Analysis of a delayed HIV/AIDS epidemic model with saturation incidence

In this paper, a delayed HIV/AIDS epidemic model with saturation incidence is proposed and analyzed. The equilibria and their stability are investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is found that if the threshold R ₀<1, then the disease-free equilibrium is globally asymptotically stable, and if the threshold R ₀>1, the system is permanent and the endemic equilibrium is asymptotically stable under certain conditions.     

Mathematical analysis of a two-patch model of tuberculosis disease with staged progression

The spread of tuberculosis is studied through a two-patch epidemiological system SE₁ ? E_nI which incorporates migrations from one patch to another just by susceptible individuals. Our model is consider with bilinear incidence and migration between two patches, where infected and infectious individuals cannot migrate from one patch to another, due to medical reasons. The existence and uniqueness of the associated endemic equilibria are discussed. Quadratic forms and Lyapunov functions are used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium (DFE) is globally asymptotically stable, and when it is greater than one there exists in each case a unique endemic equilibrium (boundary equilibria and endemic equilibrium) which is globally asymptotically stable. Numerical simulation results are provided to illustrate the theoretical results.     

Asymptotic optimality of the generalized bayes estimator in multiparameter cases

The higher order asymptotic efficiency of the generalized Bayes estimator is discussed in multiparameter cases. For all symmetric loss functions, the generalized Bayes estimator is second order asymptotically efficient in the classA ₂ of the all second order asymptotically median unbiased (AMU) estimators and third order asymptotically efficient in the restricted classD of estimators.     

Qualitative analysis of a SIR epidemic model with saturated treatment rate

On account of the effect of limited treatment resources on the control of epidemic disease, a saturated removal rate is incorporated into Hethcote’s SIR epidemiological model (Hethcote, SIAM Rev. 42:599–653, 2000). Unlike the original model, the model has two endemic equilibria when R ₀<1. Furthermore, under some conditions, both the disease free equilibrium and one of the two endemic equilibria are asymptotically stable, i.e., the model has bistable equilibria. Therefore, disease eradication not only depends on R ₀ but also on the initial sizes of all sub-populations. By the Poincaré-Bendixson theorem, Poincaré index, center manifold theorem, Hopf bifurcation theorem and Lyapunov-Lasalle theorem, etc., the existence and asymptotical stability of the equilibria, the existence, stability and direction of Hopf bifurcation are discussed, respectively.     

Asymptotic behaviour for wave equation with time-dependent coefficients

Abstract We shall find asymptotic profiles for strictly hyperbolic equations with time-dependent coefficients which are of Lipschitz class and have some stability condition. More precisely, it will be shown that there exists a solution which is not asymptotically free provided that the coefficient tends slowly to some constant. Keywords: Wave equation, Asymptotic profiles, Asymptotic integrations     

Spread of disease in a patchy environment

For a two patches SIR model, it is shown that its dynamic behavior is determined by several quantities. We have shown that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable, otherwise it is unstable. Some sufficient conditions for the local stability of boundary equilibria are obtained. Numerical simulations indicate that travel between patches can reduces oscillations in both patches; may enhances oscillations in both patches; or travel switches oscillations from one patch to another.     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

Uniform asymptotic joint normality of sample quantities in censored cases

Summary The asymptotic joint distribution of an increasing number of sample quantiles as the sample size increases, when the underlying sample is censored, is shown to be asymptotically uniformly (or type (B) _d ) normally distributed under fairly general conditions. The discussions for uncensored cases have been given by [4].     

Global dynamics of delay‐distributed HIV infection models with differential drug efficacy in cocirculating target cells

In this paper, we investigate the dynamical behaviors of three human immunodeficiency virus infection models with two types of cocirculating target cells and distributed intracellular delay. The models take into account both short‐lived infected cells and long‐lived chronically infected cells. In the two types of target cells, the drug efficacy is assumed to be different. The incidence rate of infection is given by bilinear and saturation functional responses in the first and second models, respectively, while it is given by a general function in the third model. Lyapunov functionals are constructed and LaSalle invariance principle is applied to prove the global asymptotic stability of all equilibria of the models. We have derived the basic reproduction number R₀ for the three models. For the first two models, we have proven that the disease‐free equilibrium is globally asymptotically stable (GAS) when R₀≤1, and the endemic equilibrium is GAS when R₀>1. For the third model, we have established a set of sufficient conditions for global stability of both equilibria of the model. We have checked our theoretical results with numerical simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Long-term dynamics of discrete-time predator–prey models: stability of equilibria,cycles and chaos

ABSTRACT

In [A.S. Ackleh, M.I. Hossain, A. Veprauskas, and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603.], we established conditions for the persistence and local asymptotic stability of the interior equilibrium for two discrete-time predator–prey models (one without and with evolution to resist toxicants). In the current paper, we provide a more in-depth analysis of these models, including global stability of equilibria, existence of cycles and chaos. Our main focus is to examine how the speed of evolution ν may impact population dynamics. For both models, we establish conditions under which the interior equilibrium is global asymptotically stable using perturbation analysis together with the construction of Lyapunov functions. For small ν, we show that the global dynamics of the evolutionary system are nothing but a continuous perturbation of the non-evolutionary system. However, when the speed of evolution is increased, we perform numerical studies which demonstrate that evolution may introduce rich dynamics including cyclic and chaotic behaviour that are not observed when evolution is absent.     

Asymptotically Stationary Processes on Amenable Groups

Abstract

Vector-valued, asymptotically stationary processes on σ-compact locally compact amenable groups are studied. For such processes, we introduce associated spectral measures. Using an “almost periodic Fourier transform,” one can identify the almost periodic spectrum of the asymptotically stationary covariance, and we construct a natural, consistent estimator for this almost periodic spectrum.     

Global dynamics of a cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection R₀ and for CTL response R₁ such that R₁<R₀. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if R₀?1, an infected equilibrium without immune response is globally asymptotically stable if R₁?1<R₀ and an infected equilibrium with immune response is globally asymptotically stable if R₁>1. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if R₁>1.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

Almost Sure Asymptotic Stability of Stochastic Volterra Integro-Differential Equations with Fading Perturbations

Abstract

In this note, we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations that are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal.     

Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system

In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non-autonomous analogues of the hyperbolic equilibria. We also prove, again parallel to the autonomous case, that all solutions converge as t→+∞ to one of these hyperbolic global solutions. We then show how to apply these results to systems that are asymptotically autonomous as t→−∞ and as t→+∞, and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non-autonomous case.     

Loss of information of a statistic for a family of non-regular distributions

In the non-regular case, the asymptotic loss of amount of information (extended to as Rényi measure) associated with a statistic is discussed. It is shown that the second order asymptotic loss of information in reducing to a statistic consisting of extreme values and an asymptotically ancillary statistic vanishes. This result corresponds to the fact that the statistic is second order asymptotically sufficient in the sense of Akahira (1991, Metron, 49, 133–143). Some examples on truncated distributions are also given.     

Modeling diseases with latency and nonlinear incidence rates: global dynamics of a multi‐group model

In this paper, we perform global stability analysis of a multi‐group SEIR epidemic model in which we can consider the heterogeneity of host population and the effects of latency and nonlinear incidence rates. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the basic reproduction number is defined by the theory of the next generation operator and proved to be a sharp threshold determining whether or not disease spread. Under certain assumptions, the disease‐free equilibrium is globally asymptotically stable if R₀≤1 and there exists a unique endemic equilibrium which is globally asymptotically stable if R₀>1. The proofs of global stability of equilibria exploit a matrix‐theoretic method using Perron eigenvetor, a graph‐theoretic method based on Kirchhoff's matrix tree theorem and Lyapunov functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

On the asymptotic convergence to mixed equilibria in 2×2 asymmetric games

We analyse the stability properties of mixed equilibria in 2×2 asymmetric games under evolutionary dynamics. With the standard replicator dynamics these equilibria are stable but not asymptotically stable. We modified the replicator dynamics by introducing players of two types: myopies — like in the standard replicator dynamics — and best responders. The behaviour of the latter is described by a continuos time version of the best reply dynamics. Asymptotic convergence under theModified Replicator Dynamics is proved by identifying a strictly decreasing Ljapunov function. We argue that the finding has important implications to justify the use of economic models with mixed strategy equilibria.     

GLOBAL DYNAMICS OF A PREDATOR-PREY MODEL WITH PREY REFUGE AND DISEASE

In this paper, we study a predator-prey model with prey refuge and disease. We study the local asymptotic stability of the equilibriums of the system. Further, we show that the equilibria are globally asymptotically stable if the equilibria are locally asymptotically stable. Some examples are presented to verify our main results. Finally, we give a brief discussion.     

Sharp existence results for self-similar solutions of semilinear wave equations

The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation with or in R ³×R ⁺ for small Cauchy data is proven if . A counterexample is given which shows that the lower bound on α is sharp. Received April 1999 – Accepted September 1999