Salmonella is one of the major sources of toxi-infection in humans in France and UnitedStates. The incidence of human salmonellosis has considerably increased over the past 20 years and this can be largely attributed to epidemics of S. enteritidis phage type 4 in poultry in numerous countries. In this article, we formulate and analyse a model in which the transmission of the disease is determined by contact between hens and Salmonella in the environment.

Recent technological advances including brain imaging (higher resolution in space andtime), miniaturization of integrated circuits (nanotechnologies), and acceleration ofcomputation speed (Moore’s Law), combined with interpenetration between neuroscience,mathematics, and physics have led to the development of more biologically plausiblecomputational models and novel therapeutic strategies. Today, mathematical models ofirreversible medical conditions such as Parkinson’s disease (PD) are developed andparameterised based on clinical data. How do these evolutions have a bearing on deep brainstimulation (DBS) of patients with PD? We review how the idea of DBS, a standardtherapeutic strategy used to attenuate neurological symptoms (motor, psychiatric), hasemerged from past experimental and clinical observations, and present how, over the lastdecade, computational models based on different approaches (phase oscillator models,spiking neuron network models, population-based models) have started to shed light ontoDBS mechanisms. Finally, we explore a new mathematical modelling approach based on neuralfield equations to optimize mechanisms of brain stimulation and achieve finer control oftargeted neuronal populations. We conclude that neuroscience and mathematics are crucialpartners in exploring brain stimulation and this partnership should also include otherdomains such as signal processing, control theory and ethics.

2D shallow water equations with depth-averaged k−εmodel is considered. A meshless method based on multiquadric radial basis functions isdescribed. This methods is based on the collocation formulation and does not require thegeneration of a grid and any integral evaluation. The application of this method to a flowin horizontal channel, taken as an experimental device, is presented. The results ofcomputations are compared with experimental data and are found to be satisfactory

We study the number of lattice points in ℝd ,d ≥ 2, lying inside an annulus as a function of the centre of theannulus. The average number of lattice points there equals the volume of the annulus, andwe study the L 1 and L 2 norms ofthe remainder. We say that a dimension is critical, if these norms do not have upper andlower bounds of the same order as the radius goes to infinity. In [Duke Math. J., 107 (2001), No. 2, 209–238], it was proved that in the case of the ball (instead of an annulus)the critical dimensions are d ≡ 1 mod 4. We show that the behaviour ofthe width of an annulus as a function of the radius determines which dimensions arecritical now. In particular, if the width is bounded away from zero and infinity, thecritical dimensions are d ≡ 3 mod 4; if the width goes to infinity, butslower than the radius, then all dimensions are critical, and if the width tends to zeroas a power of the radius, then there are no critical dimensions.

Relaxation oscillations are limit cycles with two clearly differenttime scales. In this article the spatio-temporal dynamics of astandard prey-predator system in the parameter region of relaxationoscillation is investigated. Both prey and predator population aredistributed irregularly at a relatively high average level between amaximal and a minimal value. However, the slowly developing complexpattern exhibits a feature of “inverse excitability”: Bothpopulations show collapses which occur erratically both in space andin time. The nature of these collapses is analysed statistically andit is shown that the model behaviour can be interpreted as aresolution of the paradox of enrichment.

Mitochondria are one of the most important organelles determining Ca2+regulatory pathway in the cell. Recent experiments suggested the existence of cytosolicmicrodomains with locally elevated calcium concentration (CMDs) in the nearest vicinity ofthe outer mitochondrial membrane (OMM). These intermediate physical connections betweenendoplasmic reticulum (ER) and mitochodria are called MAM (mitochondria-associated ERmembrane) complexes.

The aim of this paper is to take into account the direct calcium flowfrom ER to mitochondria implied by the existence of MAMs and perform detailed numericalanalysis of the influence of this flow on the type and shape of calcium oscillations.Depending on the permeability of MAMs interface and ER channels, different patterns ofoscillations appear (simple, bursting and chaotic). For some parameters the oscillatorypattern disappear and the system tends to a steady state with extremely high calcium levelin mitochondria, which can be interpreted as a crucial point at the beginning of anapoptotic pathway.

There are two mathematical models of elastic walls of healthy and atherosclerotic bloodvessels developed and studied. The models are included in a numerical model of globalblood circulation via recovery of the vessel wall state equation. The joint model allowsus to study the impact of arteries atherosclerotic disease of a set of arteries onregional haemodynamics.

We propose a mixed formulation for non-isothermal Oldroyd–Stokes problem where the bothextra stress and the heat flux’s vector are considered. Based on such a formulation, adual mixed finite element is constructed and analyzed. This finite element method enablesus to obtain precise approximations of the dual variable which are, for the non-isothermalfluid flow problems, the viscous and polymeric components of the extra-stress tensor, aswell as the heat flux. Furthermore, it has properties analogous to the finite volumemethods, namely, the local conservation of the momentum and the mass.

A mathematical model, coupling the dynamics of short-term stem-like cells and matureleukocytes in leukemia with that of the immune system, is investigated. The model isdescribed by a system of seven delay differential equations with seven delays. Threeequilibrium points E0, E1,E2 are highlighted. The stability and the existence of theHopf bifurcation for the equilibrium points are investigated. In the analysis of themodel, the rate of asymmetric division and the rate of symmetric division are veryimportant.

Previous work has shown that intracellular delay needs to be taken into account toaccurately determine the half-life of free virus from drug perturbation experiments [1]. The delayalso effects the estimated value for the infected T-cell loss rate when we assume that the drug isnot completely effective [19]. Models of virus infection that include intracellular delay are moreaccurate representations of the biological data.
We analyze a non-linear model of the human immunodeficiency virus (HIV) infection thatconsiders the interaction between a replicating virus, CD4+ T-cell and the cytotoxic-lymphocytes(CTL).We then investigate the intracellular delay effect on the stability of the endemically infectedsteady state. Criteria are given to ensure that the infected steady state is asymptotically stable forall delays. Model analysis also allows the prediction of a critical delay $\tau_c$ below which the effectorCTL can play a significant role in the immune control mechanism even when the basic reproductionnumber is high.

A computational framework for testing the effects of cytotoxic molecules, specific to agiven phase of the cell cycle, and vascular disrupting agents (VDAs) is presented. Themodel is based on a cellular automaton to describe tumour cell states transitions fromproliferation to death. It is coupled with a model describing the tumour vasculature andits adaptation to the blood rheological constraints when alterations are induced by VDAstreatment. Several therapeutic protocols in two structurally different vascular networkswere tested by varying the duration of cytotoxic drug perfusion and the periodicity oftreatment cycles. The impact of VDAs were also tested both experimentally from intravitalmicroscopy through a dorsal skinfold chamber on a mouse and numerically. Simulationresults show that combining cytotoxic treatment with a post treatment of VDA through ajudicious timing could favour the rapid eradication of the tumour. The computationalframework thus gives some insights into the outcome of cytotoxic and VDAs treatments on aqualitative basis. Future validation from our experimental setup could open up newperspectives towards Computer-Assisted Therapeutic Strategies.

Compartmentalization is a general principle in biological systems which is observable on all size scales, ranging from organelles inside of cells, cells in histology, and up to the level of groups, herds, swarms, meta-populations, and populations. Compartmental models are often used to model such phenomena, but such models can be both highly nonlinear and difficult to work with.

Fortunately, there are many significant biological systems that are amenable to linear compartmental models which are often more mathematically accessible. Moreover, the biology and mathematics is often so intertwined in such models that one can be used to better understand the other. Indeed, as we demonstrate in this paper, linear compartmental models of migratory dynamics can be used as an exciting and interactive means of introducing sophisticated mathematics, and conversely, the associated mathematics can be used to demonstrate important biological properties not only of seasonal migrations but also of compartmental models in general.

We have found this approach to be of great value in introducing derivatives, integrals, and the fundamental theorem of calculus. Additionally, these models are appropriate as applications in a differential equations course, and they can also be used to illustrate important ideas in probability and statistics, such as the Poisson distribution.

Chattopadhyay et al. [Biosystems (2003),68, pp. 5-17] proposed and analyzed an N – P model in the presenceof viral infection on phytoplankton population. They studied thedynamics under the constant nutrient input. The present paper dealswith the problem with seasonal variability on nutrient input. We usea general periodic function for nutrient input. We observe thedynamics of the system by considering (i) the infected phytoplanktonconsumes nutrient and (ii) the infected phytoplankton is not in astate to consume nutrient. Conditions for the persistence andextinction of populations are worked out. Our numerical experimentsshow that if the infected phytoplankton does not take nutrient thensusceptible phytoplankton coexists with the infected ones. But ifthe infected phytoplankton consumes nutrient then there is a chancefor extinction of susceptible phytoplankton for high rate ofinfection. We also observe that periodic nutrient input enforces thesystem to enter into chaotic region.

The seasonality hypothesis states that climates characterized by large annual cycles select for large body sizes. In order to study the effects of seasonality on the evolution of body size, we use a model that is based on physiological rules and first principles. At the ecological time scale, our model results show that both larger productivity and seasonality may lead to larger body sizes. Our model is the first dynamic and process-based model to support the seasonality hypothesis and hence demonstrates the importance of basing models on physiological processes. We focus not only on variability at the ecological time scale, but also on the temporal variations in seasonality existing at geological time scales. A particularly strong forcing of seasonality exists on the scale of 20,000-400,000 years, the scale of Milankovitch cycles. Therefore, we simulated the evolutionary response of body size to a Milankovitch-type of forcing of climate and food density. Results illustrate that for a given level of investment in reserves body size may track climatic cycles, and that below a certain seasonality threshold the body size will decrease rapidly, leading to extinction.

We establish sharp semiclassical upper bounds for the moments of some negative powers forthe eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporatedin the problem, we obtain sharp lower bounds for the moments of positive powers notexceeding one for such eigenvalues. When considering a Schrödinger operator with therelativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove thesharp Lieb-Thirring estimate for the moments of some negative powers of itseigenvalues.

The impacts of the two-beam interference heating on the number of core-shell and embeddednanoparticles and on nanostructure coarsening are studied numerically based on thenon-linear dynamical model for dewetting of the pulsed-laser irradiated, thin (< 20nm) metallic bilayers. The model incorporates thermocapillary forces and disjoiningpressures, and assumes dewetting from the optically transparent substrate atop of thereflective support layer, which results in the complicated dependence of lightreflectivity and absorption on the thicknesses of the layers. Stabilizing thermocapillaryeffect is due to the local thickness-dependent, steady-state temperature profile in theliquid, which is derived based on the mean substrate temperature estimated from theelaborate thermal model of transient heating and melting/freezing. Linear stabilityanalysis of the model equations set for Ag/Co bilayer predicts the dewetting length scalesin the qualitative agreement with experiment.

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.

We define two notions of discrete dimension based on the Minkowski and Hausdorffdimensions in the continuous setting. After proving some basic results illustrating thesedefinitions, we apply this machinery to the study of connections between the Erdős andFalconer distance problems in geometric combinatorics and geometric measure theory,respectively.

We consider the problem of frictional contact between an piezoelectric body and aconductive foundation. The electro-elastic constitutive law is assumed to be nonlinear andthe contact is modelled with the Signorini condition, nonlocal Coulomb friction law and aregularized electrical conductivity condition. The existence of a unique weak solution ofthe model is established. The finite elements approximation for the problem is presented,and error estimates on the solutions are derived

This paper is concerned with the existence and stability of travelling front solutionsfor more general autocatalytic chemical reaction systems ut = duxx − uf(v), vt = vxx + uf(v)with d > 0 and d ≠ 1, wheref(v) has super-linear or linear degeneracy atv = 0. By applying Lyapunov-Schmidt decomposition method in someappropriate exponentially weighted spaces, we obtain the existence and continuousdependence of wave fronts with some critical speeds and with exponential spatial decay ford near 1. By applying special phase plane analysis and approximatecenter manifold theorem, the existence of traveling waves with algebraic spatial decay orwith some lower exponential decay is also obtained for d > 0. Further,by spectral estimates and Evans function method, the wave fronts with exponential spatialdecay are proved to be spectrally or linearly stable in some suitable exponentiallyweighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to belocally exponentially stable in some exponentially weighted spaces and Lyapunov stable inCunif(ℝ) space, if the initial perturbation of the waves issmall in both the weighted and unweighted norms; the perturbation of the waves also stayssmall in L1(ℝ) norm and decays algebraically inCunif(ℝ) norm, if the initial perturbation is in additionsmall in L1 norm.