In this paper several models in virus dynamics with and without immune response arediscussed concerning asymptotic behaviour. The case of immobile cells but diffusing viruses andT-cells is included. It is shown that, depending on the value of the basic reproductive number R0of the virus, the corresponding equilibrium is globally asymptotically stable. If R0 < 1 then thevirus-free equilibrium has this property, and in case R0 > 1 there is a unique disease equilibriumwhich takes over this property.

Background: Community-acquired methicillin-resistant Staphylococcusaureus (CA-MRSA), a novel strain of MRSA, has recently emerged and rapidlyspread in the community. Invasion into the hospital setting with replacement of thehospital-acquired MRSA (HA-MRSA) has also been documented. Co-colonization with bothCA-MRSA and HA-MRSA would have important clinical implications given differences inantimicrobial susceptibility profiles and the potential for exchange of geneticinformation.

Methods: A deterministic mathematical model was developed to characterizethe transmission dynamics of HA-MRSA and CA-MRSA in the hospital setting and to quantifythe emergence of co-colonization with both strains

Results: The model analysis shows that the state of co-colonization becomesendemic over time and that typically there is no competitive exclusion of either strain.Increasing the length of stay or rate of hospital entry among patients colonized withCA-MRSA leads to a rapid increase in the co-colonized state. Compared to MRSAdecolonization strategy, improving hand hygiene compliance has the greatest impact ondecreasing the prevalence of HA-MRSA, CA-MRSA and the co-colonized state.

Conclusions: The model predicts that with the expanding community reservoirof CA-MRSA, the majority of hospitalized patients will become colonized with both CA-MRSAand HA-MRSA.

Scattering of electromagnetic (EM) waves by small (ka ≪ 1) impedance particleD of anarbitrary shape, embedded in a homogeneous medium, is studied. Analytic, closed form,formula for the scattered field is derived. The scattered field is of the orderO(a2 −κ), where κ ∈ [ 0,1)is a number. This field is much larger than in the case of Rayleigh-type scattering. Thenumerical results demonstrate a wide range of applicability of the analytic formula forthe scattered field. Comparison with Mie-type solution is carried out for various boundaryimpedances and radii of the particle.

Tuberculosis (TB) remains a major global health problem. A possible risk factor for TB isdiabetes (DM), which is predicted to increase dramatically over the next two decades,particularly in low and middle income countries, where TB is widespread. This study aimedto assess the strength of the association between TB and DM. We present a deterministicmodel for TB in a community in order to determine the impact of DM in the spread of thedisease. The important mathematical features of the TB model are thoroughly investigated.The epidemic threshold known as the basic reproduction number and equilibria for the modelare determined and stabilities analyzed. The model is numerically analyzed to assess theimpact of DM on the transmission dynamics of TB. We perform sensitivity analysis on thekey parameters that drive the disease dynamics in order to determine their relativeimportance to disease transmission and prevalence. Numerical simulations suggest that DMenhances the TB transmission and progression to active TB in a community. The resultssuggest that there is a need for increased attention to intervention strategies such asthe chemoprophylaxis of TB latent individuals and treatment of active TB in people withDM, which may include testing for suspected diabetes, improved glucose control, andincreased clinical and therapeutic monitoring in order to reduce the burden of thedisease.

Gaseous detonation is a phenomenon with very complicated dynamics which has been studied extensively by physicists, mathematicians and engineers for many years. Despite many efforts the problem is far from a complete resolution. Recently the theory of subsonic detonation that occurs in highly resistant porous media was proposed in [4]. This theory provides a model which is realistic, rich and suitable for a mathematical study. In particular, the model is capable of describing the transition from a slowly propagating deflagration wave to the fast detonation wave. This phenomena is known as a deflagration to detonation transition and is one of the most challenging issues in combustion theory. In this paper we will present some recent mathematical results concerning initiation of reaction in porous media, existence and uniqueness of traveling fronts, quenching and propagation.

In this work we study a nonlocal reaction-diffusion equation arising in populationdynamics. The integral term in the nonlinearity describes nonlocal stimulation ofreproduction. We prove existence of travelling wave solutions by the Leray-Schauder methodusing topological degree for Fredholm and proper operators and special a priori estimatesof solutions in weighted Hölder spaces.

Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined thewind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFDhas had its most favorable impact on the aerodynamic design of the high-speed cruiseconfiguration of a transport. This success has raised expectations among aerodynamiciststhat the applicability of CFD can be extended to the full flight envelope. However, thecomplex nature of the flows and geometries involved places substantially increased demandson the solution methodology and resources required. Currently most simulations involveReynolds-Averaged Navier-Stokes (RANS) codes although Large Eddy Simulation (LES) andDetached Eddy Suimulation (DES) codes are occasionally used for component analysis ortheoretical studies. Despite simplified underlying assumptions, current RANS turbulencemodels have been spectacularly successful for analyzing attached, transonic flows. Whetheror not these same models are applicable to complex flows with smooth surface separation isan open question. A prerequisite for answering this question is absolute confidence thatthe CFD codes employed reliably solve the continuous equations involved. Too often,failure to agree with experiment is mistakenly ascribed to the turbulence model ratherthan inadequate numerics. Grid convergence in three dimensions is rarely achieved. Evenresidual convergence on a given grid is often inadequate. This paper discusses issuesinvolved in residual and especially grid convergence.

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming ina viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity andopposite direction applied on the rigid bacterial body and on an associated region in thefluid representing the flagellar bundle. The method for solving the fluid flow and themotion of the bacteria is based on a variational formulation written on the whole domain,strongly coupling the fluid and the rigid particle problems: rigid motion is enforced bypenalizing the strain rate tensor on the rigid domain, while incompressibility is treatedby duality. This model allows to achieve an accurate description of fluid motion andhydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopicmodel is also used, in which the size of the bacteria is supposed to be much smaller thanthe elements of fluid: the perturbation of the fluid due to propulsion and motion of theswimmers is neglected, and the fluid is only subjected to the buoyant forcing induced bythe presence of the bacteria, which are denser than the fluid. Although this model doesnot accurately take into account hydrodynamic interactions, it is able to reproducecomplex collective dynamics observed in concentrated bacterial suspensions, such asbioconvection. From a mathematical point of view, both models lead to a minimizationproblem which is solved with a standard Finite Element Method. In order to ensurerobustness, a projection algorithm is used to deal with contacts between particles. Wealso reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation onthe oxygen concentration is solved in the fluid domain, with a source term accounting foroxygen consumption by the bacteria. The orientations of the individual bacteria aresubjected to random changes, with a frequency that depends on the surrounding oxygenconcentration, in order to favor the direction of the concentration gradient.

In epidemic modeling, the Susceptible-Alert-Infected-Susceptible (SAIS) model extends theSIS (Susceptible-Infected-Susceptible) model. In the SAIS model, “alert” individualsobserve the health status of neighbors in their contact network, and as a result, they mayadopt a set of cautious behaviors to reduce their infection rate. This alertness, whenincorporated in the mathematical model, increases the range of effective/relativeinfection rates for which initial infections die out. Built upon the SAIS model, this workinvestigates how information dissemination further increases this range. Informationdissemination is realized through an additional network (e.g., an online social network)sharing the contact network nodes (individuals) with different links. These “informationlinks” provide the health status of one individual to all the individuals she is connectedto in the information dissemination network. We propose an optimal informationdissemination strategy with an index in quadratic form relative to the informationdissemination network adjacency matrix and the dominant eigenvector of the contactnetwork. Numerical tools to exactly solve steady state infection probabilities andinfluential thresholds are developed, providing an evaluative baseline for our informationdissemination strategy. We show that monitoring the health status of a small but “central”subgroup of individuals and circulating their incidence information optimally enhances theresilience of the society against infectious diseases. Extensive numerical simulations ona survey–based contact network for a rural community in Kansas support these findings.

The shape and velocity of a sliding droplet are computed by solving the Navier-Stokes equation with free interface boundary conditions. The Galerkin finite element methodis implemented in a 2D computation domain discretized using an unstructured mesh withtriangular elements. The mesh is refined recursively at the corners (contact points). Thestationary sliding velocity is found to be strongly dependent on grid refinement, which isa consequence of the contact line singularity resolved through the effective slip across thefinite elements adjacent to the contact point. For small droplets, this dependence is wellapproximated by a theoretical estimates obtained using multiscale expansion and matchingtechnique in lubrication approximation, where the corner element size is used as a microscaleparameter. For larger droplets, the shape is also dependent on grid refinement. This questions the validity of numerous computations of flows with moving contact line where gridsare invariably much more coarse than molecular scales on which the singularity is resolved. Itis suggested that extrapolation to molecular scales should be used to obtain realistic results.

The stability of a one-spike solution to a general class of reaction-diffusion (RD) system with both regular and anomalous diffusion is analyzed. The method of matched asymptotic expansions is used to construct a one-spike equilibrium solution and to derive a nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O(1) time-scale. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined in terms of the roots of certain simple transcendental equations that involve two key parameters related to the choice of the nonlinear kinetics. From a rigorous analysis of these transcendental equations by using a winding number approach and explicit calculations, sufficient conditions are given to predict the occurrence of Hopf bifurcations of the one-spike solution. Our analysis determines explicitly the number of possible Hopf bifurcation points as well as providing analytical formulae for them. The analysis is implemented for the shadow limit of the RD system defined on a finite domain and for a one-spike solution of the RD system on the infinite line. The theory is illustrated for two specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation is unique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopf bifurcation.

The projection method is a simple way of constructing functions and filters byintegrating multidimensional functions and filters along parallel superplanes in the spacedomain. Equivalently expressed in the frequency domain, the projection method constructs anew function by simply taking a cross-section of the Fourier transform of amultidimensional function. The projection method is linked to several areas such as boxsplines in approximation theory and the projection-slice theorem in image processing. Inthis paper, we shall systematically study and discuss the projection method in the area ofmultidimensional framelet and wavelet analysis. We shall see that the projection methodnot only provides a painless way for constructing new wavelets and framelets but also is auseful analysis tool for studying various optimal properties of multidimensional refinablefunctions and filters. Using the projection method, we shall explicitly and easilyconstruct a tight framelet filter bank from every box spline filter having at least orderone sum rule. As we shall see in this paper, the projection method is particularlysuitable to be applied to frequency-based nonhomogeneous framelets and wavelets in anydimensions, and the periodization technique is a special case of the projection method forobtaining periodic wavelets and framelets from wavelets and framelets on Euclidean spaces.

We construct a stochastic model ofbacteriophage parasitism of a host bacteria that accounts fordemographic stochasticity of host and parasite and allows formultiple bacteriophage adsorption to host. We analyze the associateddeterministic model, identifying the basic reproductive number forphage proliferation, showing that host and phage persist when itexceeds unity, and establishing that the distribution of adsorbedphage on a host is binomial with slowly evolving mean. Notsurprisingly, extinction of the parasite or both host and parasitecan occur for the stochastic model.

Progression along the successive phases of the mammalian cell cycle is driven by anetwork of cyclin-dependent kinases (Cdks). This network is regulated by a variety ofnegative and positive feedback loops. We previously proposed a detailed, 39-variable modelfor the Cdk network and showed that it is capable of temporal self-organization in theform of sustained oscillations, which correspond to the repetitive, transient, sequentialactivation of the cyclin- Cdk complexes that govern the successive phases of the cellcycle [Gérard and Goldbeter (2009) Proc Natl Acad Sci 106, 21643-8]. Here we compare thedynamical behavior of three models of different complexity for the Cdk network driving themammalian cell cycle. The first is the detailed model that counts 39 variables and isbased on Michaelis-Menten kinetics for the enzymatic steps. From this detailed model, webuild a version based only on mass-action kinetics, which counts 80 variables. In thisversion we do not need to assume that enzymes are present in much smaller amounts thattheir substrates, which is not necessarily the case in the cell cycle. We show that thesetwo versions of the model for the Cdk network yield similar results. In particular theypredict sustained oscillations of the limit cycle type. We show that the model for the Cdknetwork can be reduced to a version containing only 5 variables, which is more amenable tostochastic simulations. This skeleton version retains the dynamic properties of the morecomplex versions of the model for the Cdk network in regard to Cdk oscillations. Theregulatory wiring of the Cdk network therefore governs its dynamic behavior, regardless ofthe degree of molecular detail. We discuss the relative advantages of each version of themodel, all of which support the view that the mammalian cell cycle behaves as a limitcycle oscillator.

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, an artificial neural network (ANN) based on hybrid algorithm combiningparticle swarm optimization (PSO) with back-propagation (BP) is proposed to forecast thedaily streamflows in a catchment located in a semi-arid region in Morocco. The PSOalgorithm has a rapid convergence during the initial stages of a global search, while theBP algorithm can achieve faster convergent speed around the global optimum. By combiningthe PSO with the BP, the hybrid algorithm referred to as BP-PSO algorithm is presented inthis paper. To evaluate the performance of the hybrid algorithm, BP neural network is alsoinvolved for a comparison purposes. The results show that the neural network model evolvedby PSO-BP algorithm has a good predictions and better convergence performances

Recent advances in high-resolution fluorescence microscopy have enabled the systematicstudy of morphological changes in large populations of cells induced by chemical andgenetic perturbations, facilitating the discovery of signaling pathways underlyingdiseases and the development of new pharmacological treatments. In these studies, though,due to the complexity of the data, quantification and analysis of morphological featuresare for the vast majority handled manually, slowing significantly data processing andlimiting often the information gained to a descriptive level. Thus, there is an urgentneed for developing highly efficient automated analysis and processing tools forfluorescent images. In this paper, we present the application of a method based on theshearlet representation for confocal image analysis of neurons. The shearletrepresentation is a newly emerged method designed to combine multiscale data analysis withsuperior directional sensitivity, making this approach particularly effective for therepresentation of objects defined over a wide range of scales and with highly anisotropicfeatures. Here, we apply the shearlet representation to problems of soma detection ofneurons in culture and extraction of geometrical features of neuronal processes in braintissue, and propose it as a new framework for large-scale fluorescent image analysis ofbiomedical data.

We present models of the dynamics ofphytoplankton aggregates. We start with an individual-based modelin which aggregates can grow, divide, joint and move randomly.Passing to infinity with the number of individuals, we obtain amodel which describes the space-size distribution of aggregates.The density distribution function satisfies a non-linear transport equation, which contains terms responsible for thegrowth of phytoplankton aggregates, their fragmentation,coagulation, and diffusion.

We study the growth rate of a cell population that follows anage-structured PDE with time-periodic coefficients. Our motivationcomes from the comparison between experimental tumor growth curvesin mice endowed with intact or disrupted circadian clocks,known to exert their influence on the cell division cycle.We compare the growth rate of the model controlled by a time-periodiccontrol on its coefficients with the growth rate of stationarymodels of the same nature, but with averaged coefficients. We firstly derive a delay differential equation which allows us to proveseveral inequalities and equalities on the growth rates. We alsodiscuss about the necessity to take into account the structure ofthe cell division cycle for chronotherapy modeling. Numerical simulationsillustrate the results.

In this paper we investigate the role of spatial effects in determining thedynamics of a subclass of signalling pathways characterised by their ability todemonstrate oscillatory behaviour. To this end, we formulate a simple spatial model of thep53 network that accounts for both a negative feedback and a transcriptional delay. We show that the formation of protein density patterns can depend on the shape of the cell, position of the nucleus, and the protein diffusion rates. Thetemporal changes in the total amounts of protein are also subject to spatial influences. The level of DNA damage required to induce sustained oscillations, forinstance, depends on the morphology of the cell. The model also provides a newinterpretation of experimentally observed undamped oscillations in p53 levels in single cells.Our simulations reveal that alternate sequences of high- and low-amplitudeoscillations can occur. We propose that the digital pulses may correspond to snap-shots ofour high-amplitude sequences. Shorter waiting-times between subsequent time-lapse fluorescence microscopy images in combination with lower detection thresholds may reveal the irregular high-frequency oscillations suggested by our spatialmodel.