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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 370968, 25 pages
http://dx.doi.org/10.1155/2015/370968
Research Article

Global Stability of Humoral Immunity HIV Infection Models with Chronically Infected Cells and Discrete Delays

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 13 May 2015; Revised 7 August 2015; Accepted 25 August 2015

Academic Editor: Zizhen Zhang

Copyright © 2015 A. M. Elaiw and N. A. Alghamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the global stability of three HIV infection models with humoral immune response. We consider two types of infected cells: the first type is the short-lived infected cells and the second one is the long-lived chronically infected cells. In the three HIV infection models, we modeled the incidence rate by bilinear, saturation, and general forms. The models take into account two types of discrete-time delays to describe the time between the virus entering into an uninfected CD4+ T cell and the emission of new active viruses. The existence and stability of all equilibria are completely established by two bifurcation parameters, and . The global asymptotic stability of the steady states has been proven using Lyapunov method. In case of the general incidence rate, we have presented a set of sufficient conditions which guarantee the global stability of model. We have presented an example and performed numerical simulations to confirm our theoretical results.

1. Introduction

During last decades, many researchers have developed and analyzed several mathematical models which describe human immunodeficiency virus (HIV) dynamics (see, e.g., [112]). HIV mainly targets the CD4+ T cells, leading to Acquired Immunodeficiency Syndrome (AIDS). Most of the HIV mathematical models presented in the literature consider only one type of infected cells called short-lived infected cells. However, it was shown that there is another source for the virus which is called long-lived chronically infected cells. This type of cells generates smaller number of viruses than the short-lived infected cells, but it lives longer [4]. The basic HIV dynamics model with long-lived chronically infected cells presented in [4] is given byHere , and are the concentrations of the uninfected CD4+ T cells, short-lived infected CD4+ T cells, long-lived chronically infected CD4+ T cells, and free virus particles, respectively. represents birth rate constant of the uninfected CD4+ T cells. is the infection rate constant. Parameters , and are the death rate constants of uninfected CD4+ T cells, short-lived infected CD4+ T cells, long-lived chronically infected CD4+ T cells, and free viruses, respectively. The fractions and with are the probabilities that, upon infection, an uninfected CD4+ T cell will become either long-lived chronically infected or short-lived infected. and denote the average numbers of free virus particles produced in the lifetime of the short-lived infected and long-lived chronically infected cells, respectively. Model (1) incorporates reverse transcriptase inhibitor drugs with drug efficacy and .

In model (1), the immune response has not been modeled. The immune response plays an important role in controlling the diseases. In reality, the immune response needs indispensable components to do its job such as antibodies, cytokines, natural killer cells, and T cells. The antibody immune response is a part of the adaptive system in which the body responds to pathogens by primarily using the antibodies which are generated by the B cells, while the other part is the Cytotoxic T Lymphocytes (CTL) immune response where the CTL attacks and kills the infected cells [3]. In malaria disease, the humoral immune response is more effective than the CTL immune response. In the virus dynamics literature, several models have considered the effect of CTL immune response [3, 13] or the humoral immune response [1416]. Obaid and Elaiw [15] proposed the following model which takes into consideration the humoral immune response:Here, the variable represents the concentration of B cells. The HIV are attacked by the B cells at rate . The terms and represent the proliferation and death rates of the B cells, respectively. In model (2), it is assumed that once the HIV contacts the CD4+ T cell, it becomes infected producing new viruses. Actually, there exists an intracellular time delay between the time the HIV contacts an uninfected CD4+ T cell and the time it becomes actively infected CD4+ T cell [17]. In the literature, several papers have proposed various HIV models with time delays [1721].

Our aim in this work is to propose three HIV dynamics models with two types of infected cells, two types of intracellular delays, and humoral immunity. Bilinear and saturated incidences have been proposed in the first and second model, respectively, while a general nonlinear incidence rate is proposed in the third model. For each model, we derive two bifurcation parameters, and , and establish the global stability using Lyapunov functional.

2. Model with Bilinear Incidence

We propose the following HIV infection model with humoral immunity, two types of infected cells and two types of intracellular delays:Parameter represents the time between HIV contact with an uninfected CD4+ T cell and the cell becoming infected but not yet producer cell. The factor represents the loss of CD4+ T cells during the interval . The parameters and represent the time necessary for producing new infectious viruses from the short-lived infected and long-lived chronically infected cells, respectively. The factors and represent the loss of short-lived and long-lived chronically infected cells during the intervals and , respectively. Here, , and are positive constants.

The initial conditions for system (2) are given bywhere , where is the Banach space of continuous functions and . We note that system (3)–(7) with initial conditions (8) has a unique solution satisfying [22].

2.1. Positive Invariance

Proposition 1. The solution of (3)–(7) with initial conditions (8) is nonnegative for and ultimately bounded.

Proof. Assume that on for some constant and , . From (3), we get and hence , for some , and sufficiently small . This leads to a contradiction; therefore, , for all . From (4), (5), and (6), we haveThis confirms that , , and , for all . By a recursive argument, we get that , , and , for all . Moreover, from (7), we obtain Clearly, , . Now, we let , , , , , , and ; then,Hence, . Since , , and , then and . Also,Hence, . Since and are nonnegative, then and . Therefore, , and are ultimately bounded.

2.2. Steady States

System (3)–(7) always admits an uninfected steady state . Let Now, we define the basic reproduction number for system (3)–(7) as The system has two other steady states, infected steady state without humoral immune response and infected steady state with humoral immune response :We note that , , , and are positive when and where , , , and and when . Now, we define humoral immune response reproduction number asClearly, . From above, we can state the following lemma.

Lemma 2. For system (3)–(7), one has the following:(i)If , then the system has only one positive steady state .(ii)If , then the system has two positive steady states and .(iii)If , then the system has three positive steady states , , and .

2.3. Global Stability Analysis

We establish the global stability of all the steady states of system (3)–(7) employing the method of Lyapunov function. Let us define

Theorem 3. For system (3)–(7), if , then is GAS.

Proof. Definewhere , are positive constants satisfying the following equations:The solution of (18) is given byThe values of , will be used through the paper. Calculating the derivative of along the solutions of system (3)–(7) and applying , we obtainThen,Therefore, if , then , for all . The solutions of system (3)–(7) limit , the largest invariant subset of [22]. We note that if and only if , , and . For each element of , we have and ,; then, andSince , then . Hence, if and only if , , , , and . It follows from LaSalle’s invariance principle (LIP) that is GAS when .

Theorem 4. For system (3)–(7), assume that ; then, is GAS.

Proof. DefineThen, is given byEquation (24) can be simplified asApplying , we getUsing the steady state conditions for ,we get andConsider the following equalities:Using (29) in case of , we obtainWe have if , then . Since , then for all , we have . We note that at . Then, from LIP, is GAS.

Theorem 5. For system (3)–(7), assume that ; then, is GAS.

Proof. We considerFunction along the trajectories of system (3)–(7) satisfiesUsing the steady state conditions for ,we get andUsing (29) in case of , we obtainSince , then , and . It is observed that if and only if , , , and . Therefore, if , then and (6) becomeswhich gives . Hence, is equal to zero at . The global stability of follows from LIP.

3. HIV Dynamics Model with Saturated Incidence

We present an HIV infection model with saturated incidence:where is the saturation incidence rate constant. Similar to the previous section, one can show that the solutions of the model are nonnegative and bounded.

3.1. Steady States

System (37) always admits an uninfected steady state , where . Now, we define the basic reproduction number for system (37) as The system has two other steady states and , whereWe note that , , , and are positive when . And and when . Now, we define another threshold parameter asClearly, . From above, we have the following result.

Lemma 6. For system (37), one has the following:(i)If , then the system has only one positive steady state .(ii)If , then the system has two positive steady states and .(iii)If , then the system has three positive steady states , , and .

3.2. Global Stability Analysis

In this subsection, we investigate the global stability of system (37) by constructing suitable Lyapunov functionals and applying LaSalle invariance principle.

Theorem 7. For system (37), if , then is GAS.

Proof. DefineCalculating the derivative of along the solutions of system (37), we obtainCollecting terms of (42), we obtainIt follows that is GAS when .

Theorem 8. For system (37), assume that ; then, is GAS.

Proof. ConsiderThen, is given bySimplifying (45), we getApplying , we get