Global Properties of a Delay-Distributed HIV Dynamics Model Including Impairment of B-Cell Functions

Abstract

In this paper, we construct an Human immunodeficiency virus (HIV) dynamics model with impairment of B-cell functions and the general incidence rate. We incorporate three types of infected cells, (i) latently-infected cells, which contain the virus, but do not generate HIV particles, (ii) short-lived productively-infected cells, which live for a short time and generate large numbers of HIV particles, and (iii) long-lived productively-infected cells, which live for a long time and generate small numbers of HIV particles. The model considers five distributed time delays to characterize the time between the HIV contact of an uninfected CD4${}^{+}$ T-cell and the creation of mature HIV. The nonnegativity and boundedness of the solutions are proven. The model admits two equilibria, infection-free equilibrium $E{P}_0$ and endemic equilibrium $E{P}_1$. We derive the basic reproduction number $R_0$, which determines the existence and stability of the two equilibria. The global stability of each equilibrium is proven by utilizing the Lyapunov function and LaSalle’s invariance principle. We prove that if $R_0<1$, then $E{P}_0$ is globally asymptotically stable, and if $R_0>1$, then $E{P}_1$ is globally asymptotically stable. These theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions, time delays, and antiviral treatment on the HIV dynamics are studied. We show that if the functions of B-cells are impaired, then the concentration of HIV is increased in the plasma. Moreover, we observe that the time delay has a similar effect to drug efficacy. This gives some impression for developing a new class of treatments to increase the delay period and then suppress the HIV replication.

1. Introduction

Human immunodeficiency virus (HIV) infects and destroys uninfected CD4${}^{+}$ T-cells, which play an essential role in the immune system. When the concentration of the uninfected CD4${}^{+}$ T-cells reaches below 200 cell/mm${}^3$, then the person is said to have progressed to acquired immunodeficiency syndrome (AIDS) [1]. Nowak and Bangham [2] proposed the following basic mathematical model, which describes the dynamics of the uninfected CD4${}^{+}$ T-cells, infected CD4${}^{+}$ T-cells producing viruses, and HIV particles:

$$\begin{array}{cc}\hfill \dot{U}\left(t\right)& =\varrho -\gamma U\left(t\right)-\omega U\left(t\right)P\left(t\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \dot{I}\left(t\right)& =\omega U\left(t\right)P\left(t\right)-\beta I\left(t\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \dot{P}\left(t\right)& =\beta MI\left(t\right)-\xi P\left(t\right),\hfill \end{array}$$

(3)

where, U, I and P are the concentrations of the uninfected CD4${}^{+}$ T-cells, infected CD4${}^{+}$ T-cells, and HIV particles, respectively. $\varrho$ is the production rate of the uninfected CD4${}^{+}$ T-cells. $M$is the average number of HIV particles generated in the lifetime of the infected CD4${}^{+}$ T-cells. The death rate constants of compartments ($U,I,P$) are give by ($\gamma ,\beta ,\xi$), respectively. The term $\omega U\left(t\right)P\left(t\right)$ represents the incidence rate of infection. Modeling of within-host HIV dynamics has received considerable attention from mathematicians during the last few decades [1,2,3,4,5]. Mathematical modeling and analysis have been essential tools to get a better systematic and quantitative understanding of viral processes that are difficult to discern through strictly experimental approaches [6]. The global stability analysis of models (1)–(3) was first investigated in [7].

Highly-active anti-retroviral therapy (HAART) consisting of several antiretroviral drugs such as reverse transcriptase inhibitor (RTI) and protease inhibitor (PI) can suppress viral replication to a low level, but cannot eradicate the virus. An important reason is that HIV provirus can reside in latently-infected CD4${}^{+}$ T-cells [8]. Latently-infected CD4${}^{+}$ T-cells live long, but can be activated to produce virus by relevant antigens. Modeling the virus dynamics with latently-infected cells was presented in [9,10]. Lyapunov functions were exploited in [11,12,13,14] to establish the global stability of virus dynamics models with latently-infected cells. It was reported in [9,15,16] that there are three classes of infected CD4${}^{+}$ T cells, (i) short-lived productively-infected cells, which live for a short time and produce large numbers of HIV, (ii) long-lived productively-infected cells, which live for a long time and produce small numbers of HIV particles, and (iii) latently-infected cells, which contain the viruses, but do not produce them until they are activated.

A major shortcoming of Models (1)–(3) is the assumption that cells produce viruses immediately after they are infected. It is commonly observed that in many biological processes, a time delay is inevitable. For HIV-1 infection, it roughly takes about one day for a newly-infected cell to become productive and then to be able to produce new virus particles [17]. Therefore, mathematicians have frequently used different types of delays to make biological models more realistic [18]. In 1996, Herz et al. [19] introduced an HIV dynamics model with intracellular delay, and they obtained the analytic expression of the viral load decline under treatment and used it to analyze the viral load decline data in patients. Nelson et al. [20] extended the analysis and showed that the time delay can affect the estimate of the death rate of infected CD4${}^{+}$ T-cells when the effectiveness of antiviral drugs is not 100%. HIV dynamics models with an intracellular time delay were used to estimate different kinetic parameters in the presence and absence of drug therapy in [17]. The global stability of delayed virus dynamics models has been studied in several works (see, e.g., [21,22,23,24,25,26,27,28,29,30,31]). HIV dynamics models with latently-infected cells and two time delays were proposed and studied in [32,33].

One of the most important extensions of models (1)–(3) is to take into account the effect of immune response. There are two main immune responses; the first is the cytotoxic T lymphocyte (CTL) immune response. The CTL cells kill the infected CD4${}^{+}$ T-cells. Viral infection models with CTL immune response have been presented in many papers (see, e.g., [2,34,35,36]). The B-cell immune response is considered as the second arm of the immune system. The function of B-cells is to produce antibodies that bind to virus particles and mark them as foreign structures for elimination by other cells of the immune system [4]. The antibodies can neutralize viruses and protect the body from infection [37]. The basic virus dynamics model with B-cell immune response was presented by Murase et al. [38]. To incorporate the time lag between the virus contacts with an uninfected cell and the production of new mature viruses, Wang and Zou [39] proposed the following model:

$$\begin{array}{cc}\hfill \dot{U}\left(t\right)& =\varrho -\gamma U\left(t\right)-\omega U\left(t\right)P\left(t\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \dot{I}\left(t\right)& =\omega U(t-{\lambda }_1)P(t-{\lambda }_1)-\beta I\left(t\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \dot{P}\left(t\right)& =\beta MI(t-{\lambda }_2)-\xi P\left(t\right)-\rho P\left(t\right)C\left(t\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \dot{C}\left(t\right)& =\epsilon P\left(t\right)C\left(t\right)-\mu C\left(t\right),\hfill \end{array}$$

(7)

where C is the concentration of B-cells. The term $\rho P\left(t\right)C\left(t\right)$ represents the neutralization rate of HIV particles. The term $\epsilon P\left(t\right)C\left(t\right)$ represents the proliferation rate of the B-cells. Parameter ${\lambda }_1$ represents the time between HIV contacts with an uninfected CD4${}^{+}$ T-cell and the cell becoming infected. The immature virus needs time ${\lambda }_2$ to be mature. Nowak and May [4] assumed a linear term for immune stimulation: B-cell abundance increases in response to free HIV particles at rate $\epsilon P$, and this leads to $\dot{C}=\epsilon P-\mu C$. Many delayed viral infection models have been developed with B-cell immune response (see, e.g., [40,41,42,43,44,45,46,47,48,49,50] ). The models presented in [2,34,36,40,41,42,43,44,45,46,47,48,49,50] were constructed under the assumption that the presence of the antigen can stimulate immunity and ignore the immune impairment. However, there are numerous experimental results suggesting that the HIV generates mutants, which escape from CTL immune responses [51,52]. Models with CTL immune impairment were studied several times (see, e.g., [53,54,55,56]).

On the other hand, there are some factors that affect the B-cell function and cause the impairment of the B-cells [57,58,59]. These factors include the following: malnutrition, tumors, cytotoxic drugs, irradiation, aging, trauma, some diseases (e.g., diabetes), and immunosuppression by microbes, e.g., malaria and the measles virus, but especially HIV [37] when the viral load is too high. Miao et al. [60] proposed a virus dynamics model with B-cell impairment, but they did not study the global stability analysis of the model. In a very recent works, Elaiw et al. [46] and Elaiw and Elnahary [47] studied the global stability of HIV dynamics models by including: B-cell immune response, time delays, and three types of infected cells (short-lived, long-lived, and latent). However, the B-cell immune impairment was ignored in [46,47].

Based on the discussion above, in this paper, we propose and analyze an HIV dynamics model taking into account the impairment of B-cell functions and five distributed time delays. The infected cells are divided into three classes, latently-infected, short-lived productively-infected, and long-lived productively-infected. The incidence rate is given by a general nonlinear function, which satisfies a set of conditions. The nonnegativity and boundedness of the solutions are proven. We derive the basic reproduction number, which determines the existence and stability of the equilibria. The global stability of all equilibria of the model is established by constructing Lyapunov functions. We also perform numerical simulations to support the global stability results and to explore the effect of time delays, impairment of the B-cell functions, and antiviral treatment on the HIV dynamics.

2. The Mathematical Model

We suggest an HIV infection model with B-cell impairment and distributed time delays as:

$$\begin{array}{cc}\hfill \dot{U}\left(t\right)& =\varrho -\gamma U\left(t\right)-({\omega }_1+{\omega }_2+{\omega }_3)\Theta (U\left(t\right),P\left(t\right)),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \dot{L}\left(t\right)& ={\omega }_1{\int }_0^{h_1}{e}^{-{\theta }_1\lambda }{f}_1\left(\lambda \right)\Theta (U(t-\lambda ),P(t-\lambda ))d\lambda -(\zeta +\nu )L\left(t\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \dot{I}\left(t\right)& ={\omega }_2{\int }_0^{h_2}{e}^{-{\theta }_2\lambda }{f}_2\left(\lambda \right)\Theta (U(t-\lambda ),P(t-\lambda ))d\lambda +\nu L\left(t\right)-\beta I\left(t\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \dot{O}\left(t\right)& ={\omega }_3{\int }_0^{h_3}{e}^{-{\theta }_3\lambda }{f}_3\left(\lambda \right)\Theta (U(t-\lambda ),P(t-\lambda ))d\lambda -\mathsf{\Lambda }O\left(t\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \dot{P}\left(t\right)& =\beta M_1{\int }_0^{h_4}{e}^{-{\theta }_4\lambda }{f}_4\left(\lambda \right)I(t-\lambda )d\lambda +\mathsf{\Lambda }{M}_2{\int }_0^{h_5}{e}^{-{\theta }_5\lambda }{f}_5\left(\lambda \right)O(t-\lambda )d\lambda -\xi P\left(t\right)-\rho P\left(t\right)C\left(t\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \dot{C}\left(t\right)& =\epsilon P\left(t\right)-\mu C\left(t\right)-\vartheta P\left(t\right)C\left(t\right),\hfill \end{array}$$

(13)

where $L\left(t\right)$, $I\left(t\right)$, and $O\left(t\right)$ represent, respectively, the concentration of the latently-infected cells, short-lived productively-infected cells, and long-lived productively-infected cells at time t. The term $({\omega }_1+{\omega }_2+{\omega }_3)\Theta (U\left(t\right),P\left(t\right))$ represents the incidence rate, where $\Theta$ is a general function. Latently-infected cells die with rate $\zeta L\left(t\right)$, and they are transmitted to short-lived productively-infected cells with rate $\nu L\left(t\right)$. Parameters $M_1$ and $M_2$ are the average number of HIV particles generated in the lifetime of the short-lived productively-infected cells and long-lived productively-infected cells, respectively. Parameter $\mathsf{\Lambda }$ is the natural death rate constant of the long-lived productively-infected cells. The B-cell impairment rate is given by $\vartheta P\left(t\right)C\left(t\right)$. All previously described parameters are positive constants. Let $e^{-{\theta }_k\lambda }{f}_k\left(\lambda \right)$ over the time interval $[0,h_k]$, $k=1,2,3$, be the probabilities that uninfected CD4${}^{+}$ T-cells contacted by HIV at time $t-\lambda$ survive $\lambda$ time units and become infected at time t. Otherwise, $e^{-{\theta }_4\lambda }{f}_4\left(\lambda \right)$ over the time interval $[0,h_4]$ and $e^{-{\theta }_5\lambda }{f}_5\left(\lambda \right)$ over the time interval $[0,h_5]$ represent, respectively, the probabilities that short-lived productively-infected cells and long-lived productively-infected cells produce new immature HIV particles at time $t-\lambda$, lose $\lambda$ time units, and become mature at time t. Here, ${\theta }_n$, $n=1,2,...,5$ are positive constants. The probability distribution function $f_n\left(\lambda \right)$, $n=1,2,...,5$ satisfies $f_n\left(\lambda \right)>0,$ and:

$${\int }_0^{h_n}{f}_n\left(\lambda \right)d\lambda =1,{\int }_0^{h_n}{e}^{l\lambda }{f}_n\left(\lambda \right)d\lambda <\infty ,$$

where $l>0$. Denote $y_n\left(\lambda \right)=e^{-{\theta }_n\lambda }{f}_n\left(\lambda \right)$ and $x_n={\int }_0^{h_n}{y}_n\left(\lambda \right)d\lambda$, $n=1,2,...,5$; thus, $0<x_n\le 1$.

We need the following Assumptions of the function $\Theta (U,P)$ [61,62,63]:

Assumption 1.

$\Theta (U,P)$ is continuously differentiable,$\Theta (U,P)>0,$ and $\Theta (0,P)=\Theta (U,0)=0$ for all $U>0$ and $P>0$,

Assumption 2.

${\displaystyle \frac{\partial \Theta (U,P)}{\partial U}}>0$, ${\displaystyle \frac{\partial \Theta (U,P)}{\partial P}}>0$, and ${\displaystyle \frac{\partial \Theta (U,0)}{\partial P}}>0$ for all $U>0$ and $P>0$,

Assumption 3.

${\displaystyle \frac{d}{dU}}\left({\displaystyle \frac{\partial \Theta (U,0)}{\partial P}}\right)>0$ for all $U>0$,

Assumption 4.

${\displaystyle \frac{\Theta (U,P)}{P}}$ is decreasing with respect to P for all $P>0$.

The initial conditions of models (8)–(13) are:

$$\begin{array}{cc}\hfill U\left(s\right)& ={\phi }_1\left(s\right),L\left(s\right)={\phi }_2\left(s\right),\hfill \\ \hfill I\left(s\right)& ={\phi }_3\left(s\right),O\left(s\right)={\phi }_4\left(s\right),\hfill \\ \hfill P\left(s\right)& ={\phi }_5\left(s\right),C\left(s\right)={\phi }_6\left(s\right),\hfill \\ \hfill {\phi }_i\left(s\right)& \ge 0,s\in [-\overline{\lambda },0],i=1,2,...,6,\hfill \end{array}$$

(14)

where $\overline{\lambda }=$ max$\{h_1,h_2,h_3,h_4,h_5\}$, and $({\phi }_1,{\phi }_2,...,{\phi }_6)\in \mathcal{C}([-\overline{\lambda },0],{\mathbb{R}}_{\ge 0}^6)$, where $\mathcal{C}$ is the Banach space of the continuous function mapping the interval $[-\overline{\lambda },0]$ into ${\mathbb{R}}_{\ge 0}^6$. By the standard theory of functional differential equations [64,65], we know that the system has a unique solution satisfying the initial conditions (14).

Proposition 1.

Let $K(t,\phi )$ be the solution of the models (8)–(13) with the initial conditions (14), then $U\left(t\right),$ $L\left(t\right),$$I\left(t\right),$$O\left(t\right),$$P\left(t\right)$, and $C\left(t\right)$ are all non-negative and ultimately bounded for all $t\ge 0$.

The proof of Proposition 1 is given in Appendix A.

Lemma 1.

Assume that Assumptions 1–4 are satisfied, then there exists a threshold parameter $R_0^G>0$ such that:

(i)

if $R_0^G\le 1$, then the model has only one equilibrium point $E{P}_0$; and

(ii)

if $R_0^G>1$, then the model has two equilibria $E{P}_0$ and $E{P}_1$.

The proof of Lemma 1 is given in Appendix A.

Global Stability of Equilibria

Define the function $G\left(u\right)=u-1-lnu$. Clearly, $G\left(u\right)\ge 0$, for $u>0$ and $G\left(1\right)=0$.

Theorem 1.

Assume that $R_0^G<1$ and Assumptions 1–4 are satisfied, then the infection-free equilibrium $E{P}_0$ of models (8)–(13) is globally asymptotically stable (G.A.S.).

Theorem 2.

Assume that $R_0^G>1$ and Assumptions 1–4 are satisfied, then the endemic equilibrium $E{P}_1$ of models (8)–(13) is G.A.S.

The proofs of Theorems 1 and 2 are given in Appendix A.

3. Numerical Simulations

We conduct numerical simulations for models (8)–(13) with the Crowley–Martin incidence function:

$$\Theta (U,P)=\frac{UP}{\left(1+{\alpha }_{1}U\right)\left(1+{\alpha }_{2}P\right)},$$

where ${\alpha }_1$ and ${\alpha }_2$ are non-negative constants. We note that if ${\alpha }_1={\alpha }_2=0,$ then we obtain bilinear incidence, if ${\alpha }_1=0$ and ${\alpha }_2\ne 0$, then we get saturated incidence, and if ${\alpha }_2=0$ and ${\alpha }_1\ne 0$, then we obtain Holling Type-II incidence. We can easily see that $\Theta (U,P)$ is a continuously differentiable function. Moreover, $\Theta (U,P)$ satisfies the following:

$\Theta (U,P)>0$, and $\Theta (0,P)=\Theta (U,0)=0$ for all $U\left(t\right)>0$ and $P\left(t\right)>0$. Thus, Assumptions 1 is satisfied.

We have:

$${\displaystyle \frac{\partial \Theta (U,P)}{\partial U}}={\displaystyle \frac{P}{{\left(1+{\alpha }_{1}U\right)}^2\left(1+{\alpha }_{2}P\right)}}>0,{\displaystyle \frac{\partial \Theta (U,P)}{\partial P}}={\displaystyle \frac{U}{\left(1+{\alpha }_{1}U\right){\left(1+{\alpha }_{2}P\right)}^2}}>0,$$

for all $U\left(t\right)>0$ and $P\left(t\right)>0$. Moreover, ${\displaystyle \frac{\partial \Theta (U,0)}{\partial P}}={\displaystyle \frac{U}{1+{\alpha }_{1}U}}>0$ for all $U\left(t\right)>0$, then Assumptions 2 is satisfied.

We also have:

$${\displaystyle \frac{d}{dU}}\left({\displaystyle \frac{\partial \Theta (U,0)}{\partial P}}\right)={\displaystyle \frac{1}{{(1+{\alpha }_{1}U)}^2}}>0,\mathrm{for}\mathrm{all}U\left(t\right)\ge 0.$$

Then, Assumptions 3 is satisfied. Finally, we have:

$${\displaystyle \frac{\partial }{\partial P}}\left({\displaystyle \frac{\Theta (U,P)}{P}}\right)={\displaystyle \frac{-{\alpha }_{2}U}{\left(1+{\alpha }_{1}U\right){\left(1+{\alpha }_{2}P\right)}^2}}<0,\mathrm{for}\mathrm{all}U\left(t\right),P\left(t\right)>0.$$

Then, Assumptions 4 is also satisfied. In addition, we take a particular form of the probability distributed function as:

$$f_n\left(\lambda \right)=\delta (\lambda -{\lambda }_n),n=1,2,...,5,$$

where $\delta (.)$ is the Dirac delta function. When $h_n\to \infty$, we have:

$${\int }_0^{\infty }{f}_n\left(\lambda \right)d\lambda =1,n=1,2,...,5.$$

We have:

$$x_n={\int }_0^{\infty }\delta (\lambda -{\lambda }_n)e^{-{\theta }_n\lambda }d\lambda =e^{-{\theta }_n{\lambda }_n},n=1,2,...,5.$$

Moreover,

$${\int }_0^{\infty }\delta (\lambda -{\lambda }_n)e^{-{\theta }_n\lambda }\Theta (U(t-\lambda ),P(t-\lambda ))d\lambda =e^{-{\theta }_n{\lambda }_n}\Theta (U(t-{\lambda }_n),P(t-{\lambda }_n)),n=1,2,3.$$

$$\begin{array}{cc}\hfill {\int }_0^{\infty }\delta (\lambda -{\lambda }_4)e^{-{\theta }_4\lambda }I(t-\lambda )d\lambda =& e^{-{\theta }_4{\lambda }_4}I(t-{\lambda }_4),\hfill \\ \hfill {\int }_0^{\infty }\delta (\lambda -{\lambda }_5)e^{-{\theta }_5\lambda }O(t-\lambda )d\lambda =& e^{-{\theta }_5{\lambda }_5}O(t-{\lambda }_5).\hfill \end{array}$$

Hence, models (8)–(13) become:

$$\begin{array}{cc}\hfill \dot{U}\left(t\right)& =\varrho -\gamma U\left(t\right)-({\omega }_1+{\omega }_2+{\omega }_3)\frac{UP}{\left(1+{\alpha }_{1}U\right)\left(1+{\alpha }_{2}P\right)},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \dot{L}\left(t\right)& ={\omega }_1{e}^{-{\theta }_1{\lambda }_1}\frac{U(t-{\lambda }_1)P(t-{\lambda }_1)}{\left(1+{\alpha }_{1}U(t-{\lambda }_1)\right)\left(1+{\alpha }_{2}P(t-{\lambda }_1)\right)}-(\zeta +\nu )L\left(t\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \dot{I}\left(t\right)& ={\omega }_2{e}^{-{\theta }_2{\lambda }_2}\frac{U(t-{\lambda }_2)P(t-{\lambda }_2)}{\left(1+{\alpha }_{1}U(t-{\lambda }_2)\right)\left(1+{\alpha }_{2}P(t-{\lambda }_2)\right)}+\nu L\left(t\right)-\beta I\left(t\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \dot{O}\left(t\right)& ={\omega }_3{e}^{-{\theta }_3{\lambda }_3}\frac{U(t-{\lambda }_3)P(t-{\lambda }_3)}{\left(1+{\alpha }_{1}U(t-{\lambda }_3)\right)\left(1+{\alpha }_{2}P(t-{\lambda }_3)\right)}-\mathsf{\Lambda }O\left(t\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \dot{P}\left(t\right)& =\beta M_1{e}^{-{\theta }_4{\lambda }_4}I(t-{\lambda }_4)+\mathsf{\Lambda }{M}_2{e}^{-{\theta }_5{\lambda }_5}O(t-{\lambda }_5)-\xi P\left(t\right)-\rho P\left(t\right)C\left(t\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \dot{C}\left(t\right)& =\epsilon P\left(t\right)-\mu C\left(t\right)-\vartheta P\left(t\right)C\left(t\right),\hfill \end{array}$$

(20)

The basic reproduction number is given by:

$$R_0=\frac{{\omega }_1\nu M_1{e}^{-{\theta }_1{\lambda }_1-{\theta }_4{\lambda }_4}+(\zeta +\nu )\left({\omega }_2{M}_1{e}^{-{\theta }_2{\lambda }_2-{\theta }_4{\lambda }_4}+{\omega }_3{M}_2{e}^{-{\theta }_3{\lambda }_3-{\theta }_5{\lambda }_5}\right)}{\xi (\zeta +\nu )(1+{\alpha }_1{U}_0)}{U}_0.$$

We let $\lambda ={\lambda }_n$, $n=1,2,...,5$. In addition, we fix the values of parameters $\varrho =10$ cells mm${}^{-3}$day${}^{-1}$, $\gamma =0.01$ day${}^{-1}$, $\zeta =0.3$ day${}^{-1}$, $\beta =0.3$ day${}^{-1}$, $\nu =0.2$day${}^{-1}$, $\epsilon =0.2$ cells virus${}^{-1}$day${}^{-1}$, $M_1=10$ virus cells${}^{-1}$, $M_2=5$ virus cells${}^{-1}$, $\xi =6$ day${}^{-1}$, $\mathsf{\Lambda }=0.1$ day${}^{-1}$, $\rho =0.4$ mm${}^3$cells${}^{-1}$day${}^{-1}$, ${\omega }_k=0.001$ mm${}^3$virus${}^{-1}$day${}^{-1}$, ${\theta }_n=0.6$day${}^{-1},$$n=1,2,...,5$, and $k=1,2,3$, and the remaining parameters will be changed. We chose the parameters of the model just to perform the numerical simulations. This is because of the difficulty of getting real data from HIV-infected patients; however, if one has real data, then the parameters of the model can be estimated and the validity of the model be established.

3.1. Stability of Equilibria

For this case, we take ${\alpha }_2=0.5$ mm${}^3$virus${}^{-1}$ and $\vartheta =0.01$ mm${}^3$virus${}^{-1}$ day${}^{-1}$, $\lambda =0.01$ day, $\mu =0.01$ day${}^{-1}$. Let us choose the following conditions:

IC1:

$U\left(s\right)=700$ cells mm${}^{-3}$, $L\left(s\right)=1$ cells mm${}^{-3}$, $I\left(s\right)=2.5$ cells mm${}^{-3}$, $O\left(s\right)=5$ cells mm${}^{-3}$, $P\left(s\right)=1$ virus mm${}^{-3}$, $C\left(s\right)=10$ cells mm${}^{-3}$,

IC2:

$U\left(s\right)=800,$$L\left(s\right)=0.8,$$I\left(s\right)=2,$$O\left(s\right)=4,$$P\left(s\right)=0.7$, $C\left(s\right)=8$,

IC3:

$U\left(s\right)=900,$$L\left(s\right)=0.4,$$I\left(s\right)=1,$$O\left(s\right)=3,$$P\left(s\right)=0.4$, $C\left(s\right)=6$, $s\in [-\lambda ,0]$.

In Figure 1, we want to confirm our global stability results given in Theorems 1 and 2, by showing that from any initial points (any disease stage) taken from a feasible set, the trajectory of the system will tend to one of the two equilibria of the system.

We suggest two different values of ${\alpha }_1$ to see its effect on the solutions of the model:

(A)

${\alpha }_1=0.003$ mm${}^3$cells${}^{-1}$, then $R_0=0.782223<1$. Figure 1 shows that, for all IC1–IC3, the solutions of the model tend to $E{P}_0=(1000,0,0,0,0,0)$. This means that, $E{P}_0$ is G.A.S., and the HIV is predicted to be completely cleared from the body.

(B)

${\alpha }_1=0.0003$ mm${}^3$cells${}^{-1}$, then we compute $R_0=2.40684>1$. Figure 1 shows that the solutions of the model converge to the equilibrium $E{P}_1=(883.713,0.770611,1.79809,3.85306,0.768011,8.68785)$ for all IC1–IC3. Then, $E{P}_1$ is G.A.S., and a chronic HIV infection is attained. Moreover, we have the following:

(i)

$E{P}_1$ is G.A.S. when $0\le {\alpha }_1<0.00213$, and

(ii)

$E{P}_0$ is G.A.S. when ${\alpha }_1>0.00213$.

Therefore, changing the parameter ${\alpha }_1$ will change the stability properties of the equilibria. Since $R_0$ does not depend on the parameter ${\alpha }_2$, then ${\alpha }_2$ has no effect on the stability of equilibria.

3.2. The Effect of the Time Delay $\lambda$ on the Stability of Equilibria

In this case, we take ${\alpha }_2=0.5$, $\vartheta =\mu =0.01$, ${\alpha }_1=0.001$, and $\lambda$ is varied. Moreover, we consider the following initial condition IC4: $U\left(s\right)=950$, $L\left(s\right)=0.2$, $I\left(s\right)=0.5,$$O\left(s\right)=1,$$P\left(s\right)=0.2,$$C\left(s\right)=4$, $s\in [-\lambda ,0]$. Figure 2 shows that as $\lambda$ is increased, the concentrations of latently-infected cells, short-lived productively-infected cells, long-lived productively-infected cells, HIV particles, and B-cells are decreased, while the concentration of uninfected CD4${}^{+}$ T-cells is increased until they reach the equilibrium point $E{P}_0$. Moreover, we have the following:

(i)

$E{P}_1$ is G.A.S. when $0\le \lambda <0.3829$.

(ii)

$E{P}_0$ is G.A.S. when $\lambda >0.3829$.

3.3. Effect of B-Cell Impairment Parameter $\vartheta$ on the HIV Dynamics

In this case, we take ${\alpha }_2=0.5$, $\lambda =\mu =0.01$, ${\alpha }_1=0.001$, and $\vartheta$ is varied. Moreover, we consider the following initial condition IC5: $U\left(s\right)=900,$ $L\left(s\right)=0.4,$ $I\left(s\right)=1,$ $O\left(s\right)=2,$ $P\left(s\right)=0.4,$ $C\left(s\right)=3,$ $s\in [-\lambda ,0]$. Figure 3 shows that as $\vartheta$ is increased, the concentrations of latently-infected cells, short-lived productively-infected cells, long-lived productively infected cells, and HIV particles are increased, while the concentrations of uninfected CD4${}^{+}$ T-cells and B-cells are decreased. We note that the parameter $\vartheta$ has no effect on the stability of equilibria.

3.4. Effect of Antiviral Treatment on the HIV Dynamics

Let us introduce the HIV dynamics model under the effect of highly-active antiretroviral therapies (HAART) as:

$$\begin{array}{cc}\hfill \dot{U}\left(t\right)& =\varrho -\gamma U\left(t\right)-(1-{ϵ}_r)({\omega }_1+{\omega }_2+{\omega }_3)\frac{UP}{\left(1+{\alpha }_{1}U\right)\left(1+{\alpha }_{2}P\right)},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \dot{L}\left(t\right)& =(1-{ϵ}_r){\omega }_1{e}^{-{\theta }_1{\lambda }_1}\frac{U(t-{\lambda }_1)P(t-{\lambda }_1)}{\left(1+{\alpha }_{1}U(t-{\lambda }_1)\right)\left(1+{\alpha }_{2}P(t-{\lambda }_1)\right)}-(\zeta +\nu )L\left(t\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \dot{I}\left(t\right)& =(1-{ϵ}_r){\omega }_2{e}^{-{\theta }_2{\lambda }_2}\frac{U(t-{\lambda }_2)P(t-{\lambda }_2)}{\left(1+{\alpha }_{1}U(t-{\lambda }_2)\right)\left(1+{\alpha }_{2}P(t-{\lambda }_2)\right)}+\nu L\left(t\right)-\beta I\left(t\right),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \dot{O}\left(t\right)& =(1-{ϵ}_r){\omega }_3{e}^{-{\theta }_3{\lambda }_3}\frac{U(t-{\lambda }_3)P(t-{\lambda }_3)}{\left(1+{\alpha }_{1}U(t-{\lambda }_3)\right)\left(1+{\alpha }_{2}P(t-{\lambda }_3)\right)}-\mathsf{\Lambda }O\left(t\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \dot{P}\left(t\right)& =(1-{ϵ}_p)\beta M_1{e}^{-{\theta }_4{\lambda }_4}I(t-{\lambda }_4)+(1-{ϵ}_p)\mathsf{\Lambda }{M}_2{e}^{-{\theta }_5{\lambda }_5}O(t-{\lambda }_5)-\xi P\left(t\right)-\rho P\left(t\right)C\left(t\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \dot{C}\left(t\right)& =\epsilon P\left(t\right)-\mu C\left(t\right)-\vartheta P\left(t\right)C\left(t\right),\hfill \end{array}$$

(26)

where ${ϵ}_r\in [0,1]$ is the efficacy of the reverse transcriptase inhibitor drug, while ${ϵ}_p\in [0,1]$ is the efficacy of the protease inhibitor drug. If ${ϵ}_r={ϵ}_p=0$, then the HAART has no effect, and if ${ϵ}_r={ϵ}_p=1$, the HIV growth is completely stopped. Let $ϵ={ϵ}_r={ϵ}_p$; consequently, the parameter $R_0\left(ϵ\right)$ is given by:

$$R_0\left(ϵ\right)=\frac{{(1-ϵ)}^2\left[{\omega }_1\nu M_1{e}^{-{\theta }_1{\lambda }_1-{\theta }_4{\lambda }_4}+(\zeta +\nu )\left({\omega }_2{M}_1{e}^{-{\theta }_2{\lambda }_2-{\theta }_4{\lambda }_4}+{\omega }_3{M}_2{e}^{-{\theta }_3{\lambda }_3-{\theta }_5{\lambda }_5}\right)\right]}{\xi (\zeta +\nu )(1+{\alpha }_1{U}_0)}{U}_0.$$

Since the goal is to clear the HIV from the body, then we have to determine the drug efficacies that make $R_0\left(ϵ\right)<1$ for models (21)–(26). Then,

$${(1-ϵ)}^2<\frac{1}{R_0\left(0\right)}.$$

Since, $0\le ϵ\le 1$, then for ${ϵ}^{min}<ϵ\le 1$, $E{P}_0$ is G.A.S. where ${ϵ}^{min}=max\left\{0,1-\frac{1}{\sqrt{R_0\left(0\right)}}\right\}$.

We take ${\alpha }_1=0.0001$, ${\alpha }_2=0.005$, $\vartheta =\lambda =0.01$, $\mu =0.7$. Then, ${ϵ}^{min}=0.4071$, and we have the following:

(i)

if $0.4071<ϵ\le 1$, then $R_0\left(ϵ\right)<1$, and $E{P}_0$ is G.A.S.;

(ii)

if $0\le ϵ<0.4071$, then $R_0\left(ϵ\right)>1$, and $E{P}_1$ is G.A.S.

We consider the following initial condition IC6: $U\left(s\right)=600,$ $L\left(s\right)=3$, $I\left(s\right)=5,$ $O\left(s\right)=10,$ $P\left(s\right)=2,$ $C\left(s\right)=1$, $s\in [-\lambda ,0]$ to show in Figure 4 the solution trajectories of models (21)–(26) for different values of $ϵ$. Clearly, from the figure, the increasing of $ϵ$ will increase the concentration of the uninfected CD4${}^{+}$ T-cells and decrease the concentrations of latently-infected cells, short-lived productively-infected cells, long-lived productively-infected cells, HIV particles, and B-cells. One can observe that the time delay $\lambda$ has a similar effect as drug efficacy $ϵ$. This gives some impression for developing a new class of treatment to increase the delay period and then suppress the HIV replication.

In practice, HAART cannot completely eradicate the HIV from the body. Therefore, one can design effective antiviral drugs to reduce the concentration of the HIV particles to a lower level. Let the endemic equilibrium of models (21)–(26) as a function of $ϵ$ be given as: $E{P}_1(U_1\left(ϵ\right),L_1\left(ϵ\right),I_1\left(ϵ\right),O_1\left(ϵ\right),P_1\left(ϵ\right),C_1\left(ϵ\right))$. Then, the objective is to design $ϵ$ such that:

$$0<P_1\left(ϵ\right)\le P_{min},\mathrm{for}\mathrm{all}{ϵ}_{min}\le ϵ<{ϵ}_{max}.$$

(27)

Let us fix the parameters ${\alpha }_1={\alpha }_2=0.0$, $\lambda =0.1$, $\mu =0.01$, and $P_{min}=0.5$. We calculate ${ϵ}_{min}$and ${ϵ}_{max}$ using two models:

• Model (I), which is given by Equations (21)–(26) and neglects the B-cell immune impairment, i.e., $\vartheta =0$. Then, we find that Inequality (27) is satisfied when $0.3486\le ϵ<0.6439.$
• Model (II), which is given by Equations (21)–(26) with B-cell immune impairment. We take $\vartheta =0.05$, then we find that Inequality (27) is satisfied when $0.5473\le ϵ<0.6439.$

Therefore, if we apply treatment with efficacy $ϵ$ such that $0.3486\le ϵ\le 0.5473$, this guarantees that $P_1\left(ϵ\right)\le 0.5$ for Model (I), but $P_1\left(ϵ\right)>0.5$ for Model (II). Therefore, more accurate drug efficacy, which is required to reduce the concentration of the HIV to a lower value is calculated by using Model (II). This shows the importance of considering the effect of immune impairment in the HIV dynamics model.

4. Discussion

In the literature, several mathematical models of HIV infection have considered the impairment of CTL functions. However, it has been reported in several papers that during HIV infection, the B-cell can lose its functions [57,58,59]. This article is an extension of the work discussed by Miao et al. [60], where only the local stability of the equilibria was discussed. We have shown that the solutions of our proposed model are nonnegative and bounded, which ensures the well-posedness of the model. We derived the basic reproduction number $R_0$, which fully determines the existence and stability of the two equilibria of the model. We investigated the global stability of the two equilibria of the model by using the Lyapunov method and LaSalle’s invariance principle. We proved that: (i) if $R_0<1$, then the infection-free equilibrium $E{P}_0$ is globally asymptotically stable, and the HIV is predicted to be completely cleared from the HIV infected individuals; (ii) if $R_0>1$, then the endemic equilibrium $E{P}_1$ is globally asymptotically stable, and a chronic HIV infection is attained. We conducted numerical simulations and showed that both the theoretical and numerical results were consistent (see Figure 1). Moreover, we studied the effect of time delays, the impairment of the B-cell functions, and antiviral treatment on the HIV dynamics. The results show that, when the B-cells lose their functions during HIV infection, the number of antibodies produced from the B-cells is decreased, and then, the number of HIV particles is increased (Figure 3). Therefore, HAART is needed to improve the health of the HIV-infected patient. We showed that more accurate drug efficacy, which is required to reduce the concentration of the HIV to a lower value, is calculated by using our proposed HIV dynamics model with immune impairment. In addition, we observed that the time delay had a similar effect as the drug efficacy (see Figure 2 and Figure 4). This gives some impression for developing a new class of treatment to increase the delay period and then suppress the HIV replication.

Looking ahead to further developments, an interesting perspective would be introducing a stochastic internal variable, as in [66], to account for virus mutations. We mention that our HIV dynamics models assumed that HIV particles and cells are equally distributed and ignored their spatial mobility. Recently, many authors have argued that the virus moves freely in the body and follows Fickian diffusion (see, e.g., [67,68,69]). Therefore, it is more reasonable to study reaction-diffusion versions of our model.