A nonautonomous epidemic model on time scales

We obtain conditions for permanence and extinction of the infection for a nonautonomous SIQR model defined on an arbitrary time scale. The threshold conditions are given by some numbers that play the role of the basic reproduction number in this setting. As a particular case of our result, we recover several threshold conditions obtained in the literature, on discrete or continuous time, for autonomous, periodic models and general nonautonomous models and we also discuss some new situations, including an aperiodic time scale.     

Identification and Inference for Multivariate Cointegrated and Ergodic Gaussian Diffusions

Inference is considered in the multivariate continuous time Gaussian Ornstein-Uhlenbeck (OU) model on the basis of observations in discrete time. Under the hypothesis of ergodicity as well as cointegration, the classical identification or ‘aliasing’ problem is re-addressed and new results given. Exact conditions are given for (i) identification of individual parameters, as well as results for, (ii) identification of rank and cointegration parameters, and, furthermore (iii) for the existence of a continuous time OU process which embeds a discrete time vector autoregression. Estimation and cointegration rank inference are discussed. An empirical illustration is given in which the ‘cost-of-carry’ hypothesis is investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Software reliability assessment models based on cumulative bernoulli trial processes

The binomial software reliability growth model (SRGM) contains most existing SRGMs proposed in earlier work as special cases, and can describe every software failure-occurrence pattern in continuous time. In this paper, we propose generalized binomial SRGMs in both continuous and discrete time, based on the idea of cumulative Bernoulli trials. It is shown that the proposed models give some new unusual discrete models as well as the well-known continuous SRGMs. Through numerical examples with actual software failure data, two estimation methods for model parameters with grouped data are provided, and the predictive model performance is examined quantitatively.     

On the Relations Between Discrete and Continuous Complexity Theory

Relations between discrete and continuous complexity models are considered. The present paper is devoted to combine both models. In particular we analyze the 3-Satisfiability problem. The existence of fast decision procedures for this problem over the reals is examined based on certain conditions on the discrete setting. Moreover we study the behaviour of exponential time computations over the reals depending on the real complexity of 3-Satisfiability. This will be done using tools from complexity theory over the integers.     

A general method for finding the optimal threshold in discrete time

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem. The optimal threshold is given by the threshold of the myopic stopping time in the auxiliary problem. Supplying also a sufficient condition for our approach to work, we obtain solutions for many prominent examples in the literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev, and the American put in option pricing under general conditions. As a further application we show that for underlying random walks (and Lévy processes in continuous time), general monotone and log-concave reward functions g lead to one-sided stopping problems.     

On exact solutions to epidemic dynamic models

In this study, we address an SIR (susceptible-infected-recovered) model that is given as a system of first order differential equations and propose the SIR model on time scales which unifies and extends continuous and discrete models. More precisely, we derive the exact solution to the SIR model and discuss the asymptotic behavior of the number of susceptibles and infectives. Next, we introduce an SIS (susceptible-infected-susceptible) model on time scales and find the exact solution. We solve the models by using the Bernoulli equation on time scales which provides an alternative method to the existing methods. Having the models on time scales also leads to new discrete models. We illustrate our results with examples where the number of infectives in the population is obtained on different time scales.     

Oscillation theorems and Rayleigh principle for linear Hamiltonian and symplectic systems with general boundary conditions

The aim of this paper is to establish the oscillation theorems, Rayleigh principle, and coercivity results for linear Hamiltonian and symplectic systems with general boundary conditions, i.e., for the case of separated and jointly varying endpoints, and with no controllability (normality) and strong observability assumptions. Our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with Dirichlet boundary conditions, and the problem with jointly varying endpoints into a problem with separated endpoints. These more general results on time scales then provide new results for the continuous time linear Hamiltonian systems as well as for the discrete symplectic systems. This paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions. Furthermore, the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories.     

Problems with the estimation of stochastic differential equations using structural equations models

The present paper deals with the identification and maximum likelihood estimation of systems of linear stochastic differential equations using panel data. So we only have a sample of discrete observations over time of the relevant variables for each individual. A popular approach in the social sciences advocates the estimation of the “exact discrete model” after a reparameterization with LISREL or similar programs for structural equations models. The “exact discrete model” corresponds to the continuous time model in the sense that observations at equidistant points in time that are generated by the latter system also satisfy the former. In the LISREL approach the reparameterized discrete time model is estimated first without taking into account the nonlinear mapping from the continuous to the discrete time parameters. In a second step, using the inverse mapping, the fundamental system parameters of the continuous time system in which we are interested, are inferred. However, some severe problems arise with this “indirect approach”. First, an identification problem may arise in multiple equation systems, since the matrix exponential function denning some of the new parameters is in general not one‐to‐one, and hence the inverse mapping mentioned above does not exist. Second, usually some sort of approximation of the time paths of the exogenous variables is necessary before the structural parameters of the system can be estimated with discrete data. Two simple approximation methods are discussed. In both approximation methods the resulting new discrete time parameters are connected in a complicated way. So estimating the reparameterized discrete model by OLS without restrictions does not yield maximum likelihood estimates of the desired continuous time parameters as claimed by some authors. Third, a further limitation of estimating the reparameterized model with programs for structural equations models is that even simple restrictions on the original fundamental parameters of the continuous time system cannot be dealt with. This issue is also discussed in some detail. For these reasons the “indirect method” cannot be recommended. In many cases the approach leads to misleading inferences. We strongly advocate the direct estimation of the continuous time parameters. This approach is more involved, because the exact discrete model is nonlinear in the original parameters. A computer program by Hermann Singer that provides appropriate maximum likelihood estimates is described.     

Discrete time queues and matrix-analytic methods

This is an expository paper dealing with discrete time analysis of queues using matrix-analytic methods (MAM). Discrete time analysis queues has always been popular with engineers who are very keen on obtaining numerical values out of their analyses for the sake of experimentation and design. As telecommunication systems are based more on digital technology these days than analog the need to use discrete time analysis for queues has become more important. Besides, we find that several queues which are difficult to analyse by the continuous time approach are sometimes easier to analyse using discrete time method. Of course, there are some queueing problems which are easier to analyse using continuous time approach instead of discrete time. We discuss, in this paper, both the advantages and disadvantages of discrete time analysis. The paper focusses on setting up several queueing systems as discrete time quasi-birth-and-death processes and then shows how to use matrix-geometric method (MGM), a class of MAM, to analyse the problems. We point out that there are other methods for analysing such queues but MGM provides a much simpler approach for setting up the problems in order to obtain semi-explicit results for computational tractability. We also point out some of the shortcomings of MGM. The paper mainly focusses on the Geo/Geo/1, PH/PH/1, GI/G/1 and GI/G/1/K systems and some of the related problems, including vacation models with time-limited visits.     

The Likelihood of the Parameters of a Continuous Time Vector Autoregressive Model

This paper provides a method of constructing the likelihood function of the parameters of a continuous time vector autoregressive model on the basis of discrete data without requiring the restrictions extant methods impose on the data that are capable of being rejected by a statistical test. In particular, the method does not rely on a steady-state assumption that can rule out unit root processes; it allows for weak assumptions on the innovations; and it allows for a mixture of skip-sampled and temporally-aggregated data. This revised version was published online in August 2006 with corrections to the Cover Date.     

Discrete and continuous time representations and mathematical models for large production scheduling problems: A case study from the pharmaceutical industry

The underlying time framework used is one of the major differences in the basic structure of mathematical programming formulations used for production scheduling problems. The models are either based on continuous or discrete time representations. In the literature there is no general agreement on which is better or more suitable for different types of production or business environments. In this paper we study a large real-world scheduling problem from a pharmaceutical company. The problem is at least NP-hard and cannot be solved with standard solution methods. We therefore decompose the problem into two parts and compare discrete and continuous time representations for solving the individual parts. Our results show pros and cons of each model. The continuous formulation can be used to solve larger test cases and it is also more accurate for the problem under consideration.     

Existence and Uniqueness of Solutions to Singular Higher Order Two-point BVPs on Time Scales

This paper investigates the existence and uniqueness of solutions for singular higher order boundary value problems on time scales by using mixed monotone method.The theorems obtained are very general.For the different time scale,the problem may be the corresponding continuous or discrete boundary value problem.     

A dynamical systems framework for intermittent data assimilation

We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.     

Existence of periodic solutions to a system with functional response on time scales

This paper investigates the existence of periodic solutions of a three-species food-chain diffusive system with Beddington-DeAngelis functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the system. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the method is unified to provide the existence of the desired solutions for continuous differential equations and discrete difference equations.     

Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and ρ:-convex functions

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth generalized fractional programming problems containing ρ-convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing two parametric and four parameter-free duality models and proving appropriate duality theorems. Several classes of generalized fractional programming problems, including those with arbitrary norms, square roots of positive semidefinite quadratic forms, support functions, continuous max functions, and discrete max functions, which can be viewed as special cases of the main problem are briefly discussed. The optimality and duality results developed here also contain, as special cases, similar results for nonsmooth problems with fractional, discrete max, and conventional objective functions which are particular cases of the main problem considered in this paper     

均值-方差准则下的多期最优保险投资(英文)

近年来,最优保险投资问题吸引了越来越多的注意。一般这个问题是在连续时间框架下来研究的。本文针对这一问题建立离散时间的最优控制模型。应用动态规划原理求解模型对应的近似问题,得到了最优投资策略和投资有效边界的解析表达形式。本文得到的最优投资策略和投资有效边界均依赖于承保参数。通过数值例子分析了承保参数对最优投资策略和有效边界的影响。     

On the semi-discretization of optimal control problems for networks of elastic strings: global optimality systems and domain decomposition

We consider a star-graph as an examplary network, with elastic strings stretched along the edges. The network is allowed to perform out-of-the plane displacements. We consider such networks as being controlled at its simple nodes via Dirichlet conditions. The objective is to steer given initial data to final target data in a given time T with minimal control costs. This problem is discussed in the continuous as well as in the discrete case. We discuss an iterative domain decomposition technique and its discrete analogue. We prove convergence and show some numerical results.     

A System Dynamics Model of Submarine Operations and Maintenance Schedules

The extent to which a proposed military force will achieve operational objectives is a prime concern of defence planners. This paper discusses the problem in the context of the exercise of sea power in distant waters and shows that a model of the whole problem would require a feedback analysis, for which an appropriate approach would be system dynamics. Such models have, in general, been continuous, but ships are discrete objects. The paper therefore addresses the construction of discrete system dynamics models as the basis for a model of the whole problem. Two models of a submarine force are presented. The first deals with the construction and major refit programmes, to evaluate the periods of fleet service availability achievable from a submarine force of a given size. The second examines unit usage during periods of fleet service.     

多用户多准则随机系统最优与最优收费

针对固定交通需求量和出行者的时间价值为离散分布的多准则随机交通均衡,分别研究了依费用度量和依时间度量的多用户多准则随机系统最优和最优收费问题.分别建立了基于费用和基于时间的随机系统最优的最优化模型,阐述了该模型解的唯一性条件及等价的变分不等式问题.运用变分不等式方法,研究了一阶最优收费的可行性,即能否依边际定价原则,通过收取与出行者类别无关的道路收费使多用户多准则随机均衡流与随机系统最优流一致.一阶最优收费不适用于依时间度量的随机系统最优情况,因而建立了一个最优化模型来得到此时的非歧视性道路收费.最后给出了具体算例.