On contemporary mortality models for actuarial use I: practice

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

As mentioned, the transition of a continuous time model into a discrete time model is not an easy issue. We discuss here various discretization procedures to turn continuous time into discrete time models. There are many methods to convert continuous time models into discrete time variants. The main discretization methods are the Euler method, the Milstein method and a new local linearization method. All those will be illustrated here to obtain discrete-time approximate models.

A fifth-order dynamic continuous model of a linear induction motor (LIM), without considering “end effects” and considering attraction force, was developed. The attraction force is necessary in considering the dynamic analysis of the mechanically loaded linear induction motor. To obtain the circuit parameters of the LIM, a physical system was implemented in the laboratory with a Rapid Prototype System. The model was created by modifying the traditional three-phase model of a Y-connected rotary induction motor in a d–q stationary reference frame. The discrete-time LIM model was obtained through the continuous time model solution for its application in simulations or computational solutions in order to analyze nonlinear behaviors and for use in discrete time control systems. To obtain the solution, the continuous time model was divided into a current-fed linear induction motor third-order model, where the current inputs were considered as pseudo-inputs, and a second-order subsystem that only models the currents of the primary with voltages as inputs. For the discrete time model, the current-fed model is discretized by solving a set of differential equations, and the subsystem is discretized by a first-order Taylor series. Finally, a comparison of the continuous and discrete time model behaviors was shown graphically in order to validate the discrete time model.

We present two approaches, discrete time and continuous time models, for individuals which propagate through binary fission. The volumes of the two daughters are a fixed part of that of the mother, not necessarily the half, and their growth rates may differ. The discrete time approach gives more insight into the results obtained with the continuous time model. We define classes in the continuous time model such that the total number of individuals in these classes at specific moments in time is equal to the unknown number in a discrete time model. Then the discrete time model is homologous to the continuous one in the sense of having the same solutions at specific moments. Population matrix theory applies when the ratio of the inter-division times of the two daughters is rational. There is inter-class convergence but no intra-class convergence. The latter feature implies that there is no convergence of the size distribution in the continuous time model either. When the ratio is irrational the continuous time model holds and there is convergence but the rate of convergence can become infinitesimally small. This phenomenon is linked with quasi-periodicity on a 2-dimensional torus.

On the Basic Assumptions in the Identification of Continuous Time Systems

NUMERICAL ILLUSTRATIONS OF THE RELEVANCE OF DIRECT CONTINUOUS-TIME MODEL IDENTIFICATION

Internet of things (IoT) systems are composed of variety of units from different domains. While developing a complete IoT system, different professionals from different domains may have to work in collaboration. In this paper we provide a framework which allows using discrete and continuous time modeling and simulation approaches in combination for IoT systems. The proposed framework demonstrates on how to model Ad-hoc and general IoT systems for software engineering purpose. We demonstrate that model-based software engineering on one hand can provide a common platform to overcome communication gaps among collaborating stakeholders whereas, on the other hand can model and integrate heterogeneous components of IoT systems. While modeling heterogeneous IoT systems, one of the major challenges is to apply continuous and discrete time modeling on intrinsically varying components of the system. Another difficulty may be how to compose these heterogeneous components into one whole system. The proposed framework provides a road-map to model discrete, continuous, Ad-hoc, general systems along with composition mechanism of heterogeneous subsystems. The framework uses a combination of Agent-based modeling, Aspect-oriented modeling, contract-based modeling and services-oriented modeling concepts. We used this framework to model a scenario example of a service-oriented IoT system as proof of concept. We analyzed our framework with existing systems and discussed it in details. Our framework provides a mechanism to model different viewpoints. The framework also enhances the completeness and consistency of the IoT software models.

This paper provides the description of main existing optimal control theory problems. Also the key features of an optimal control problem for spacecraft and launch vehicles are identified. The authors pose the problem of formation of linear discrete-time mathematical model based on continuous-time model described by linear inhomogeneous nonstationary differential equations system. A first-stage launch vehicle “Soyuz-2.1v” perturbed motion model with additional degrees of freedom (flexible body fluctuations and slosh modes fluctuations in the fuel tanks) in pitch plane is considered. Continuous-time model in vector-matrix explicit canonical form is composed. The sampling of continuous-time model is performed, sampling method is described. Comparative response analysis of the continuous-time and discrete-time models is produced.

In this paper two approximate linear time-invariant dynamic models, a continuous-time model and a discrete-time one, are proposed for a planar sandwich structure, constituted by two thin piezoelectric elements, mounted on a thin elastic support. The continuous-time model is determined by applying the integral Hamilton principle, once the kinematics of the structure has been approximated by a polynomial function; the discrete-time model is obtained via the direct discretization of the action functional. The non-interacting control and tracking of reference trajectories are stated and solved, both in the continuous-time case and in the discrete-time one. Some simulation results are reported in the paper. >

An analysis of 15 years of hourly observations of cloud cover at an airport location near Copenhagen, Denmark shows that the main part of the variations can be described by first-order homogeneous Markov models. Models in both discrete and continuous time are considered. Special emphasis is laid upon the representation of the variation by a Markov model in continuous time. The physical restrictions for transitions of cloud cover are investigated and it is shown that the natural restrictions imply a very simple structure for the matrix of transition rates corresponding to the embedded Markov process in continuous time. The maximum likelihood estimate of the matrix of transition rates is found by numerical methods and an indication of the estimation error under the model is given.

Day-to-day dynamics with advanced traveler information

In this paper, we discretize and analyze a stoichiometric optimal foraging model where the grazer's feeding effort depends on the producer's nutrient quality. We systematically make comparisons of the dynamical behaviors between the discrete-time model and the continuous-time model to study the robustness of model predictions to time discretization. When the maximum growth rate of producer is low, both model types admit similar dynamics including bistability and deterministic extinction of the grazer caused by low nutrient quality of the producer. Especially, the grazer is benefited from optimal foraging similarly in both discrete-time and continuous-time models. When the maximum growth rate of producer is high, dynamics of the discrete-time model are more complex including chaos. A phenomenal observation is that under extremely high light intensities, the grazer in the continuous-time model tends to perish due to poor food quality, however, the grazer in the discrete-time model persists in regular or irregular oscillatory ways. This significant difference indicates the necessity of studying discrete-time models which naturally include species' generations and are thus more popular in theoretical biology. Finally, we discuss how the shape of the quality-based feeding function regulates the beneficial or restraint effect of optimal foraging on the grazer population.

Discrete versus continuous time models: Local martingales and singular processes in asset pricing theory

Discrete-time modelling (DTM) dominates the systems engineering literature in the applications of block-oriented modelling. The discrete environment of computer-based process control systems and discrete sampling are two major reasons [1]. Also, a DTM is easier to obtain because all input changes are approximated by piecewise step input sequences. Nonetheless, DTM has (potentially) two critical drawbacks relative to continuous-time modelling (CTM). DTM requires constant and frequent sampling and can only predict at those points. DTM is not potentially as accurate as CTM because, at best, it can only approximate continuous-time processes. For Hammerstein and Wiener CTM, this article proposes compact CTM algorithms under sinusoidal input sequences for Hammerstein and Wiener modelling. The proposed method depends only on the most previous input changes and provides exact solutions that are applicable to multiple-input, multiple-output systems as demonstrated.

The aim of this work is to formulate and analyze a new and generalized discrete-time population dynamics model for a two-stage species with recruitment and capture factors. This model is derived from a well-known continuous-time population dynamics model of a two-stage species with recruitment and capture developed by Ladino and Valverde and the nonstandard finite difference (NSFD) methodology proposed by Mickens. We establish positivity and asymptotic stability of the proposed discrete-time population dynamics model. As an important consequence, the population dynamics of the new discrete-time model is determined fully. Also, a set of numerical examples is conducted to illustrate the theoretical results and to demonstrate advantages of the new model. The theoretical results and numerical examples show that the proposed discrete-time model not only preserves correctly the population dynamics of the continuous one but is also easy to be implemented. However, some discrete-time models based on the standard Runge–Kutta methods fail to preserve the population dynamics of the continuous-time model. As a result, they generate numerical approximations which are not only non-negative but also unstable.