Discrete Mathematical Model Research Articles

Mathematical modelling of binge drinking from social interactions

Binge drinking is a multi-factorial problem where peer influence in the immediate environment has a key role in the appearance of the first episode and the recurrence. However, information regarding the interaction of variables that mediate the influence among people is limited. Binge drinking pattern is generally derived from interrelationships with other drinking states in a manner analogous to the susceptible-infectious relationship. Nevertheless, the classic modelling approach is not appropriate because the periodicity of consumption is associated with specific social contexts where social drinkers adopt binge drinking. We analyse the temporal evolution of binge drinkers, formulating a discrete mathematical model that incorporates the effect of the size of the influence group, the incitement-to-use and the refractoriness to recurrence as parameters. Our results allow evaluating the potential impact of interventions according to the individual and social characteristics of the susceptible population. New horizons of epidemiological research are established to prevent alcoholism.

Variational approach to the construction of discrete mathematical model of the pendulum motion with vibrating suspension with friction

The main purpose of this work is, first, a construction of the indirect Hamilton’s variational principle for the problem of motion of a pendulum with a vibration suspension with friction, oscillating along a straight line making a small angle with the vertical line. Second, the construction on its basis of the difference scheme. Third, to carry out its investigation by methods of numerical analysis. Methods. The problem of motion of the indicated pendulum is considering as a particular case of the given boundary problem for a nonlinear second order differential equations. For the solution of problem of its variational formulation there is used the criterion of potentiality of operators — the symmetry of the Gateaux derivative of nonlinear ˆ operator of the given problem. This criterion is also used for the construction of variational multiplier and the corresponding Hamilton’s variational principle. On its basis there is constructed and investigated a discrete analog of the given boundary problem and a problem of motion of the pendulum. Results. It is proved that the operator of the given boundary problem is not potential with respect to the classical bilinear form. There is found a variational multiplier and constructed the corresponding indirect Hamilton’s variational principle. On its basis there is obtained a discrete analog of the given boundary problem and its solution is found. As particular cases one can deduce from that the corresponding results for the problem of motion of the pendulum. There are performed numerical experiments, establishing the dependence of solutions of the problem of motion of the pendulum on the change of parameters. Conclusion. There is worked out a variational approach to the construction of two difference schemes for the problem of a pendulum with a suspension with friction, oscillating along a straight line making a small angle with the vertical line. There are presented results of numerical simulation under different parameters of the problem. Numerical results show that under sufficiently small amplitude and sufficiently big frequency of the oscillations of the point of suspension the pendulum realizes a periodical motion.

Modeling and simulation of high energy density lithium-ion battery for multiple fault detection

Lithium-ion battery, a high energy density storage device has extensive applications in electrical and electronic gadgets, computers, hybrid electric vehicles, and electric vehicles. This paper presents multiple fault detection of lithium-ion battery using two non-linear Kalman filters. A discrete non-linear mathematical model of lithium ion battery has been developed and Unscented Kalman filter (UKF) is employed to estimate the model parameter. Occurrences of multiple faults such as over-charge, over-discharge and short circuit faults between inter cell power batteries, affects the parameter variation of system model. Parallel combinations of some UKF (bank of filters) compare the model parameter variation between the normal and faulty situation and generates residual signal indicating different fault. Simulation results of multiple numbers of statistical tests have been performed for residual based fault diagnosis and threshold calculation. The performance of UKF is then compared with Extended Kalman filter (EKF) with same battery model and fault scenario. The simulation result proves that UKF model responses better and quicker than that of EKF for fault diagnosis.

A Discrete Mathematical Model of the Variable State of the Pandemic

The paper describes and analyzes a discrete mathematical model of the variable state of the pandemic, which is important for determining production quantities of vaccines and antiviral drugs, predicting the number of infected persons, and the intensity of the process of disseminating information or new ideas to the public. According to the system of differential equations of the pandemic, a discrete mathematical model in vector-matrix form was developed and the equilibrium of the model in the space state was proved. As a result of the implementation of the pandemic model, the discrete dynamic curves of the variable state were obtained in a Matlab package.

Discrete spread model for COVID‐19: the case of Lebanon

BackgroundMathematical models are essential to predict the likely outcome of an epidemic. Various models have been proposed in the literature for disease spreads. Some are individual based models and others are compartmental models. In this study, discrete mathematical models are developed for the spread of the coronavirus disease 2019 (COVID‐19).MethodsThe proposed models take into account the known special characteristics of this disease such as the latency and incubation periods, and the different social and infectiousness conditions of infected people. In particular, they include a novel approach that considers the social structure, the fraction of detected cases over the real total infected cases, the influx of undetected infected people from outside the borders, as well as contact‐tracing and quarantine period for travelers. The first model is a simplified model and the second is a complete model.ResultsFrom a numerical point of view, the particular case of Lebanon has been studied and its reported data have been used to estimate the complete discrete model parameters using optimization techniques. Moreover, a parameter analysis and several prediction scenarios are presented in order to better understand the role of the parameters.ConclusionsUnderstanding the role of the parameters involved in the models help policy makers in deciding the appropriate mitigation measures. Also, the proposed approach paves the way for models that take into account societal factors and complex human behavior without an extensive process of data collection.

A discrete Huber-Braun neuron model: from nodal properties to network performance.

The online version contains supplementary material available at 10.1007/s11571-022-09806-1.

An Overview on Modelling of Complex Interconnected Nonlinear Systems

This paper proposes new mathematical models of representation, which can describe the dynamic behavior of large-scale nonlinear systems, such as an extended mathematical model of Volterra series, interconnected Hammerstein structures, and interconnected Wiener structures. In this research, we focus on the class of large-scale nonlinear systems, which are composed of several interconnected nonlinear subsystems. In this context, a discrete nonlinear mathematical model with unknown time-varying parameters, mono-variable, characterizes each interconnected subsystem operating in a deterministic or stochastic environment. An illustrative numerical simulation example of two interconnected nonlinear processes is provided to prove the validity and the performance of the developed theoretical results.

Dynamical behaviors of a k-order fuzzy difference equation

Abstract Difference equations are often used to create discrete mathematical models. In this paper, we mainly study the dynamical behaviors of positive solutions of a nonlinear fuzzy difference equation: x n + 1 = x n A + B x n − k ( n = 0 , 1 , 2 , … ) , {x}_{n+1}=\frac{{x}_{n}}{A+B{x}_{n-k}}\hspace{0.33em}\left(n=0,1,2,\ldots ), where parameters A , B A,B and initial value x − k , x − k + 1 , … , x − 1 , x 0 {x}_{-k},{x}_{-k+1},\ldots ,{x}_{-1},{x}_{0} , k ∈ { 0 , 1 , … } k\in \{0,1,\ldots \} are positive fuzzy numbers. We investigate the existence, boundedness, convergence, and asymptotic stability of the positive solutions of the fuzzy difference equation. At last, we give numerical examples to intuitively reflect the global behavior. The conclusion of the global stability of this paper can be applied directly to production practice.

Simulation Research on Deadbeat Direct Torque and Flux Control of Permanent Magnet Synchronous Motor

Direct torque control (DTC) is widely used in a permanent-magnet synchronous motor (PMSM), but it has its own shortcomings caused by high torque ripple. Deadbeat-direct torque and flux control (DB-DTFC) is a new torque and flux method compared with DTC. However, the traditional DB-DTFC is often based on rotor-flux-oriented control. The reference voltage of the stator is computed in a rotor-flux-oriented coordinate system, and the solution involves solving quadratic equations, which will increase the burden of computational processing. To improve the computation of the reference voltages and the control performance, this paper proposes a new DB-DTFC algorithm and introduces its basic principles. First, the proposed DB-DTFC algorithm uses the forward Euler equation to solve the reference voltage in a stator-flux-oriented coordinate system. Second, the discrete mathematical model is used to predict the next control current to achieve deadbeat control. Third, the structural model of the proposed DB-DTFC is constructed. Finally, the simulation model of the proposed DB-DTFC algorithm is built with a MATLAB/Simulink platform. The simulation results prove that the proposed DB-DTFC algorithm can achieve better control performance in torque and flux control compared with the DTC algorithm and SVM-based direct torque and flux control (SVM-DTFC) algorithm. In particular, the torque index of DB-DTFC is reduced about 6% in a limited speed range in comparison with the DTC algorithm.

Construction of COVID-19 Epidemic Prevention and Control and Public Health Emergency Response System Based on Discrete Stochastic Mathematical Model.

This research was aimed at exploring the construction and evaluation method of the comprehensive emergency response system for public health emergencies under the COVID-19 (coronavirus disease 2019) epidemic situation based on discrete stochastic mathematical model. The response of the Centers for Disease Control and Prevention (CDCP) of Taiyuan city in the COVID-19 epidemic situation was taken as an example. A new discrete stochastic COVID-19 epidemic spread mathematical model which integrated public health intervention and input cases was proposed. The model was parameterized by multisource data, and the impact of different flow patterns on the risk of secondary outbreak was analyzed. The advantages and disadvantages of its emergency system construction were analyzed. Additionally, the improvement measures and suggestions for the existing problems were proposed. Results suggested that there was only one specialized disease prevention and control institution in Taiyuan, and there were only 11 centers for disease prevention and control, accounting for 6.2% (11/177) of the total in Shanxi Province. Through the analysis, it was found that the current public health emergency response system in Taiyuan city had imperfect management coordination mechanism, incomplete plan type, serious shortage of public health personnel, poor information communication efficiency, insufficient early warning efficiency of the epidemic detection system, and weak logistics material security links. Therefore, it was proposed to establish a sound coordination system of emergency health management and vigorously promote the construction of emergency health management institutions. Thus, a public health emergency management system integrating management coordination system, plan system, emergency team building system, material reserve management, and other functions was formed. The application of discrete stochastic mathematical model suggests that intermittent population flow and effective isolation of infected people in transient population can effectively reduce the risk of secondary outbreak. The system analysis here also provides theoretical basis for improving the construction of public health emergency response system in Taiyuan.

Optimal harvest strategy based on a discrete age-structured model with monthly fishing effort for chub mackerel, Scomber japonicus, in South Korea

Currently, the catch of fishery resources in South Korea has fallen more than 40%, from 1.73 million tons in 1986 to 1.01 million tons in 2018. In particular, the amount of chub mackerel, one of the most favored fish species by South Koreans, caught in 2017 was 103,870 tons, which was only 25% of the total catch, compared with 415,003 tons in 1996. The total allowable catch system and the closed season are currently in place as a countermeasure to the continued decline in fishery resources; however, their effectiveness is questionable. Our study aims to maximize fishing profits by controlling the monthly fishing effort while maintaining the amount of chub mackerel. Discrete age-structured mathematical model was established with an optimal harvest control system. Density-dependent mortality was applied at the juvenile stage to reflect fish mortality because it is highly dependent on population density in the underage phase. The conditions of the parameters guaranteeing the existence of optimal harvest strategies were demonstrated and obtained using Pontryagin's maximum principle. We compared an optimal harvest strategies and the actual harvest strategies of fishing effort from July 2010 to June 2020. In addition, we compared the profit and biomass of the optimal harvest strategies when designating one month in a year as closed season. We also conducted sensitivity analysis by varying the price of chub mackerel and cost of fishing effort. As a result, the optimal harvest strategies are to have the highest amount of fishing effort from August to September and gradually reduce the amount of effort. The optimal harvest strategies could improve the maximum economic yield and the maximum sustainable yield by 11.9% and 10.9%, respectively, over the observed strategy. To the best of our knowledge, forecasting analysis of the maximum economic yield and closed season conducted based on optimal harvest control system for Chub mackerel in South Korea. The appropriate allocation of monthly fishing effort can increase sustainable catch and profit. In addition, it is better to implement a closed season in July to maximize the profit of fishing and in June to the resource recovery.

Nonlinear Seepage Mathematical Model of Fractured Tight Stress Sensitive Reservoir and Its Application

A tight reservoir can only have industrial production value after fracturing. After volume fracturing, there are artificial fracturing zones and fracturing unmodified zones composed of a matrix, microfractures, and main fractures. There are also pressure-sensitive effects and nonlinear seepage in the matrix. There is an unsteady channeling flow between the matrix and microfractures and main fractures and a threshold pressure gradient in the matrix. The fluid flow between the three media and the transfer flow between the media are very complex. Aiming at the problem of poor simulation accuracy because the above characteristics of tight reservoir fracturing development are not considered in the current commercial numerical simulation software Eclipse, CMG, and VIP, the triple medium coupling nonlinear mathematical model of tight reservoir fracturing development matrix/microfractures/main fractures was established in this study, and the discrete mathematical model was constructed using the full implicit numerical solution method. The model considered the effects of threshold fracturing gradient and nonlinear seepage on the reverse side of seepage. In terms of pressure-sensitive effect, the latest generalized pressure-sensitive effect calculation model, considering rock physical parameters established by authors, was adopted. In terms of matrix/fracture transfer flow, the nonlinear matrix/fracture transfer flow model considering threshold pressure gradient and pressure-sensitive effect established by authors was adopted. Finally, a tight reservoir numerical simulation software was developed based on a triple medium coupling nonlinear model. The software was used to simulate the development effect of turbidite tight reservoir in Wang 587 Block of Jiyang depression, Shengli Oilfield. The results showed that the fitting accuracy of the new model was 13.3% higher than that of Eclipse, and the calculation results were more in line with the production practice of tight reservoirs. The new model established an important theoretical basis for scientifically and effectively guiding the development of tight reservoirs.

Dynamic Discrete Inventory Control Model with Deterministic and Stochastic Demand in Pharmaceutical Distribution

This paper presents an inventory control problem in a private pharmaceutical distribution company from the Republic of Serbia. The company realizes that distribution within nine neighbouring countries and inventory control in the pharmaceutical supply chain is centralized. In order to constitute a conceptual model of the problem, we propose the modern control theory concept. The conceptual model is based on the specific practical assumptions and constraints of the supply chain. Thereafter, a dynamic discrete mathematical model of inventory control is formulated to reflect elements of the system and their relations. The model considers multiple pharmaceutical products, variable lead time, realized stochastics and deterministic demand, and different ordering policies (Lot for Lot and Fixed Order Quantity). Deterministic demand is represented as a sales forecast for each product per month, while stochastic demand is generated as a random variation of sales forecast in a range of ±20%. Two objective functions are defined as the maximization of the difference between planned average inventory level and realized average inventory level, and the minimization of stock-out situations. We develop a procedure for the determination of reorder points and the number of deliveries to achieve proposed objective functions. The model overcomes shortages of theoretically-based distribution requirements planning models and offers solutions to the limitations in inventory control practice. Real-life data, collected over two years, are used for the validation of the proposed model and the solution procedure. Numerical examples illustrate the model application and behaviour.

Application of Discrete Mathematical Model in Edge Distortion Correction of Moving Image

The existing edge distortion correction methods of moving images ignore the consistency of human subjective vision in the restoration process. Therefore, this study proposes an edge distortion correction method of moving images using a discrete mathematical model. According to the image binarization, the moving target region is segmented, the motion fuzzy degradation model is constructed by using the discrete mathematical model, the global structure information of the image is considered by the structure similarity, the image gradient is transformed from the rectangular coordinate system to the polar coordinate system, the image deconvolution algorithm of the edge distribution of the natural image is used to enhance the image edge features, and the edge distortion correction of the moving image is realized. The experimental results show that the astringency of the experimental scattergram is obviously improved, the fitting line is almost a straight line, and it has good recognition of the jump of the gray level of noise points, which can meet the requirements of the consistency of the subjective vision of human eyes.

A discrete mathematical modeling and optimal control of diseases caused by Fusarium Oxysporum

A discrete mathematical modeling and optimal control of diseases caused by Fusarium Oxysporum

A discrete mathematical model SEIR with the evolution of the regions

In this study, we provide a discrete mathematical SEIR model that depicts the evolution of an infectious disease while introducing the novel idea of taking regional infection spread into account. To reduce the disease’s ability to spread among people and places, we suggest three control measures. The optimal controls are defined using the Pontryagin maximum principle, and the optimality system is solved using an iterative method. Finally, MATLAB-based numerical simulations are performed to check the results of the theoretical analysis.

Simulink modeling and dynamic characteristics of discrete memristor chaotic system

<sec>In the last two years, the discrete memristor has been proposed, and it is in the early stages of research. Now, it is particularly important to use various simulation softwares to expand the applications of the discrete memristor model. Based on the difference operator, in this paper, a discrete memristor model with quadratic nonlinearity is constructed. The addition, subtraction, multiplication and division of the discrete memristor mathematical model are clarified, and the charge <i>q</i> is obtained by combining the discrete-time summation module, thereby realizing the Simulink simulation of the discrete memristor. The simulation results show that the designed memristor meets the three fingerprints of memristor, indicating that the designed discrete memristor belongs to generalized memristor.</sec><sec>Using memristors to construct chaotic systems is one of the current research hotspots, but most of the literature is about the introduction of continuous memristors into continuous chaotic systems. In this paper, the obtained discrete memristor is introduced into a three-dimensional chaotic map which is mentioned in a Sprott’s book titled as <i>Chaos and Time-Series Analysis</i>, and a new four-dimensional memristor chaotic map is designed. Meanwhile, the Simulink model of the chaotic map is established. It is found that attractors with different sizes and shapes can be observed by changing the parameters in the Simulink model, indicating that the changes of system parameters and memristor parameters can change the dynamic behavior of the system. The analyses of equilibria and equilibrium stability show that the four-dimensional chaotic map has infinite equilibrium points. The Lyapunov exponent spectra and bifurcation diagrams of the circuit imply that the map can transform between weak chaotic state, chaotic state, and hyperchaotic state. Meanwhile, the multistability and coexisting attractors are analyzed under different initial conditions. Moreover, by comparing the results of measuring the complexity, it is found that the chaotic map with discrete memristor has richer dynamical behaviors and higher complexity than the original map.</sec><sec>From the perspective of system modeling, in this paper the discrete memristor modeling and discrete memristor map designing are discussed based on the Matlab/Simulink. It further verifies the realizability and lays a foundation for the future applications of discrete memristor.</sec>

A Local Iterative Linear Approximation Framework to Integrate Discrete Choice and Mathematical Optimization Models

A Local Iterative Linear Approximation Framework to Integrate Discrete Choice and Mathematical Optimization Models

A Dynamic Programming Approach to the Design of Composite Aircraft Wings

A light and reliable aircraft has been the major goal of aircraft designers. It is imperative to design the aircraft wing skins as efficiently as possible since the wing skins comprise more than fifty percent of the structural weight of the aircraft wing. The aircraft wing skin consists of many different types of material and thickness configurations at various locations. Selecting a thickness for each location is perhaps the most significant design task. In this paper, we formulate discrete mathematical programming models to determine the optimal thicknesses for three different criteria: maximize reliability, minimize weight, and achieve a trade-off between maximizing reliability and minimizing weight. These three model formulations are generalized discrete resource-allocation problems, which lend themselves well to the dynamic programming approach. Consequently, we use the dynamic programming method to solve these model formulations. To illustrate our approach, an example is solved in which dynamic programming yields a minimum weight design as well as a trade-off curve for weight versus reliability for an aircraft wing with thirty locations (or panels) and fourteen thickness choices for each location.

Mathematical modeling and optimal control strategy for the obesity epidemic

The aim of this work is to propose a discrete mathematical model to study the behavioral dynamics of a population affected by the disease of obesity. Thus, the population under study is divided into six compartments: susceptible (S), exposed (E), slightly obese (I1), moderately obese (I2), very obese (I3), and recovered (R). To fight this disease, we used four controls: Awareness through education and media, food and sports programs, medical treatment with drugs, and treatment with surgical intervention. The discrete time Pontryagin maximum principle is used to characterize the optimal controls. The numerical simulation via MATLAB confirms the performance of theoretical results.