Conditions For Local Stability Research Articles

Mathematical model of mosquito population dynamics with constants and periodic releases of Wolbachia-infected males

In this article, we propose three mathematical models of mosquito dynamics. The first model describes the dynamics of wild mosquitoes in combination with mechanical control and larvicide. For this model, we derive a threshold parameter N , called the basic number of offspring, and show that the trivial equilibrium is globally asymptotically stable when N ≤ 1 , while if N > 1 , the non-trivial equilibrium is globally asymptotically stable. In the second model, we evaluated a constant release of Wolbachia-infected males in the wild mosquito population. Our analysis identified a critical threshold for releasing Wolbachia-infected male mosquitoes M i crit to reduce or eliminate the wild mosquito population. In the third model, we assessed wild mosquito dynamics by periodically releasing Wolbachia-infected male mosquitoes. The analysis of this model produced two thresholds, T ∗ the threshold on the release period and Λ ∗ the threshold on the release quantity. The existence of trivial periodic solutions is obtained, and the corresponding local and global stability conditions are proved by Floquet's theory and Lyapunov's stability theorem, respectively. We also prove the existence conditions for non-trivial periodic solutions and their local stability. Finally, we perform some numerical simulations to illustrate our theoretically pertinent results.

Optimal control of rotavirus infection in breastfed and non-breastfed children

According to the World Health Organization’s guidelines, a two-year mandatory breastfeeding period is strongly recommended, supported by research demonstrating its significant benefits in reducing childhood illnesses and mortality rates. The present study is designed to address this recommendation by focusing on a specific infection, namely rotavirus. Acknowledging this evidence, researchers recognize the need for a deeper understanding of the dynamics of rotavirus disease. In response, our study focuses on constructing and analyzing an SIR epidemic model uniquely designed for transmitting rotavirus in children. The model partitions the children population into three compartments: susceptible, infected, and recovered. To enhance precision, we further categorize susceptible children based on breastfeeding status and infected individuals based on infectious or non-infectious nature. This detailed categorization enables a thorough examination of rotavirus transmission dynamics. The proposed model analyzes two equilibria, disease-free and endemic, revealing local and global stability conditions determined by the basic reproduction number (R0). Our exploration extends to local stability analysis for the endemic equilibrium when R0>1 and global stability assessment using the Lyapunov theory under specific conditions. Subsequently, we employ optimal control theory through Pontryagin’s maximum principle to minimize the cost burden associated with disease control in children. Then, our work provides a sensitivity analysis of different parameters for the basic reproduction number to enhance our understanding of model dynamics. Finally, numerical simulations conducted with MATLAB and Python software validate our analytical findings, offering a comprehensive and practical assessment of the proposed model’s implications for controlling rotavirus transmission in children.

On fractional ring neural networks with multiple time delays: Stability and Hopf bifurcation analysis

Stability and Hopf bifurcation analysis of neural networks have become popular topics in recent years. In this paper, a large-scale fractional ring neural network with multiple time delays is considered. The model comprises three rings that share a single node. The effects of time delay and fractional order on its stability and Hopf bifurcation have been investigated. Firstly, the characteristic equation is obtained by using the Laplace transform. Then, the time delay and fractional order are set as a bifurcation parameter respectively. Correspondingly, the local stability and Hopf bifurcation conditions are derived by analyzing the characteristic equations. Finally, two examples are given to support the theoretical analysis. In fact, this article provides an effective method for analyzing the stability and Hopf bifurcation of fractional ring neural networks.

A holistic exploration of the optimal control strategies on an enhanced mathematical model for the co-infection of HIV/AIDS and varicella-zoster

Because of its high contagiousness and correlation with HIV/AIDS complaints, the virus that causes varicella-zoster virus and its interactions have major consequences for a considerable portion of people worldwide. The primary aim of this work is to suggest and examine optimal control methods for managing the transmission dynamics of HIV/AIDS and Varicella-Zoster co-infection, using an integer model approach. The mathematical analyses of the proposed integer order model places particular emphases on the boundedness and non-negativity of the model solutions, scrutinizing equilibrium points, determining the models basic reproduction ratios (the models basic reproduction numbers) through the next-generation matrix operator method, and assessing the model equilibrium points existences and stabilities in local approach by considering the local stability conditions of Routh and Hurwitz. Additionally, it incorporates an optimal control framework to enhance our understanding of the dynamics involved in the spreading of HIV/AIDS and Varicella-Zoster co-infection within a considered population. This entails determining preventative measures that can be deliberately put into place to lessen the effects of these co-infections. The solutions of the HIV/AIDS and Varicella-Zoster co-infection model converges to the co-infection endemic equilibrium point whenever the associated basic reproduction number is greater than unity, as verified by numerical simulation results. Including optimal management gives the research an innovative viewpoint and helps identify tactical ways to mitigate the negative effects of this co-infection on the public health. The results highlight how crucial it is to address these complex structures in order to protect and improve public health outcomes. Implementing the proposed protection measures and treatment measures simultaneously has most effective result to minimize and eliminate the HIV/AIDS and Varicella-Zoster co-infection disease throughout the population.

Co-infection dynamics between HIV-HTLV-I disease with the effects of Cytotoxic T-lymphocytes, saturated incidence rate and study of optimal control

The spreading of HIV or HTLV-I among the cells has received the great attention in recent modelling study to explore the virus infection dynamics. The co-infection of HIV and HTLV-I with the effect of Cytotoxic T-lymphocytes (CTLs) immune response is also important from epidemiological point of view. To identify the co-infection scenario of HIV and HTLV-I with the CTLs effect we proposed in this paper a six compartmental ODE-model with uninfected, HIV-infected, HTLV-I infected CD4+T cells and free HIV virus particles with HIV specific CTLs and HTLV-I specific CTLs. The rates of infection of the cases are considered here saturated type and proliferation rate of uninfected and HIV infected CD4+ T-cells are of logistic terms. To establish the well-posedness of the model we have shown that the solution of the proposed model is non-negative and bounded. We obtain the basic reproduction number which is the maximum of the HIV-related reproduction and the HTLV-I related reproduction number. Along with the disease free equilibrium point the system contains other seven endemic equilibrium points containing infection by single disease or both. Analytically, we establish the local and global stability conditions of the equilibrium points and also we establish that the system experiences transcritical bifurcation by the generation of only HIV or HTLV-I infected endemic equilibrium point. Using numerical simulations, we validate the theoretical results and found two infection paths, one initiating with HIV and other with HTLV-I, both cases ultimately become co-infected. Finally, using the optimal control analysis we found the optimal policy for treatment using AVR, RTI & PI for HIV or AZT for HTLV-I control and lastly concluded by some recommendations.

Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system’s solutions, we investigate stochasticity’s influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

Mathematical analysis of the role of host-to-host transmission of Maize Streak Virus Disease with Atangana-Baleanu derivative

In this article a deterministic model for Maize Streak Virus Disease (MSVD) using a fractional-order differential equation with the Atangana-Baleanu Caputo-type operator is developed. Focusing on the role of host-to-host transmission, the it is shown that the presence and stability of equilibria depends on maize field carrying capacity and half-saturation constant of susceptible maize. The MSVD-free equilibrium is globally asymptotically stable when the basic reproduction number is below unity. Local stability conditions for endemic equilibria are established using Lyapunov second technique and Routh-Hurwitz criteria. Matrix-based formulae are also presented for determining bifurcation and it is shown that the model exhibits forward bifurcation. Sensitivity analysis reveals the significant impact of the probability of infection between hosts on MSVD spread. A two-point Lagrange interpolation polynomial is developed for numerical solutions of the model, enabling exploration of theoretical findings and assessment of how epidemiological factors influence MSVD propagation. The paper contributes a comprehensive understanding of the disease's behavior under various conditions.

Optimal control of red palm weevil model incorporating sterile insect technique, mechanical injection, and pheromone traps

The red palm weevil is one of the most dangerous pests affecting date palm trees in the Gulf region and the Middle East. This paper investigates the control of the red palm weevil (RPW) in date palms using several techniques, namely stem injection, sex pheromone traps, and the Sterile Insect Technique. A mathematical model describing the spread of the red palm weevil is presented, and the boundedness of the solution for the RPW system is studied. The conditions for local stability, forward bifurcation, and Hopf bifurcation are also obtained. Additionally, this research highlights the significance of optimal control in managing RPW and simulates the model using the Forward-Backward Sweep numerical method. It is observed that the parameters of the sterile insect technique, the pheromone trap, and the mechanical injection play important roles in controlling the RPW. Mechanical Injection may be more effective in controlling the palm weevil compared to the other two methods.

A robust Study of the Treatment Delay Effect on the Dynamics of Epidemic Disease

A general treatment function with an epidemic model that involves the delay in the treatment period has been proposed and studied in this work. This model contains two compartments, namely susceptible denoted by and infected denoted by . The existence of all the fixed points has been determined. The system has two equilibrium points, namely the uninfected equilibrium point (UIEP) and the endemic equilibrium point (EEP). The conditions for local stability and Hopf bifurcation have been discussed. The stability of the periodic solutions and the direction of the Hopf bifurcation properties have been studied analytically and numerically.

Lost life expectancy following a hypothetical urban radiological incident

The consequences of mass radiological events, particularly those involving the activation of a radiological dispersion device (RDD), have been extensively studied by scientific groups. However, the critical initial period of such an event, usually spanning the first 100 h, can be characterized by a scarcity of information, potentially leading to delays in mitigating strategies. In response, a research group utilized computer simulations to generate solid, conservative analytical details that can aid decision-making and guide the prioritization of initial care based on variables such as age, sex, location, and local atmospheric stability conditions. The study estimates the Lost Life Expectancy (LLE) and provides relevant information to increase support for decision-making and allow evaluation of data closer to the lay public. The research team behind the study has been granted funding by the Brazilian National Council for Scientific and Technological Development (CNPq), and further simulations will be conducted utilizing codes that implement numerical models, specifically in atmospheric data forecasting. The methodology used to evaluate the LLE can be applied to any location, provided that the relevant variables are updated accordingly. Overall, this study offers critical insights into the impact of mass radiological events and enhances simulations' predictive capacity and precision.

On the Local and String Stability Analysis of Traffic Collision Risk

Conventional traffic stability studies primarily concentrate on the evolution of disturbances in vehicle motion but seldom consider how collision risk changes spatially and temporally. This study bridges the gap by extending the principles of traffic stability analysis to the field of traffic safety, focusing specifically on the temporal and spatial dynamics of collision risk. Leveraging the concepts of local and string stability, we formulate conditions under which collision risk behaves in a stable manner over time and space through the transfer function approach. A comparative analysis between conventional traffic stability and the newly introduced concept of collision risk stability reveals that while conditions for local stability are largely aligned in both domains, the criteria for string stability differ. These theoretical insights are substantiated through microscopic simulations using a variety of car-following models. The simulations also indicate that the consistency between theoretical and simulation outcomes diminishes as the disturbance magnitude increases, which is attributed to the linearization errors inherent in applying the transfer function in the theoretical derivations.

The Synchronisation Problem of Chaotic Neural Networks Based on Saturation Impulsive Control and Intermittent Control

This paper primarily focuses on the chaos synchronisation analysis of neural networks (NNs) under a hybrid controller. Firstly, we design a suitable hybrid controller with saturated impulse control, combined with time-dependent intermittent control. Both controls are low-energy consumption and discrete, aligning well with industrial development needs. Secondly, the saturation function in the chaotic neural network is addressed using the polyhedral representation method and the sector nonlinearity method, respectively. By integrating the Lyapunov stability theory, Jensen’s inequality, the mathematical induction method, and the inequality reduction technique, we establish suitable time-dependent Lyapunov generalised equations. This leads to the estimation of the domain of attraction and the derivation of local exponential stability conditions for the error system. The validity of the achieved theoretical criteria is eventually demonstrated through numerical experiment simulations.

Stability analysis of fuzzy differential equation for prey–predator system under Z-number fuzzy and granular differentiability concept

The stability analysis is one of the basic problems in the field of system, prey–predator, production inventory control, signal processing etc. The goal of stability analysis for prey–predator and production inventory control model is to determine the region of the different variables and parameters at which the system is stable or unstable. This paper explores an uncertain prey–predator model where both prey and predator populations are fuzzy variables. Also we have considered all the prey–predator parameters of the system are imprecise. This fuzzy prey–predator model is converted to an equivalent crisp model using granular differentiability approach. The condition of local stability is obtained mathematically by analysis the eigenvalues of characteristic equation. And also we introduced a production inventory model as a case study of fuzzy dynamical system to illustrate the efficiency of the result and practical aspects.

Qualitative behavior of a highly non-linear Cutaneous Leishmania epidemic model under convex incidence rate with real data.

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.

Reasonable design method of box crane girder by Taguchi method

Overhead cranes are widely used in industrial systems. In this study, the research object is the main box girder of the overhead crane. The research objective is to find the parameters to obtain a lighter structure, which reduces the market price of the crane. The article studies the method of calculating crane girders and sets up the optimization algorithm. The study will use the Taguchi method, and ANOVA analysis to evaluate the influence of box girder parameters. The girder weight, stress, local stability, static displacement, and vibration frequency are response values. Constraint conditions are evaluated by examining each factor with response value according to the orthogonal matrix L16. Analysis of the Signal to Noise ratio and ANOVA by Minitab software will select the optimal parameters that satisfy the constraints, the goal is to reduce the volume compared to the original design. The test results of the crane with a lifting capacity of 250 tons, an aperture of 31 m, the girder weight reduced by 24.33 %, while the stress increased by only 6.16 %. The new design ensures suitable local stability conditions by making better use of the material's capabilities. With the new parameters, the technical criteria are guaranteed.

Transient Dynamics of a Food Chain Model with Vigilance in the Biparameter Spaces

This article focuses on the asymptotic and transient behavior of a food chain model with vigilance in the biparameter spaces. Firstly, the local asymptotic stability conditions of the positive equilibrium point are obtained. Then, the transient dynamical behavior of the positive equilibrium point is analyzed based on eight parameters. Using numerical simulation, the stable regions of the model with single and double parameters are plotted, as well as the value changes of resilience, reactivity, and amplification envelope within the stable regions. The results show that the coexistence equilibrium point is always reactive, and the corresponding parameter ranges for system stability are large, but the resilience values are small. The vigilance parameter of the middle- and lower-level organisms has the greatest impact on the system’s resilience and reactivity, and the lowest vigilance corresponds to the maximum resilience and reactivity. Our results provide important references for effective management and protection of wild animal populations by the potential for transient amplification, disturbance growth time, and amplitude.

Dynamics of a diffusive two predators- one prey system

This paper analyses a diffusive predator-prey model consisting of a single prey species and two predator species with modified Leslie-Gower term Holling type II functional response subject to the homogeneous Neumann boundary condition. Local stability condition is derived by the application of Routh-Hurwitz criterion. Global asymptotic stability of the unique positive steady state is shown by constructing a suitable Lyapunov function when self diffusion is allowed where as non-constant positive steady states can exist due to the presence of cross-diffusion, that means, cross-diffusion can induce stationary pattern. Taking the cross diffusion as a bifurcation parameter, one can show the existence of positive non-constant solutions with the help of bifurcation theory. A brief conclusion completes the paper.

Analyzing Dynamical Behavior of Tuberculosis Transmission

In this article, we investigated the dynamical behavior of a fuzzy mathematical model with tuberculosis (TB) treatment function. The considered model analyzes fuzzy basic reproduction numbers and stability analysis for both local and global stability conditions around disease-free equilibrium points, as well as its local stability conditions around endemic equilibrium points. We examine the TB model’s sensitivity analysis with a focus on the variation of each of its parameters. We propose predicting the extension of the virus load in TB using fuzzy bifurcation. At the end, we have given a numerical simulation to illustrate the outcomes by considering some special types of treatment functions.

Design and implementation for the state time-delay and input saturation compensator of gas turbine aero-engine control system

In order to obtain desired steady-state and transient performance of a gas turbine aero-engine during its operation, a usual way is to control the engine to operate in a manner close to its physical limits. The thermodynamic performance of a gas turbine aero-engine varies greatly within a wide working envelope, affected by factors such as engine performance degradation, actuator response lag and combustion time-delay. When the engine works in extreme conditions, for example rapid acceleration, some phenomena such as over-speed, over-temperature and over-pressure may occur. For this reason, various restrictions are often set in the engine control system to ensure safety during its operation, which are often realized by the form of saturation. In this paper, a compensation structure for gas turbine aero-engine state time-delay and input saturation is proposed. At first, the problems frequently occurred in the gas turbine aero-engine control system are investigated. And the gas turbine aero-engine control model is derived based on a thermodynamic non-linear model and its linearization model is established, together with its designed compensation structure. Differing from the existing technologies, a combination of dynamic system input saturation feedback and static state time-delay feedback is adopted in the proposed model. Then, the saturated nonlinearity and time-delay characteristic are introduced as convex combinations in the stability analysis. The local stability conditions of the closed-loop system are obtained by using Lyapunov–Krasovskii functionals and linear matrix inequalities (LMIs). Finally, hardware-in-loop (HIL) experiments are conducted to verify the feasibility of the proposed compensation method.

Distributed adaptive control architectures for uncertain multiagent system in the presence of coupled dynamics

This paper studies the analysis and synthesis of distributed adaptive control architectures for uncertain multiagent systems with coupled dynamics in a leader-follower setting, where overall closed-loop system stability is established, and sufficient stability conditions are defined. Specifically, we first analyse a standard distributed adaptive control law and reveal local stability conditions that guarantee the boundedness of the closed-loop system trajectories. We next synthesise fixed and adaptive robustifying terms to show relaxed local stability conditions. We also show that the fixed robustifying term can yield high-gain parameters in the resulting control signals, where the adaptive robustifying term removes this high-gain requirement. Finally, an illustrative numerical example is further presented to compare the proposed methods and demonstrate our theoretical contributions.