigation in developmental processes such as plant morphology, genetics, plant physiology, molecular biology and epigenetics. As knowledge on the fundamentals of plant embryogenesis, growth and development is constantly improving, we gather here the latest data on genetic, molecular and hormonal regulation of plant development together with the basic background knowledge. Special emphasis is placed on the regulation of cell cycle progression, on the role of the signal molecules phytohormones in plant development and on the details of plant meristems (loci containing plant stem cells) function. We also explore several proposed biological models regarding regulating plant development. The information presented here could be used as a basis for mathematical modeling and computer simulation of developmental processes in plants.

We study the spectral stability of solitary wave solutions to the nonlinear Diracequation in one dimension. We focus on the Dirac equation with cubic nonlinearity, knownas the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model.Presented numerical computations of the spectrum of linearization at a solitary wave showthat the solitary waves are spectrally stable. We corroborate our results by findingexplicit expressions for several of the eigenfunctions. Some of the analytic results holdfor the nonlinear Dirac equation with generic nonlinearity.

Cell motility is an integral part of a diverse set of biological processes. The quest formathematical models of cell motility has prompted the development of automated approachesfor gathering quantitative data on cell morphology, and the distribution of molecularplayers involved in cell motility. Here we review recent approaches for quantifying cellmotility, including automated cell segmentation and tracking. Secondly, we present our ownnovel method for tracking cell boundaries of moving cells, the Electrostatic ContourMigration Method (ECMM), as an alternative to the generally accepted level set method(LSM). ECMM smoothly tracks regions of the cell boundary over time to compute localmembrane displacements using the simple underlying concept of electrostatics. It offerssubstantial speed increases and reduced computational overheads in comparison to the LSM.We conclude with general considerations regarding boundary tracking in the context ofmathematical modelling.

This review aims at presenting asynoptic, if not exhaustive, point of view on some of the problemsencountered by biologists and physicians who deal with naturalcell proliferation and disruptions of its physiological control incancer disease. It also aims at suggesting how mathematicians arenaturally challenged by these questions and how they might help,not only biologists to deal theoretically with biologicalcomplexity, but also physicians to optimise therapeutics, on whichlast point the focus will be set here. To this purpose,mathematical modelling should represent proliferating cellpopulation dynamics with natural built-in control targets (whichimplies modelling the cell division cycle), together with thedistribution of drugs in the organism and their molecular actionson different targets at the cell level on proliferation, i.e.,molecular pharmacokinetics-pharmacodynamics of antiproliferativedrugs. This should make possible optimal control of drug deliverywith constraints to be determined according to the mainpharmacological issues encountered in the clinic: unwanted toxicside-effects, occurrence of drug resistance. Mathematicalmodelling should also take into account physiological determinantsof cell and tissue proliferation, such as intervention of theimmune system, circadian control on cell cycle checkpointproteins, and activity of intracellular drug processing enzymestogether with individual variations in the activities of theseproteins (genetic polymorphism). Taking these points into accountwill add to the rich scenery of normal or disrupted cell andtissue regulations, and their corrections by drugs, a naturalenvironmental, whole body physiological, frame. It is necessaryindeed to consider such a framework if one wants to eventually beactually helpful to clinicians who routinely treat by combinationsof drugs living Humans with their complex whole body regulations,often dependent on genotypic variations, and not isolated cells ortissues.

Peristaltic pumping of fluid is a fundamental method of transport in many biologicalprocesses. In some instances, particles of appreciable size are transported along with thefluid, such as ovum transport in the oviduct or kidney stones in the ureter. In some ofthese biological settings, the fluid may be viscoelastic. In such a case, a nonlinearconstitutive equation to describe the evolution of the viscoelastic contribution to thestress tensor must be included in the governing equations. Here we use an immersedboundary framework to study peristaltic transport of a macroscopic solid particle in aviscoelastic fluid governed by a Navier-Stokes/Oldroyd-B model. Numerical simulations ofperistaltic pumping as a function of Weissenberg number are presented. We examine thespatial and temporal evolution of the polymer stress field, and also find that theviscoelasticity of the fluid does hamper the overall transport of the particle in thedirection of the wave.

Darwin illustrated his theory about emergence and evolution of biological species with adiagram. It shows how species exist, evolve, appear and disappear. The goal of this workis to give a mathematical interpretation of this diagram and to show how it can bereproduced in mathematical models. It appears that conventional models in populationdynamics are not sufficient, and we introduce a number of new models which take intoaccount local, nonlocal and global consumption of resources, and models with space andtime dependent coefficients.

This paper presents mathematical models for tuberculosis and its dynamics under theimplementation of the direct observation therapy strategy (DOTS) in Nigeria. The models establishconditions for the eradication of tuberculosis in Nigeria based on the fraction of detectedinfectious individuals placed under DOTS for treatment. Both numerical and qualitative analysisof the models were carried out and the effect of the fraction of detected cases of active TB on thevarious epidemiological classes is investigated. The results showed that, provided that the fractionof detected infectious individuals exceeded a critical value, there exists a globally stable diseasefree equilibrium. However, if this critical detection level is not reached, the disease-free equilibriumwill be unstable even with the very high probability successful treatment under DOTS. Theresults showed that DOTS expansion in Nigeria must include significant increase in the detectionrate of infectious individuals; otherwise the effect in reducing the incidence in Nigeria will notbe achieved disregarding the tremendous efforts in any other direction, and the huge number ofundetected cases will make DOTS insignificant with respect to tuberculosis control.

Although age-related heterogeneity of infection has been addressed in variousepidemic models assuming a demographically stationary population, only a few studies haveexplicitly dealt with age-specific patterns of transmission in growing or decreasing population.To discuss the threshold principle realistically, the present study investigates an age-duration-structured SIR epidemic model assuming a stable host population, as the first scheme toaccount for the non-stationality of the host population. The basic reproduction number R0is derived using the next generation operator, permitting discussions over the well-knowninvasion principles. The condition of endemic steady state is also characterized by usingthe effective next generation operator. Subsequently, estimators of R0 are offered which canexplicitly account for non-zero population growth rate. Critical coverages of vaccination arealso shown, highlighting the threshold condition for a population with varying size. Whenquantifying R0 using the force of infection estimated from serological data, it should beremembered that the estimate increases as the population growth rate decreases. On thecontrary, given the same R0, critical coverage of vaccination in a growing population would behigher than that of decreasing population. Our exercise implies that high mass vaccinationcoverage at an early age would be needed to control childhood vaccine-preventable diseasesin developing countries.

Intra-specific competition in population dynamics can be described by integro-differentialequations where the integral term corresponds to nonlocal consumption of resources by individualsof the same population. Already the single integro-differential equation can show theemergence of nonhomogeneous in space stationary structures and can be used to model the processof speciation, in particular, the emergence of biological species during evolution [S. Genieys et al., Math. Model. Nat. Phenom. 1 (2006), no. 1, 65-82], [S. Genieys et al., Comptes Rendus Biologies, 329 (11), 876-879 (2006)]. Onthe other hand, competition of two different species represents a well known and well studiedmodel in population dynamics. In this work we study how the intra-specific competition can influencethe competition between species. We will prove the existence of travelling waves for the casewhere the support of the kernel of the integral is sufficiently narrow. Numerical simulations willbe carried out in the case of large supports.

Parameterization of zooplankton functional response is crucial for constructing plankton models. Theoreticalstudies predict enhancing of system stability in case the response is of sigmoid type. Experiments on feeding in laboratories tell us infavor of non-sigmoid types for most herbivorous zooplankton species. However, recent field observations show that the overall functionalresponse of zooplankton in the whole euphotic zone can exhibit a sigmoid behavior even when the response for the same species in laboratorymesocosms is non-sigmoid. Here we propose a simple model explaining the observed alterations of functional response. We divide the euphoticzone into a number of layers and take into account the food-dependent migration of zooplankton. In each layer, the functional response (localresponse) is suggested to be non-sigmoid. We show that the overall response of zooplankton exhibits different behavior compared to thepatterns of the local response. In particular, the model predicts emergence of a sigmoid type as a result of zooplankton accumulation andfeeding in layers with high phytoplankton density. We show the importance of light attenuation by phytoplankton on the alteration offunctional response. The modelling results allow us to hypothesize that the sigmoid functional response in real communities should emergemore often than it was suggested earlier based only on experimental studies on zooplankton feeding.

In vitro transmesothelial migration assays of ovarian cancer cells, isolated oraggregated in multicellular spheroids, are reproduced deducing suitable Cellular PottsModels (CPM). We show that the simulations are in good agreement with the experimentalevidence and that the overall process is regulated by the activity of matrixmetalloproteinases (MMPs) and by the interplay of the adhesive properties of the cellswith the extracellular matrix and between cells, both of the same type and of differenttypes. In particular, the process depends on the ability of the cell to induce theloosening of cadherin-mediated junctions. Coherently with experiments, it is found thatsingle cell invasion is more conservative with a crucial role played by MMPs. A similarimportant role is played in cell spheroid invasion, which in comparison is moredisruptive. It achieves monofocal or multifocal characteristics according to the relativeadhesion affinity among cells or between them and the mesothelial layer.

We display a gallery of Lorenz-like attractors that emerge in a class ofthree-dimensional maps. We review the theory of Lorenz-like attractors for diffeomorphisms(as opposed to flows), define various types of such attractors, and find sufficientconditions for three-dimensional Henon-like maps to possess pseudohyperbolic Lorenz-likeattractors. The numerically obtained scenarios of the creation and destruction of theseattractors are also presented.

Tuberculosis is the leading cause of death due to infectious diseases in the world today,and it is increasing due to co-infection with HIV-1, the causative agent of AIDS. Here, weexamine the impact that HIV-1 infection has on persons with latent tuberculosis. Based on previouswork, we develop a mathematical model of an adaptive immune response in the lung whichconsiders relevant immune effectors such as macrophages, various sub-populations of T-cells, andkey cytokines to predict which mechanisms are important to HIV-1 infection induced reactivationof tuberculosis. Our results indicate that persons latently infected with TB who are subsequentlyco-infected with HIV-1 will suffer reactive TB. The mechanisms that contribute to this are essentiallyrelated to a completely different cytokine environment at the onset of HIV-1 infection dueto the presence of Mycobacterium tuberculosis. Our analysis suggests that macrophages play animportant role during co-infection and decreases in macrophage counts are coupled to a decline inCD4+ T-cells and increased viral loads. These mechanisms are also coupled to lower recruitmentof T-cells and macrophages, compromising protective immunity in the lung and eventually leadingto TB reactivation. These results point to potential targets for drug and vaccine therapies.