Stability Analysis Of Equilibrium Points Research Articles

Modelling the dynamics of anti-microbial resistant cholera infection with reinfection

Cholera continues to pose a significant global public health challenge. The occurrence of severe weather events, including floods, droughts, and cyclones, has led to a surge in new cholera outbreaks and exacerbated both epidemic and endemic conditions in numerous countries. We propose a ten-state ordinary differential system of equations to model antimicrobial resistant cholera infections, considering both direct and indirect transmission pathways. Antimicrobial resistant cholera infection is considered to occur in some infected individuals receiving antibiotic treatment. The model integrates key interventions such as vaccination, screening, and treatment to curb cholera spread. Loss of infection-acquired immunity for the recovered individuals is also considered in the model. A threshold quantity referred to as the model reproduction number Rc is computed and stability analysis of equilibrium points for the full model and submodels is performed. Sensitivity analysis is performed and specific measures to reduce parameters associated with cholera transmission and enhance parameters related to control measures are suggested. Numerical simulations are carried out to support theoretical findings. The results obtained places a significant emphasis on antimicrobial resistance and its impact on cholera control. The importance of minimizing antimicrobial resistance through targeted interventions is highlighted. Therefore, there is an urgent need to implement policies aimed at mitigating the spread of antimicrobial resistance.

Effect of nonlinear stiffness on single link flexible joint robotic manipulator and its control

This study aims to investigate the significant dynamics of a Single Link Flexible Joint (SLFJ) Robotic Manipulator and implement chaotic control using the Adaptive Sliding Mode Control (ASMC) algorithm. The mathematical model of the system incorporates stiffness nonlinearity by introducing the function (x-x3) Various dynamical tools such as stability analysis of equilibrium points, sensitivity to initial conditions, orbit diagrams for parameter variations, Lyapunov spectrum, and frequency spectrum analysis are employed. The ASMC algorithm is utilized to suppress the chaotic behavior of the system. The equilibrium points are determined for critical parameter values, with the Jacobian matrix and corresponding eigenvalues exhibiting conjugate complex with negative real parts, indicating stability resembling a spiral saddle. Phase portraits illustrate the location and range of the chaotic attractor. Investigation of parameter ′a′ using orbit diagrams and Lyapunov spectrum reveals the transition from periodic to chaotic oscillations. The frequency spectrum confirms the chaotic nature of the system. Parameter estimation, state trajectories, and sliding surface analysis demonstrate that the ASMC scheme can effectively control the system in less than 0.3 s. Nonlinear stiffness has not been previously considered in SLFJ Robotic Manipulators, highlighting a novel aspect of this study. The results unveil new chaotic ranges for parameter variation previously undocumented. Frequency spectrum analysis is employed to distinguish between noise and chaotic oscillations, addressing a significant challenge in real-time systems. The ASMC algorithm is specifically designed to control chaotic oscillations, demonstrating its effectiveness in this context.

Analyzing the spatial motion of a rigid body subjected to constant body-fixed torques and gyrostatic moment

This paper aims to explore the rotatory spatial motion of an asymmetric rigid body (RB) under constant body-fixed torques and a nonzero first component gyrostatic moment vector (GM). Euler's equations of motion are used to derive a set of dimensionless equations of motion, which are then proposed for the stability analysis of equilibrium points. Specifically, this study develops 3D phase space trajectories for three distinct scenarios; two of them are applied constant torques that are directed on the minor and major axes, while the third one is the action of applied constant torque on the body’s middle axis. Novel analytical and simulation results for both scenarios of constant torque applied along the minor and middle axes are provided in the context of separatrix surfaces, equilibrium manifolds, periodic or non-periodic solutions, and periodic solutions’ extreme. Concerning the scenario of a directed torque on the major axis, a numerical solution for the problem is presented in addition to a simulation of the graphed results for the angular velocities' trajectories in various regions. Moreover, the influence of GM is examined for each case and a full modeling for the body's stability has been present. The exceptional impact of these results is evident in the development and assessment of systems involving asymmetric RBs, such as satellites and spacecraft. It may serve as a motivating factor to explore different angles within the GM in similar cases, thereby influencing various industries, including engineering and astrophysics applications.

A novel fractional-order stochastic epidemic model to analyze the role of media awareness in the spread of conjunctivitis

This study introduces a novel fractional-order stochastic epidemic model to analyze the spread of conjunctivitis, a prevalent ocular infection, while accounting for the influence of media awareness on disease transmission. The model incorporates fractional derivatives to capture memory effects and non-local interactions inherent in epidemic processes, allowing for a more accurate representation of disease dynamics. The stability analysis of equilibrium points is carried out based on the basic reproduction number ℛ0 and fractional-order α. Further, the Hopf bifurcation phenomenon is discussed in this paper. Stochasticity accounts for the randomness in transmission events. The findings of this study provide insights into the complex interrelationship between disease dynamics and media influence, shedding light on the role of public awareness in mitigating or exacerbating conjunctivitis outbreaks. The implications of this work extend to public health policy formulation, highlighting the importance of targeted communication strategies in controlling and preventing the spread of conjunctivitis and similar infectious diseases.

Dynamical analysis of a novel discrete fractional lumpy skin disease model

Lumpy skin disease is a viral disease that affects cattle and is caused by the lumpy skin disease virus. This work is devoted to presenting and analyzing the nonlinear dynamics of a novel discrete fractional model for lumpy skin disease. The equilibrium points of the proposed discrete fractional model are found. The stability analysis of equilibrium points is carried out. The influences of key parameters in the model are investigated, and then the stability regions of a disease-free steady state in the space of parameters are obtained. A proposed efficient control scheme is implemented to stabilize the disease-free equilibrium point when it is unstable. The influences of fractional-order parameters on the applied control scheme are explored. Finally, numerical simulations are performed to verify the theoretical findings obtained and confirm the effectiveness of the employed control scheme.

Mathematical Modelling of Drug Abuse Reduction Strategies taking into account the Treatment Type and Risks Level

Drug abuse is one of the global issues and has spread among teenagers. Drugs may lead to subordination, health problems and even death. There are several policies made in each country related to the problem of drug abuse, both punishment and treatment. In this paper, we discuss the treatment and strategy to reduce the number of drug users. Drug users can recover themselves by undergoing rehabilitation in the form of inpatient or outpatient care. We first conduct qualitative analyses including stability analysis of equilibrium points of the model, the basic reproduction number and parameter sensitivity analysis. Mathematical model of drug abuse reduction by concerning type of treatment along with risk level without control has two equilibrium points, namely non-endemic or drug-free equilibrium and endemic equilibrium. Sensitivity analysis is provided to investigate which parameter that most affects the dynamical behaviour of the drug abuse model in terms of stability of the non-endemic and endemic equilibrium point. Then we impose an anti-drug campaign on the model as strategy control to reduce the number of drug abusers. Simulation results show that the anti-drug campaign has a significant effect in reducing both the number of drug abusers who received any treatment and do not get any treatment.

Qualitative analysis of HIV and AIDS disease transmission: impact of awareness, testing and effective follow up.

Background: Since the early 1980s, human immunodeficiency virus (HIV) and its accompanying acquired immunodeficiency syndrome (AIDS) have spread worldwide, becoming one of the world's major global health issues. From the beginning of the epidemic until 2020, about 79.3 million people became infected, with 36.3 million deaths due to AIDS illnesses.This huge figure is a result of those unaware of their status due to stigmatization and invariably spreading the virus unknowingly. Methods: Qualitative analysis through a mathematical model that will address HIV unaware individuals and the effect of an increasing defaulter on the dynamics of HIV/AIDS was investigated. The impact of treatment and the effect of inefficient follow-up on the transmission of HIV/AIDS were examined. The threshold for the effective reduction of the unaware status of HIV through testing, in response to awareness, and the significance of effective non-defaulting in treatment commonly called defaulters loss to follow-up as these individuals contribute immensely to the spread of the virus due to their increase in CD4+ count was determined in this study. Stability analysis of equilibrium points is performed using the basic reproduction number $R_0$, an epidemiological threshold that determines disease eradication or persistence in viral populations.We tested the most sensitive parameters in the basic reproduction numbers. Results: The results portray that early identification and treatment only are inadequate for the illness to be eradicated, but effectively used of condom, strict adherence to treatment and counseling of and testing of pregnant women contribute to a decrease in infected HIV individuals. Conclusions: Other control techniques, such as treatment adherence and effective condom usage, and reduction in vertical transmission cannot be over-emphasis to lessen the disease's burden. Policymakers must address these strategies through a series of public awareness campaigns about the dangers of not adhering to treatment procedures and patterns.

Sustainable implementation of the carbon-labeling policy with customer participation and government supervision

China is vigorously pursuing carbon neutrality targets to combat global warming. As a powerful tool for encouraging individuals to adjust their consumption patterns and fostering the advancement of green consumption, the overall planning of carbon-labeling policy is accelerating accordingly. To better comprehend and strengthen the sustainable implementation of carbon-labeling policy, this study constructs a tripartite mainstay game to explore the interactive behavior of carbon-labeled enterprises and customers, with the substantial involvement of governmental regulators. First, the evolutionary stability strategy (ESS) is determined by solving replicated dynamic equations and stability analysis of equilibrium points. Then, the practicability and rationality of the evolutionary game model are assessed ESSs corresponding to various scenarios in the carbon-labeling scheme. Finally, the first Chinese television manufacturer to acquire carbon-labeled certification, TCL Group, is utilized as evidence to validate the theoretical findings and support the subsequent arguments: There are eight equilibrium points and three potential ESSs in the game model, and the selection of each ESS is primarily determined by the trade-off between costs and revenues for each stakeholder; the level of carbon-labeled enterprises' green R&D effort has a beneficial effect on the implementation of carbon-labeling policy, while customers' identification of carbon-labeled products will potentially affect operational orientation; and governments' stringent oversight is a vital assurance that both the carbon-labeled enterprises and customers adhere to green initiatives. The results thus not only elaborate on the efficient approach and insights for the sustainable implementation of carbon-labeling policy with the participation of multiple stakeholders, but also offer suggestions for enhancing incentives to improve regulatory regimes and market outcomes.

Examining of a tumor system with Caputo derivative

Cancer is a disease that many people are exposed to, which results in the recovery of some and the death of others. For this reason, A system reflecting the relationship between immune system and tumor growth in this study is examined. This system is handled with the traditional Caputo fractional derivative. The stability analysis of equilibrium points and solution properties of this system is searched. Then, the conditions about the existence and uniqueness of the solution for this system are given. In conclusion, the fractional system is solved benefiting from Grünwald-Letnikov scheme.

Stability Analysis of Equilibrium Point and Limit Cycle of Two-Dimensional Nonlinear Dynamical Systems—A Tutorial

The equilibrium state of a dynamical system can be divided into the equilibrium point and limit cycle. In this paper, the stability analysis of the equilibrium point and limit cycle of dynamical systems are presented through different and all possible approaches, and those approaches are compared as well. In particular, the author presented the stability analysis of the equilibrium point through phase plane approach, Lyapunov–LaSalle energy-based approach, and linearization approach, respectively, for two-dimensional nonlinear system, while the stability analysis of the limit cycle is analyzed by using the LaSalle local invariant set theorem and Poincaré–Bendixson theorem, which is only valid in two-dimensional systems. Different case studies are used to demonstrate the stability analysis of equilibrium point and limit cycle.

Discretization and Stability Analysis for a Generalized Type Nonlinear Pharmacokinetic Models

Estimating the effects of drugs at different stages is directly proportional to the duration of recovery and the duration of pulling through with the disease. It is very important to estimate the effects of drugs at different stages. For this reason, solving Pharmacokinetic models which investigate these effects are very important. In this study, numerical solutions of one, two, and three-compartment nonlinear Pharmacokinetic models have been studied. Distributed order differential equations have been used for the solution. Numerical solutions have been found with the density function contained in distributed order differential equations and different values of this function. A nonstandard finite difference scheme has been used for numerical solutions. Finally, stability analyses of equilibrium points of the obtained discretized system have also been researched with the help of the Matignon criterion.

Qualitative analysis of HIV and AIDS disease transmission: impact of awareness, testing and effective follow up.

Background: Since the early 1980s, human immunodeficiency virus (HIV) and its accompanying acquired immunodeficiency syndrome (AIDS) have spread worldwide, becoming one of the world's major global health issues. From the beginning of the epidemic until 2020, about 79.3 million people became infected, with 36.3 million deaths due to AIDS illnesses. This huge figure is a result of those unaware of their status due to stigmatization and invariably spreading the virus unknowingly. Methods: Qualitative analysis through a mathematical model that will address HIV unaware individuals and the effect of an increasing defaulter on the dynamics of HIV/AIDS was investigated. The impact of treatment and the effect of inefficient follow-up on the transmission of HIV/AIDS were examined. The threshold for the effective reduction of the unaware status of HIV through testing, in response to awareness, and the significance of effective non-defaulting in treatment commonly called defaulters loss to follow-up as these individuals contribute immensely to the spread of the virus due to their increase in CD4+ count was determined in this study. Stability analysis of equilibrium points is performed using the basic reproduction number $R_0$, an epidemiological threshold that determines disease eradication or persistence in viral populations. We tested the most sensitive parameters in the basic reproduction numbers. The model of consideration in this study is based on the assumption that information (awareness) and non-stigmatization can stimulate change in the behaviours of infected individuals, and can lead to an increase in testing and adherence to treatment. This will in turn reduce the basic reproduction number, and consequently, the spread of the virus. Results: The results portray that the early identification and treatment are inadequate for the illness to be eradicated. Conclusions: Other control techniques, such as treatment adherence and effective condom usage, should be investigated in order to lessen the disease's burden.

Bifurcation analysis for Shil'nikov Chaos Electro-dissolution of Copper.

This paper is devoted to study the local bifurcations and stability of three dimensional systems that representing a Shil'nikov chaos during copper electro-dissolution. The local stability analysis of equilibrium points has been studied. It is shown that transcritical bifurcation can appears in the system. Also, the existence of Hopf bifurcation of the system around the equilibrium points is studied when the parameter passes through the critical value. Normal form theory is used to study bifurcating periodic solutions.

Floquet stability analysis of equilibrium points and numerical exploration of pulsating zero-velocity curves and Newton–Raphson basins of attraction

An important and pivot aspect of a dynamical system is the stability property and its range, which play a crucial role towards the stabilisation of a mission design. Present paper deals with Floquet stability analysis of equilibrium points and estimation of pulsating zero velocity curves and Newton–Raphson basins of attraction associated to these equilibrium points under the radiation pressure effect in the photo-gravitational planar elliptic restricted four body problem. Floquet stability test for the equilibrium points is performed with the help of the transition curves, which are generated under the influence of radiation pressure. It is noticed that stability range of the respective equilibrium points have deviated due to radiation pressure. To observe the possible regions of motion for restricted body, the pulsating zero velocity curves are numerically explored after establishing an invariant relation. It is found that not only the radiation parameter but also eccentricity and true anomaly reflect a considerable impact on the shape and size of forbidden regions. To see the tendency of randomly selected points in the phase space, Newton–Raphson basins of attraction associated to each attractor are estimated and effects of perturbing parameters are analysed. It is observed that effects of radiation and mass parameters are considerable however, that of eccentricity and true anomaly are negligible. Moreover, from probability distribution bar diagram, most probable number of iterations is 10 for the case when two primaries are of equal masses and third one is different, whereas it is 20 for the case when all primaries are of identical mass. These results will be helpful to study the generalised problem along with other kind of perturbations.

A Simple Conservative Chaotic Oscillator with Line of Equilibria: Bifurcation Plot, Basin Analysis, and Multistability

Here, a novel conservative chaotic oscillator is presented. Various dynamics of the oscillator are examined. Studying the dynamical properties of the oscillator reveals its unique behaviors. The oscillator is multistable with symmetric dynamics. Equilibrium points of the oscillator are investigated. Bifurcations, Lyapunov exponents (LEs), and the Poincare section of the oscillator’s dynamics are analyzed. Also, the oscillator is investigated from the viewpoint of initial conditions. The study results show that the oscillator is conservative and has no dissipation. It also has various dynamics, such as equilibrium point and chaos. The stability analysis of equilibrium points shows there are both stable and unstable fixed points.

Tripartite evolutionary game analysis of governance mechanism in Chinese WEEE recycling industry

The rapidly increase in waste electrical and electronic equipment (WEEE) poses challenges to the government to manage the WEEE recycling industry. In order to better understand and improve the governance mechanism in the WEEE recycling industry, taking China as an example, we construct a tripartite evolutionary game model consisting of the manufacturer, the collector, and the government by underscoring product eco-design and qualified disassembly. First, we obtain the evolutionary stability strategy (ESS) by calculating the replicator dynamic equation and the stability analysis of equilibrium points. Then, we examine the feasibility and rationality of the evolutionary game model by analyzing ESSs corresponding to different stages of the WEEE recycling industry. Finally, we explore the ramifications of the governance mechanism on the behaviors of the manufacturer’s product eco-design and the collector’s qualified disassembly through a numerical example. We find that the evolutionary game model exists eleven equilibrium points and six possible ESSs, and the choice of each ESS depends mainly on the trade-off between costs and revenues for each stakeholder. Through the numerical study, we observe that the current Chinese WEEE governance mechanism is not effective. We also observe that the manufacturer’s willingness to implement product eco-design is decreasing in the tax rate but increasing in the environmental tex, and the collector’s willingness to recycle WEEE through a compliance process is increasing in both the subsidy and the penalty. We suggest that the government focuses on improving the reward-penalty mechanism, establishing appropriate WEEE disposal fund levy and subsidy standards, and removing informal collectors to promote the healthy development of WEEE recycling industry.

Stability and bifurcation analysis of a contaminated sir model with saturated type incidence rate and Holling type-III treatment function

The paper deals with an epidemic model which is contaminated. Logistic growth input of vulnerable is taken in our SIR model. Saturated type incidence rate alongside Holling type III treatment function and time delay in growth component has been considered. The stability analysis of equilibrium points is done and the basic reproduction number (R 0 ) is obtained. Using (R 0 ), Local Stability around disease-free equilibrium point has been acquired. When R 0 is less than one, it is locally stable which means disease doesn’t exist in the environment anymore and when R0 is greater than one, it is unstable which means disease still exists in the environment. We have shown that at R 0 =1 the model exhibits forward bifurcation. The conditions for the local stability of endemic equilibrium point have also been obtained. Conditions for the existence of Hopf bifurcation along with its direction and stability has been determined. Numerical simulations have been performed utilizing MATLAB to confirm the logical outcomes acquired. We have seen that to diminish the effect of illness, contamination ought to be decreased. Also, suitable strategies should be chosen to expand the capacity of treatment to decrease the effect of infection.

Families’ influence on romantic relationship and its reconstruction

Families’ influence on romantic relationship is investigated from theoretical and practical perspectives. Theoretically, an ordinary differential equation based model is proposed to describe the romantic relationship between two partners, where the influence received by each individual from its family is taken into account. The introduction of the families’ opinions lead to rich and interesting structure in the dynamical process. With the decreasing response of a partner to its own family, two bifurcations occur, separating the dynamical behavior into three types, i.e., the transition from damping oscillation to one of four stable states, two of stable states, and limit cycles. These findings are explained with the stability analysis of equilibrium points. Practically, for each individual the opinion from its partner’s family is an interesting but hidden variable. The reservoir computing is adopted to discover the hidden variable from the activities of the individual, its family, and its partner. The model and the discovering method can be extended easily to investigate the relationship between two social groups such as the lateral negotiation, where the two representatives play game under the guidance from their own groups each.

Stability analysis of delay induced tumor and immune system interaction model

This paper consists of interactions among T-helper cells (resting cells), Cytotoxic-T lymphocytes (hunting cells) and tumor cells with time delay. Stability analysis of equilibrium points and existence of hopf bifurcation is studied. Time delay is utilized as bifurcation parameter. Numerical simulations depict the dynamical performance of the system with different time delay values, which illustrate the phenomenon of long term tumor decline.

A theoretical analysis of a SEAIJR model of Spanish flu with fractional derivative

A nonlinear system of ordinary differential equations comprised with six classes depicting the spread of the 1918–1920 Spanish flu has been considered in this work. Specific analysis including the well-poseness of the model, equilibrium points, stability analysis of equilibrium points have been presented. A Lyapunov analysis was presented also. Using the linear growth and the Lipschitz conditions, we established the conditions under which the model has a unique system of solutions. While the model was extended as the time derivative was converted to different fractional differential operators, a numerical scheme based on the Lagrange polynomial interpolation was used to solve numerically the obtained models and numerical simulations were performed for different values of fractional orders.