Conditions For Global Asymptotic Stability Research Articles

ANALYSIS OF A NONAUTONOMOUS HIV/AIDS EPIDEMIC MODEL WITH DISTRIBUTED TIME DELAY

In this paper, we have considered a nonautonomous stage‐structured HIV/AIDS epidemic model through vertical and horizontal transmissions of infections, having two stages of the period of infection according to the developing progress of infection before AIDS defined would be detected, with varying total population size and distributed time delay to become infectious (through horizontal transmission) due to intracellular delay between initial infection of a cell by HIV and the release of new virions. The infected people in the different stages have different ability of transmitting disease. We have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected people. We have introduced some new threshold values _Ro and R* and further obtained that the disease will be permanent when _Ro > 1 and the disease will be going to extinct when R* < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings.

Permanence and extinction for a nonautonomous avian-human influenza epidemic model with distributed time delay

In this paper we have constructed a mathematical model to interpret the spread of wild avian influenza from the birds to the humans, after the emergence of mutant avian influenza, with nonautonomous ordinary differential equations and distributed time delay due to the intracellular delay between initial infection of a cell and the release of new virus particles. The bird population is divided into two classes: susceptible birds and infected birds with wild avian influenza, by a S I compartmental model. The human population is divided into four classes, namely susceptible humans, humans infected with wild avian influenza, humans infected with mutant avian influenza, and recovered humans from mutant avian influenza, by a S I R compartmental model. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

Parameter condition for global asymptotic stability of FAST TCP in the presence of cross traffics

We consider the global asymptotic stability of FAST TCP in the presence of cross traffics as well as network feedback delays. Based on a continuous-time dynamic model that describes the FAST TCP source and the queuing delay dynamics with cross traffics, we achieve a sufficient condition for the global asymptotic stability. The sufficient condition is expressed in terms of the tuning parameter α independently of the update gain γ. The simulation results demonstrate the validity of the sufficient condition for the global asymptotic stability of FAST TCP.

PERMANENCE AND EXTINCTION OF A NONAUTONOMOUS STAGE-STRUCTURED EPIDEMIC MODEL WITH DISTRIBUTED TIME DELAY

In this paper, we have considered a nonautonomous stage-structured epidemic model having two stages of the period of infection according to the progressing process of some infectious diseases (e.g. Chagas' disease, hepatitis C, etc.) with varying total population size and distributed time delay to become infectious. The infected persons in the different stages have different ability of transmitting disease. We have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R0 and R* and further obtained that the disease will be permanent when R0 > 1 and the disease will be going to extinct when R* < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

ADVANCED IMPULSIVE DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

We prove the existence and uniqueness of the solutions of a class of first order nonhomogeneous advanced impulsive differential equations with piecewise constant arguments. We also study the conditions of periodicity, oscillation, nonoscillation and global asymptotic stability for some special cases.

Conditions for Global Dynamic Stability of a Class of Resource-Bounded Model Ecosystems

This paper studies a class of dynamical systems that model multi-species ecosystems. These systems are 'resource bounded' in the sense that species compete to utilize an underlying limiting resource or substrate. This boundedness means that the relevant state space can be reduced to a simplex, with coordinates representing the proportions of substrate utilized by the various species. If the vector field is inward pointing on the boundary of the simplex, the state space is forward invariant under the system flow, a requirement that can be interpreted as the presence of non-zero exogenous recruitment. We consider conditions under which these model systems have a unique interior equilibrium that is globally asymptotically stable. The systems we consider generalize classical multi-species Lotka-Volterra systems, the behaviour of which is characterized by properties of the community (or interaction) matrix. However, the more general systems considered here are not characterized by a single matrix, but rather a family of matrices. We develop a set of 'explicit conditions' on the basis of a notion of 'uniform diagonal dominance' for such a family of matrices, that allows us to extract a set of sufficient conditions for global asymptotic stability based on properties of a single, derived matrix. Examples of these explicit conditions are discussed.

Global stability analysis of computer networks with arbitrary topology and time-varying delays

In this paper, we determine the delay-dependent conditions of global asymptotic stability for a class of multi-dimensional nonlinear time-delay systems with application to computer communication networks. A nonlinear delayed model is considered for a rate-based congestion control system of a heterogeneous network with arbitrary topology and time-varying delays. We propose a Lyapunov-based method to obtain a sufficient condition under which global asymptotic stability of the equilibrium is guaranteed. The main contribution of the paper lies in considering time variations of delays in a heterogeneous network which may be applicable in actual networks. Moreover, we obtain conditions for Internet-style networks with multi-source multi-link topology. We first prove the stability for a class of nonlinear time-delay systems. Then, we apply the results to a Kelly’s rate-based approximation of the congestion control system.

Analysis of a delay nonautonomous predator-prey system with disease in the prey

In this paper we have considered a nonautonomous predator-prey model with time delay due to gestation, in which a disease that can be transmitted by contact spreads among the prey only. Here, we have established some sufficient conditions on the permanence of the system by using inequality analytical technique. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. We have observed that the time delay has no effect on the permanence of the system but it has an effect on the global asymptotic stability of this model. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness.

Polynomial relaxation-based conditions for global asymptotic stability of equilibrium points of genetic regulatory networks

An important problem in systems biology consists of establishing whether an equilibrium point of a genetic regulatory network (GRN) is stable or not. This article investigates this problem for GRNs with SUM or PROD regulatory functions. It is shown that sufficient conditions for global asymptotical stability of an equilibrium point of these networks can be derived in terms of convex optimisations with linear matrix inequality constraints. These conditions are obtained by looking for a Lyapunov function through the use of suitable polynomial relaxations, and do not introduce approximations of the nonlinearities present in the GRNs. The benefit of these conditions is that their conservatism can be decreased by increasing the degree of the introduced polynomial relaxations. Numerical examples illustrate the usefulness of the proposed conditions.

Analysis of a Nonautonomous HIV/AIDS Model

In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R 0 and R ∗ and further obtained that the disease will be permanent when R 0 > 1 and the disease will be going to extinct when R ∗ < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

Comments on Fuzzy Control Systems Design via Fuzzy Lyapunov Functions

This paper considers the work entitled "Fuzzy control systems design via fuzzy Lyapunov functions" and published in IEEE Transactions on Systems, Man, and Cybernetics-Part B , where the authors try to extend the work of Rhee and Won. Nevertheless, the results proposed by Li have been obtained without taking into account a necessary path independency condition to ensure the line integral function to be a Lyapunov function candidate, and consequently, the proposed global asymptotic stability and stabilization conditions are unsuitable.

New delay-dependent global asymptotic stability condition for Hopfield neural networks with time-varying delays

This paper deals with the global asymptotic stability problem for Hopfield neural networks with time-varying delays. By resorting to the integral inequality and constructing a Lyapunov-Krasovskii functional, a novel delay-dependent condition is established to guarantee the existence and global asymptotic stability of the unique equilibrium point for a given delayed Hopfield neural network. This criterion is expressed in terms of linear matrix inequalities (LMIs), which can be easily checked by utilizing the recently developed algorithms for solving LMIs. Examples are provided to demonstrate the effectiveness and reduced conservatism of the proposed condition.

Global Asymptotic Robust Stability and Global Exponential Robust Stability of Neural Networks with Time-Varying Delays

In this paper, based on nonnegative matrix theory, the Halanay's inequality and Lyapunov functional, some novel sufficient conditions for global asymptotic robust stability and global exponential robust stability of neural networks with time-varying delays are presented. It is shown that our results improve and generalize several previous results derived in the literatures. From the obtained results, some linear matrix inequality criteria are derived. Finally, a simulation is given to show the effectiveness of the results.

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.

Global stability analysis scheme for a class of nonlinear time delay systems

We consider a class of nonlinear time delay systems created by generalizing the model for FAST TCP, an Internet congestion control algorithm. We achieve a time delay independent sufficient condition for the global asymptotic stability of the class of systems. The sufficient condition is verified by constructing two sequences that represent the lower and upper bound variations of the system trajectory in time, and showing that the two sequences converge to the equilibrium point of the system. The simulation results exemplify that the sufficient condition is valid for global asymptotic stability, and that the sufficient condition is a close approximation to the unknown necessary and sufficient condition for global asymptotic stability.

Asymptotic Stability in Hybrid Systems via Nested Matrosov Functions

A theorem on nested Matrosov functions is extended to time-varying hybrid systems. It provides sufficient conditions for uniform global asymptotic stability of a compact set. An application to parameter identification with state resets is made and illustrated on an example.

Novel delay-dependent global asymptotic stability condition of Hopfield neural networks with delays

In this paper, the global asymptotic stability of Hopfield neural networks with delays is investigated. Distinct differences from other analytical approaches lie in transforming to an equivalent system by using a parameterized transformation which allows free variables in an operator. A novel, less conservative and restrictive criterion than those established in the earlier references is given in terms of several matrix inequalities by utilizing the Lyapunov theory and matrix inequality framework. The results are related to the size of delays. Numerical examples are given to show the effectiveness of our proposed method.

Asymptotic Stability Problem for Discrete-time Delay Systems with Saturated State Feedback

摘要: 讨论具有饱和状态反馈的离散时滞线性系统的渐近稳定性问题. 给出了在一定假设下, 判断系统全局渐近稳定和局部渐近稳定的充分条件, 并通过数值算例验证了所给出的条件是有效的. 关键词: 离散时滞线性系统 / 饱和状态反馈 / 全局(局部)渐近稳定 / 不变吸引椭球体 / 饱和函数

Stability of one-leg θ-methods for nonlinear neutral differential equations with proportional delay

The stability properties of one-leg θ-methods for nonlinear neutral differential equations with proportional delay is investigated. In recent years, the stability of one-leg θ-methods for this class of equations on a quasi-geometric mesh is investigated. Instead, in the present paper, the focus is on stability of one-leg θ-methods for the neutral differential equations with constant delay obtained by applying the approach of transformation to the proportional delay equations. Some sufficient conditions for global stability and asymptotic stability are established. Two numerical examples are also included.

Oscillations in I/O Monotone Systems Under Negative Feedback

Oscillatory behavior is a key property of many biological systems. The small-gain theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium, and hence its violation is a necessary condition for the existence of periodic solutions. One advantage of the use of the monotone SGT technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low-dimensional (in many interesting examples, just one-dimensional) model reduction. This robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters. However, verifying the conditions needed in order to apply the SGT is not always easy. This paper provides an approach to the verification of the needed properties and illustrates the approach through an application to a classical model of circadian oscillations, as a nontrivial "case study," and provides a theorem in the converse direction of predicting oscillations when the SGT conditions fail.