Global analysis of a periodic epidemic model on cholera in presence of bacteriophage

We propose and analyze a recurrent epidemic model of cholera in the presence of bacteriophage. The model is extended by general periodic incidence functions for low‐infectious bacterium and high‐infectious bacterium, respectively. A general periodic shedding function for two infected class (phage‐positive and phage‐negative) and a generalized contact and intrinsic growth function for susceptible class are also considered. Under certain biological assumptions, we derive the basic reproduction number (R₀) in a periodic environment for the proposed model. We also observe the global stability of the disease‐free equilibrium, existence, permanence, and global stability of the positive endemic periodic solution of our proposed model. Finally, we verify our results with specific functional form. Copyright © 2016 John Wiley & Sons, Ltd.     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Within‐host and synaptic transmissions: contributions to the spread of HIV infection

We study the contributions of within‐host (virus‐to‐cell) and synaptic (cell‐to‐cell) transmissions in a mathematical model for human immunodeficiency virus epidemics. The model also includes drug resistance. We prove the local and global stability of the disease‐free equilibrium and the local stability of the endemic equilibrium. We analyse the effect of the cell‐to‐cell transmission rate on the value of the reproduction number, R₀. Moreover, we show evidence of a qualitative change in the models' dynamics, subjected to the value of the drug efficacy. In the end, important inferences are drawn. Copyright © 2016 John Wiley & Sons, Ltd.     

On the stabilizing effect of convection in three‐dimensional incompressible flows

We investigate the stabilizing effect of convection in three‐dimensional incompressible Euler and Navier‐Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three‐dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three‐dimensional Euler or Navier‐Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier‐Stokes equations. We will present numerical evidence that seems to support that the three‐dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three‐dimensional model and how the convection term in the full Euler and Navier‐Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. © 2008 Wiley Periodicals, Inc.     

A model of thermal dissipation for a one‐dimensional viscous reactive and radiative gas

We consider a model of thermal dissipation for a Stefan–Boltzmann model of viscous and reactive gas in a bounded interval. We prove the existence of a global‐in‐time solution, and we give the asymptotic behaviour for the corresponding Dirichlet problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number

We study asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the global attractors to the Boussinesq system for Rayleigh‐Bénard convection converge to that of the infinite‐Prandtl‐number model for convection as the Prandtl number approaches infinity. This offers partial justification of the infinite‐Prandtl‐number model for convection as a valid simplified model for convection at large Prandtl number even in the long‐time regime. © 2006 Wiley Periodicals, Inc.     

Justifying the small‐world phenomenon via random recursive trees

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three‐fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank‐based selection model, the Aiello‐Chung‐Lu models, the generalized linear preference model, directed scale‐free graphs, the Cooper‐Frieze model, and random unordered increasing k‐trees. Our results shed light on why the small‐world phenomenon is observed in so many real‐world graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 201–224, 2017     

On smoothed analysis in dense graphs and formulas

We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman and colleagues (Random Struct Algor 3 ; Random Struct Algor 4 ). Here we obtain a sharp threshold for the appearance of a fixed subgraph and for certain Ramsey properties. We also consider a related model of random k‐SAT formulas, where an instance is obtained by adding random k‐clauses to a fixed formula with a given number of clauses, and derive tight bounds for the non‐satisfiability of the thus‐obtained random formula. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

COPAR—multivariate time series modeling using the copula autoregressive model

The analysis of multivariate time series is a common problem in areas like finance and economics. The classical tools for this purpose are vector autoregressive models. These however are limited to the modeling of linear and symmetric dependence. We propose a novel copula‐based model that allows for the non‐linear and non‐symmetric modeling of serial as well as between‐series dependencies. The model exploits the flexibility of vine copulas, which are built up by bivariate copulas only. We describe statistical inference techniques for the new model and discuss how it can be used for testing Granger causality. Finally, we use the model to investigate inflation effects on industrial production, stock returns and interest rates. In addition, the out‐of‐sample predictive ability is compared with relevant benchmark models. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence and uniqueness of measure valued solutions to a Vlasov‐type equation with local alignment

We use a particle method to study a Vlasov‐type equation with local alignment, which was proposed by Sebastien Motsch and Eitan Tadmor [J. Statist. Phys., 141(2011), pp. 923‐947]. For N‐particle system, we study the unconditional flocking behavior for a weighted Motsch‐Tadmor model and a model with a “tail”. When N goes to infinity, global existence and stability (hence uniqueness) of measure valued solutions to the kinetic equation of this model are obtained. We also prove that measure valued solutions converge to a flock. The main tool we use in this paper is Monge‐Kantorovich‐Rubinstein distance.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

On the asymptotics of fault probability in least‐recently‐used caching with Zipf‐type request distribution

We consider the least‐recently‐used cache replacement rule with a Zipf‐type page request distribution and investigate an asymptotic property of the fault probability with respect to an increase of cache size. We first derive the asymptotics of the fault probability for the independent‐request model and then extend this derivation to a general dependent‐request model, where our result shows that under some weak assumptions the fault probability is asymptotically invariant with regard to dependence in the page request process. In a previous study, a similar result was derived by applying a Poisson embedding technique, where a continuous‐time proof was given through some assumptions based on a continuous‐time modeling. The Poisson embedding, however, is just a technique used for the proof and the problem is essentially on a discrete‐time basis; thus, it is preferable to make assumptions, if any, directly in the discrete‐time setting. We consider a general dependent‐request model and give a direct discrete‐time proof under different assumptions. A key to the proof is that the numbers of requests for respective pages represent conditionally negatively associated random variables. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Rate‐independent behavior of ferromagnets

We review properties of a model allowing for rate‐independent evolution ofbulk ferromagnets. The model is based on Brown's micromagnetic theory specialized to a large‐body limit by DeSimone [2] which is enriched by a rate‐independent dissipative mechanism. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bankruptcy prediction by generalized additive models

We compare several accounting‐based models for bankruptcy prediction. The models are developed and tested on large data sets containing annual financial statements for Norwegian limited liability firms. Out‐of‐sample and out‐of‐time validation shows that generalized additive models significantly outperform popular models like linear discriminant analysis, generalized linear models and neural networks at all levels of risk. Further, important issues like default horizon and performance depreciation are examined. We clearly see a performance depreciation as the default horizon is increased and as time goes by. Finally a multi‐year model, developed on all available data from three consecutive years, is compared with a one‐year model, developed on data from the most recent year only. The multi‐year model exhibits a desirable robustness to yearly fluctuations that is not present in the one‐year model. Copyright © 2006 John Wiley & Sons, Ltd.     

Stabilized nodal‐based finite element methods for the magnetic induction problem

We consider the time‐dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal‐based finite element approximations of the induction model with inf‐sup stable finite elements for the magnetic field and the magnetic pseudo‐pressure are investigated. Based on a residual‐based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal‐based finite element method for the numerical solution. Error estimates are given for the semi‐discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Well‐posedness for the FENE dumbbell model of polymeric flows

We prove local and global well‐posedness for the FENE dumbbell model for a very general class of potentials. Indeed, in prior local or global well‐posedness results, conditions on the strength of the singularity (or on the parameter b) were made. Here we give a proof in the general case. We also prove global existence results if the data is small or if we restrict to the co‐rotational model in dimension 2. © 2008 Wiley Periodicals, Inc.     

Nonexistence of coexisting steady‐state solutions of Dirichlet problem for a cross‐diffusion model

We study a special case of Shigesada–Kawasaki–Teramoto (SKT) model for two competing species with the Dirichlet boundary conditions. In our case, one of the species is not influenced by self‐diffusion or cross‐diffusion. We specify the explicit range of parameters by contradiction such that there are no coexisting steady‐state solutions to the model.     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

Convergence of a Force‐Based Hybrid Method in Three Dimensions

We study a force‐based hybrid method that couples an atomistic model with the Cauchy‐Born elasticity model. We show that the proposed scheme converges to the solution of the atomistic model with second‐order accuracy, since the ratio between lattice parameter and the characteristic length scale of the deformation tends to 0. Convergence is established for the three‐dimensional system without defects, with general finite‐range atomistic potential and simple lattice structure. The proof is based on consistency and stability analysis. General tools for stability analysis are developed in the framework opseudodifference operators in arbitrary dimensions. © 2012 Wiley Periodicals, Inc.     

Dynamical study of fractional order differential equations of predator‐pest models

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.