A mathematical model of the tumour growth along a blood vessel is proposed. Themodel employs the mixture theory approach to describe a tissue which consists of cells, extracellularmatrix and liquid. The growing tumour tissue is supposed to be surrounded by the hosttissue. Tumours where complete oxydation of glucose prevails are considered. Special attention ispaid to consistent description of oxygen consumption and growth processes based on the energybalance. A finite difference numerical method is proposed. The level set method is used to trackan interface between the tissues. The simulations show localization of the tumour within a limiteddistance from the vessels and constant expansion velocity along the vessels.

Recent discovery of cancer stem cells in tumorigenic tissues has raised many questionsabout their nature, origin, function and their behavior in cell culture. Most of currentexperiments reporting a dynamics of cancer stem cell populations in culture show theeventual stability of the percentages of these cell populations in the whole population ofcancer cells, independently of the starting conditions. In this paper we propose amathematical model of cancer stem cell population behavior, based on specific features ofcancer stem cell divisions and including, as a mathematical formalization of cell-cellcommunications, an underlying field concept. We compare the qualitative behavior ofmathematical models of stem cells evolution, without and with an underlying signal. Inabsence of an underlying field, we propose a mathematical model described by a system ofordinary differential equations, while in presence of an underlying field it is describedby a system of delay differential equations, by admitting a delayed signal originated byexisting cells. Under realistic assumptions on the parameters, in both cases (ODE withoutunderlying field, and DDE with underlying field) we show in particular the stability ofpercentages, provided that the delay is sufficiently small. Further, for the DDE case (inpresence of an underlying field) we show the possible existence of, either damped orstanding, oscillations in the cell populations, in agreement with some existingmathematical literature. The outcomes of the analysis may offer to experimentalists a toolfor addressing the issue regarding the possible non-stem to stem cells transition, bydetermining conditions under which the stability of cancer stem cells population can beobtained only in the case in which such transition can occur. Further, the provideddescription of the variable corresponding to an underlying field may stimulate furtherexperiments for elucidating the nature of “instructive" signals for cell divisions,underlying a proper pattern of the biological system.

We present a user-friendly version of a double operator integration theory which stillretains a capacity for many useful applications. Using recent results from the lattertheory applied in noncommutative geometry, we derive applications to analogues of theclassical Heinz inequality, a simplified proof of a famous inequality ofBirman-Koplienko-Solomyak and also to the Connes-Moscovici inequality. Our methods aresufficiently strong to treat these inequalities in the setting of symmetric operator normsin general semifinite von Neumann algebras.

An overview of recent results pertaining to the hydrodynamic description (both Newtonianand non-Newtonian) of granular gases described by the Boltzmann equation for inelasticMaxwell models is presented. The use of this mathematical model allows us to get exactresults for different problems. First, the Navier–Stokes constitutive equations withexplicit expressions for the corresponding transport coefficients are derived by applyingthe Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheologicalproperties in the uniform shear flow (USF) are obtained in the steady state as well as inthe transient unsteady regime. Next, an exact solution for a special class of Couetteflows characterized by a uniform heat flux is worked out. This solution shares the samerheological properties as the USF and, additionally, two generalized transportcoefficients associated with the heat flux vector can be identified. Finally, the problemof small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like methodand generalized (tensorial) transport coefficients are obtained.

A numerical model to compute the dynamics of glaciers is presented. Ice damage due tocracks or crevasses can be taken into account whenever needed. This model allowssimulations of the past and future retreat of glaciers, the calving process or thebreak-off of hanging glaciers. All these phenomena are strongly affected by climatechange.

Ice is assumed to behave as an incompressible fluid with nonlinear viscosity, so that thevelocity and pressure in the ice domain satisfy a nonlinear Stokes problem. The shape ofthe ice domain is defined using the volume fraction of ice, that is one in the ice regionand zero elsewhere. The volume fraction of ice satisfies a transport equation with asource term on the upper ice-air free surface accounting for ice accumulation or melting.If local effects due to ice damage must be taken into account, the damage functionD is introduced, ranging between zero if no damage occurs and one.Then, the ice viscosity μ in the momentum equation must be replaced by(1 − D)μ. The damage function Dsatisfies a transport equation with nonlinear source terms to model cracks formation orhealing.

A splitting scheme allows transport and diffusion phenomena to be decoupled. Two fixedgrids are used. The transport equations are solved on an unstructured grid of small cubiccells, thus allowing numerical diffusion of the volume fraction of ice to be reduced asmuch as possible. The nonlinear Stokes problem is solved on an unstructured mesh oftetrahedrons, larger than the cells, using stabilized finite elements.

Two computations are presented at different time scales. First, the dynamics ofRhonegletscher, Swiss Alps, are investigated in 3D from 2007 to 2100 using severalclimatic scenarios and without considering ice damage. Second, ice damage is taken intoaccount in order to reproduce the calving process of a 2D glacier tongue submerged bywater.

A one-dimensional lattice of SIR (susceptible/infected/removed) epidemic centres isconsidered numerically and analytically. The limiting solutions describing the behaviour of thestandard SIR model with a small number of initially infected individuals are derived, and expressionsfound for the duration of an outbreak. We study a model for a weakly mixed populationdistributed between the interacting centres. The centres are modelled as SIR nodes with interactionbetween sites determined by a diffusion-type migration process. Under the assumption of fastmigration, a one-dimensional lattice of SIR nodes is studied numerically with deterministic andrandom coupling, and travelling wave-like solutions are found in both cases. For weak coupling,the main part of the travelling wave is well approximated by the limiting SIR solution. Explicitformulae are found for the speed of the travelling waves and compared with results of numericalsimulation. Approximate formulae for the epidemic propagation speed are also derived whencoupling coefficients are randomly distributed, they allow us to estimate how the average speed inrandom media is slowed down.

The release of chemicals following herbivore grazing on primary producers may providefeeding cues to carnivorous predators, thereby promoting multitrophic interactions. Inparticular, chemicals released following grazing on phytoplankton by microzooplanktonherbivores have been shown to elicit a behavioural foraging response in carnivorouscopepods, which may use this chemical information as a mechanism to locate and remainwithin biologically productive patches of the ocean. In this paper, we use a 1D spatialreaction-diffusion model to simulate a tri-trophic planktonic system in the water column,where predation at the top trophic level (copepods) is affected by infochemicals releasedby the primary producers forming the bottom trophic level. The effect of theinfochemical-mediated predation is investigated by comparing the case where copepodsforage randomly to the case where copepods adjust their vertical position to follow thedistribution of grazing-induced chemicals. Results indicate that utilization ofinfochemicals for foraging provides fitness benefits to copepods and stabilizes the systemat high nutrient load, whilst also forming a possible mechanism for phytoplankton bloomformation. We also investigate how the copepod efficiency to respond to infochemicalsaffects the results, and show that small increases (2%) in the ability of copepods tosense infochemicals can promote their persistence in the system. Finally we argue thateffectively employing infochemicals for foraging can be an evolutionarily stable strategyfor copepods.

A numerical study of a two-dimensional model for premixed gas combustion in a narrow, semi-infinite channel with no-slip boundary condition is performed. The work is motivated by recent theoretical advances revealing the major role of hydraulic resistance in deflagration-to-detonation transition, one of the central yet still inadequately understood phenomena of gaseous combustion. The work is a continuation and extension of recently reported results over non-isothermal boundary conditions, wider channels, and lower incipient flame velocities, closer to those of real life explosives.

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We derive the fractional Fokker-Planck equation for the density of cells and apply this equation to the anomalous chemotaxis problem. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensionalsuperdiffusive Brusselator model are analyzed. The superdiffusive Brusselator modeldiffers from its regular counterpart in that the Laplacian operator of the regular modelis replaced by ∂ α /∂|ξ|α , 1 < α< 2, an integro-differential operator that reflects the nonlocal behavior ofsuperdiffusion. The order of the operator, α, is a measure of the rate ofsuperdiffusion, which, in general, can be different for each of the two components. Aweakly nonlinear analysis is used to derive two coupled amplitude equations describing theslow time evolution of the Turing and Hopf modes. We seek special solutions of theamplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed modesolution, and analyze their stability to long-wave perturbations. We find that thestability criteria of all three solutions depend greatly on the rates of superdiffusion ofthe two components. In addition, the stability properties of the solutions to theanomalous diffusion model are different from those of the regular diffusion model.Numerical computations in a large spatial domain, using Fourier spectral methods in spaceand second order Runge-Kutta in time are used to confirm the analysis and also to findsolutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime.Specifically, we find a large number of steady state patterns consisting of a localizedregion or regions of stationary stripes in a background of time periodic cellular motion,as well as patterns with a localized region or regions of time periodic cells in abackground of stationary stripes. Each such pattern lies on a branch of such solutions, isstable and corresponds to a different initial condition. The patterns correspond to thephenomenon of pinning of the front between the stripes and the time periodic cellularmotion. While in some cases it is difficult to isolate the effect of the diffusionexponents, we find characteristics in the spatiotemporal patterns for anomalous diffusionthat we have not found for regular (Fickian) diffusion.

This special issue of Mathematical Modelling of Natural Phenomena on biomathematics education shares the work of fifteen groups at as many different institutions that have developed beautiful biological applications of mathematics that are different in three ways from much of what is currently available. First, many of these selections utilize current research in biomathematics rather than the well-known textbook examples that are at least a half-century old. Second, the selections focus on modules that are intended for instant classroom adoption, adaptation, and implementation. Instead of focusing on how to overcome the challenges of implementing mathematics into biology or biology into mathematics or on educational research on the effectiveness of some small implementation, the authors develop individual biological models sufficiently well such that they can be easily adopted and adapted for use in both mathematics and biology classrooms. A third difference in this collection is the substantive inclusion of discrete mathematics and innovative pedagogies. Because calculus-based models have received the majority of the biomathematics modeling attention until very recently, the focus on discrete models may seem surprising. The examples range from DNA nanostructures through viral capsids to neuronal processes and ecosystem problems. Furthermore, a taxonomy of quantitative reasoning and the role of modeling per se as a different practice are contextualized in contemporary biomathematics education.

Motivated by structured parasite populations in aquaculture we consider a class ofsize-structured population models, where individuals may be recruited into the populationwith distributed states at birth. The mathematical model which describes the evolution ofsuch a population is a first-order nonlinear partial integro-differential equation ofhyperbolic type. First, we use positive perturbation arguments and utilise results fromthe spectral theory of semigroups to establish conditions for the existence of a positiveequilibrium solution of our model. Then, we formulate conditions that guarantee that thelinearised system is governed by a positive quasicontraction semigroup on the biologicallyrelevant state space. We also show that the governing linear semigroup is eventuallycompact, hence growth properties of the semigroup are determined by the spectrum of itsgenerator. In the case of a separable fertility function, we deduce a characteristicequation, and investigate the stability of equilibrium solutions in the general case usingpositive perturbation arguments.

An overview of a general approach for mathematical modeling of evolving heterogeneouspopulations using a wide class of selection systems and replicator equations (RE) ispresented. The method allows visualizing evolutionary trajectories of evolvingheterogeneous populations over time, while still enabling use of analytical tools ofbifurcation theory. The developed theory involves introducing escort systems of auxiliary“keystone" variables, which reduce complex multi-dimensional inhomogeneous models tolow dimensional systems of ODEs that in many cases can be investigated analytically. Inaddition to a comprehensive theoretical framework, a set of examples of the method’sapplicability to questions ranging from preventing the tragedy of the commons to cancertherapy is presented.

In this work, we introduce a new software created to study hematopoiesis at the cellpopulation level with the individually based approach. It can be used as an interface between theoreticalworks on population dynamics and experimental observations. We show that this softwarecan be useful to study some features of normal hematopoiesis as well as some blood diseases suchas myelogenous leukemia. It is also possible to simulate cell communication and the formation ofcell colonies in the bone marrow.

In proteomics study, Imaging Mass Spectrometry (IMS) is an emerging and very promisingnew technique for protein analysis from intact biological tissues. Though it has showngreat potential and is very promising for rapid mapping of protein localization and thedetection of sizeable differences in protein expression, challenges remain in dataprocessing due to the difficulty of high dimensionality and the fact that the number ofinput variables in prediction model is significantly larger than the number ofobservations. To obtain a complete overview of IMS data and find trace features based onboth spectral and spatial patterns, one faces a global optimization problem. In thispaper, we propose a weighted elastic net (WEN) model based on IMS data processing needs ofusing both the spectral and spatial information for biomarker selection andclassification. Properties including variable selection accuracy of the WEN model arediscussed. Experimental IMS data analysis results show that such a model not only reducesthe number of side features but also helps new biomarkers discovery.

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 consider Schrödinger operators Hα given by equation(1.1) below. We study the asymptotic behavior of the spectral density E(Hα,λ)for λ → 0 andthe L1 →L∞ dispersive estimates associated to theevolution operator e−itHα.In particular we prove that for positive values of α, the spectral densityE(Hα,λ)tends to zero as λ →0 with higher speed compared to the spectral density of Schrödingeroperators with a short-range potential V. We then show how the long time behavior ofe−itHαdepends on α.More precisely we show that the decay rate of e−itHαfor t → ∞ canbe made arbitrarily large provided we choose α large enough and consider a suitable operatornorm.

A model of tumor growth in a spatial environment is analyzed. The model includesproliferating and quiescent compartments of tumor cells indexed by successively mutated cell phenotypesof increasingly proliferative aggressiveness. The model incorporates spatial dependencedue to both random motility and directed movement haptotaxis. The model structures tumor cellsby both cell age and cell size. The model consists of a system of nonlinear partial differentialequations for the compartments of tumor cells, extracellular matrix, matrix degradative enzyme,and oxygen. The existence, uniqueness, positivity, regularity, and growth characteristics of thesolutions are investigated.

The paper is devoted to the method of computer simulation of protein interactions takingpart in photosynthetic electron transport reactions. Using this method we have studiedkinetic characteristics of protein-protein complex formation for four pairs of proteinsinvolved in photosynthesis at a variety of ionic strength values. Computer simulationsdescribe non-monotonic dependences of complex formation rates on the ionic strength as theresult of long-range electrostatic interactions. Calculations confirm that the decrease inthe association second order rate constant at low values of the ionic strength is causedby the protein pairs spending more time in “wrong” orientations which do not satisfy thedocking conditions and so do not form the final complex capable of the electrontransfer.

In this work we review the aggregation of variables method for discrete dynamicalsystems. These methods consist of describing the asymptotic behaviour of a complex systeminvolving many coupled variables through the asymptotic behaviour of a reduced systemformulated in terms of a few global variables. We consider population dynamics modelsincluding two processes acting at different time scales. Each process has associated a mapdescribing its effect along its specific time unit. The discrete system encompassing bothprocesses is expressed in the slow time scale composing the map associated to the slow oneand the k-th iterate of the map associated to the fast one. In the linear case a result isstated showing the relationship between the corresponding asymptotic elements of bothsystems, initial and reduced. In the nonlinear case, the reduction result establishes theexistence, stability and basins of attraction of steady states and periodic solutions ofthe original system with the help of the same elements of the corresponding reducedsystem. Several models looking over the main applications of the method to populationsdynamics are collected to illustrate the general results.