In this work, we present some concepts recently introduced in the analysis and control ofdistributed parameter systems: Spreadability,vulnerability and protector control. These conceptspermit to describe many biogeographical phenomena, as those of pollution, desertificationor epidemics, which are characterized by a spatio-temporal evolution

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.

We are interested in the stability of the localized stationary solutions of a three-component reaction-diffusion system with one activator and two inhibitors. We show that depending on control parameters, solutions in form of moving and breathing localized structures can be observed in the vicinity of the codimension-two bifurcation point. We analyze this situation performing multiple scale perturbation expansion in the vicinity of the bifurcation point and derive a set of order parameter equations, explicitly describing the dynamics of the single localized structure. Numerical simulations are carried out, showing good agreement with the analytical predictions.

We introduce new classes of modulation spaces over phase space. By means of theKohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operatorsthat bound their operator norms on Lp–spaces,Sobolev spaces, and modulation spaces.

A reaction–diffusion replicator equation is studied. A novel method to apply theprinciple of global regulation is used to write down a model with explicit spatialstructure. Properties of stationary solutions together with their stability are analyzedanalytically, and relationships between stability of the rest points of thenon-distributed replicator equation and the distributed system are shown. In particular,we present the conditions on the diffusion coefficients under which the non-distributedreplicator equation can be used to describe the number and stability of the stationarysolutions to the distributed system. A numerical example is given, which shows that thesuggested modeling framework promotes the system’s persistence, i.e., a scenario ispossible when in the spatially explicit system all the interacting species survive whereassome of them go extinct in the non-distributed one.

Many tumours undergo periods in whichthey apparently do not grow but remain at a roughly constant sizefor extended periods. This is termed tumour dormancy. Themechanisms responsible for dormancy include failure to develop aninternal blood supply, individual tumour cells exiting the cellcycle and a balance between the tumour and the immune response toit. Tumour dormancy is of considerable importance in the naturalhistory of cancer. In many cancers, and in particular in breastcancer, recurrence can occur many years after surgery to removethe primary tumour, following a long period of occult disease.Mathematical modelling suggested that continuous growth of tumourswas incompatible with data of the times of recurrence in breastcancer, suggesting that tumour dormancy was a common phenomenon.Modelling has also been applied to understanding the mechanismsresponsible for dormancy, how they can be manipulated and theimplications for cancer therapy. Here, the literature onmathematical modelling of tumour dormancy is reviewed. Inconclusion, promising future directions for research arediscussed.

In this note, we consider a nonlinear diffusion equation with a bistable reaction termarising in population dynamics. Given a rather general initial data, we investigate itsbehavior for small times as the reaction coefficient tends to infinity: we prove ageneration of interface property.

Propagation of polymerization fronts with liquid monomer and liquid polymer is consideredand the influence of vibrations on critical conditions of convective instability isstudied. The model includes the heat equation, the equation for the concentration and theNavier-Stokes equations considered under the Boussinesq approximation. Linear stabilityanalysis of the problem is fulfilled, and the convective instability boundary is founddepending on the amplitude and on the frequency of vibrations

The paper combines analytic and numeric tools to investigate a nonlinear optimal control problem relevant to the economics of climate change. The problem describes optimal investments into pollution mitigation and environmental adaptation at a macroeconomic level. The steady-state analysis of this problem focuses on the optimal ratio between adaptation and mitigation. In particular, we analytically prove that the long-term investments into adaptation are profitable only for economies above certain efficiency threshold. Numerical simulation is provided to estimate how the economic efficiency and capital deterioration affect the optimal policy.

We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magneticfield which is described by the magnetic Schrödinger operator with a periodic potentialplus a finitely supported perturbation. We describe all eigenvalues and resonances of thisoperator, and theirs dependence on the magnetic field. The proof is reduced to theanalysis of the periodic Jacobi operators on the half-line with finitely supportedperturbations.

We study the destabilization mechanism in a unidirectional ring of identical oscillators,perturbed by the introduction of a long-range connection. It is known that for ahomogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivialequilibrium undergoes a sequence of Hopf bifurcations eventually leading to thecoexistence of multiple stable periodic states resembling the Eckhaus scenario. We showthat this destabilization scenario persists under small non-local perturbations. In thiscase, the Eckhaus line is modulated according to certain resonance conditions. In the casewhen the shortcut is strong, we show that the coexisting periodic solutions split up intotwo groups. The first group consists of orbits which are unstable for all parametervalues, while the other one shows the classical Eckhaus behavior.

We study the existence and some properties of travelling waves in partially degenerate reaction-diffusion systems. Such systems may for example describe intracellular calcium dynamics in the presence of immobile buffers. In order to prove the wave existence, we first consider the non degenerate case and then pass to the limit as some of the diffusion coefficient converge to zero. The passage to the limit is based on a priori estimates of solutions independent of the values of the diffusion coefficients. The wave uniqueness is also proved.

In this paper we present a new hybrid method, called SASP method. We propose thehybridization of two methods, the simulated annealing (SA), which belong to the class ofglobal optimization based on the principles of thermodynamics, and the descent method werewe estimate the gradient using the simultaneous perturbation. This hybrid method givesbetter results. We use the Normal Boundary Intersection approach (NBI) based on the SASPmethod to solve a portfolio optimization problem. Such problem is a multi-objectiveoptimization problem, in order to solve this problem we use three statistical quantities:the expected value, the variance and the Conditional Value-at-Risk (CVaR). The purpose ofthis work is to find the efficient boundary of the considered multi-objective problemusing the NBI method based on the SASP method

Seasonal forcing is identified as a key pattern generating mechanism in an epidemic model of rabies dispersal. We reduce an established individual-based high-detail modeldown to a deterministic conceptual model. The characteristic wave pattern characterized byhigh densities of infected individuals is maintained throughout the reduction process. In ourmodel it is evident that seasonal forcing is the dominant factor that drives pattern formation.In particular we show that seasonal forcing can cause splitting of epidemic wave fronts.

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

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.

We analyse a reduced version of the Grodins et al. control model [14] of respiration involving only CO2, andshow that it can be dramatically simplified by the use of judicious approximations. Inparticular, we show that the conceptual basis of the popular model of Mackey and Glass[20] is at odds with the important transportprocesses of the Grodins model. Despite this, a realistic approximation of the Grodinsmodel yields a Mackey-Glass type model with almost the same criterion for the onset ofCheyne-Stokes breathing.

While the reduced Grodins model does apparently provide a realistic mechanism forinstability, consideration of the buffering effect of the blood-brain barrier appears tomake it unlikely. We conclude that a realistic physiological model of Grodins type toexplain Cheyne–Stokes breathing is not yet in place, and raise the question whether thebicarbonate buffering system has a rôle to play.

In this paper, we consider a system of nonlinear delay-differential equations(DDEs) which models the dynamics of the interaction between chronic myelogenousleukemia (CML), imatinib, and the anti-leukemia immune response. Because of thechaotic nature of the dynamics and the sparse nature of experimental data, welook for ways to use computation to analyze the model without employing directnumerical simulation. In particular, we develop several tools usingLyapunov-Krasovskii analysis that allow us to test the robustness of the modelwith respect to uncertainty in patient parameters. The methods developed in thispaper are applied to understanding which model parameters primarily affect thedynamics of the anti-leukemia immune response during imatinib treatment. Thegoal of this research is to aid the development of more efficient modelingapproaches and more effective treatment strategies in cancer therapy.

We propose a matrix population modellingapproach in order to describe the dynamics of a grayling(Thymallus thymallus, L. 1758) population living in the Ainriver (France). We built a Leslie like model, which integrates theclimate changes in terms of temperature and discharge. First, weshow how temperature and discharge can be related to life historytraits like survival and reproduction. Second, we show how to usethe population model to precisely examine the life cycle ofgrayling : estimated numbers of individuals at the beginning ofthe reproduction season, estimated number of individuals insummer, comparison between observed and simulated numbers ofindividuals. Finally, we discuss our results in the framework ofthe assessment of impact of possible climate changes on graylingpopulations.