Dynamically-consistent non-standard finite difference method for an epidemic model

This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss–Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens’ non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered.     

The Sinc-Galerkin method for parameter-dependent self-adjoint problems

The Sinc-Galerkin method is being applied to a growing number of diverse problems in ordinary and partial differential equations including both forward and inverse (parameter recovery) problems. As a result of these continuing extensions, the treatment of parameter-dependent problems needs to be thoroughly investigated. Two specific questions considered here are the incorporation of various nonhomogeneous boundary conditions and the treatment of a variable parameter. The latter topic is particularly important for inverse problems that arise when numerically estimating physical parameters. The point of view taken emphasizes the maintenance of the classical exponential convergence rate. The techniques described are suitable both for the direct problem and for the parameter estimation problem. Numerical results are presented to substantiate the accuracy of the method.     

Optimal control applied on an HIV‐1 within‐host model

The treatment of human immunodeficiency virus (HIV) remains a major challenge, even if significant progress has been made in infection treatment by ‘drug cocktails’. Nowadays, research trend is to minimize the number of pills taken when treating infection. In this paper, an HIV‐1 within host model where healthy cells follow a simple logistic growth is considered. Basic reproduction number of the model is calculated using next generation matrix method, steady states are derived; their local, as well as global stability, is discussed using the Routh–Hurwitz criteria, Lyapunov functions and the Lozinskii measure approach. The optimal control policy is formulated and solved as an optimal control problem. Numerical simulations are performed to compare several cases, representing a treatment by Interleukin2 alone, classical treatment by multitherapy drugs alone, then both treatments at the same time. Objective functionals aim to (i) minimize infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood. Copyright © 2015 John Wiley & Sons, Ltd.     

Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA

In this paper, we consider the case of downside risk measures with cardinality and bounding constraints in portfolio selection. These constraints limit the amount of capital to be invested in each asset as well as the number of assets composing the portfolio. While the standard Markowitz’s model is a convex quadratic program, this new model is a NP-hard mixed integer quadratic program. Realizing the computational intractability for this class of problems, especially large-scale problems, we first reformulate it as a DC program with the help of exact penalty techniques in Difference of Convex functions (DC) programming and then solve it by DC Algorithms (DCA). To check globality of computed solutions, a global method combining the local algorithm DCA with a Branch-and-Bound algorithm is investigated. Numerical simulations show that DCA is an efficient and promising approach for the considered problem.      

A Method of Dealing with Certain Non-Linear Allocation Problems Using the Transportation Technique

An investigation was carried out to establish a method of determining the allocation between certain factories of a known production requirement so as to minimize the total expense of production. Each factory was made up of a number of departments or services whose total expense varied non-linearly with production level.The method was to divide the production range of each department in each factory into regions of approximate linear expense, and consider the associated total company expense connected with the optimum allocation for each possible group of regions (one region in each department). In theory the simplex technique could be applied to each group to find its optimum allocation, but as the number of groups is prodigious for quite simple problems this is not practicable. (In the actual problem for which the method was produced there was of the order of 100,000 groups.)A procedure was devised using the transportation technique together with various lemmas proved in the text and feasibility considerations, by which it was possible to reduce rapidly the number of groups needed to be considered in detail to just a few.Other problems should be amenable to this method provided they are similar to the one above in two respects.(1) The departmental variation of expense with production from one region to the next is in general continuous with decreasing rate of change of expense, i.e. the variation is concave.(2) It should be possible to put the requirements into one of the units of capacity usage in which there are important physical or maximum capacity restrictions.In addition, the method is particularly suitable when the number of factories is not large.As the method required repeated applications of the transportation model, a technique was devised for obtaining an optimum solution to this model more quickly than by the usual techniques. This technique will be described in another paper.An example of a whole problem is given after the Appendix. A full solution of this problem is shown in which occur most of the points dealt with in the text.The problem for which the method was devised, was one of minimizing the total expense of production of twenty common products between three factories. Each factory had seven departments or services, in most of which two regions were required to describe satisfactorily a department's variation of expense with production, in linear terms. A complete solution was obtained in a few days using the method described without recourse to an electronic computer (not counting the initial computations of costs, etc. required).     

Numerical method for the inverse heat transfer problem in composite materials with Stefan-Boltzmann conditions

In this paper, one specific kind of heat transfer problem with nonlinear Stefan-Boltzmann conditions are considered in a three dimensional multi-layer domain. Theoretical results for forward and inverse problems are presented. Numerical simulations of specific models from applications are provided to demonstrate the heat transfer process in the composite materials of forward problem. One reconstruction method is proposed to find the corrosion part, and the numerical examples show that the reconstruction algorithm is effective.     

A combined molecular dynamics-phase-field modelling approach to fracture

In order to better understand and ease the determination of material and model parameters required for the macroscopic modelling of brittle fracture, a bottom-up comparative study between molecular dynamics (MD) simulations and the continuum phase-field modelling (PFM) is carried out. In particular, based on the MD simulations of fracture of a highly brittle material, a number of PFM parameters such as the width of the transition zone between the damaged and the undamaged material, the crack resistance and the elasticity modulus are estimated. This study opens the door for an efficient way for multi-scale modelling of fracture. To illustrate this approach, a comparative two-dimensional numerical initial-boundary-value problem (IBVP) for the highly brittle aragonite (CaCO₃) is presented. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-coloring and job-scheduling with assignment and incompatibility costs

Consider a scheduling problem (P) which consists of a set of jobs to be performed within a limited number of time periods. For each job, we know its duration as an integer number of time periods, and preemptions are allowed. The goal is to assign the required number of time periods to each job while minimizing the assignment and incompatibility costs. When a job is performed within a time period, an assignment cost is encountered, which depends on the involved job and on the considered time period. In addition, for some pairs of jobs, incompatibility costs are encountered if they are performed within common time periods. (P) can be seen as an extension of the multi-coloring problem. We propose various solution methods for (P) (namely a greedy algorithm, a descent method, a tabu search and a genetic local search), as well as an exact approach. All these methods are compared on different types of instances.     

Application and modification of the POD method and the POD-DEIM for model reduction in porous-media simulations

Researchers with a continuum-mechanical background typically use a multi-phasic and multi-component modelling approach for materials with a saturated porous microstructure. Therefore, the mechanical behaviour is considered in a continuum-mechanical manner and solved using the finite-element method (FEM). The developed models need to be complex enough to capture the relevant properties of the considered materials, what often results in expensive simulations with a very large number of degrees of freedom (DOF). The aim of the present contribution is to reduce the computing time of these simulations through model-reduction methods, while the accuracy of the solution needs to be maintained. Therefore, the method of proper-orthogonal decomposition (POD) for linear problems and the discrete-empirical-interpolation method (DEIM) in combination with the POD method (POD-DEIM) for nonlinear problems are investigated. Using the POD method, a given data set is approximated with a low-dimensional subspace. To generate this data set, the vector of unknowns of the FE simulation is stored in a pre-computation in the full (unreduced) system in each time-step (so-called “snapshots” of the system). Dealing with porous-media problems, the primary variables are the solid displacement, the pore pressure and, depending on the particular problem, other primary variables. Following this, the primary variables have entries with very huge differences in their absolute values. As a result, non-negligible rounding errors may occur when applying the POD method. To overcome this problems, modifications of the classical POD method need to be performed for such problems. The present contribution discusses this issue and presents results for the reduced simulations of porous media. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于惩罚似然比的高维空间相关过程EWMA质量监控模型

变量选择控制图是高维统计过程监控的重要方法。针对传统变量选择控制图较少考虑高维过程空间相关性而造成监控效率低的问题,提出一种基于Fused-LASSO的高维空间相关过程监控模型。首先,利用Fused LASSO算法对似然比检验进行改进;然后,推导出基于惩罚似然比的监控统计量;最后,通过仿真模拟和真实案例分析所提监控模型的性能。仿真实验和真实案例均表明:在高维空间相关过程中,当相邻监控变量同时发生异常时,利用所提监控方法能够准确识别潜在异常变量,取得较好的监控效果。     

Integrated lot-sizing and one-dimensional cutting stock problem with usable leftovers

This paper addresses the integration of the lot-sizing problem and the one-dimensional cutting stock problem with usable leftovers (LSP-CSPUL). This integration aims to minimize the cost of cutting items from objects available in stock, allowing the bringing forward production of items that have known demands in a future planning horizon. The generation of leftovers, that will be used to cut future items, is also allowed and these leftovers are not considered waste in the current period. Inventory costs for items and leftovers are also considered. A mathematical model for the LSP-CSPUL is proposed to represent this problem and an approach, using the simplex method with column generation, is proposed to solve the linear relaxation of this model. A heuristic procedure, based on a relax-and-fix strategy, was also proposed to find integer solutions. Computational tests were performed and the results show the contributions of the proposed mathematical model, as well as, the quality of the solutions obtained using the proposed method.

     

Spreading and vanishing in a West Nile virus model with expanding fronts

We study a simplified version of a West Nile virus(WNv) model discussed by Lewis et al.(2006),which was considered as a first approximation for the spatial spread of WNv. The basic reproduction number R_0 for the non-spatial epidemic model is defined and a threshold parameter R_0~D for the corresponding problem with null Dirichlet boundary condition is introduced. We consider a free boundary problem with a coupled system, which describes the diffusion of birds by a PDE and the movement of mosquitoes by an ODE. The risk index R_0~F(t) associated with the disease in spatial setting is represented. Sufficient conditions for the WNv to eradicate or to spread are given. The asymptotic behavior of the solution to the system when the spreading occurs is considered. It is shown that the initial number of infected populations, the diffusion rate of birds and the length of initial habitat exhibit important impacts on the vanishing or spreading of the virus. Numerical simulations are presented to illustrate the analytical results.     

Forward dynamics simulation of human body under tilting perturbations

Human body uses different strategies to maintain its stability and these strategies vary from fixed-foot strategies to strategies which foot is moved in order to increase the support base. Tilting movement of foot is one type of the perturbations usually is exposed to human body. In the presence of such perturbations human body must employ appropriate reactions to prevent threats like falling. But it is not clear that how human body maintains its stability by central nervous system (CNS). At present study it is tried that by presenting a musculoskeletal model of human lower extremity with four links, three degrees of freedom (DOF) and eight skeletal muscles, the level of muscle activations causes the maintenance of stability, be investigated. Using forward dynamics solution, leads to a more general problem, rather than inverse dynamics. Hence, forward dynamics solution by forward optimization has been used for solving this highly nonlinear problem. To this end, first the system’s equations of motion has been derived using lagrangian dynamics. Eight Hill-type muscles as actuators of the system were modeled. Because determination of muscle forces considering their number is an undetermined problem, optimization of an appropriate goal function should be practiced. For optimization problem, the characteristics of genetic algorithms as a method based on direct search, and the direct collocation method, has been profited. Also by considering requirements of problem, some constraints such as conservation of model stability are entered into optimization procedure. Finally to investigate validation of model, the results from optimization and experimental data are compared and good agreements are obtained.     

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.     

Fast approximate truncated SVD

This paper presents a new method for the computation of truncated singular value decomposition (SVD) of an arbitrary matrix. The method can be qualified as deterministic because it does not use randomized schemes. The number of operations required is asymptotically lower than that using conventional methods for nonsymmetric matrices and is at a par with the best existing deterministic methods for unstructured symmetric ones. It slightly exceeds the asymptotical computational cost of SVD methods based on randomization; however, the error estimate for such methods is significantly higher than for the presented one. The method is one‐pass, that is, each value of the matrix is used just once. It is also readily parallelizable. In the case of full SVD decomposition, it is exact. In addition, it can be modified for a case when data are obtained sequentially rather than being available all at once. Numerical simulations confirm accuracy of the method.     

A boundary integral formulation for elastically deformable particles in a viscous fluid

This paper reports a formulation and implementation of a mixed (both direct and indirect) boundary element method using the double layer and its adjoint in a form suitable for solving Stokes flow problems involving elastically deformable particles. The formulation is essentially the Completed Double Layer Boundary Element Method for solving an exterior traction problem for the surrounding fluid or solid phase, followed by an interior displacement, and a mobility problem (if required) for the elastic particles. At the heart of the method is a deflation procedure that allows iterative solution strategies to be adopted, effectively opens the way for large-scale simulations of suspensions of deformable particles to be performed. Several problems are considered, to illustrate and benchmark the method. In particular, an analytical solution for an elastic sphere in an elongational flow is derived. The stresslet calculations for an elastic sphere in shear and elongational flows indicate that elasticity of the inclusions can potentially lead to positive second normal stress difference in shear flow, and an increase in the tensile resistance in elongational flow.This work is supported by a grant from the Australian Research Grant Council. X-J F wishes to acknowledge the support of the National Natural Science Foundation of China.     

CFD simulation of a packed-bed solid-state fermentation bioreactor

In recent years, Solid-State Fermentation (SSF) has shown much promise for the development of bioprocesses and products. SSF involves the growth of microorganisms within a bed of moist solid particles permeated by a continuous gas phase and a minimum of visible water. SSF offers potential advantages over submerged culture: since the concentrations of products are often higher, smaller bioreactors can be used, reducing operational costs. However, there is a major challenge in obtaining adequate heat and mass transfer when this fermentation method is used at large scales. Mathematical models and computer simulations are useful tools for designing strategies to overcome this challenge; use of these tools can reduce costs of experimental development programs at pilot-scale and production scale, by reducing the number of fermentation experiments required. In the current work, we used the commercial CFD software ANSYS FLUENT® 16.0 to develop a mathematical model for heat and mass transfer in a pilot-scale packed-bed bioreactor. The model was used to simulate two different experiments that had been carried out previously in the bioreactor: first, the cooling of a bed of soybeans and, second, the heating of a bed composed of a mixture of wheat bran and sugarcane bagasse. The simulations considered the dynamics of airflow in the porous substrate bed and the non-equilibrium transfer of heat and moisture between the solid and gas phases. The second simulation considered a heterogeneous distribution of porosity within the substrate bed. Even though the two experiments were quite different, the same mathematical model was able to represent the temperature profiles observed experimentally. In the second simulation, the average temperature difference between the experimental and predicted values was 0.07°C. A third simulation was done for the growth of the filamentous fungus Aspergillus niger in SSF, with the predictions being compared to the results of a traditional mathematical model based on differential equations. Our work provides the basis for the development of a suitable and reliable mathematical model for testing operating conditions and control strategies for large-scale cultivation of microorganisms in SSF bioreactors.     

Bi-criterion procedures to support logistics decision making: cost and uncertainty

In practical decision making, one often is interested in solutions that balance multiple objectives. In this study we focus on generating efficient solutions for optimization problems with two objectives and a large but finite number of feasible solutions. Two classical approaches exist, being the constraint method and the weighting method, for which a specific implementation is required for this problem class. This paper elaborates specific straightforward implementations and applies them to a practical allocation problem, in which transportation cost and risk of shortage in supplied livestock quality are balanced. The variability in delivered quality is modelled using a scenario-based model that exploits historical farmer quality delivery data. The behaviour of both implementations is illustrated on this specific case, providing insight in (i) the obtained solutions, (ii) their computational efficiency. Our results indicate how efficient trade-offs in bi-criterion problems can be found in practical problems.     

Gradient-based approximation methods applied to the optimal design of vehicle suspension systems using computational models with severe inherent noise

The principal aim of this paper is to evaluate the feasibility of using gradient-based approximation methods for the optimisation of the spring and damper characteristics of an off-road vehicle, for both ride comfort and handling. The Sequential Quadratic Programming algorithm and the relatively new Dynamic-Q method are the two successive approximation methods used for the optimisation. The determination of the objective function value is performed using computationally expensive numerical simulations that exhibit severe inherent numerical noise. The use of forward finite differences and central finite differences for the determination of the gradients of the objective function within Dynamic-Q is also investigated. This is done in investigating methods for overcoming the difficulties associated with the optimisation of noisy objective functions.A recreational off-road vehicle is modelled in ADAMS, and coupled to MATLAB for the execution of the optimisation process. The full vehicle ADAMS model includes suspension kinematics, a load-dependent tyre model, as well as non-linear springs and dampers. Up to four design variables are considered in modelling the suspension characteristics.It is found that both algorithms perform well in optimising handling. However, difficulties are encountered in obtaining improvements in the design process when ride comfort is considered. Nevertheless, meaningful design configurations are still achievable through the proposed optimisation process, at a relatively low cost in terms of the number of simulations that have to be performed.     

Optimal control of switched systems with time delay

In this note, we develop a computational method for solving an optimal control problem which is governed by a switched dynamical system with time delay. Our approach is to parameterize the switching instants as a new parameter vector to be optimized. Then, we derive the required gradient of the cost function which is obtained via solving a number of delay differential equations forward in time. On this basis, the optimal control problem can be solved as a mathematical programming problem.