Dynamics of stochastic heroin epidemic model with L\""evy jumps

People have paid the surge of attention to the prevention and the control of the heroin epidemic for the number of drug addicts is increasing dramatically. In the study of the heroin epidemic, modeling is an important tool. So far many heroin epidemic models are often characterized by ordinary differential equations (ODEs) and many results about them have been obtained. But unfortunately, there is little literature of stochastic heroin epidemic model with jumps. Based on this point, this paper establishes a class of heroin epidemic models---stochastic heroin epidemic model with L\"evy jumps. Under some given conditions, the existence of the global positive solution of such model is first obtained. We then study the asymptotic behavior of this model by applying the Lyapunov technique.     

Asymptotic stability of one prey and two predators model with two functional responses

We formulate a mathematical model to study the complex dynamical behavior of a three dimensional model consisting of one prey and two predators involving Beddington–DeAngelis and Crowley–Martin functional responses. The existence and stability conditions of the equilibrium points are analyzed. The global asymptotic stability of the interior equilibrium point, if exists, is proved by considering Lyapunov function. Several numerical simulations are performed to illustrate the theoretical analysis. The multiple states of stability are observed in one example whereas another example exhibits the global stability of interior equilibrium point.

     

Dynamic behaviour for a nonautonomous heroin epidemic model with time delay

In this paper, we have modified the White and Comiskey heroin epidemic model (White and Comiskey in Math. Biosci. 208:312–324, 2007) into a nonautonomous heroin epidemic model with distributed time delay. We have introduced some new threshold values R _* and R ^* and further obtained that the heroin-using career will be permanent when R _*>1 and the heroin-using career will be going to extinct when R ^*<1. Using the method of Lyapunov functional, some sufficient conditions are derived for the global asymptotic stability of the system. The aim of this modification is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.     

A note on global stability for a heroin epidemic model with distributed delay

By using the direct Lyapunov method and constructing appropriate Lyapunov functional, the global stability for the heroin epidemic model with distributed delay is investigated. It is shown that the disease endemic equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [J. Liu, T. Zhang, Global behavior of a heroin epidemic model with distributed delay, Appl. Math. Lett. 24 (2011) 1685–1692].     

Stochastic dynamics of an HIV/AIDS epidemic model with treatment

Abstract

We investigate a stochastic HIV/AIDS epidemic model with treatment. The model allows for two stages of infection namely the asymptomatic phase and the symptomatic phase. We prove existence of global positive solutions. We show that the solutions are stochastically ultimately bounded and stochastically permanent. We also study asymptotic behaviour of the solution to the stochastic model around the disease-free equilibrium of the underlying deterministic model. Our theoretical results are illustrated by way of numerical simulations.     

On global subdifferentials with applications in nonsmooth optimization

The notions of global subdifferentials associated with the global directional derivatives are introduced in the following paper. Most common used properties, a set of calculus rules along with a mean value theorem are presented as well. In addition, a diversity of comparisons with well-known subdifferentials such as Fréchet, Dini, Clarke, Michel–Penot, and Mordukhovich subdifferential and convexificator notion are provided. Furthermore, the lower global subdifferential is in fact proved to be an abstract subdifferential. Therefore, the lower global subdifferential satisfies standard properties for subdifferential operators. Finally, two applications in nonconvex nonsmooth optimization are given: necessary and sufficient optimality conditions for a point to be local minima with and without constraints, and a revisited characterization for nonsmooth quasiconvex functions.

     

Generalised linear model trees with global additive effects

Model-based trees are used to find subgroups in data which differ with respect to model parameters. In some applications it is natural to keep some parameters fixed globally for all observations while asking if and how other parameters vary across subgroups. Existing implementations of model-based trees can only deal with the scenario where all parameters depend on the subgroups. We propose partially additive linear model trees (PALM trees) as an extension of (generalised) linear model trees (LM and GLM trees, respectively), in which the model parameters are specified a priori to be estimated either globally from all observations or locally from the observations within the subgroups determined by the tree. Simulations show that the method has high power for detecting subgroups in the presence of global effects and reliably recovers the true parameters. Furthermore, treatment–subgroup differences are detected in an empirical application of the method to data from a mathematics exam: the PALM tree is able to detect a small subgroup of students that had a disadvantage in an exam with two versions while adjusting for overall ability effects.

     

Identification of mechanical properties of arteries with certification of global optimality

In this study, we consider identification of parameters in a non-linear continuum-mechanical model of arteries by fitting the models response to clinical data. The fitting of the model is formulated as a constrained non-linear, non-convex least-squares minimization problem. The model parameters are directly related to the underlying physiology of arteries, and correctly identified they can be of great clinical value. The non-convexity of the minimization problem implies that incorrect parameter values, corresponding to local minima or stationary points may be found, however. Therefore, we investigate the feasibility of using a branch-and-bound algorithm to identify the parameters to global optimality. The algorithm is tested on three clinical data sets, in each case using four increasingly larger regions around a candidate global solution in the parameter space. In all cases, the candidate global solution is found already in the initialization phase when solving the original non-convex minimization problem from multiple starting points, and the remaining time is spent on increasing the lower bound on the optimal value. Although the branch-and-bound algorithm is parallelized, the overall procedure is in general very time-consuming.

     

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT

We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.     

WELL-POSEDNESS AND GLOBAL ATTRACTORS FOR LIQUID CRYSTALS ON RIEMANNIAN MANIFOLDS

ABSTRACT

We study the coupled Navier-Stokes Ginzburg-Landau model of nematic liquid crystals introduced by F.H. Lin, which is a simplified version ofthe Ericksen-Leslie system. We generalize the model to compact n-dimensional Riemannian manifolds, deriving the system from a variational principle, and provide a very simple proof of local well-posedness for this coupled system using a contraction mapping argument. We then prove that this system is globally well-posed and has compact global attractors when the dimension of the manifold M is two. A small data result in n dimensions follows easily. Finally, we introduce the Lagrangian averaged liquid crystal equations, which arise from averaging the Navier-Stokes fluid motion over small spatial scales in the variational principle. We show that this averaged system is globally well-posed and has compact global attractors even when M is three-dimensional.     

Variational time discretizations of higher order and higher regularity

We consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied to Dahlquist’s stability problem, the presented methods provide the same stability properties as dG or cGP methods. Provided that suitable quadrature rules of Hermite type are used to evaluate the integrals in the variational conditions, the variational time discretization methods are connected to special collocation methods. For this case, we present error estimates, numerical experiments, and a computationally cheap postprocessing that allows to increase both the accuracy and the global smoothness by one order.

     

Dimension pure globale des anneaux commutatifs

For every (n,m) ? IN^?× IN ^?, verifying l We study direct systems of rings and give a bound for the pure - global - dimension of their limits, not depending on the ring cardinality. Using this result and a theorem of C0UCH0T we give examples of pure -heriditary group- rings (ie. of pure -global - dimension one) of large cardinality.

Finaly we prove - using a result of VASC0NCEL0S and a theorem of JENSEN - that some countably products of noetherian rings are coherent rings and that their pure - global - dimension is exactly two.     

A modified modulus-based matrix splitting iteration method for solving implicit complementarity problems

In this paper, a modified modulus-based matrix splitting iteration method is established for solving a class of implicit complementarity problems. The global convergence conditions are given when the system matrix is a positive definite matrix or an H₊-matrix, respectively. In addition, some numerical examples show that the proposed method is efficient.

     

Global Existence of Classical Solutions of the Nordström–Vlasov System in Two Space Dimensions

ABSTRACT

The dynamics of a self-gravitating ensemble of collisionless particles is modeled by the Nordström–Vlasov system in the framework of the Nordström scalar theory of gravitation. For this system in two space dimensions, integral representations of the first-order derivatives of the field are derived. Using these representations we show global existence of smooth solutions for large data.     

Study of a Tritrophic Food Chain Model with Non-differentiable Functional Response

We study a model of three interacting species in a food chain composed by a prey, an specific predator and a generalist predator. The capture of the prey by the specific predator is modelled as a modified Holling-type II non-differentiable functional response. The other predatory interactions are both modelled as Holling-type I. Moreover, our model follows a Leslie-Gower approach, in which the function that models the growth of each predator is of logistic type, and the corresponding carrying capacities depend on the sizes of their associated available preys. The resulting model has the form of a set of nonlinear ordinary differential equations which includes a non-differentiable term. By means of topological equivalences and suitable changes of parameters, we find that there exists an Allee threshold for the survival of the prey population in the food chain, given, effectively, as a critical level for the generalist predator. The dynamics of the model is studied with analytical and computational tools for bifurcation theory. We present two-parameter bifurcation diagrams that contain both local phenomena (Hopf, saddle-node transcritical, cusp, Bogdanov-Takens bifurcations) and global events (homoclinic and heteroclinic connections). In particular, we find that two types of heteroclinic cycles can be formed, both of them containing connections to the origin. One of these cycles is planar involving the absence of the specific predator. In turn, the other heteroclinic cycle is formed by connections in the full three-dimensional phase space.

     

ON THE STRUCTURE OF NON-COMMUTATIVE REGULAR LOCAL RINGS OF DIMENSION TWO

ABSTRACT

In this paper we conjecture that the center of a non-commutative complete regular local ring of global dimension two is a formal power series ring in two variables. We prove this conjecture in the special case of Ore extensions.     

On the order of global branches

We give both a topological and an algebraic definition of the order of a global branch of a variety along a subvariety. Then we show that these two definitions agree. Finally we compare this order with the orders of the related local branches.

---

Sunto Diamo due definizioni, una topologica e una algebrica, di ordine di un ramo globale di una varietà lungo una sottovarietà. Poi mostriamo che queste due definizioni coincidono. Infine confrontiamo questo ordine con gli ordini dei relativi rami locali.

---

---

Partially supported by CNR funds.     

Lévy noise perturbation for an epidemic model with impact of media coverage

ABSTRACT

This work is devoted to study the existence and uniqueness of global positive solution for a stochastic epidemic model with media coverage driven by Lévy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Numerical simulations are presented to confirm the theoretical results.     

Far Casting Cross-Validation

Cross-validation has long been used for choosing tuning parameters and other model selection tasks. It generally performs well provided the data are independent, or nearly so. Improvements have been suggested which address ordinary cross-validation’s (OCV) shortcomings in correlated data. Whereas these techniques have merit, they can still lead to poor model selection in correlated data or are not readily generalizable to high-dimensional data.

The proposed solution, far casting cross-validation (FCCV), addresses these problems. FCCV withholds correlated neighbors in every aspect of the cross-validation procedure. The result is a technique that stresses a fitted model’s ability to extrapolate rather than interpolate. This generally leads to better model selection in correlated datasets.

Whereas FCCV is less than optimal in the independence case, our improvement of OCV applies more generally to higher dimensional error processes and to both parametric and nonparametric model selection problems. To facilitate introduction, we consider only one application, namely estimating global bandwidths for curve estimation with local linear regression. We provide theoretical motivation and report some comparative results from a simulation experiment and on a time series of annual global temperature deviations. For such data, FCCV generally has lower average squared error when disturbances are correlated.

Supplementary materials are available online.