We develop a stage-structured model that describes the dynamics of two competing species each of which have sexual and clonal reproduction. This is typical of many plants including irises. We first analyze the dynamical behavior of a single species model. We show that when the inherent net reproductive number is smaller than one then the population will go to extinction and if it is larger than one then an interior equilibrium exists and it is globally asymptotically stable. Then we analyze the two-species model and establish conditions on the reproduction and survivorship rates that lead to competitive exclusion. We show that the winner species is the one that attains higher density at which its net reproductive number equals unity. Numerical results corroborating the theoretical ones are also presented.

Tuberculosis (TB) is the leading cause of death among individuals infected with thehepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidablemathematical challenges due to the fact that the models of transmission are quitedistinct. We formulate and analyze a deterministic mathematical model which incorporatesof the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only andTB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has aglobally-asymptotically stable disease-free equilibrium whenever the associatedreproduction number is less than unity, the TB-only sub-model undergoes the phenomenon ofbackward bifurcation, where a stable disease-free equilibrium co-exists with a stableendemic equilibrium, for a certain range of the associated reproduction number less thanunity. Thus, for TB, the classical requirement of having the associated reproductionnumber to be less than unity, although necessary, is not sufficient for its elimination.It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcationphenomenon. Through simulations, we mainly find that i) the two diseases will co-existwhenever their partial reproductive numbers exceed unity; (ii) the increased progressionrate due to exogenous reinfection from latent to active TB in co-infected individuals mayplay a significant role in the rising prevalence of TB; and (iii) the increasedprogression rates from acute stage to chronic stage of HBV infection have increased theprevalence levels of HBV and TB prevalences.

Detailed descriptions of cardiacgeometry and architecture are necessary for examining andunderstanding structural changes to the myocardium that are theresult of pathologies, for interpreting the results ofexperimental studies of propagation, and for use as athree-dimensional orthotropically anisotropic model for thecomputational reconstruction of propagation during arrhythmias.Diffusion tensor imaging (DTI) provides a means to reconstructfibre and sheet orientation throughout the ventricles. Wereconstruct and quantify canine cardiac architecture in selectedregions of the left and right ventricular free walls and theinter-ventricular septum. Fibre inclination angle rotates smoothlythrough the wall in all regions, from positive in the endocardiumto negative in the epicardium. However, fibre transverse and sheetangles show large variability in basal regions. Additionally,regions where two populations (positive and negative) of sheetstructure merge are identified. From these data, we conclude thata single DTI-derived atlas model of ventricular architectureshould be applicable to modelling propagation in wedges from theequatorial and apical left ventricle, and allow comparisons toexperimental studies carried out in wedge preparations. However,due to inter-individual variability in basal regions, individual(rather than atlas) DTI models of basal wedges or of the wholeventricles will be required.

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.

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systemsdepending rationally on the spectral parameter is treated and its applications tononlinear equations are given. Explicit solutions of direct and inverse problems forDirac-type systems, including systems with singularities, and for the system auxiliary tothe N-wave equation are reviewed. New results on explicit construction ofthe wave functions for radial Dirac equation are obtained.

Directional multiscale representations such as shearlets and curvelets have gainedincreasing recognition in recent years as superior methods for the sparse representationof data. Thanks to their ability to sparsely encode images and other multidimensionaldata, transform-domain denoising algorithms based on these representations are among thebest performing methods currently available. As already observed in the literature, theperformance of many sparsity-based data processing methods can be further improved byusing appropriate combinations of dictionaries. In this paper, we consider the problem of3D data denoising and introduce a denoising algorithm which uses combined sparsedictionaries. Our numerical demonstrations show that the realization of the algorithmwhich combines 3D shearlets and local Fourier bases provides highly competitive results ascompared to other 3D sparsity-based denosing algorithms based on both single and combineddictionaries.

We perform a sensitivity analysis for a thus far unstudied mathematical model for theformation, growth and lysis of clots in vitro. The sensitivity analysis procedure uses anensemble standard deviation for species concentrations, and is equivalent to a variancedecomposition procedure also available in the literature. Our analysis shows that fibrinproduction is most sensitive to the rate constant governing activation of prothrombin tothrombin. Further, the time-averaged sum of all species’ concentrations is most sensitiveto the rate constants governing the inactivation of VIIIa (intrinsic as well as by APC).We therefore conclude that the rate constants for VIIIa inactivation affect the model thegreatest: this conclusion must be experimentally verified to determine if such is indeedthe case for hemostasis.

A stenosis is the narrowing of the artery, this narrowing is usually the result of theformation of an atheromatous plaque infiltrating gradually the artery wall, forming a bumpin the ductus arteriosus. This arterial lesion falls within the general context ofatherosclerotic arterial disease that can affect the carotid arteries, but also thearteries of the heart (coronary), arteries of the legs (PAD), the renal arteries... It cancause a stroke (hemiplegia, transient paralysis of a limb, speech disorder, sailing beforethe eye). In this paper we study the blood-plaque and blood-wall interactions using afluid-structure interaction model. We first propose a 2D analytical study of thegeneralized Navier-Stokes equations to prove the existence of a weak solution forincompressible non-Newtonian fluids with non standard boundary conditions. Then, coupled,based on the results of the theoretical study approach is given. And to form a realisticmodel, with high accuracy, additional conditions due to fluid-structure coupling areproposed on the border undergoing inetraction. This coupled model includes (a) a fluidmodel, where blood is modeled as an incompressible non-Newtonian viscous fluid, (b) asolid model, where the arterial wall and atherosclerotic plaque will be treated as nonlinear hyperelastic solids, and (c) a fluid-structure interaction (FSI) model whereinteractions between the fluid (blood) and structures (the arterial wall and atheromatousplaque) are conducted by an Arbitrary Lagrangian Eulerian (ALE) method that allowsaccurate fluid-structure coupling.

Early studies of the novel swine-origin 2009 influenza A (H1N1) epidemic indicateclinical attack rates in children much higher than in adults. Non-medical interventionssuch as school closings are constrained by their large socio-economic costs. Here wedevelop a mathematical model to ascertain the roles of pre-symptomatic influenzatransmission as well as symptoms surveillance of children to assess the utility of schoolclosures. Our model analysis indicates that school closings are advisable whenpre-symptomatic transmission is significant or when removal of symptomatic children isinefficient. Our objective is to provide a rational basis for school closings decisionsdependent on virulence characteristics and local surveillance implementation, applicableto the current epidemic and future epidemics.

Stimuli-responsive polymers result inon-demand regulation of properties and functioning of variousnanoscale systems. In particular, they allow stimuli-responsivecontrol of flow rates through membranes and nanofluidic deviceswith submicron channel sizes. They also allow regulation of drugrelease from nanoparticles and nanofibers in response totemperature or pH variation in the surrounding medium. In thepresent work two relevant mathematical models are introduced toaddress precipitation-driven deposition of surface layers ofstimuli-responsive polymers and describe a possible effect ofswelling-shrinkage transition of such polymers on drug release.

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.

The interplay between dissipation and long-range repulsive/attractive forces inhomogeneous, dilute, mono-disperse particle systems is studied. Thepseudo-Liouville operator formalism, originallyintroduced for hard-sphere interactions, is modified such that it provides very goodpredictions for systems with weak long-range forces at low densities, with the ratio ofpotential to fluctuation kinetic energy as control parameter. By numerical simulations,the theoretical results are generalized with empirical, density dependentcorrection-functions up to moderate densities.

The main result of this study on dissipative cooling is an analytical prediction for the reduced cooling rate due torepulsive forces and for the increased rate due to attractive forces. In the latter case,surprisingly, for intermediate densities, similar cooling behavior is observed as insystems without long-range interactions. In the attractive case, in general, dissipationleads to inhomogeneities earlier and faster than in the repulsive case.

Influenza has been responsible for human suffering and economic burden worldwide. Isolation is one of the most effective means to control the disease spread. In this work, we incorporate isolation into a two-strain model of influenza. We find that whether strains of influenza die out or coexist, or only one of them persists, it depends on the basic reproductive number of each influenza strain, cross-immunity between strains, and isolation rate. We propose criteria that may be useful for controlling influenza. Furthermore, we investigate how effective isolation is by considering the host’s mean age at infection and the invasion rate of a novel strain. Our results suggest that isolation may help to extend the host’s mean age at infection and reduce the invasion rate of a new strain. When there is a delay in isolation, we show that it may lead to more serious outbreaks as compared to no delay.

An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103–112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopf-bifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.

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.

An optimal control problem is studiedfor a Lotka-Volterra system of three differential equations. Itmodels an ecosystem of three species which coexist. The speciesare supposed to be separated from each others. Mathematically,this is modeled with the aid of two control variables. Somenecessary conditions of optimality are found in order to maximizethe total number of individuals at the end of a given timeinterval.

A two dimensional two-delays differential system modeling the dynamics of stem-like cellsand white-blood cells in Chronic Myelogenous Leukemia is considered. All three types ofstem cell division (asymmetric division, symmetric renewal and symmetric differentiation)are present in the model. Stability of equilibria is investigated and emergence ofperiodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventuallyshown. The stability of these limit cycles is studied using the first Lyapunovcoefficient.

In this paper we build and analyze networks using the statistical and programmingenvironment R and the igraph package. We investigate random, small-world, and scale-freenetworks and test a standard problem of connectivity on a random graph. We then develop amethod to study how vaccination can alter the structure of a disease transmission network.We also discuss a variety of other uses for networks in biology.

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.