The purpose of this paper is to apply particle methods to the numerical solution of theEPDiff equation. The weak solutions of EPDiff are contact discontinuities that carrymomentum so that wavefront interactions represent collisions in which momentum isexchanged. This behavior allows for the description of many rich physical applications,but also introduces difficult numerical challenges. We present a particle method for theEPDiff equation that is well-suited for this class of solutions and for simulatingcollisions between wavefronts. Discretization by means of the particle method is shown topreserve the basic Hamiltonian, the weak and variational structure of the originalproblem, and to respect the conservation laws associated with symmetry under the Euclideangroup. Numerical results illustrate that the particle method has superior features in bothone and two dimensions, and can also be effectively implemented when the initial data ofinterest lies on a submanifold.

Implicit sampling is a sampling scheme for particle filters, designed to move particles one-by-one so that they remain in high-probability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations.

We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.

In this work we discuss and analyze spiking patterns in a generic mathematical model oftwo coupled non-identical nonlinear oscillators supplied with a spike-timing dependentplasticity (STDP) mechanism. Spiking patterns in the system are shown to converge to aphase-locked state in a broad range of parameters. Precision of the phase locking, i.e.the amplitude of relative phase deviations from a given reference, depends on the naturalfrequencies of oscillators and, additionally, on parameters of the STDP law. Thesedeviations can be optimized by appropriate tuning of gains (i.e. sensitivity tospike-timing mismatches) of the STDP mechanisms. The deviations, however, can not be madearbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights.Thus if accurate phase-locking in the system is required then an additional tuningmechanism is generally needed. We found that adding a very simple adaptation dynamics inthe form of slow fluctuations of the base line in the STDP mechanism enables accuratephase tuning in the system with arbitrary high precision. The scheme applies to systems inwhich individual oscillators operate in the oscillatory mode. If the dynamics ofoscillators becomes bistable then relative phase may fail to converge to a given valuegiving rise to the emergence of complex spiking sequences.

The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vriesequations over a slowly varying random bottom is rigorously studied. One motivation forstudying such a system is better understanding the unidirectional motion of interactingsurface and internal waves for a fluid system that is formed of two immiscible layers. Itwas shown recently by Craig-Guyenne-Sulem [1] thatin the regime where the internal wave has a large amplitude and a long wavelength, thedynamics of the surface of the fluid is described by the Schrödinger equation, while thatof the internal wave is described by the Korteweg-de Vries equation. The purpose of thisletter is to show that in the presence of a slowly varying random bottom, the coupledwaves evolve adiabatically over a long time scale. The analysis covers the cases when thesurface wave is a stable bound state or a long-lived metastable state.

Existence and stability of periodic solutions are studied for a system of delaydifferential equations with two delays, with periodic coefficients. It models theevolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenousleukemia under a periodic treatment that acts only on mature cells. Existence of a guidingfunction leads to the proof of the existence of a strictly positive periodic solution by atheorem of Krasnoselskii. The stability of this solution is analysed.

National policies regarding the BCG vaccine for tuberculosis vary greatly throughout the international community and several countries are currently considering discontinuing universal vaccination. Detractors of BCG point to its uncertain effectiveness and its interference with the detection and treatment of latent tuberculosis infection (LTBI).

In order to quantify the trade-off between vaccination and treatment of LTBI, a mathematical model was designed and calibrated to data from Brazil, Ghana, Germany, India, Mexico, Romania, the United Kingdom and the United States. Country-specific thresholds for when LTBI treatment outperforms mass vaccination were found and the consequences of policy changes were estimated.

Our results suggest that vaccination outperforms LTBI treatment in all settings but with greatly reduced efficiency in low incidence countries. While national policy statements emphasize BCG’s interference with LTBI detection, we find that reinfection should be more determinant of a country’s proper policy choice.

The dynamic model of plant growth GreenLab describes plant architecture and functionalgrowth at the level of individual organs. Structural development is controlled by formalgrammars and empirical equations compute the amount of biomass produced by the plant, andits partitioning among the growing organs, such as leaves, stems and fruits. The number oforgans initiated at each time step depends on the trophic state of the plant, which isevaluated by the ratio of biomass available in plant to the demand of all the organs. Thecontrol of the plant organogenesis by this variable induces oscillations in the simulatedplant behaviour. The mathematical framework of the GreenLab model allows to compute theconditions for the generation of oscillations and the value of the period according to theset of parameters. Two case-studies are presented, corresponding to emergence ofoscillations associated to fructification and branching.

Similar alternating patterns arecommonly reported by botanists. In this article, two examples were selected: alternatepatterns of fruits in cucumber plants and alternate appearances of branches inCecropia trees. The model was calibrated from experimental datacollected on these plants. It shows that a simple feedback hypothesis of trophic controlon plant structure allows the emergence of cyclic patterns corresponding to the observedones.

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.

Two-dimensional inviscid channel flow of an incompressible fluid is considered. It isshown that if the flow is steady and features no horizontal stagnation, then the flow mustnecessarily be a parallel shear flow.

The measurement of CFSE dilution by flow cytometry is a powerful experimental tool tomeasure lymphocyte proliferation. CFSE fluorescence precisely halves after each celldivision in a highly predictable manner and is thus highly amenable to mathematicalmodelling. However, there are several biological and experimental conditions that canaffect the quality of the proliferation data generated, which may be important to considerwhen modelling dye dilution data sets. Here we overview several of these variablesincluding the type of fluorescent dye used to monitor cell division, dye labellingmethodology, lymphocyte subset differences, in vitro versus in vivo experimental assays,cell autofluorescence, and dye transfer between cells.

A linear, uniformly stratified ocean model is used to investigate propagation of baroclinic Kelvin waves in a cylindrical basin. It is found that smaller wave amplitudes are inherent to higher mode individual terms of the obtained solutions that are also evanescent away of a costal line toward the center of the circular basin. It is also shown that the individual terms if the obtained solutions can be visualized as spinning patterns in rotating stratified fluid confined in a circular basin. Moreover, the fluid patterns look rotating in an anticlockwise sense looking above the North Pole and that spinning is more intensive for smaller mode numbers. Finally, we observe the existence of the oceanic region where the pressure increases relatively rapidly with the depth.

We analyze the problem of shear-induced electrokinetic lift on a particle freely suspended near a solid wall, subject to a homogeneous (simple) shear. To this end, we apply the large-Péclet-number generic scheme recently developed by Yariv et al. (J. Fluid Mech., Vol. 685, 2011, p. 306). For a force- and torque-free particle, the driving flow comprises three components, respectively describing (i) a particle translating parallel to the wall; (ii) a particle rotating with an angular velocity vector normal to the plane of shear; and (iiii) a stationary particle in a shear flow. Symmetry arguments reveal that the electro-viscous lift, normal to the wall, is contributed by Maxwell stresses accompanying the induced electric field, while electro-viscous drag and torque corrections, parallel to the wall, are contributed by the Newtonian stresses accompanying the induced flow. We focus upon the near-contact limit, where all electro-viscous contributions are dominated by the intense electric field in the narrow gap between the particle and the wall. This field is determined by the gap-region pressure distributions associated with the translational and rotational components of the driving Stokes flow, with the shear-component contribution directly affecting only higher-order terms. Owing to the similarity of the corresponding pressure distributions, the induced electric field for equal particle–wall zeta potentials is proportional to the sum of translation and rotation speeds. The electro-viscous loads result in induced particle velocities, normal and tangential to the wall, inversely proportional to the second power of particle–wall separation.

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovskyequation (ut + cux + α uux + α1u2ux + βuxxx)x = γuwhich is also known as the extended rotation-modified Korteweg–de Vries(KdV) equation. This equation is used for the description of internal oceanic wavesaffected by Earth’ rotation. Particular versions of this equation with zero some ofcoefficients, α, α1, β, orγ are also known in numerous applications. The proposed numericalscheme is a further development of the well-known finite-difference scheme earlier usedfor the solution of the KdV equation. The scheme is of the second order accuracy both ontemporal and spatial variables. The stability analysis of the scheme is presented forinfinitesimal perturbations. The conditions for the calculations with the appropriateaccuracy have been found. Examples of calculations with the periodic boundary conditionsare presented to illustrate the robustness of the proposed scheme.

About twenty five years ago the first discrete mathematical model of the immune systemwas proposed. It was very simple and stylized. Later, many other computational models havebeen proposed each one adding a certain level of sophistication and detail to thedescription of the system. One of these, the Celada-Seiden model published back in 1992,was already mature at its birth, setting apart from the topic-specific nature of the othermodels. This one was not just a model but rather a framework with which one couldimplement his own immunological theories.

Here we describe this computational framework, developed to perform simulations ofdifferent pathologies that are directly or indirectly connected to the immune system. Webriefly describe the system first, then we report on few applications so to give thereader a clear idea of its practical utility in clinical research problems.

The invasive capability is fundamental in determining the malignancy of a solid tumor.Revealing biomedical strategies that are able to partially decrease cancer invasiveness istherefore an important approach in the treatment of the disease and has given rise tomultiple in vitro and in silico models. We here developa hybrid computational framework, whose aim is to characterize the effects of thedifferent cellular and subcellular mechanisms involved in the invasion of a malignantmass. In particular, a discrete Cellular Potts Model is used to represent the populationof cancer cells at the mesoscopic scale, while a continuous approach of reaction-diffusionequations is employed to describe the evolution of microscopic variables, as the nutrientsand the proteins present in the microenvironment and the matrix degrading enzymes secretedby the tumor. The behavior of each cell is then determined by a balance of forces, such ashomotypic (cell-cell) and heterotypic (cell-matrix) adhesions and haptotaxis, and ismediated by the internal state of the individual, i.e. its motility. The resultingcomposite model quantifies the influence of changes in the mechanisms involved in tumorinvasion and, more interestingly, puts in evidence possible therapeutic approaches, thatare potentially effective in decreasing the malignancy of the disease, such as thealteration in the adhesive properties of the cells, the inhibition in their ability toremodel and the disruption of the haptotactic movement. We also extend the simulationframework by including cell proliferation which, following experimental evidence, isregulated by the intracellular level of growth factors. Interestingly, in spite of theincrement in cellular density, the depth of invasion is not significantly increased, asone could have expected.

In this paper we present an epidemic model affecting an age-structured population. Weshow by numerical simulations that this demographic structure can induce persistentoscillations in the epidemic. The model is then extended to encompass a stage-structureddisease within an age-dependent population. In this case as well, persistent oscillationsare observed in the infected as well as in the whole population.