Stability Of Equilibrium Point Research Articles

Dynamic analysis of human papillomavirus transmission model under vaccine intervention: a case study of cervical cancer patients from Hungary

Nearly 80% of women are estimated to have at least one HPV infection from high-risk types. Although the majority can clear the infection through their immune system, some are at risk for cervical cancer. Persistent high-risk HPV infections are the leading cause of cervical cancer worldwide. This paper proposes a mathematical model that examines the effects of HPV transmission on cervical cancer patients under vaccine intervention. The model’s fundamental properties are investigated, including the stability of equilibrium points and the existence of forward bifurcations. Subsequently, based on cervical cancer patient data collected from Hungary between 2000 and 2020, the model’s optimal parameter values are identified using a nonlinear least squares method. Further, we perform a sensitivity analysis of the key cervical cancer progression parameters. Our results indicate that both direct HPV vaccination in susceptible populations and additional vaccination in individuals who have recovered can improve immune responses and reduce the risk of cervical cancer. In addition, the study of the effects of intervention measures on cervical cancer patients in Hungary from 2000 to 2030 reveals that reducing the contact rate is conducive in the short term to curbing the development of cervical cancer; however, in the long term, relying solely on this measure is not sufficient to lead to a significant decrease in the number of cervical cancer cases.

Multistability of state-dependent switched neural networks with Mexican-hat-type activation functions

This paper investigates the issue of multistability in state-dependent switched neural networks (SSNNs) with Mexican-hat-type activation functions (AFs). It establishes the coexistence and stability of multiple equilibrium points (EPs). Initially, the state space is partitioned based on the geometric characteristics of the Mexican-hat-type AF, enabling to determine the positions of the EPs. Secondly, the coexistence of 9h17h25h33h4 EPs for n-neurons SSNNs under specific sufficient conditions is proved with the Brouwer’s fixed-point theorem. Next, by using diagonally dominant matrix theory and Gershgorin circle theorem, it is proven that there are 5h14h23h32h4 asymptotically stable EPs under some conditions, where h1,h2,h3 and h4 are nonnegative integers satisfying 0≤h1+h2+h3+h4≤n. Therefore, we can obtain that SSNNs can have larger storage capacity by selecting the appropriate parameters hi,i=1,2,3,4. Finally, the correctness of the results in this paper is verified through two numerical examples.

Dynamical analysis, control, boundedness and prediction for a fractional-order financial risk system

Abstract This paper delves into the dynamical analysis, chaos control, Mittag-Leffler Boundedness (MLB) and forecasting a Fractional-Order Financial Risk (FOFR) system through an absolute function term. To this end, the FOFR system is first proposed, and the Adomian Decomposition Method (ADM) is employed to resolve this fractional-order system.The stability of equilibrium points and the corresponding control schemes are assessed, and several classical tools such as Lyapunov Exponents (LE), Bifurcation Diagrams, Complexity Analysis (CA), and 0-1 test are further extended to analyze the dynamical behaviors of FOFR. Then the global Mittag-Leffler Attractive Set (MLAS) and Mittag-Leffler Positive Invariant Set (MLPIS) for the proposed Financial Risk (FR) system are discussed. Finally, a proficient reservoir-computing (RC) method is applied to forecast the temporal evolution of the complex dynamics for the proposed system, and some simulations are carried out to show the effectiveness and feasibility of present scheme.

Stability enhancement for seamless control in networked microgrids with energy storage: Nonlinear spatio-temporal control perspective

The integration of distributed energy resources into modernized networked microgrids, combined with the increasing variability in load dynamics, presents significant stability challenges. This research offers a comprehensive analysis of the stability of cascaded interactions in nonlinear multi-timescale systems, including grid-following and grid-forming inverters. The proposed grid-forming controller, integrated with energy storage systems and a nonlinear Lyapunov function, facilitates seamless control and stabilization of these inverters. This approach addresses challenges related to nonlinear plant and network dynamics, uncertainties in state variables, and unmodeled system dynamics. The closed-loop control system ensures asymptotic spatio-temporal dynamical stability through constrained time state convergence to stable equilibrium points while minimizing control oscillations. Comparative analysis and high-fidelity hardware-in-loop emulations demonstrate the superior performance of the proposed controller over traditional ones in diverse dynamic scenarios. Its ability to rapidly reject disturbances, minimize overshoot, converge efficiently, and enhance power quality confirms its effectiveness and highlights its potential for real-world applications.

Optimization of SOX2 Expression for Enhanced Glioblastoma Stem Cell Virotherapy

The Zika virus has been shown to infect glioblastoma stem cells via the membrane receptor αvβ5, which is activated by the stem-specific transcription factor SOX2. Since the expression level of SOX2 is an important predictive marker for successful virotherapy, it is important to understand the fundamental mechanisms of the role of SOX2 in the dynamics of cancer stem cells and Zika viruses. In this paper, we develop a mathematical ODE model to investigate the effects of SOX2 expression levels on Zika virotherapy against glioblastoma stem cells. Our study aimed to identify the conditions under which SOX2 expression level, viral infection, and replication can reduce or eradicate the glioblastoma stem cells. Analytic work on the existence and stability conditions of equilibrium points with respect to the basic reproduction number are provided. Numerical results were in good agreement with analytic solutions. Our results show that critical threshold levels of both SOX2 and viral replication, which change the stability of equilibrium points through population dynamics such as transcritical and Hopf bifurcations, were observed. These critical thresholds provide the optimal conditions for SOX2 expression levels and viral bursting sizes to enhance therapeutic efficacy of Zika virotherapy against glioblastoma stem cells. This study provides critical insights into optimizing Zika virus-based treatment for glioblastoma by highlighting the essential role of SOX2 in viral infection and replication.

Tripartite evolutionary game and simulation of solvency supervision under C‐ROSS II based on prospect theory

AbstractThis paper focuses on the implementation of “C‐ROSS II” and utilizes an evolutionary game model to investigate regulatory issues. Based on prospect theory, a three‐party evolutionary game model is constructed among regulatory agencies, insurance companies, and consumers with incomplete rationality, examining evolutionary stability strategies. Meanwhile, considering the different attitudes of policyholders in the face of loss and return, the heterogeneous risk preference is analyzed by changing the prospect parameters. The results show that increases in penalty amounts, positive incentives, and consumer sensitivity to losses will promote the evolution of the system to the optimal stable equilibrium point. However, rises in brand incomes and rectification costs, as well as decreases in capital costs, will decrease the probability of regulatory authorities enforcing strict supervision.

Multistability and fixed-time multisynchronization of switched neural networks with state-dependent switching rules

This paper presents theoretical results on the multistability and fixed-time synchronization of switched neural networks with multiple almost-periodic solutions and state-dependent switching rules. It is shown herein that the number, location, and stability of the almost-periodic solutions of the switched neural networks can be characterized by making use of the state-space partition. Two sets of sufficient conditions are derived to ascertain the existence of 3n exponentially stable almost-periodic solutions. Subsequently, this paper introduces the novel concept of fixed-time multisynchronization in switched neural networks associated with a range of almost-periodic parameters within multiple stable equilibrium states for the first time. Based on the multistability results, it is demonstrated that there are 3n synchronization manifolds, wherein n is the number of neurons. Additionally, an estimation for the settling time required for drive–response switched neural networks to achieve synchronization is provided. It should be noted that this paper considers stable equilibrium points (static multisynchronization), stable almost-periodic orbits (dynamical multisynchronization), and hybrid stable equilibrium states (hybrid multisynchronization) as special cases of multistability (multisynchronization). Two numerical examples are elaborated to substantiate the theoretical results.

Analysis of a Bifurcation and Stability of Equilibrium Points for Jeffrey Fluid Flow through a Non-Uniform Channel

This study aims to analyze how the parameter flow rate and amplitude of walling waves affect the peristaltic flow of Jeffrey’s fluid through an irregular channel. The movement of the fluid is described by a set of non-linear partial differential equations that consider the influential parameters. These equations are transformed into non-dimensional forms with appropriate boundary conditions. The study also utilizes dynamic systems theory to analyze the effects of the parameters on the streamline and to investigate the position of critical points and their local and global bifurcation of flow. The research presents numerical and analytical methods to illustrate the impact of flow rate and amplitude changes on fluid transport. It identifies three types of streamline patterns that occur: backwards, trapping, and augmented flow resulting from changes in the value of flow rate parameters.

Smart waterborne disease control for a scalable population using biodynamic model in IoT network

We propose a biodynamic model for managing waterborne diseases over an Internet of Things (IoT) network, leveraging the scalability of LoRa IoT technology to accommodate a growing human population. The model, based on fractional order derivatives (FOD), enables smart prediction and control of pathogens that cause waterborne diseases using IoT infrastructure. The human-pathogen-based biodynamic FOD model utilises epidemic parameters (SVIRT: susceptibility, vaccination, infection, recovery, and treatment) transmitted over the IoT network to predict pathogenic contamination in water reservoirs and dumpsites in Iji-Nike, Enugu, the study community in Nigeria. These pathogens contribute to person-to-person, water-to-person, and dumpsite-to-person transmission of disease vectors. Five control measures are proposed: potable water supply, treatment, vaccination, adequate sanitation, and health education campaigns. A stable disease-free equilibrium point is found when the effective reproduction number of the pathogens, R0eff<1 and unstable if R0eff>1. While other studies showed a 98.2% reduction in infections when using IoT alone, this paper demonstrates that combining the SVIRT epidemic control parameters (such as potable water supply and health education campaign) with IoT achieves a 99.89% reduction in infected human populations and a 99.56% reduction in pathogen populations in water reservoirs. Furthermore, integrating treatment with sanitation results in a 99.97% reduction in infected populations. Finally, combining these five control strategies nearly eliminates infection and pathogen populations, demonstrating the effectiveness of multifaceted approaches in public health and environmental management. This study provides a blueprint for governments to plan sustainable smart cities for a growing population, ensuring potable water free from pathogenic contamination,in line with the United Nations Sustainable Development Goals #6 (Clean Water and Sanitation) and #11 (Sustainable Cities and Communities).

Mathematical modeling of allelopathic stimulatory phytoplankton species using fractal–fractional derivatives

In the current study, we employ the novel fractal–fractional operator in the Atangana–Baleanu sense to investigate the dynamics of an interacting phytoplankton species model. Initially, we utilize the Picard-Lindelöf theorem to validate the uniqueness and existence of solutions for the model. We then explore equilibrium points within the phytoplankton model and conduct Hyers–Ulam stability analysis. Additionally, we present a numerical scheme utilizing the Newton polynomial to validate our analytical findings. Numerical simulations illustrate the dynamical behavior of the model across various fractal and fractional parameter values, visualized through graphical representations. Our simulations reveal that the stability of equilibrium points is not significantly impacted with the long-term memory effect, which is characterized by fractal–fractional order values. However, an increase in fractal–fractional parameters accelerates the convergence of solutions to their intended equilibrium states.

Transient Synchronous Stability Analysis of Grid-Following Converter Considering Outer-Loop Control with Current Limiting

With the evolution of modern power systems, inverter-based resources have become increasingly prevalent. As critical energy conversion interfaces, grid-following converters exhibit dynamic performances, presenting challenges for system security and stability. This paper focuses on the transient synchronization stability of converters after disturbances, highlighting differences in mechanisms compared to synchronous generators. Although previous studies on the transient synchronization stability of converters have been conducted, they primarily concentrate on the dynamics of the phase-locked loop, with limited consideration of the effects of outer-loop control. This has created a cognitive bottleneck in understanding the transient synchronization mechanisms of converters. To address these challenges, this paper models a grid-following voltage source converter system, incorporating detailed converter control strategies and current-limiting control. The stability regions of the stable equilibrium point under various fault severities are first analyzed. Then, the impacts of outer-loop control, including PI control and current-limiting control, on transient synchronization are examined. The study systematically elucidates the influence of outer-loop control on the transient synchronization stability of converters. Finally, the validity of the proposed theory is confirmed through simulations conducted in PSCAD/EMTDC.

Analysis and Simulation of SEIR Mathematical Model of Stunting Case in Indonesia

Stunting is one of the focuses of the target for improving nutrition in the world until 2025. The government of Indonesia targets the prevalence of stunting to decrease to 14% in 2024. There are some factor affect stunting, some of them are sanitation, health care and parenting of child. In this research, we constructed a SEIR mathematical model of stunting case in Indonesia. There are four compartements in this model : the population of newborn are likely to be exposed to stunting (S), the population of children were having an early symptom stunting (E), the population of affected child stunting permanent (I) and the population of children showing symptom stunting but no caught stunting (free stunting) (R). The disease free and endemic equilibrium point are the equilibrium of the formed model. The local stability of equilibrium points have been analyzed, determined the basic reproduction number, and conducted a sensitivity analysis. If then is asymptotically stable If then is asymptotically stable. Sensitivity analysis revealed that the parameters and significantly contribute to the increase in the basic reproduction number . It means that the good sanitation, good healthcare, good parenting of child and decrease the rate of transmission indirectly from the person affected stunting are the most important thing to reduce the rate of stunting in Indonesia.

Stability of equilibrium states for a stochastic difference equation of exponential form

It is well know that many process and problems in different fields of sciences and engineering can be modelled by linear and nonlinear difference equations. Particularly, in case of systems biology, ecology, biochemistry, genetics and physiology dynamics, many population models are governed by exponential difference equations, and lot of papers were published in this matter. The goal of this work is to study a nonlinear second order difference equation of exponential form: Xn+1 = bXn + cXne−σXn−1, n ≥ 0 where the parameters b, c have arbitrary values, σ &gt; 0 and the initial values, X0, X−1 are arbitrary positive numbers. Our equation has an equilibrium points and is exposed to additive stochastic perturbations that are assumed a sequence of independent random variables with zero mean, unit variance and these perturbations are proportional to the deviation of the system state Xn from equilibrium points, and the result is stochastic difference equation. We cannot find the solutions of nonlinear stochastic difference equations of exponential form in all cases. So, one can study the behavior of solutions by asymptotic stability of equilibrium points. Additionally, with the general method of Lyapunov functional constructions for stochastic difference equations with discrete time, we provide the necessary and sufficient conditions for the asymptotic mean square stability of the two equilibrium points (zero and positive) within the linear approximation of the stochastic difference equation. These conditions also serve as sufficient conditions for the stability in probability of the equilibrium points of the initial nonlinear equation. Finally, to demonstrate the validity of the obtained results, some numerical examples and simulations of equations solutions are made and numerous graphical illustrations of stability trajectories of solutions are plotted.

Servo Control of a Current-Controlled Attractive-Force-Type Magnetic Levitation System Using Fractional-Order LQR Control

Recent research on fractional-order control laws has introduced the fractional calculus concept into the field of control engineering. As described herein, we apply fractional-order linear quadratic regulator (LQR) control to a current-controlled attractive-force-type magnetic levitation system, which is a strongly nonlinear and unstable system, to investigate its control performance through experimentation. First, to design the controller, a current-controlled attractive-force-type magnetic levitation system expressed as an integer-order system is extended to a fractional-order system expressed using fractional-order derivatives. Then, target value tracking control of a levitated object is achieved by adding states, described by the integrals of the deviation between the output and the target value, to the extended system. Next, a fractional-order LQR controller is designed for the extended system. For state-feedback control, such as fractional-order servo LQR control, which requires the information of all states, a fractional-order state observer is configured to estimate fractional-order states. Simulation results demonstrate that fractional-order servo LQR control can achieve equilibrium point stabilization and enable target value tracking. Finally, to verify the fractional-order servo LQR control effectiveness, experiments using the designed fractional-order servo LQR control law are conducted with comparison to a conventional integer-order servo LQR control.

Research on Division of Labor Decision and System Stability of Swarm Robots Based on Mutual Information.

In rational decision-making processes, the information interaction among individual robots is a critical factor influencing system stability. We establish a game-theoretic model based on mutual information to address division of labor decision-making and stability issues arising from differential information interaction among swarm robots. Firstly, a mutual information model is employed to measure the information interaction among robots and analyze its influence on the behavior of individual robots. Secondly, employing the Cournot model and the Stackelberg model, we model the diverse decision-making behaviors of swarm robots influenced by discrepancies in mutual information. The intricate decision dynamics exhibited by the system under the disparity mutual information values during the game process, along with the stability of Nash equilibrium points, are analyzed. Finally, dynamic complexity simulations of the game models are simulated under the disparity mutual information values: (1) When ν1 of the game model varies within a certain range, the Nash equilibrium point loses stability and enters a chaotic state. (2) As I(X;Y) increases, the decision-making pattern of robots transitions gradually from the Cournot game to the Stackelberg game. Concurrently, the sensitivity of swarm robotics systems to changes in decision parameter decreases, reducing the likelihood of the system entering a chaotic state.

A secure communication scheme based on generalized modified projective synchronization of a new 4-D fractional-order hyperchaotic system

This paper focuses on the exploration of the chaotic behavior of a new 4-D fractional-order hyperchaotic system with five nonlinearities. The Adomian decomposition method is employed to solve the fractional-order hyperchaotic system. The stability of equilibrium points in this system is analyzed. Through bifurcation diagrams, Lyapunov exponent spectra, chaotic attractors, 0-1 test, C 0 complexity, and spectral entropy, the chaotic dynamics of the proposed system are investigated. Using these tools, we demonstrate the fractional-order system’s sensitivity to variation in both the derivative order and initial conditions. Moreover, a modified generalized projective synchronization is developed to implement chaos synchronization between two coupled fractional-order hyperchaotic systems. Furthermore, this work presents an application of synchronization scheme in secure communication. Numerical simulations were implemented to validate the effectiveness of the proposed secure communication scheme.

Unraveling the importance of early awareness strategy on the dynamics of drug addiction using mathematical modeling approach.

A drug is any substance capable of altering the functioning of a person's body and mind. In this paper, a deterministic nonlinear model was adapted to investigate the behavior of drug abuse and addiction that incorporates intervention in the form of awareness and rehabilitation. In the mathematical analysis part, the positivity and boundedness of the solution and the existence of drug equilibria have been ascertained, which shows that the model consists of two equilibria: a drug-free equilibrium and a drug endemic equilibrium point. The drug-free equilibrium was found to be both globally and locally asymptotically stable if the effective reproduction number is less than or equal to one (Rc≤1). Furthermore, we were able to show the existence of a unique drug endemic equilibrium whenever Rc>1. Global asymptotic stability of a drug endemic equilibrium point has been ascertained using a nonlinear Lyapunov function of Go-Volterra type, which reveals that the drug endemic equilibrium point is globally asymptotically stable if an effective reproduction number is greater than unity and if there is an absence of a reversion rate of mended individuals (i.e., ω=0). In addition, an optimal control problem was formulated to investigate the optimal strategy for curtailing the spread of the behavior using control variables. The control variables are massive awareness and rehabilitation intervention of both public and secret addicted individuals. The optimal control simulation shows that massive awareness control is the best to control drug addiction in a society. In sensitivity analysis section, the proportion of those who are exposed publicly shows to be a must sensitive parameter that can reduce the reproduction number, and the effective contact rate shows to be a must sensitive parameter to increase the reproduction number. Numerical simulations reveal that the awareness rate of exposed publicly and the rehabilitation rate of addicted publicly are very important parameters to control drug addiction in a society.

Analysis of a Resistive-Capacitive Shunted Josephson Junction with Topologically Nontrivial Barrier Coupledto a RLC Resonator

This paper presents the resistive-capacitive shunted Josephson junction (RCSJJ) with a topologically nontrivial barrier (TNB) coupled to a linear RLC resonator. The rate equations describing RCSJJ with TNB coupled to the linear RLC resonator are established via Kirchhoff’s current and voltage laws. The model exhibits four, two, or no equilibrium points depending on the external direct current (DC) source and the fractional parameter. The stability analysis of the equilibrium points with credit to the Routh-Hurwitz stability criterion reveals that the stability of equilibrium points depends on the DC source and the fractional parameter. Current-voltage characteristic reveals the presence of a birhythmicity zone which is sensitive to the fractional parameter m. As the fractional parameter increases, the coexistence of the resonant state is destroyed, which is followed simultaneously by the appearance of a new resonance state. Depending on initial conditions, birhythmic behaviour is characterized by the existence of a limit cycle. The projection of the phase space in the specific plane and the time evolution of charge is predicted in which the amplitude of attractors reported is sensitive to the parameter m. Lastly, with a defined fractional parameter, the amplitude of the branch locked to the resonator is greater than the unlocked branch.

When and why microbial-explicit soil organic carbon models can be unstable

Abstract. Microbial-explicit soil organic carbon (SOC) cycling models are increasingly being recognized for their advantages over linear models in describing SOC dynamics. These models are known to exhibit oscillations, but it is not clear when they yield stable vs. unstable equilibrium points (EPs) – i.e., EPs that exist analytically but are not stable in relation to small perturbations and cannot be reached by transient simulations. The occurrence of such unstable EPs can lead to unexpected model behavior in transient simulations or unrealistic predictions of steady-state soil organic carbon (SOC) stocks. Here, we ask when and why unstable EPs can occur in an archetypal microbial-explicit model (representing SOC, dissolved OC (DOC), microbial biomass, and extracellular enzymes) and some simplified versions of it. Further, if a model formulation allows for physically meaningful but unstable EPs, can we find constraints in the model parameters (i.e., environmental conditions and microbial traits) that ensure stability of the EPs? We use analytical, numerical, and descriptive tools to answer these questions. We found that instability can occur when the resupply of a growth substrate (DOC) is (via a positive feedback loop) dependent on its abundance. We identified a conservative, sufficient condition in terms of model parameters to ensure the stability of EPs. Principally, three distinct strategies can avoid instability: (1) neglecting explicit DOC dynamics, (2) biomass-independent uptake rate, or (3) correlation between parameter values to obey the stability criterion. While the first two approaches simplify some mechanistic processes, the third approach points to the interactive effects of environmental conditions and parameters describing microbial physiology, highlighting the relevance of basic ecological principles for the avoidance of unrealistic (i.e., unstable) simulation outcomes. These insights can help to improve the applicability of microbial-explicit models, aid our understanding of the dynamics of these models, and highlight the relation between mathematical requirements and (in silico) microbial ecology.

Stability analysis of a nonlinear malaria transmission epidemic model using an effective numerical scheme

Malaria is a fever condition that results from Plasmodium parasites, which are transferred to humans by the attacks of infected female Anopheles mosquitos. The deterministic compartmental model was examined using stability theory of differential equations. The reproduction number was obtained to be asymptotically stable conditions for the disease-free, and the endemic equilibria were determined. More so, the qualitatively evaluated model incorporates time-dependent variable controls which was aimed at reducing the proliferation of malaria disease. The reproduction number R o\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(o\\right)$$\\end{document} was determined to be an asymptotically stable condition for disease free and endemic equilibria. In this paper, we used various schemes such as Runge–Kutta order 4 (RK-4) and non-standard finite difference (NSFD). All of the schemes produce different results, but the most appropriate scheme is NSFD. This is true for all step sizes. Various criteria are used in the NSFD scheme to assess the local and global stability of disease-free and endemic equilibrium points. The Routh–Hurwitz condition is used to validate the local stability and Lyapunov stability theorem is used to prove the global asymptotic stability. Global asymptotic stability is proven for the disease-free equilibrium when R0≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\le 1$$\\end{document}. The endemic equilibrium is investigated for stability when R0≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}\\ge 1$$\\end{document}. All of the aforementioned schemes and their effects are also numerically demonstrated. The comparative analysis demonstrates that NSFD is superior in every way for the analysis of deterministic epidemic models. The theoretical effects and numerical simulations provided in this text may be used to predict the spread of infectious diseases.