Equilibrium Point Research Articles

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Dynamical analysis of a modified Leslie–Gower predator–prey model with disease in prey incorporating a prey refuge and treatment

In this paper, we present and assess a predator–prey Leslie–Gower model including disease, refuge and treatment in prey population. There are two groups of prey: those who are susceptible and infected. It is hypothesized that prey population is affected by diseases and refuge, and grows logistically in the absence of predators. Infected prey population receives treatment. The predators’ growth rate is governed by the modified Leslie–Gower dynamics. The dynamical attributes of the resulting system are boundedness, positivity of solutions, extinction criteria, existence and (local and global) stability. Biology uses mathematical analysis to identify the possible attributes of equilibrium points. The focus of this study is to assess how treatment and refuge affect the populations of ill prey, susceptible prey, predators and treated prey. The numerical simulation indicates that the influence of treatment, and refuge change the dynamics of the system (2.1). Extensive numerical simulations were performed to validate our analytical findings by using the Mathematica and MATLAB software.

Complexity analysis of technology spillover supply chain with dual business objectives

ABSTRACT Based on the existence of technology spillover effect of manufacturers, considering the manufacturer's commercial goals of profit and market share, a dynamic price game model of technology spillover supply chain with dual commercial goals is established. Based on game theory and nonlinear dynamics theory, the equilibrium point and local stability of the system are analyzed. In addition, the effects of price adjustment rate, balance coefficient of business objectives, and technology spillover on model complexity, the change of manufacturer price and profit were analyzed through numerical simulation. The results show that the price adjustment rate, the balance coefficient of business objectives, and the technology spillover have an impact on the stability and complexity of the manufacturer's price competition model. Rationally controlling the price adjustment rate, and enhancing technological spillovers are beneficial to the healthy development of manufacturers and the healthy competition among manufacturers, maintaining the stability of product market prices, and effectively avoiding market chaos. Moreover, when considering market share, manufacturers can expand the system stability domain.

The dynamics of hybrid-immune and immunodeficient susceptible individuals and the three stages of COVID-19 vaccination

The World Health Organization has disclosed that the hybrid-immune and immunodeficient individuals are two distinct classes of individuals susceptible to the COVID-19 virus. To model this unique characteristics of two distinct categories of susceptible individuals and the dynamics of the three phases of a vaccination program implemented by the World Health Organization in Malaysia, which have not been accounted for in previous studies, a twelve compartmental SSVEIHQR-D epidemiological model was developed. This model aimed to accurately capture the spread of the COVID-19 virus by fitting real-life data to the model and obtaining updated estimates of the reproduction number. The study also focused on assessing current control measures and exploring strategies to eradicate the virus and mitigate future outbreaks. Mathematical analyses of the model included investigations into stability, equilibrium points, the basic reproduction number R0, optimal control strategies, and sensitivity analyses. Estimation and fitting of the model parameters were conducted using daily situation reports from the Ministry of Health of Malaysia. The obtained value of the basic reproduction number, based on fitted parameter values, indicated stability and reflected the current pandemic situation more realistically. Additionally, the herd immunity threshold was calculated and interpreted in the context of the study findings. Finally, practical insights and recommendations derived from the model’s results were provided to inform government agencies, public health organizations, and intervention bodies, aimed at controlling the spread of the COVID-19 virus.

Considering individual game behavior and time delay in the multi-channel rumor propagation model

The rapid development of the Internet has accelerated the spread of rumors, posing challenges to social cohesion and stability. To address this, a multi-channel rumor propagation model incorporating individual game behavior and time delay is proposed. It depicts individuals strategically choosing propagation channels in the rumor spread process, capturing real-world intricacies more faithfully. Specifically, the model allowing spreaders to choose between text and video information base channels. Strategy adoption hinges on benefits versus costs, with payoffs dictating strategy and the propagation process determining an individual’s state. By theoretical analysis of the model, the propagation threshold and equilibrium points are obtained. Then the stability of the model is further demonstrated based on Routh–Hurwitz judgment and Descartes’ Rule of Signs. Numerical simulations are conducted to verify the correctness of the theoretical results and the sensitivity of the model to key parameters. The outcomes reveal that increasing the propagation cost of spreaders can significantly curb the spread of rumors. In contrast to the classical ISR model, rumors spread faster and more widely in the improved multi-channel rumor propagation model in this paper, which is a feature more aligned with real-world scenarios. Finally, the validity and predictive ability of the model are verified by using real rumor propagation data sets, indicating that the improved multi-channel rumor propagation model has good practical application and predictive value.

Dynamics of a Delayed Fractional-Order Predator–Prey Model with Cannibalism and Disease in Prey

In this study, a class of delayed fractional-order predation models with disease and cannibalism in the prey was studied. In addition, we considered the prey stage structure and the refuge effect. A Holling type-II functional response function was used to describe predator–prey interactions. First, the existence and uniform boundedness of the solutions of the systems without delay were proven. The local stability of the equilibrium point was also analyzed. Second, we used the digestion delay of predators as a bifurcation parameter to determine the conditions under which Hopf bifurcation occurs. Finally, a numerical simulation was performed to validate the obtained results. Numerical simulations have shown that cannibalism contributes to the elimination of disease in diseased prey populations. In addition, the size of the bifurcation point τ0 decreased with an increase in the fractional order, and this had a significant effect on the stability of the system.

Dynamic vaccination strategies in dual-strain epidemics: A multi-agent-based game-theoretic approach on scale-free hybrid networks

This paper proposes a multi-agent-based game-theoretic framework to explore the dynamics of decision-making among agents in a situation involving a dual-strain epidemic, where the emergence of the second strain is directly related to the first strain. It considers the impact of imperfect vaccination on dual strains within the SIRV framework, in a hybrid network that combines scale-free and grid-based structures. This research presents three models based on the level of decision-making of agents: Individual Decision Making (IDM), Collective Decision Making (CDM), and Stochastic Decision Making (SDM) model. In the IDM model, agents can continuously update their strategies based on the infection and vaccination statuses of their immediate neighbors. The CDM model offers a macro-level approach where agents’ willingness to vaccinate varies uniformly, while the SDM model maintains a constant level of willingness for vaccination across agents. This study demonstrates that agents using the IDM model tend to initiate vaccination earlier in scenarios of high infection rates, effectively mitigating the spread of the disease in subsequent epidemic seasons. However, when both infection and vaccination rates heavily influence decision-making, there is a noticeable decrease in vaccination uptake, thereby aggravating the spread of infection. Additionally, the authors found that in scenarios of high infection, an over reliance on vaccination without adequate initial uptake can jeopardize disease control efforts. Another important finding of this study is that despite the CDM and SDM models showing a quicker emergence of the second strain and reaching equilibrium points faster than the IDM model, the overall infection rate remains lower in the IDM model. This research underscores the complexity of managing dual-strain epidemics through vaccination strategies and highlights the significance of micro-level decision-making in epidemic control.

Mathematical Model of the Dynamics of three Different Species in Predator-Prey System

To understand how organisms interact with themselves and their surroundings, it is necessary to use mathematical models to explore the science of multispecies cohabitation. This paper studied the interaction between three species of animals (Lion, Leopard, and Hare) in an environment using a formulated predator-prey model of system of three nonlinear differential equations of first order with five equilibrium points. The equilibrium points were found using the Routh Hurwitz Stability Criterion to be locally asymptotically stable. It was also established that the solution of the model exist and are all positive for any t≥0. Some numerical simulations were carried out to show the analytical solution of the model and the results presented. The results show that the predator-prey presented in this work can be used perfectly to study any species interaction with the same nature with the ones considered in this paper. For further studies, this model can be extended to include four or more species interactions

Mathematical Analysis of Substance Abuse and its Effect

In this paper, a model for substance abuse was formulated. A compartmentalization approach was adopted to develop the model, and five nonlinear equations were developed based on the number of compartments. The model was analyzed qualitatively and quantitatively. The Invariant region, positivity, equilibrium points (substance-free and substance-present), basic reproduction number R0, the sensitivity analysis, and the model’s local and global asymptotic stability were obtained. Numerical simulations were conducted to see the changes in the population dynamics of different classes in the sub-population. The results authenticated that the recovery rate of moderate substance users and the recovery rate of rehabilitation/treatment of individuals are indications that they help reduce the number of substance abuse individuals. Also, the recruitment and contact rates impact the population of substance abuse.

Mathematical Model and Optimal Control Strategy for the Dynamics of Hepatitis B Virus Disease Incorporating Treatment Failure and Advanced Stage Compartments

In this paper, mathematical model for the dynamic of Hepatitis B virus infection was developed taking into cognizance the effect of rate of progression to the end stage (cirrhosis and HCC) by individuals who experience treatment failure in the treatment class and chronically infected class. In order to track the transmission of these diseases, the basic reproduction number which determines the rate of new secondary infection when an infected individual is introduced into a totally susceptible population was computed by next generation matrix method (NGM). The stability of the disease-free equilibrium point was also investigated using Routh-Hurwitz Criterion and the result proven to be Locally Asymptotically Stable (LAS) when the basic reproduction number less than unity. The global stability was also investigated by Castillo-Chavez method and the results show that the model is Globally Asymptotically (GAS) when reproduction number is less than unity. Sensitivity analysis was carried out on the parameters of the reproduction number, using forward normalized sensitivity index to ascertain those parameters of high impact on the reproduction number to aid the incorporation of the control variables appropriately. Numerical simulation was also carried using Forward-backward Runge-Kutta fourth order scheme to study the effect of these control strategies on the dynamics of the model and the results shows that the best strategy to implement to help eradicate HBV disease in the system is strategy E which is the combination of effective condom use, HBV vaccination, HBV Treatment and Behavioral change

Lagrange points and regionally conserved quantities

Lagrange points are the equilibrium points within a restricted three-body system, epitomized by the Trojan asteroids near the L4 and L5 points of the Sun–Jupiter system. They also play a crucial role in some space missions, including the James Webb Space Telescope which is located at the Sun–Earth L2 point. While the existence of five Lagrange points is a well-known feature of the restricted three-body problem, the equations describing the precise location of all five points are not extensively documented. This work presents a derivation of all Lagrange points using polar coordinates and a new normalization scheme that offers a simpler interpretation of solutions compared to prior analyses. A subtle issue concerning the treatment of angular momentum in the potential formulation of this problem is addressed and resolved. The supplementary material to this work contains additional mathematical details and discussion.

Distributed Nash equilibrium seeking in noncooperative game with partial decision information of neighbors.

In the real world, individuals may conceal some of their real decision information to their neighbors due to competition. It is a challenge to explore the distributed Nash equilibrium when individuals play the noncooperative game with partial decision information in complex networks. In this paper, we investigate the distributed Nash equilibrium seeking problem with partial decision information of neighbors. Specifically, we construct a two-layer network model, where players in the first layer engage in game interactions and players in the second layer exchange estimations of real actions with each other. We also consider the case where the actions of some players remain unchanged due to the cost of updating or personal reluctance. By means of the Lyapunov function method and LaSalle's invariance principle, we obtain the sufficient conditions in which the consensus of individual actions and estimations can be achieved and the population actions can converge to the Nash equilibrium point. Furthermore, we investigate the case with switched topologies and derive the sufficient conditions for the convergence of individual actions to Nash equilibrium by the average dwell time method. Finally, we give numerical examples for cases of fixed and switched topologies to verify our theoretical results.

Optimal control strategies for toxoplasmosis disease transmission dynamics via harmonic mean-type incident rate

Toxoplasma infection in humans is considered due to direct contact with infected cats. Toxoplasma infection (an endemic disease) has the potential to affect various organs and systems (brain, eyes, heart, lungs, liver, and lymph nodes). Bilinear incidence rate and constant population (birth rate is equal to death rate) are used in the literature to explain the dynamics of Toxoplasmosis disease transmission in humans and cats. The goal of this study is to consider the mathematical model of Toxoplasma disease with harmonic mean type incident rate and also consider that the population of humans and cats is not equal (birth rate and the death rate are not equal). In examining Toxoplasma transmission dynamics in humans and cats, harmonic mean incidence rates are better than bilinear incidence rates. The disease dynamics are first schematically illustrated, and then the law of mass action is applied to obtain nonlinear ordinary differential equations (ODEs). Analysis of the boundedness, positivity, and equilibrium points of the system has been analyzed. The reproduction number is calculated using the next-generation matrix technique. The stability of disease-free and endemic equilibrium are analyzed. Sensitivity analysis is also done for reproduction number. Numerical simulation shows that the infection is spread in the population when the contact rate βh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta_{h}$$\\end{document} and βc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta_{c}$$\\end{document} increases while the infection is reduced when the recovery rate δh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta_{h}$$\\end{document} increases. This study investigates the impact of various optimal control strategies, such as vaccinations for the control of disease and the awareness of disease awareness, on the management of disease.

Mathematical modeling and simulation for malaria disease transmission using the CF fractional derivative

A major global health problem continues to be malaria, a disease that can be fatal brought on by Plasmodium parasites and spread by the bite of infected Anopheles mosquitoes. We provide a deterministic mathematical model in this study to simulate the dynamics of malaria transmission between humans and mosquitoes. We present a new compartment for hospitalized patients as well as fractional calculus. The next-generation matrix technique is used to obtain the fundamental reproduction number, R0, and stability conditions for the model’s equilibrium points are derived. Both locally and globally, the sensitivity analysis for the fundamental reproduction number R0 is satisfied. In MAPLE, the Runge–Kutta fourth-order approach is used to simulate the model.

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.

Mathematical model to assess the impact of contact rate and environment factor on transmission dynamics of rabies in humans and dogs

This paper presents a mathematical model to understand how rabies spreads among humans, free-range, and domestic dogs. By analyzing the model, we discovered that there are equilibrium points representing both disease-free and endemic states. We calculated the basic reproduction number, R0 using the next generation matrix method. When R0<1, the disease-free equilibrium is globally stable, whereas when R0≥1, the endemic equilibrium is globally stable. To identify the most influential parameters in disease transmission, we used the normalized forward sensitivity index. The simulations revealed that the contact rates between the infectious agent and humans, free-range dogs, and domestic dogs, have the most significant impact on rabies transmission. The study also examines how periodic changes in transmission rates affect the disease dynamics, emphasizing the importance of transmission frequency and amplitude on the patterns observed in rabies spread. To reduce disease sensitivity, one should prioritize effective disease control measures that focus on keeping both free-range and domestic dogs indoors. This is a crucial factor in preventing the spread of disease and should be implemented as a primary disease control measure.

A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics

This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like difference operator, is employed to establish the fractional form of the map with short memory. The numerical simulation results highlight its excellent dynamical behavior, revealing that the addition of the piecewise fractional order makes the memristive-based Duffing map even more chaotic. It is characterized by distinct features, including the absence of an equilibrium point and the presence of multiple hidden chaotic attractors.

Analyzing a Dynamical System with Harmonic Mean Incidence Rate Using Volterra–Lyapunov Matrices and Fractal-Fractional Operators

This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics.

A comprehensive analysis of COVID-19 nonlinear mathematical model by incorporating the environment and social distancing

This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. The study develops a sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V–L) matrices method. To understand the dynamic behavior of COVID-19, numerical simulations are essential. For this purpose, the study employs a robust numerical technique known as the non-standard finite difference (NSFD) method, introduced by Mickens. Various results are visually presented through graphical representations across different parameter values to illustrate the impact of environmental factors and social distancing measures.

Mathematical Modeling of Infectious Disease and Prey-Predator Interaction with Optimal Control

In this paper, the impact of viral illnesses on the predator-prey relationship with an optimal control analysis is studied. An ecoepidemiological model of four compartments, namely, susceptible prey, susceptible predator, infected prey, and infected predator populations, in the interaction of the prey-predator system is formulated. The fundamental tenet of our ecoepidemiology model is that sick predators do not engage in predation. It is confirmed that the system’s solution exists, is positive, and is bounded. The system’s equilibrium points are determined and computed. Lyapunov functions and a linearizing form are used for local and global stability analysis, respectively. The next generation matrix approach is used to calculate the threshold value for diseased predators and prey at the disease-free equilibrium point. Optimal treatment options for vulnerable and infected populations are established by applying optimal control theory to the ecoepidemiology model of a prey-predator system. MATLAB software is utilized to obtain numerical simulations that validate the analytical outcomes. The optimal control problem simulations demonstrate that the number of infected populations in a given prey-predator system can be decreased by implementing control measures.