Global properties of a class of HIV infection models with Beddington–DeAngelis functional response

In this paper, the global properties of a class of human immunodeficiency virus (HIV) models with Beddington–DeAngelis functional response are investigated. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of three HIV infection models. The first model considers the interaction process of the HIV and the CD4 ⁺ T cells and takes into account the latently and actively infected cells. The second model describes two co‐circulation populations of target cells, representing CD4 ⁺ T cells and macrophages. The third model is a two‐target‐cell model taking into account the latently and actively infected cells. We have proven that if the basic reproduction number R₀ is less than unity, then the uninfected steady state is globally asymptotically stable, and if R₀ > 1, then the infected steady state is globally asymptotically stable. Copyright © 2012 John Wiley & Sons, Ltd.     

Bifurcation analysis of the HIV‐1 within host model

In this paper, a bifurcation solution's analysis is proposed for an HIV‐1 within the host model around its chronic equilibrium point, this is carried out based on Lyapunov–Schmidt approach. It is shown that the coefficient b, which represents the healthy CD4⁺ T‐cells growth rate, is a bifurcation parameter; this means that the rate of multiplication of healthy cells can have serious effects on the qualitative dynamical properties and structural stability of the infection evolution dynamics. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of the dynamics of a delayed HIV pathogenesis model

In this paper, considering full Logistic proliferation of CD4⁺ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay τ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results.     

Stability of HIV/HTLV-I co-infection model with delays

In this paper, we formulate a within-host dynamics model for HIV/HTLV-I co-infection under the influence of cytotoxic T lymphocytes (CTLs). The model incorporates silent HIV-infected CD4⁺T cells and silent HTLV-infected CD4⁺T cells. The model includes two routes of HIV transmission, virus to cell (VTC) and cell to cell (CTC). It also incorporates two modes of HTLV-I transmission, horizontal transmission via direct CTC contact and vertical transmission through mitotic division of Tax-expressing HTLV-infected cells. The model takes into account five types of distributed-time delays. We analyze the model by proving the nonnegativity and boundedness of the solutions, calculating all possible equilibria, deriving a set of key threshold parameters, and proving the global stability of all equilibria. The global asymptotic stability of all equilibria is established by utilizing Lyapunov function and LaSalle's invariance principle. We present numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we discuss the effect of HTLV-I infection on the HIV dynamics and vice versa.     

一类带有治愈率的HIV感染的CD4＋T细胞模型的动力学性质

本文主要介绍一类带有治愈率的HIV感染的CD4 T细胞模型的动力学性质,同时证明了如果基本再生数R0＜1,HIV感染消失;如果R0＞1,HIV感染持续.然后进行数值模拟,给出了地方性平衡点E·全局稳定的参数域,得到了地方性平衡点E·不稳定时周期解存在.     

Stabilization of HIV/AIDS model by receding horizon control

This work concerns the stabilization of uninfected steady state of an ordinary differential equation system modeling the interaction of the HIV virus and the immune system of the human body. The control variable is the drug dose, which, in turn, affects the rate of infection of CD4⁺ T cells by HIV virus. The feedback controller is constructed by a variant of the receding horizon control (RHC) method. Simulation results are discussed.     

A fractional-order model of HIV infection with drug therapy effect

In this paper, the fractional-order model that describes HIV infection of CD4⁺ T cells with therapy effect is given. Generalized Euler Method (GEM) is employed to get numerical solution of such problem. The fractional derivatives are described in the Caputo sense.     

Approximating solutions of Hammerstein type equations in Banach spaces

Abstract

Let X be a real Banach space and X^? be its dual. Let F: X → X^? and K: X^? → X be Lipschitz monotone mappings. In this paper an explicit iterative scheme is constructed for approximating solutions of the Hammerstein type equation, 0 = u + KF u, when they exist. Strong convergence of the scheme is obtained under appropriate conditions. Our results improve and unify many of the results in the literature.     

Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces

This paper deals with the existence of mild L-quasi-solutions to the initial value problem for a class of semilinear impulsive evolution equations in an ordered Banach space E. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. An example is also given.     

A differential equation model of HIV infection of CD4T-cells with cure rate

A differential equation model of HIV infection of CD4⁺T-cells with cure rate is studied. We prove that if the basic reproduction number R₀<1, the HIV infection is cleared from the T-cell population and the disease dies out; if R₀>1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R₀>1. Furthermore, we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

New results for the periodic boundary value problem for impulsive integro-differential equations

This paper discusses the periodic boundary value problem for first-order nonlinear impulsive integro-differential equations. We prove some new comparison principles, and then establish new existence results for extremal solutions by using these principles and the monotone iterative technique. These results improve and extend all known relevant conclusions from the literature. Some examples are also given to illustrate the advantage of our results.     

A delay differential equation model of HIV infection of CD4+ T-cells with cure rate

A delay differential equation as a mathematical model that described HIV infection of CD4⁺ T-cells is analyzed. When the constant death rate of infected but not yet virus-producing cells is equal to zero, the stability of the non-negative equilibria and the existence of Hopf bifurcation are investigated. A stability switch in the system due to variation of delay parameter has been observed, so is the phenomena of Hopf bifurcation and stable limit cycle. The estimation of the length of delay to preserve stability has been calculated. Further, when the constant death rate of infected but not yet virus-producing cells is not equal to zero, by using the geometric stability switch criterion in the delay differential system with delay dependent parameters, we present that stable equilibria become unstable as the time delay increases. Numerical simulations are carried out to explain the mathematical conclusions.     

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.     

Existence and iteration of positive solutions for a singular two-point boundary value problem with a <Emphasis Type="Italic">p</Emphasis>-Laplacian operator

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation (ϕ _p (u′))′+q(t)f(u) = 0, 0 < t < 1, where ϕ _p (s):= |s| ^p−2 s, p > 1, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0; 1.     

Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate

In this paper, we investigate a disease transmission model of SIRS type with latent period τ?0 and the specific nonmonotone incidence rate, namely, . For the basic reproduction number R₀>1, applying monotone iterative techniques, we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in [Hai-Feng Huo, Zhan-Ping Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 459-468]. Moreover, combining both monotone iterative techniques and the Lyapunov functional techniques to an SIR model by perturbation, we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium.     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells

The studies of nonlinear models in epidemiology have generated a deep interest in gaining insight into the mechanisms that underlie AIDS-related cancers, providing us with a better understanding of cancer immunity and viral oncogenesis. In this article, we analyze an HIV-1 model incorporating the relations between three dynamical variables: cancer cells, healthy CD4 ⁺ T lymphocytes, and infected CD4 ⁺ T lymphocytes. Recent theoretical investigations indicate that these cells interactions lead to different dynamical outcomes, for instance to periodic or chaotic behavior. Firstly, we analytically prove the boundedness of the trajectories in the system’s attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. Our calculations reveal that the highest observable variable is the population of cancer cells, thus indicating that these cells could be monitored in future experiments in order to obtain time series for attractor’s reconstruction. We identify different dynamical behaviors of the system varying two biologically meaningful parameters: r ₁, representing the uncontrolled proliferation rate of cancer cells, and k ₁, denoting the immune system’s killing rate of cancer cells. The maximum Lyapunov exponent is computed to identify the chaotic regimes. Considering very recent developments in the literature related to the homotopy analysis method (HAM), we calculate the explicit series solutions of the cancer model and focus our analysis on the dynamical variable with the highest observability index. An optimal homotopy analysis approach is used to improve the computational efficiency of HAM by means of appropriate values for the convergence control parameter, which greatly accelerate the convergence of the series solution. The approximated analytical solutions are used to compute density plots, which allow us to discuss additional dynamical features of the model.     

Bifurcation analysis of a multidelayed HIV model in presence of immune response and understanding of in‐host viral dynamics

In this paper, we proposed a multidelayed in‐host HIV model to represent the interaction between human immunodeficiency virus and immune response. One delay was considered to incorporate the time required by the virus for various intracellular events to make a host cell productively infective, and the second delay was introduced to take into account the time required for adaptive immune system to respond against infection. We extensively analyzed this multidelayed model analytically and numerically. We show that delay may have both destabilizing and stabilizing effects even when the system contains a single immune response delay. It happens when there exists two sequences of critical values of this delay. If the system starts with stable state in absence of delay, then the smallest value of these critical delays causes instability and the next higher value causes stability. The system may also show stability switching for different values of the virus replication factor. These results demonstrate the possible reasons of intrapatients and interpatients variability of CD4⁺ T cells and virus counts in HIV‐infected patients.