Equilibrium Point Research Articles

Parameter identification of a reaction-diffusion predator-prey system based on optimal control theory

This paper takes a reaction-diffusion predator-prey system with ratio-dependent Holling III functional response function and Leslie-Gower term into consideration. First of all, the system model is proposed on the basis of basic biological assumptions and previous work, and the existence conditions of the equilibrium point of the system are discussed. Secondly, under the assumption of the existence of equilibrium point, the Turing instability necessary conditions induced by diffusion are investigated. Thirdly, optimal control theory is derived, and the adjoint system and the first-order optimization condition are established. Fourthly, parameter identification based on optimal control is studied, and the technique is extended to network structure. Finally, extensive numerical simulations, including Turing pattern, Normalized Population High Distribution Area (NPHDA) diagram and parameter identification, are carried out to illustrate and validate the analytical results. For the system dynamics phenomena, the results from different perspectives effectively demonstrate that the theoretical findings, numerical simulations and natural reality are identical. In terms of the parameter identification of continuous model and network model, the efficiency and accuracy of various algorithms are fully tested.

Numerical Analysis on Performance Improvement of a Vertical Plate Indirect Evaporative Cooler with Baffles

The performance of the Plate Indirect Evaporative Cooler (PIEC) can be effectively improved by incorporating baffles in the dry channel. However, in the dimensional influence of the baffles on PIEC performance there remains a research gap. In order to investigate the impact of baffle dimensions on the wet bulb efficiency, namely the average heat transfer coefficient and the cooling capacity of the PIEC, this paper proposed and verified a three-dimensional numerical model and method based on the species transport model and the Euler wall film model. At the same time, in order to obtain the equilibrium point between the enhanced heat transfer performance and the additional resistance induced by baffles, a comprehensive performance evaluation index is introduced. The results indicate that, under the same conditions, (1) the baffle effect on PIEC performance is significant at a lower inlet air velocity, and the wet bulb efficiency of the PIEC with baffles can be improved by 22.8%; (2) the baffle effect on PIEC performance is negative if its relative length exceeds 60% or the primary air inlet velocity surpasses 4 m/s under the conditions specified in this paper; and (3) the baffle effect on PIEC performance is significant when its channel height is lower and its channel width is larger, and the wet bulb efficiency of the PIEC with baffles can be improved by 29.3%.

Pattern formation and delay-induced instability in a Leslie–Gower type prey–predator system with Smith growth function

In this article, we have investigated the spatiotemporal dynamics and delay-induced instability of a Leslie–Gower type prey–predator model under the influence of environmental toxicants with Smith growth function. This growth function is more realistic than logistic growth as it better describes the growth of the biological population. It has been used where the growth limitations are based on the proportion of available resources not utilized. A few works of Smith’s growth models are reported in the literature. Therefore, spatiotemporal dynamics and pattern formation with delay effect remain an exciting area of research, which motivates the present work. This work has studied two types of dynamical systems: (i) an ordinary differential temporal system with time delay and (ii) a reaction–diffusion system with time delay. The existence of equilibrium points and their stability conditions are discussed. Hopf bifurcation emerges in both proposed systems with respect to delay parameter. The stability and direction of Hopf bifurcation and delay–diffusion-driven instability have been investigated for the reaction–diffusion system. Numerical simulation is performed to support the analytical results and theorems. Moreover, the existence of Hopf and delay-induced instability are proved numerically. Interesting one-dimensional regular and irregular stripe patterns are obtained for increased values of the time delay parameter. Also, the presence of natural toxicants has a negative impact on the growth of prey–predator species.

Data-driven economic MPC with asymptotic stability and strong duality verification using Hankel matrix

We consider the problem of dynamic regulation with an economic cost function to control unknown linear systems, in which improving the economic performance and guaranteeing the stability of economical optimal equilibrium point are control objectives. A data-driven economic MPC scheme is presented using measured input-output trajectories without a prior system identification step. Our method uses Hankel matrices which include one input-output data trajectory for prediction in economic MPC, while persistently exciting of the input generating the data is needed. One of the novelties of the presented framework is directly verifying the strong duality property from input-output trajectory with the general cost function, considered as the supply rate. This is used to find a Lyapunov function for data-driven economic MPC. Under the strong duality assumption, asymptotic stability of the economical optimal equilibrium point for the closed-loop system with terminal equality constraint is guaranteed. The proposed data-driven economic MPC approach needs only persistently exciting data trajectory along with an upper bound on the system order and need no model description and no online parameter estimation. The proposed scheme applicability compared to the existing model-based economic MPC and data-driven MPC is illustrated for continuous stirred tank reactor (CSTR) and a numerical example and the robustness of the proposed scheme is evaluated in the case of measurement noise, as well as nonlinear model for CSTR system.

Fractional mathematical modeling of breast cancer stages with true data from Saudi Arabia

This research focuses on the development, analysis, and simulation of fractional mathematical models to investigate the transmission dynamics of different phases of breast cancer. The suggested breast cancer model incorporates three often-used fractional operators in epidemiology: Caputo, Caputo–Fabrizio–Caputo, and Atangana–Baleanu–Caputo operators. In this study, the determination of the equilibrium point and its stability analysis is conducted using the Routh–Hurwitz criterion. Additionally, we examine the existence and uniqueness of solutions for the fractional system using Krasnoselskii’s and Banach fixed-point theory. Moreover, the global stability is discussed via the Ulam–Hyres criterion. Furthermore, the fractional models are being verified using reported occurrences of stage IV breast cancer among females in Saudi Arabia from 2004 to 2016. The real data is used to determine the values of the parameters that are fitted using the least squares error-minimizing methodology. Also, residuals and efficiency rates are computed for the integer as well as fractional-order models. Graphical representations are used to illustrate numerical results by examining different choices of fractional order parameters. Then, the dynamic characteristics of various stages of breast cancer are analyzed to demonstrate the impact of fractional order on breast cancer progression and how the rate of chemotherapy influences its behavior. We provide graphical results for a breast cancer model with effective parameters, resulting in fewer future incidences in the population of stages III and IV. Chemotherapy often raises the risk of cardiotoxicity, and our proposed model output reflects this. The goal of this study is to reduce the incidence of cardiotoxicity in chemotherapy patients while also increasing the pace of patient recovery. This research has the potential to significantly improve outcomes for patients and provide information on treatment strategies for breast cancer patients.

Modeling and analysis of a fractional order spatio-temporal SEIR model: Stability and prediction

This study introduces a novel fractional-order spatio-temporal SEIR model for epidemic modeling, providing an advanced approach to understanding disease dynamics. Our model, categorizing the population into Susceptible (S), Exposed (E), Infected (I), and Recovered (R), incorporates fractional calculus to accurately reflect the complex, non-linear nature of infectious diseases. Key findings include the confirmation of the existence and uniqueness of the model’s solutions, ensuring reliability for epidemiological predictions. Through rigorous stability analysis at both disease-free and endemic equilibrium points, we identified critical parameters influencing epidemic outcomes. Numerical simulations reveal that the fractional order significantly impacts disease progression, offering valuable insights for intervention strategies.

The Impact of U.S. Carbon Tariff on Chinas Exports and Welfare Based on Empirical Analysis and Game Models

This paper will begin by formulating an optimal tariff model between China and the U.S., leading us to an equilibrium point. From this vantage point, we will deduce the consequences of U.S. tariffs on China's exports and the ramifications of U.S. carbon tariffs on China's societal welfare. Next, we will develop an economic model considering the existing China-U.S. tariffs and their trading relationship. This model will help determine the influence of U.S. tariffs on China's export volume and societal welfare. In addition, we will compute the impact of U.S. tariffs on China's societal welfare and export to the U.S., factoring in China's carbon emissions per 100 yuan of GDP, and finally, concern the industrial impact of such carbon tariffs. Third, using two distinct game models, we would probe into the viability and possibility of the U.S. levying a carbon tariff on Chinese exports. The evolutionary game models reflect real-world scenarios, which point out that the U.S. will eventually apply a carbon tariff on China. Next, the dynamic game model considers different parties, Chinas government and firms, and different factors, environmental costs, and green subsidies. The model provides the circumstances in which the U.S. will levy a carbon tariff, whether China should provide a green subsidy, and whether the company should implement green production technology. Finally, we would discuss potential strategies China might adopt to navigate the challenges of the U.S. carbon tariff and examine the broader implications of such a move.

Effect analysis on passive and active regeneration performance of catalytic diesel particulate filters at the equilibrium point in the presence of ash layer

AbstractCatalytic diesel particulate filters (CDPF) form the main post‐processing devices for the diesel engines. Particulates trapped by CDPF can be oxidized by nitrogen dioxide (NO2) and oxygen (O2). The CDPF passive and active regeneration model on platinum‐based catalyst in the presence of ash layer, including the mass, momentum, and enthalpy balances of the gas and the solid phase, the gas diffusion, and the kinetic of the soot oxidation reaction is performed and validated by the bench test. The effects of different temperatures on CDPF passive and active regeneration at the equilibrium point are analyzed. Further, the effective analysis of ash loading and ash layer distribution on the equilibrium state is conducted. The research on CDPF passive and active regeneration at the equilibrium point in the presence of ash layer provides a theoretical basis for CDPF application.

Optimal control on ABC fractal fractional order model of micro-plastic pollution in soil and its effect on the nutrient cycle

Micro-plastic pollution is the major ultimatum to contaminate agricultural soils and empirical situation to lessening the soil quality and fertility. In this work the soil pollution due to micro-plastic in the Atangana-Baleanu-Caputo fractal dimension and fractional order inquiry with four compartments namely, Microplastic (P(t)), Soil pollution (S(t)), Remediate or recycling (R(t)) and Soil Nutrients cycle (N(t)). The qualitative analysis of the steady states is explained as locally stable, globally stable and Hyer Ulam stable. The expected equilibrium point appears in a pollution-free equilibrium point and pollution-extinct equilibrium point. Two significant best control strategies (u1(t) and u2(t)) are found to reduce soil pollution. Sensitivity analyses bring off the nature of various deportment of Basic reproduction numbers. Numerical approximation extracted with the help of Newton's Polynomial method. The purpose of precise expected results is demonstrated as a numerical simulation for various orders (0≤α≤0.6) and dimensions.

Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic.

Energy flow investigations of Rayleigh-Plesset equation for cavitation simulations

Nonlinear Rayleigh-Plesset Equation (RPE) for cavitation simulations is investigated using energy-flow theory. Nondimensional RPE affected by Reynolds-number, surface-tension, bubble-pressure, and liquid static-dynamic pressure ratio is derived to examine its equilibrium points with stability, possible periodical/chaotic motions by the energy-flow criteria. An example is numerically analyzed to illustrate the developed method. Five cases of the example reveal: 1) It is a damped system, where initial disturbances are gradually reduced with its phase point tends to its stable equilibrium-point; 2) Reynolds-number affects the damping of system, large one corresponds small damping; 3) Bubble-pressure, surface-tension and liquid static-dynamic pressure ratio affect the position of equilibrium point; 4) Periodical orbit appears only in forced vibrations, in which free vibration is reduced with time forward and the system finally shows a stable periodical oscillation; 5) Energy flow criteria for chaotic motions is not reached, and there are no chaotic motions for the cases of example. Numerical simulations confirm the developed energy-flow means with available computer code is effective to investigate generalized RPEs to reveal their inherent characteristics affected by required parameters in engineering cavitation analysis and designs, such as considering mass transports across the boundary of bubble by evaporation or condensation of liquids.

Mathematical Modeling of COVID-19 with Chronic Patients and Sensitivity Analysis

Human health is constantly threatened by the appearance and resurgence of several diseases, as shown by recent epidemics. COVID-19 was one of the epidemics that left its mark on the world in terms of economic and human damages. In the search for solution to this pandemic, the scientific community is involved in all its diversity. Mathematicians are taking part in the fight through mathematical modeling in various approaches. Ordinary derivative compartmental modeling approache is one of the techniques widely used in epidemiological modeling. This paper presents a mathematical contribution to fight against COVID-19 using a compartmental SQEICRS model. This model takes into account five stages. In particular, the role of chronic diseases on the dynamique of COVID-19, is focused. A mathematical analysis of the model has been carried out, and shows that the model is well-posed in the biological and mathematical sense. Aspects such as existence, equilibrium points and their stability, the basic reproduction number R0and sensitivity anlysis have been discussed. Sensitivity analysis allowed us to identify the parameters which contribute to the spread of the disease, including the chronicity rate due to chronic diseases. The direction of disease propagation was also determined according to <I>R</I><sub>0</sub>. Finally, the numerical results with Matlab are in conformity with theoretical results.

Understanding COVID-19 propagation: a comprehensive mathematical model with Caputo fractional derivatives for Thailand

This research introduces a sophisticated mathematical model for understanding the transmission dynamics of COVID-19, incorporating both integer and fractional derivatives. The model undergoes a rigorous analysis, examining equilibrium points, the reproduction number, and feasibility. The application of fixed point theory establishes the existence of a unique solution, demonstrating stability in the model. To derive approximate solutions, the generalized Adams-Bashforth-Moulton method is employed, further enhancing the study's analytical depth. Through a numerical simulation based on Thailand's data, the research delves into the intricacies of COVID-19 transmission, encompassing thorough data analysis and parameter estimation. The study advocates for a holistic approach, recommending a combined strategy of precautionary measures and home remedies, showcasing their substantial impact on pandemic mitigation. This comprehensive investigation significantly contributes to the broader understanding and effective management of the COVID-19 crisis, providing valuable insights for shaping public health strategies and guiding individual actions.

Study on SEAI Model of COVID-19 Based on Asymptomatic Infection

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when R0<1, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R0>1, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease.

Estimating Markov Chain Mixing Times: Convergence Rate Towards Equilibrium of a Stochastic Process Traffic Assignment Model

Network equilibrium models have been extensively used for decades. The rationale for using equilibrium as a predictor is essentially that (i) a unique and globally stable equilibrium point is guaranteed to exist and (ii) the transient period over which a system adapts to a change is sufficiently short in time that it can be neglected. However, we find transport problems without a unique and stable equilibrium in the literature. Even if it exists, it is not certain how long it takes for the system to reach an equilibrium point after an external shock onto the transport system, such as infrastructure improvement and damage by a disaster. The day-to-day adjustment process must be analysed to answer these questions. Among several models, the Markov chain approach has been claimed to be the most general and flexible. It is also advantageous as a unique stationary distribution is guaranteed in mild conditions, even when a unique and stable equilibrium does not exist. In the present paper, we first aim to develop a methodology for estimating the Markov chain mixing time (MCMT), a worst-case assessment of the convergence time of a Markov chain to its stationary distribution. The main tools are coupling and aggregation, which enable us to analyse MCMTs in large-scale transport systems. Our second aim is to conduct a preliminary examination of the relationships between MCMTs and critical properties of the system, such as travellers’ sensitivity to differences in travel cost and the frequency of travellers’ revisions of their choices. Through analytical and numerical analyses, we found key relationships in a few transport problems, including those without a unique and stable equilibrium. We also showed that the proposed method, combined with coupling and aggregation, can be applied to larger transport models. History: This paper has been accepted for the Transportation Science Special Issue on the 25th International Symposium on Transportation and Traffic Theory. Funding: This study was financially supported by the Japan Society for the Promotion of Science [Grant-in-Aid 20H00265]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2024.0523 .

Analyses of the SIR Epidemic Model Including Treatment and Immigration

This paper aims to examine the dynamics of a variation of a nonlinear SIR epidemic model. We analyze the complex dynamic nature of the discrete-time SIR epidemic model by discretizing a continuous SIR epidemic model subject to treatment and immigration effects with the Euler method. First of all, we show the existence of equilibrium points in the model by reducing the three-dimensional system to the two-dimensional system. Next, we show the stability conditions of the obtained positive equilibrium point and the visibility of flip bifurcation. A feedback control strategy is applied to control the chaos occurring in the system after a certain period of time. We also perform numerical simulations to support analytical results. We do all these analyses for models with and without immigration and show the effect of immigration on dynamics.

Comparative analysis of a fractional co-infection model using nonstandard finite difference and two-step Lagrange polynomial methods

In this work, Caputo fractional model for co-dynamics of two-strains (Alpha and Delta variants) of SARS-CoV-2 and tuberculosis (TB) is formulated. We investigates the basic mathematical analysis, existence of unique solution, stability analysis, sensitivity analysis and simulations of the proposed model. In the basic mathematical analysis, it is proved that the model’s solutions are non-negative, bounded and locally asymptotically stable at disease free equilibrium (DFE) point. The existence of unique solution is proved by using Banach fixed point theorem (BFPT). Stability analysis is carried out by employing Hyers–Ulam stability criteria. Sensitivity analysis is performed to assess the impact of the model parameters on the general dynamics of the model by implementing partial rank correlation coefficient (PRCC) technique. Simulations of the proposed model obtained from non-standard finite difference scheme (NSFDs) are compared with the simulations obtained from two-step Lagrange polynomial method (TLPM). It is concluded that NSFDs gives more accurate and realistic results than TLPM. To assess some control measures necessary for reducing the co-spread of both diseases, comparative simulations are carried out for the infected classes. Also, numerical assessments showed that preventive efforts against SARS-CoV-2 variants could results in the reduction of TB prevalence and the co-infection of both diseases.

Unsteady MHD oblique stagnation point slip flow of maxwell fluid over an oscillating–stretching cylinder with Cattaneo–Christov double diffusion model

The unsteady oblique stagnation point slip flow of a Maxwell fluid over an oscillating–stretching cylinder with convective heat transfer is investigated. The evaluation of this phenomenon is facilitated by employing the Cattaneo–Christov double diffusion model. According to the oblique stagnation point flow characteristics, an ordinary differential equation about pressure is derived, and pressure terms can be corrected by solving this equation. The equations are subjected to the similarity transformation, resulting in a set of dimensionless equations. The homotopy analysis method (HAM) is subsequently utilized to obtain their solutions. The effects of various parameters on the velocity, temperature, and concentration are analyzed. It is observed that slip boundary conditions accelerate fluid flow by reducing resistance. Elevating the Schmidt number means a reduction in the mass diffusion, leading to a diminished concentration. The temperature distributions of different thermal relaxation time parameters intersect at a thermal equilibrium point. Moreover, three-dimensional plots depicting the evolution of velocity and temperature over time demonstrate excellent agreement with the flow behavior and boundary conditions. They contribute to the comprehension and prediction of time-dependent flow and heat transfer phenomena.

A novel model predictive controller for the drifting vehicle to track a circular trajectory

ABSTRACT In rally racing, a high sideslip cornering technique known as drifting is employed to minimise the lap time. However, for autonomous vehicles, tracking the trajectory while drifting is a challenging control problem due to the conflict between the two objectives. This paper proposes a novel model predictive controller (MPC) with two operation modes to address this conflict. In Mode 1, fuzzy expert strategies combined with the lateral error are designed to calculate the drift equilibrium point. Subsequently, the linear MPC controller is utilised to achieve quick adjustments of the lateral error. In Mode 2, terminal state constraints and error constraints are added to the MPC optimisation problem. The stability of the MPC controller is guaranteed to keep the vehicle within the expected error range. Finally, a comparison between the existing expert controller and the proposed controller is carried out on both the CarSim platform and a 1:10 scale race car. The results demonstrate that the proposed controller outperforms the existing expert controller in terms of tracking performance and drift stability.

Evolutionary game and system dynamics for analysis on stakeholder strategies of regional high-speed rail project in investment decision stage

This paper studies the strategic choices under the conflict among stakeholders of regional high-speed rail project at the investment decision stage in the context of the ‘high-speed rail controversy’. This study develops a tripartite evolutionary game model among the central government, the local government and China Railway, analyzes the evolutionary stability strategy, discusses the influence of multiple factors on the choice of tripartite strategy, and further calculates and analyzes the stability of the equilibrium point in the tripartite evolutionary game system. Then the dynamic change trend of strategy choice brought by the change of each factor is investigated through system dynamics simulation. The results show that the lower the intervention cost and reward to the other two parties, and the increase of the opportunity cost of non-intervention and punishment to the other two parties, the faster the convergence of the central government's choice of intervention strategy. The increase of the rewards and punishments, and the increase of negotiation profits, non-negotiation costs, and loss of reputation for non-negotiation, and the decrease in the negotiation costs and non-negotiation profits all contribute to the shortening of the convergence time of the local government's choice of negotiation strategy. The increase of the rewards and punishments, the increase of support profits, the loss of reputation for non-support, and the decrease of the profits for non-support and the extra cost for support will promote the convergence of China Railway's choice of the support strategy. Finally, the simulation results are discussed so as to target feasible coordination strategies, which provides a theory basis for the practice of conflict coordination among stakeholders of regional high-speed rail project at the investment decision stage.