Analysis of General Humoral Immunity HIV Dynamics Model with HAART and Distributed Delays

Abstract

This paper deals with the study of an HIV dynamics model with two target cells, macrophages and CD4${}^{+}$ T cells and three categories of infected cells, short-lived, long-lived and latent in order to get better insights into HIV infection within the body. The model incorporates therapeutic modalities such as reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). The model is incorporated with distributed time delays to characterize the time between an HIV contact of an uninfected target cell and the creation of mature HIV. The effect of antibody on HIV infection is analyzed. The production and removal rates of the ten compartments of the model are given by general nonlinear functions which satisfy reasonable conditions. Nonnegativity and ultimately boundedness of the solutions are proven. Using the Lyapunov method, the global stability of the equilibria of the model is proven. Numerical simulations of the system are provided to confirm the theoretical results. We have shown that the antibodies can play a significant role in controlling the HIV infection, but it cannot clear the HIV particles from the plasma. Moreover, we have demonstrated that the intracellular time delay plays a similar role as the Highly Active Antiretroviral Therapies (HAAT) drugs in eliminating the HIV particles.

1. Introduction

Modeling the HIV dynamics has received considerable attention from mathematicians during the recent decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. The first HIV dynamics model is proposed by Nowak and Bangham [1] as:

$$\begin{array}{cc}\hfill \dot{s}\left(t\right)& =\rho -\delta s\left(t\right)-\overline{\lambda }s\left(t\right)p\left(t\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \dot{y}\left(t\right)& =\overline{\lambda }s\left(t\right)p\left(t\right)-\eta y\left(t\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \dot{p}\left(t\right)& =\overline{N}\eta y\left(t\right)-gp\left(t\right),\hfill \end{array}$$

(3)

where s, y and p are the concentrations of the CD4${}^{+}$ T cells, infected cells and HIV particles; $\rho$, $\delta$ and $\overline{\lambda }$ represent the production, death and infection rates of the uninfected CD4${}^{+}$ T cells, respectively; $\eta$ and g are the death rate constants of the infected cells and free HIV, respectively; $\overline{N}$ is the average number of HIV particles generated in the lifetime of the infected cells. A class of latently infected cells has been modeled in the HIV dynamics in [23,24,25,26]. Elaiw et al. [27] have extended model (1)–(3) by considering distributed time delays, B cells (x), and three categories of infected cells, short-lived (y), long-lived (u) and latent (w) as:

$$\begin{array}{cc}\hfill \dot{s}& =\pi \left(s\left(t\right)\right)-\lambda {\rm Y}(s\left(t\right),p\left(t\right)),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \dot{w}& ={\lambda }_1\underset{0}{\overset{h_1}{\int }}{f}_1\left(\tau \right)e^{-{\mu }_1\tau }{\rm Y}(s\left(t-\tau \right),p\left(t-\tau \right))d\tau -(\alpha +\beta ){\psi }_1\left(w\left(t\right)\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \dot{y}& ={\lambda }_2\underset{0}{\overset{h_2}{\int }}{f}_2\left(\tau \right)e^{-{\mu }_2\tau }{\rm Y}(s\left(t-\tau \right),p\left(t-\tau \right))d\tau +\alpha {\psi }_1\left(w\left(t\right)\right)-\eta {\psi }_2\left(y\left(t\right)\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \dot{u}& ={\lambda }_3\underset{0}{\overset{h_3}{\int }}{f}_3\left(\tau \right)e^{-{\mu }_3\tau }{\rm Y}(s\left(t-\tau \right),p\left(t-\tau \right))d\tau -\nu {\psi }_3\left(u\left(t\right)\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \dot{p}& =N\eta {\psi }_2\left(y\left(t\right)\right)+M\nu {\psi }_3\left(u\left(t\right)\right)-g{\psi }_4\left(p\left(t\right)\right)-\mu {\psi }_4\left(p\left(t\right)\right){\psi }_5\left(x\left(t\right)\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \dot{x}& =r{\psi }_4\left(p\left(t\right)\right){\psi }_5\left(x\left(t\right)\right)-\omega {\psi }_5\left(x\left(t\right)\right),\hfill \end{array}$$

(9)

where $f_i\left(\tau \right)e^{-{\mu }_i\tau }$ over the time interval $\left[0,h_i\right],$ $i=1,2,3$ represents the probabilities that uninfected cells contacted by HIV at time $t-\tau$ survived $\tau$ time units and become infected at time t; ${\rm Y},\pi ,{\psi }_j,j=1,\dots ,5$ are general nonlinear functions. The probability distribution function $f_i\left(\tau \right)$ satisfies $f_i\left(\tau \right)>0$ and

$$\underset{0}{\overset{h_i}{\int }}{f}_i\left(\tau \right)d\tau =1,\underset{0}{\overset{h_i}{\int }}{f}_i\left(\eta \right)e^{\upsilon \eta }d\eta <\infty ,i=1,2,3,$$

where $\upsilon >0$. Let us denote ${\Theta }_i\left(\tau \right)=f_i\left(\tau \right)e^{-{\mu }_i\tau }$ and $F_i=\underset{0}{\overset{h_i}{\int }}{\Theta }_i\left(\tau \right)d\tau$, thus $0<F_i\le 1$, $i=1,2,3$.

Model (4)–(9) assumes that the HIV infects one category of target cells, CD4${}^{+}$ T cells. However, Perleson et al. have observed that after the rapid first phase of decay during the initial 1–2 weeks of an antiretroviral treatment, plasma virus levels declined at a considerably slower rate [28]. This second phase of viral decay was attributed to the turnover of infected macrophages. Therefore, the HIV model with two categories of target cells, CD4${}^{+}$ T cells and macrophages is more accurate than that model with only one category target cells, CD4${}^{+}$ T cells. As a result, more accurate drug efficacy can be determined when using the HIV model with two classes of target cells. Recently, many efforts have been devoted to the analysis of various mathematical models of HIV dynamics with two categories of target cells (see, e.g., [29,30,31,32,33,34,35,36,37]). However, in these papers, the production and removal rates of the HIV particles and cells are given by linear functions; moreover, only one or two classes of infected cells are considered.

The aim of the present paper is to propose and analyze an HIV dynamics model which extends model (4)–(9) and describes the dynamics of HIV with two categories of target cells, CD4${}^{+}$ T cells and macrophages. We study the basic and global properties of the model. Using Lyapunov function and LaSalle’s invariance principle, we have established the global asymptotic stability of the three equilibria of the model. We have shown that the antibodies can reduce the HIV level, but they cannot clear the HIV particles completely. The effect of HAART drugs and intracellular time delays in stabilizing the HIV dynamics system around the infection-free equilibrium are shown. The importance of considering the second class of target cells (macrophages) in the HIV dynamics is also shown.

The rest of the paper is organized as follows: in Section 2, we present the HIV model and study its basic properties. In Section 3, the global stability of three equilibria is established. In Section 4, we offer some numerical simulations to confirm our analytical results. The paper ends with some conclusions and discussion in Section 5.

2. Mathematical Model

We propose the following model:

$$\begin{array}{cc}\hfill {\dot{s}}_i& ={\pi }_i\left(s_i\left(t\right)\right)-{\lambda }_i{{\rm Y}}_i(s_i\left(t\right),p\left(t\right)),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\dot{w}}_i& ={\lambda }_{1i}\underset{0}{\overset{h_{1i}}{\int }}{f}_{1i}\left(\tau \right)e^{-{\mu }_{1i}\tau }{{\rm Y}}_i(s_i\left(t-\tau \right),p\left(t-\tau \right))d\tau -({\alpha }_i+{\beta }_i){\psi }_{1i}\left(w_i\left(t\right)\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\dot{y}}_i& ={\lambda }_{2i}\underset{0}{\overset{h_{2i}}{\int }}{f}_{2i}\left(\tau \right)e^{-{\mu }_{2i}\tau }{{\rm Y}}_i(s_i\left(t-\tau \right),p\left(t-\tau \right))d\tau +{\alpha }_i{\psi }_{1i}\left(w_i\left(t\right)\right)-{\eta }_i{\psi }_{2i}\left(y_i\left(t\right)\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\dot{u}}_i& ={\lambda }_{3i}\underset{0}{\overset{h_{3i}}{\int }}{f}_{3i}\left(\tau \right)e^{-{\mu }_{3i}\tau }{{\rm Y}}_i(s_i\left(t-\tau \right),p\left(t-\tau \right))d\tau -{\nu }_i{\psi }_{3i}\left(u_i\left(t\right)\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \dot{p}& ={\displaystyle \sum _{i=1}^2}\left(N_i{\eta }_i\underset{0}{\overset{h_{4i}}{\int }}{f}_{4i}\left(\tau \right)e^{-{\mu }_{4i}\tau }{\psi }_{2i}\left(y_i\left(t-\tau \right)\right)d\tau +M_i{\nu }_i\underset{0}{\overset{h_{5i}}{\int }}{f}_{5i}\left(\tau \right)e^{-{\mu }_{5i}\tau }{\psi }_{3i}\left(u_i\left(t-\tau \right)\right)d\tau \right)\hfill \\ & -g{\psi }_{41}\left(p\left(t\right)\right)-\mu {\psi }_{41}\left(p\left(t\right)\right){\psi }_{42}\left(x\left(t\right)\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \dot{x}& =r{\psi }_{41}\left(p\left(t\right)\right){\psi }_{42}\left(x\left(t\right)\right)-\omega {\psi }_{42}\left(x\left(t\right)\right),\hfill \end{array}$$

(15)

where $i=1$for the CD4${}^{+}$ T cells and $i=2$ for the macrophages. We have ${\lambda }_{m1}=(1-{\epsilon }_1){\overline{\lambda }}_{m1},{\lambda }_{m2}=(1-f{\epsilon }_1){\overline{\lambda }}_{m2}$, $m=1,2,3$, $N_1=(1-{\epsilon }_2){\overline{N}}_1$, $M_1=(1-{\epsilon }_2){\overline{M}}_1,N_2=(1-h{\epsilon }_2){\overline{N}}_2$, $M_2=(1-h{\epsilon }_2){\overline{M}}_2,$ ${\lambda }_i={\lambda }_{1i}+{\lambda }_{2i}+{\lambda }_{3i}$, and $f,h\in (0,1)$. Let $f_{ji}\left(\tau \right)e^{-{\mu }_{ji}\tau }$ over the time interval $\left[0,h_{ji}\right],$ $j=1,\dots ,5$, $i=1,2,$ are the probabilities that uninfected target cells contacted by HIV at time $t-\tau$ survived $\tau$ time units and became infected at time t. Denote ${\Theta }_{ji}\left(\tau \right)=f_{ji}\left(\tau \right)e^{-{\mu }_{ji}\tau }$ and $F_{ji}=\underset{0}{\overset{h_{ji}}{\int }}{\Theta }_{ji}\left(\tau \right)d\tau$; thus $0<F_{ji}\le 1$, $j=1,\dots ,5,i=1,2$. We assume that:

Hypothesis 1 (H1).

(i) There exists$s_i^0$such that${\pi }_i\left(s_i^0\right)=0$, ${\pi }_i\left(s_i\right)>0$for$s_i\in [0,s_i^0)$, (ii)${\pi }_i^{\prime }\left(s_i\right)<0$for$s_i>0$; (iii) there are$b_i>0$and${\overline{b}}_i>0$such that${\pi }_i\left(s_i\right)\le b_i-{\overline{b}}_i{s}_i$for$s_i\ge 0$.

Hypothesis 2 (H2).

(i)${{\rm Y}}_i(s_i,p)>0$and${{\rm Y}}_i(0,p)={{\rm Y}}_i(s_i,0)=0$for$s_i,p>0$, (ii)$\frac{\partial {{\rm Y}}_i(x_i,p)}{\partial s_i}>0,$$\frac{\partial {{\rm Y}}_i(s_i,p)}{\partial p}>0$and, $\frac{\partial {{\rm Y}}_i(s_i,0)}{\partial p}>0$for all$s_i,p>0$, (iii)$\frac{d}{d{s}_i}\left({\displaystyle \frac{\partial {{\rm Y}}_i(s_i,0)}{\partial p}}\right)>0$for$s_i>0$.

Hypothesis 3 (H3).

(i)${\psi }_{ji}\left(\eta \right)>0$ for $\eta >0$, ${\psi }_{ji}\left(0\right)=0,$ $j=1,\dots ,4,$ $i=1,2$, (ii) ${\psi }_{ji}^{\prime }\left(\eta \right)>0,{\psi }_{42}^{\prime }\left(\eta \right)>0$ for $\eta >0,$ $j=1,2,3,$ $i=1,2,$ ${\psi }_{41}^{\prime }\left(\eta \right)>0,$ for $\eta \ge 0$; (iii) there are ${\alpha }_{ji}>0,$ $j=1,\dots ,4,$ $i=1,2,$ such that ${\psi }_{ji}\left(\eta \right)\ge {\alpha }_{ji}\eta$ for $\eta \ge 0.$

Hypothesis 4 (H4).

${\displaystyle \frac{{{\rm Y}}_i(s_i,p)}{{\psi }_{41}\left(p\right)}}$is decreasing function w.r.t.p, for$p>0.$

Remark 1.

From H1–H4, we have

$$\left({{\rm Y}}_i(s_i,p)-{{\rm Y}}_i(s_i,p^{*})\right)\left(\frac{{{\rm Y}}_i(s_i,p)}{{\psi }_{41}\left(p\right)}-\frac{{{\rm Y}}_i(s_i,p^{*})}{{\psi }_{41}\left(p^{*}\right)}\right)\le 0,$$

which gives

$$\left(1-\frac{{{\rm Y}}_i(s_i,p^{*})}{{{\rm Y}}_i(s_i,p)}\right)\left(\frac{{{\rm Y}}_i(s_i,p)}{{{\rm Y}}_i(s_i,p^{*})}-\frac{{\psi }_{41}\left(p\right)}{{\psi }_{41}\left(p^{*}\right)}\right)\le 0.$$

We consider systems (10)–(15) with the initial conditions:

$$\begin{array}{cc}\hfill s_1\left(t\right)& ={\phi }_1\left(\theta \right),s_2\left(t\right)={\phi }_2\left(\theta \right),w_1\left(t\right)={\phi }_3\left(\theta \right),w_2\left(t\right)={\phi }_4\left(\theta \right),y_1\left(t\right)={\phi }_5\left(\theta \right),\hfill \\ \hfill y_2\left(t\right)& ={\phi }_6\left(\theta \right),u_1\left(t\right)={\phi }_7\left(\theta \right),u_2\left(t\right)={\phi }_8\left(\theta \right),p\left(t\right)={\phi }_9\left(\theta \right),x\left(t\right)={\phi }_{10}\left(\theta \right),\hfill \\ \hfill {\phi }_j\left(\theta \right)& \ge 0,\theta \in \left[-\varsigma ,0\right],j=1,\dots ,10,\hfill \end{array}$$

(16)

where $\varsigma =max\left\{h_{11},h_{12},h_{21},h_{22},h_{31},h_{32},h_{41},h_{42},h_{51},h_{52}\right\}$ and denoted by C is the Banach space of continuous functions mapping the interval $\left[-\varsigma ,0\right]$ into ${\mathbb{R}}_{\ge 0}$ and $\left({\phi }_1\left(\theta \right),\dots ,{\phi }_{10}\left(\theta \right)\right)\in C\left(\left[-\varsigma ,0\right],{\mathbb{R}}_{\ge 0}^{10}\right)$. According to [38], there exists a unique solution for system (10)–(15) with initial (16).

2.1. Properties of Solutions

Lemma 1.

Suppose that H1–H3 hold, then the solutions of system (10)–(15) are non-negative and ultimately bounded.

The proof Lemma 1 is given in Appendix A.

Lemma 1 shows that

$$\begin{array}{cc}\hfill \Omega & =\left\{(s_1,s_2,w_1,w_2,y_1,y_2,u_1,u_2,p,x)\in C^{10}:∥s_i∥\le M_{1i},∥w_i∥\le M_{1i},\right.\hfill \\ & \left.∥y_i∥\le M_{2i},∥u_i∥\le M_{3i},∥p∥\le M_{41},∥x∥\le M_{42}\right\}\hfill \end{array}$$

is positively invariant with respect to system (10)–(15).

2.2. Equilibria

The basic reproduction number $R_0$ of system (10)–(15) is given by:

$$R_0={\displaystyle \sum _{i=1}^2}\frac{{\gamma }_i}{{\psi }_{41}^{\prime }\left(0\right)}\frac{\partial {{\rm Y}}_i(s_i^0,0)}{\partial p}.$$

The equilibria of system (10)–(15) satisfy the following equations:

$$\begin{array}{cc}\hfill 0& ={\pi }_i\left(s_i\right)-{\lambda }_i{{\rm Y}}_i(s_i,p),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill 0& ={\lambda }_{1i}{F}_{1i}{{\rm Y}}_i(s_i,p)-({\alpha }_i+{\beta }_i){\psi }_{1i}\left(w_i\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill 0& ={\lambda }_{2i}{F}_{2i}{{\rm Y}}_i(s_i,p)+{\alpha }_i{\psi }_{1i}\left(w_i\right)-{\eta }_i{\psi }_{2i}\left(y_i\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill 0& ={\lambda }_{3i}{F}_{3i}{{\rm Y}}_i(s_i,p)-{\nu }_i{\psi }_{3i}\left(u_i\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill 0& ={\displaystyle \sum _{i=1}^2}\left(N_i{\eta }_i{F}_{4i}{\psi }_{2i}\left(y_i\right)+M_i{\nu }_i{F}_{5i}{\psi }_{3i}\left(u_i\right)\right)-g{\psi }_{41}\left(p\right)-\mu {\psi }_{41}\left(p\right){\psi }_{42}\left(x\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill 0& =r{\psi }_{41}\left(p\right){\psi }_{42}\left(x\right)-\omega {\psi }_{42}\left(x\right).\hfill \end{array}$$

(22)

From Equation (22), we have $x=0$ or $p={\psi }_{41}^{-1}(\omega /r)$. Let us define

$$\begin{array}{cc}\hfill {\Lambda }_{1i}\left(s_i\right)& ={\psi }_{1i}^{-1}\left(\frac{{\lambda }_{1i}{F}_{1i}}{{\lambda }_i({\alpha }_i+{\beta }_i)}{\pi }_i\left(s_i\right)\right),{\Lambda }_{2i}\left(s_i\right)={\psi }_{2i}^{-1}\left(\frac{{\alpha }_i{\lambda }_{1i}{F}_{1i}+({\alpha }_i+{\beta }_i){\lambda }_{2i}{F}_{2i}}{{\lambda }_i{\eta }_i({\alpha }_i+{\beta }_i)}{\pi }_i\left(s_i\right)\right),\hfill \\ \hfill {\Lambda }_{3i}\left(s_i\right)& ={\psi }_{3i}^{-1}\left(\frac{{\lambda }_{3i}{F}_{3i}}{{\lambda }_i{\nu }_i}{\pi }_i\left(s_i\right)\right),{\Lambda }_4\left(s_i\right)={\psi }_{41}^{-1}\left({\displaystyle \sum _{i=1}^2}\frac{{\gamma }_i}{{\lambda }_i}{\pi }_i\left(s_i\right)\right),\hfill \end{array}$$

(23)

where ${\gamma }_i={\displaystyle \sum _{i=1}^2}\frac{N_i{F}_{4i}({\alpha }_i{\lambda }_{1i}{F}_{1i}+({\alpha }_i+{\beta }_i){\lambda }_{2i}{F}_{2i})+M_i{F}_{3i}{F}_{5i}{\lambda }_{3i}({\alpha }_i+{\beta }_i)}{g({\alpha }_i+{\beta }_i)}$. It follows from Equations (17)–(21) that:

$$w_i={\Lambda }_{1i}\left(s_i\right),y_i={\Lambda }_{2i}\left(s_i\right),u_i={\Lambda }_{3i}\left(s_i\right),p={\Lambda }_4\left(s_i\right).$$

(24)

Obviously, ${\Lambda }_{1i}\left(s_i\right),{\Lambda }_{2i}\left(s_i\right),{\Lambda }_{3i}\left(s_i\right),{\Lambda }_4\left(s_i\right)>0$ for $s_i\in [0,s_i^0)$ and ${\Lambda }_{1i}\left(s_i^0\right)={\Lambda }_{2i}\left(s_i^0\right)={\Lambda }_{3i}\left(s_i^0\right)={\Lambda }_4\left(s_i^0\right)=0,$ $i=1,2$. From Equations (17), (23), and (24), we obtain

$${\displaystyle \sum _{i=1}^2}{\gamma }_i{{\rm Y}}_i(s_i,{\Lambda }_4\left(s_i\right))-{\psi }_{41}\left({\Lambda }_4\left(s_i\right)\right)=0.$$

(25)

Equation (25) has two possible solutions, ${\Lambda }_4=0$ and ${\Lambda }_4\ne 0$. The solution ${\Lambda }_4=0$ implies $s_i=s_i^0$ which gives the infection-free equilibrium ${\Pi }_0(s_1^0,s_2^0,0,0,0,0,0,0,0,0)$. The other solution ${\Lambda }_4\ne 0$ admits an antibody-inactivated infection equilibrium ${\Pi }_1({\tilde{s}}_1,{\tilde{s}}_2,{\tilde{w}}_1,{\tilde{w}}_2,{\tilde{y}}_1,{\tilde{y}}_2,{\tilde{u}}_1,{\tilde{u}}_2,\tilde{p},0)$, where

$$\begin{array}{cc}\hfill {\pi }_i\left({\tilde{s}}_i\right)& ={\lambda }_i{{\rm Y}}_i({\tilde{s}}_i,\tilde{p}),{\lambda }_{1i}{F}_{1i}{{\rm Y}}_i({\tilde{s}}_i,\tilde{p})=({\alpha }_i+{\beta }_i){\psi }_{1i}\left({\tilde{w}}_i\right),\hfill \\ \hfill {\eta }_i{\psi }_{2i}\left({\tilde{y}}_i\right)& ={\lambda }_{2i}{F}_{2i}{{\rm Y}}_i({\tilde{s}}_i,\tilde{p})+{\alpha }_i{\psi }_{1i}\left({\tilde{w}}_i\right),\hfill \\ \hfill {\nu }_i{\psi }_{3i}\left({\tilde{u}}_i\right)& ={\lambda }_{3i}{F}_{3i}{{\rm Y}}_i({\tilde{s}}_i,\tilde{p}),g{\psi }_{41}\left(\tilde{p}\right)={\displaystyle \sum _{i=1}^2}\left(N_i{\eta }_i{F}_{4i}{\psi }_{2i}\left({\tilde{y}}_i\right)+M_i{\nu }_i{F}_{5i}{\psi }_{3i}\left({\tilde{u}}_i\right)\right).\hfill \end{array}$$

(26)

Now, we consider the other solution of Equation (22) is $\overline{p}={\psi }_{41}^{-1}\left({\displaystyle \frac{\omega }{r}}\right)>0.$ Substitute $p=\overline{p}={\psi }_{41}^{-1}\left({\displaystyle \frac{\omega }{r}}\right)$ in Equation (17) and let ${\Delta }_i\left(s_i\right)={\pi }_i\left(s_i\right)-{\lambda }_i{{\rm Y}}_i(s_i,\overline{p})=0$. Using H1 and H2, we have ${\Delta }_i$ is strictly decreasing, ${\Delta }_i\left(0\right)={\pi }_i\left(0\right)>0$ and ${\Delta }_i\left(s_i^0\right)=-{\lambda }_i{{\rm Y}}_i(s_i^0,\overline{p})<0$. Thus, there exists unique ${\overline{s}}_i\in (0,s_i^0)$ such that ${\Delta }_i\left({\overline{s}}_i\right)=0$. It follows from Equations (21) and (24) that

$$\begin{array}{cc}\hfill {\overline{w}}_i& ={\Lambda }_{1i}\left({\overline{s}}_i\right)>0,{\overline{y}}_i={\Lambda }_{2i}\left({\overline{s}}_i\right)>0,{\overline{u}}_i={\Lambda }_{3i}\left({\overline{s}}_i\right)>0,\hfill \\ \hfill \overline{p}& ={\psi }_{41}^{-1}\left({\displaystyle \frac{\omega }{r}}\right)>0,\overline{x}={\psi }_{42}^{-1}\left(\frac{g}{\mu }\left({\displaystyle \sum _{i=1}^2}{\gamma }_i\frac{{{\rm Y}}_i({\overline{s}}_i,\overline{p})}{{\psi }_{41}\left(\overline{p}\right)}-1\right)\right).\hfill \end{array}$$

Thus, $\overline{x}>0$ when ${\displaystyle \sum _{i=1}^2}{\gamma }_i\frac{{{\rm Y}}_i({\overline{s}}_i,\overline{p})}{{\psi }_{41}\left(\overline{p}\right)}>1$. Define $R_1$ as follows:

$$R_1={\displaystyle \sum _{i=1}^2}{\gamma }_i\frac{{{\rm Y}}_i({\overline{s}}_i,\overline{p})}{{\psi }_{41}\left(\overline{p}\right)},$$

which represents the antibody immune response activation number.

If $R_1>1$, then $\overline{x}={\psi }_{42}^{-1}\left(\frac{g}{\mu }(R_1-1)\right)>0,$ and there exists an antibody-activated infection equilibrium ${\Pi }_2({\overline{s}}_1,{\overline{s}}_2,{\overline{w}}_1,{\overline{w}}_2,{\overline{y}}_1,{\overline{y}}_2,{\overline{u}}_1,{\overline{u}}_2,\overline{p},\overline{x})$. Clearly, from H2 and H4, we have

$$\begin{array}{cc}\hfill R_1& ={\displaystyle \sum _{i=1}^2}{\gamma }_i\frac{{{\rm Y}}_i({\overline{s}}_i,\overline{p})}{{\psi }_{41}\left(\overline{p}\right)}\le {\displaystyle \sum _{i=1}^2}\underset{p\to 0^{+}}{lim}{\gamma }_i\frac{{{\rm Y}}_i({\overline{s}}_i,p)}{{\psi }_{41}\left(\overline{p}\right)}\le {\displaystyle \sum _{i=1}^2}\frac{{\gamma }_i}{{\psi }_{41}^{\prime }\left(0\right)}\frac{\partial {{\rm Y}}_i({\overline{s}}_i,0)}{\partial p}\le {\displaystyle \sum _{i=1}^2}\frac{{\gamma }_i}{{\psi }_{41}^{\prime }\left(0\right)}\frac{\partial {{\rm Y}}_i(s_i^0,0)}{\partial p}\hfill \\ & =R_0.\hfill \end{array}$$

3. Global Stability

Denote $\left(s_i,w_i,y_i,u_i,p,x\right)=\left(s_i\left(t\right),w_i\left(t\right),y_i\left(t\right),u_i\left(t\right),p\left(t\right),x\left(t\right)\right).$

Theorem 1.

Let hypotheses H1–H4 be valid and $R_0\le 1$; then, ${\Pi }_0$ is globally asymptotically stable.

Lemma 2.

Suppose that $R_0>1$ and H1–H4 hold; then,

$$sgn(R_1-1)=sgn(\tilde{p}-\overline{p})=sgn({\overline{s}}_i-{\tilde{s}}_i).$$

Theorem 2.

Suppose that hypotheses H1–H4 are satisfied, ${\Pi }_1$ exists and $R_1\le 1$, then ${\Pi }_1$ is globally asymptotically stable.

Theorem 3.

Let hypotheses H1–H4 hold true and $R_1>1$, then ${\Pi }_2$ is globally asymptotically stable.

The proofs of Lemma 2 and Theorems 1–3 are given in Appendix A.

4. Numerical Simulations

We choose

$$\begin{array}{cc}\hfill {\pi }_1\left(s_1\left(t\right)\right)& ={\rho }_1-{\beta }_{11}{s}_1\left(t\right)+B{s}_1\left(t\right)\left(1-\frac{s_1\left(t\right)}{s_{max}}\right),{\pi }_2\left(s_2\left(t\right)\right)={\rho }_2-{\beta }_{12}{s}_2\left(t\right),\hfill \\ \hfill {{\rm Y}}_i(s_i\left(t\right),p\left(t\right))& =s_i\left(t\right)p\left(t\right),{\psi }_{ji}\left(\theta \right)=\theta ,j=1,\dots ,4,i=1,2,\hfill \end{array}$$

where $B<{\beta }_{11}$. Clearly, ${\pi }_i\left(0\right)={\rho }_i>0$ and ${\pi }_i\left(s_i^0\right)=0$, where

$$s_1^0=\frac{s_{max}}{2B}\left(B-{\beta }_{11}+\sqrt{{(B-{\beta }_{11})}^2+\frac{4{\rho }_{1}B}{s_{max}}}\right),s_2^0=\frac{{\rho }_2}{{\beta }_{12}}.$$

We have

$${\pi }_1^{\prime }\left(s_1\right)=-{\beta }_{11}+B-\frac{2B{s}_1}{s_{max}}<0,{\pi }_2^{\prime }\left(s_2\right)=-{\beta }_{12}.$$

Clearly, ${\pi }_i\left(s_i\right)>0$ for $s_i\in [0,s_i^0)$ and

$$\begin{array}{cc}\hfill {\pi }_1\left(s_1\right)& ={\rho }_1-({\beta }_{11}-B)s_1-B\frac{s_1^2}{s_{max}}\le {\rho }_1-({\beta }_{11}-B)s_1,\hfill \\ \hfill {\pi }_2\left(s_2\right)& ={\rho }_2-{\beta }_{12}{s}_2.\hfill \end{array}$$

Then, H1 is satisfied. Clearly ${{\rm Y}}_i(s_i,p)>0,$ ${{\rm Y}}_i(0,p)={\rm Y}(s_i,0)=0$ for $s_i$, $p\in (0,\infty )$, and

$$\frac{\partial {{\rm Y}}_i(s_i,p)}{\partial s_i}=p,\frac{\partial {{\rm Y}}_i(s_i,p)}{\partial p}=s_i,\frac{\partial {{\rm Y}}_i(s_i,0)}{\partial p}=s_i.$$

Then, $\frac{\partial {{\rm Y}}_i(s_i,p)}{\partial s_i}>0,\frac{\partial {{\rm Y}}_i(s_i,p)}{\partial p}>0$ and $\frac{\partial {{\rm Y}}_i(s_i,0)}{\partial p}>0$ for all $s_i,p\in (0,\infty )$. Therefore, H1 is satisfied. In addition

$${\left(\frac{\partial {{\rm Y}}_i(s_i,0)}{\partial p}\right)}^{\prime }=1>0\mathrm{for}\mathrm{all}{s}_i>0.$$

It follows that, H2 is satisfied. One can show that function ${\psi }_{ji}$ satisfies H3. Moreover,

$$\frac{\partial }{\partial p}\left(\frac{{{\rm Y}}_i(s_i,p)}{{\psi }_{41}\left(p\right)}\right)=0.$$

Therefore, H4 holds true.

In addition, we take a particular form of the probability distributed function as:

$$f_{ji}\left(\tau \right)=\delta \left(t-{\tau }_{ji}\right),j=1,\dots ,5,i=1,2,$$

where $\delta \left(.\right)$ is the Dirac delta function and ${\tau }_{ji}\in \left[0,h_{ji}\right],j=1,\dots ,5,i=1,2$ are constants. When $h_{ji}\to \infty$, we have:

$$\underset{0}{\overset{\infty }{\int }}{f}_{ji}\left(\tau \right)d\tau =1,j=1,\dots ,5,i=1,2.$$

We have

$$F_{ji}=\underset{0}{\overset{\infty }{\int }}\delta \left(t-{\tau }_{ji}\right)e^{-{\mu }_{ji}\tau }d\tau =e^{-{\mu }_{ji}{\tau }_{ji}},j=1,\dots ,5,i=1,2.$$

Moreover,

$$\underset{0}{\overset{\infty }{\int }}\delta \left(t-{\tau }_{ji}\right)e^{-{\mu }_{ji}\tau }{s}_i\left(t-\tau \right)p\left(t-\tau \right)d\tau =e^{-{\mu }_{ji}{\tau }_{ji}}{s}_i\left(t-{\tau }_{ji}\right)p\left(t-{\tau }_{ji}\right),j=1,2,3,i=1,2,$$

$$\begin{array}{cc}\hfill \underset{0}{\overset{\infty }{\int }}\delta \left(t-{\tau }_{4i}\right)e^{-{\mu }_{4i}\tau }{y}_i\left(t-\tau \right)d\tau & =e^{-{\mu }_{4i}{\tau }_{4i}}{y}_i\left(t-{\tau }_{4i}\right),\hfill \\ \hfill \underset{0}{\overset{\infty }{\int }}\delta \left(t-{\tau }_{5i}\right)e^{-{\mu }_{5i}\tau }{u}_i\left(t-\tau \right)d\tau & =e^{-{\mu }_{5i}{\tau }_{5i}}{u}_i\left(t-{\tau }_{5i}\right),i=1,2.\hfill \end{array}$$

Hence, model (10)–(15) becomes:

$$\begin{array}{cc}\hfill {\dot{s}}_1\left(t\right)& ={\rho }_1-{\delta }_1{s}_1\left(t\right)+B{s}_1\left(t\right)\left(1-\frac{s_1\left(t\right)}{s_{max}}\right)-(1-{\epsilon }_1){\overline{\lambda }}_1{s}_1\left(t\right)p\left(t\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\dot{s}}_2\left(t\right)& ={\rho }_2-{\delta }_2{s}_2\left(t\right)-(1-f{\epsilon }_1){\overline{\lambda }}_2{s}_2\left(t\right)p\left(t\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\dot{w}}_1\left(t\right)& =(1-{\epsilon }_1)\overline{\lambda }{}_{11}{e}^{-{\mu }_{11}{\tau }_{11}}{s}_1\left(t-{\tau }_{11}\right)p\left(t-{\tau }_{11}\right)-({\alpha }_1+{\beta }_1)w_1\left(t\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\dot{w}}_2\left(t\right)& =(1-f{\epsilon }_1)\overline{\lambda }{}_{12}{e}^{-{\mu }_{12}{\tau }_{12}}{s}_2\left(t-{\tau }_{12}\right)p\left(t-{\tau }_{12}\right)-({\alpha }_2+{\beta }_2)w_2\left(t\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\dot{y}}_1\left(t\right)& =(1-{\epsilon }_1)\overline{\lambda }{}_{21}{e}^{-{\mu }_{21}{\tau }_{21}}{s}_1\left(t-{\tau }_{21}\right)p\left(t-{\tau }_{21}\right)+{\alpha }_1{w}_1\left(t\right)-{\eta }_1{y}_1\left(t\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\dot{y}}_2\left(t\right)& =(1-f{\epsilon }_1)\overline{\lambda }{}_{22}{e}^{-{\mu }_{22}{\tau }_{22}}{s}_2\left(t-{\tau }_{22}\right)p\left(t-{\tau }_{22}\right)+{\alpha }_2{w}_2\left(t\right)-{\eta }_2{y}_2\left(t\right),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\dot{u}}_1\left(t\right)& =(1-{\epsilon }_1)\overline{\lambda }{}_{31}{e}^{-{\mu }_{31}{\tau }_{31}}{s}_1\left(t-{\tau }_{31}\right)p\left(t-{\tau }_{31}\right)-{\nu }_1{u}_1\left(t\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\dot{u}}_2\left(t\right)& =(1-f{\epsilon }_1)\overline{\lambda }{}_{32}{e}^{-{\mu }_{32}{\tau }_{32}}{s}_2\left(t-{\tau }_{32}\right)p\left(t-{\tau }_{32}\right)-{\nu }_2{u}_2\left(t\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \dot{p}\left(t\right)& =(1-{\epsilon }_2)\overline{N_1}{\eta }_1{e}^{-{\mu }_{41}{\tau }_{41}}{y}_1\left(t-{\tau }_{41}\right)\hfill \\ & +(1-h{\epsilon }_2)\overline{N_2}{\eta }_2{e}^{-{\mu }_{42}{\tau }_{42}}{y}_2\left(t-{\tau }_{42}\right)\hfill \\ & +(1-{\epsilon }_2)\overline{M_1}{\nu }_1{e}^{-{\mu }_{51}{\tau }_{51}}{u}_1\left(t-{\tau }_{51}\right)\hfill \\ & +(1-h{\epsilon }_2)\overline{M_2}{\nu }_2{e}^{-{\mu }_{52}{\tau }_{52}}{u}_2\left(t-{\tau }_{52}\right)-gp\left(t\right)-\mu p\left(t\right)x\left(t\right),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \dot{x}\left(t\right)& =rp\left(t\right)x\left(t\right)-\omega x\left(t\right).\hfill \end{array}$$

(36)

The parameters $R_0$ and $R_1$ for this application are given by:

$$\begin{array}{cc}\hfill R_0& ={\displaystyle \sum _{i=1}^2}\frac{(\left\{N_i{A}_i{e}^{-{\mu }_{4i}{\tau }_{4i}}+M_i{\lambda }_{3i}({\alpha }_i+{\beta }_i)e^{-{\mu }_{3i}{\tau }_{3i}-{\mu }_{5i}{\tau }_{5i}}\right\}}{g({\alpha }_i+{\beta }_i)}{s}_i^0,\hfill \\ \hfill R_1& ={\displaystyle \sum _{i=1}^2}\frac{(\left\{N_i{A}_i{e}^{-{\mu }_{4i}{\tau }_{4i}}+M_i{\lambda }_{3i}({\alpha }_i+{\beta }_i)e^{-{\mu }_{3i}{\tau }_{3i}-{\mu }_{5i}{\tau }_{5i}}\right\}}{g({\alpha }_i+{\beta }_i)}{\overline{s}}_i,\hfill \\ \hfill A_i& ={\alpha }_i{\lambda }_{1i}{e}^{-{\mu }_{1i}{\tau }_{1i}}+({\alpha }_i+{\beta }_i){\lambda }_{2i}{e}^{-{\mu }_{2i}{\tau }_{2i}}.\hfill \end{array}$$

In

Table 1. The data of example (27)–(36).

Parameter	Value	Parameter	Value	Parameter	Value	Parameter	Value
${\rho }_1$	10	${\rho }_2$	$0.03198$	${\eta }_1$	$0.35$	${\eta }_2$	$0.03$
${\delta }_1$	$0.01$	${\delta }_2$	$0.002$	${\nu }_1$	$0.03$	${\nu }_2$	$0.01$
B	$0.0002$	$x_{max}$	1200	g	3	$\mu$	$0.5$
${\alpha }_1$	$0.2$	${\alpha }_2$	$0.01$	$\omega$	$0.1$	${\mu }_e$	1
${\beta }_1$	$0.02$	${\beta }_2$	$0.001$	f	$0.3$	h	$0.3$
${\overline{N}}_1$	30	${\overline{N}}_2$	5	${\overline{M}}_1$	10	${\overline{M}}_2$	2
${\epsilon }_1$	varied	${\epsilon }_2$	varied	r	varied	$\tau$	varied
${\overline{\lambda }}_{i1}$	$0.0000625$	${\overline{\lambda }}_{i2}$	$0.0000625$	-	-	-	-

, we present the values of parameters of system (27)–(36). We let ${\tau }_{ij}=\tau ,{\mu }_{ij}={\mu }_e,i=1,2,\dots ,5$, $j=1,2$. We solve the system of delay differential Equations (27)–(36) with constant delays by using dde23 program in MATLAB (version 7).

4.1. Stability of the Equilibria of the System

In this part of simulation, we choose three different initial conditions:

IC1: 

$(s_1,s_2,w_1,w_2,y_1,y_2,u_1,u_2,p,x)\left(\theta \right)=(450,8,6,0.15,6,0.18,35,0.3,10,6),$

IC2: 

$(s_1,s_2,w_1,w_2,y_1,y_2,u_1,u_2,p,x)\left(\theta \right)=(650,10,4,0.1,5,0.06,25,0.1,7,4),$

IC3: 

$(s_1,s_2,w_1,w_2,y_1,y_2,u_1,u_2,p,x)\left(\theta \right)=(850,12,2,0.05,3,0.02,15,0.02,4,2),$$\theta \in \left[-\tau ,0\right].$

We study three cases by choosing ${\epsilon }_1,{\epsilon }_2,\tau$ and r. In Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, we want to confirm our global stability results given in Theorems 1–3, by showing that any initial points taken from a feasible set the trajectory of the system will tend to one of the three equilibria of the system.

Case (I): Choose ${\epsilon }_1=0.7,{\epsilon }_2=0.7,\tau$$=0.85$ and $r=0.009,$ which gives $R_0=0.2778<1$ and $R_1=0.1156<1$. Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 show that the concentrations of the uninfected CD4${}^{+}$ T cells and macrophages are increasing and reach the values $s_1^0=1003.3$ and $s_2^0=16$, respectively. In other words, concentrations of short-lived, long-lived and latently infected cells as well as HIV particles and B cells converge to zero. This confirms the result of Theorem 1, which is ${\Pi }_0$ is globally asymptotically stable. As a result, the HIV is removed from the plasma.

Case (II): We take ${\epsilon }_1=0.2,{\epsilon }_2=0.3,\tau =0.5$ and $r=0.004$. For these values, $R_1=0.6692<1<R_0=$ $3.1775$. From Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, we can see that for IC1–IC3 the state ${\Pi }_1=(346.147,7.549,6.054,0.31,7.265,0.217,44.397,0.341,12.688,0.0)$ is reached, where the HIV infection is chronic and the antibody immune system is inactive. Hence, Theorem 2 is confirmed.

Case (III): ${\epsilon }_1=0.2,{\epsilon }_2=0.3,\tau$$=0.5$ and $r=0.05$. Then, we calculate $R_0=3.1775>1$ and $R_1=2.4497>1.$ From Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, we see that for IC1–IC3, the state ${\Pi }_2=(773.46,13.594,2.132,0.088,2.559,0.062,15.638,0.097,2,7.405)$ is reached, where the HIV infection is chronic and the antibody immune system is active. Thus, Theorem 3 is confirmed.

We mention that, $R_0$ does not depends on the parameters $\omega$, r and $\mu$. This fact seems to suggest that antibody immune response does not play a role in clearing the HIV particles. From above, we can see that $R_1$ can be increased by increasing the value of r. When we compare the Cases (II) and (III), we can see from the figures that, when the antibody immune response is activated (i.e., $R_1>1$), it reduces the concentrations of HIV free particles and infected (short-lived, long-lived and latent) cells and increases the concentration of uninfected target cells. It means that the antibody immune response can play a significant role in controlling the infection.

Effect of the Drug Efficacy on the Stability of the System

We take ${\epsilon }_1={\epsilon }_2=\epsilon ,$ $\tau =0.5,$ and $r=0.05$. In

Table 2. The values of the steady states, $R_0$ and $R_1$ for model (27)–(36) with different values of $\epsilon$.

$\mathit{\epsilon }$	Equilibria	${\mathit{R}}_0$	${\mathit{R}}_1$
0	${\Pi }_2=(731.427,13.4656,2.5206,0.0928,3.0248,0.065,18.4847,0.1021,2,16.6341)$	$5.6732$	$4.1361$
$0.2$	${\Pi }_2=(773.4604,13.5944,2.1324,0.0881,2.5589,0.0617,15.6376,0.0969,2,9.3198)$	$3.6313$	$2.7995$
$0.5$	${\Pi }_2=(846.3064,13.7922,1.4583,0.0808,1.7499,0.0566,10.6940,0.0889,2,0.5505)$	$1.4191$	$1.1970$
$0.54636$	${\Pi }_1=(941.6053,15.0676,0.5739,0.0339,0.6887,0.0237,4.2086,0.0372,0.7797,0)$	$1.1683$	1
$0.55$	${\Pi }_1=(956.889,15.2913,0.4318,0.0256,0.5182,0.0179,3.1666,0.0282,0.582,0)$	$1.1496$	$0.9852$
$0.58032$	${\Pi }_0=\left(1003.3,16,0,0,0,0,0,0,0,0\right)$	1	$0.8654$
$0.7$	${\Pi }_0=\left(1003.3,16,0,0,0,0,0,0,0,0\right)$	$0.5114$	$0.4602$

, we present the values of $R_0$ and $R_1$ for selected values of $\epsilon$. It is seen that the values of $R_0$ and $R_1$ are decreased as $\epsilon$ is increased. Using the values of the parameters given in Table 1, we obtain the following cases:

(i)

if $0\le \epsilon <0.54636$, then $R_1>1$, ${\Pi }_2$ exists and it is globally asymptotically stable,

(ii)

if $0.54636\le \epsilon <0.58032$, then $R_1\le 1$, ${\Pi }_1$ exists and it is globally asymptotically stable and

(iii)

if $0.58032\le \epsilon \le 1$, then $R_0\le 1$ and ${\Pi }_0$ is globally asymptotically stable.

Thus, the results of Theorems 1–3 and the numerical results are compatible. Therefore, we can say that treatment with sufficient drug efficacy can successfully clear the virus from the plasma.

4.2. Effect of the Time Delay on the Stability of the System

Choose ${\epsilon }_1={\epsilon }_2=0,$ and $r=0.05$. The initial conditions are considered to be $(s_1,s_2,w_1,w_2,y_1,y_2,u_1,u_2,p,x)\left(\theta \right)=(800,14,4,0.1,4,0.05,20,0.015,6,5)$, $\theta \in \left[-\tau ,0\right]$. Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 and

Table 3. The values of steady states, $R_0$ and $R_1$ for model (27)–(36) with different values of $\tau$.

$\mathit{\tau }$	Equilibria	${\mathit{R}}_0$	${\mathit{R}}_1$
$0.1$	${\Pi }_2=(731.43,13.47,3.76,0.14,4.51,0.1,27.58,0.15,2,44.37)$	$11.70$	$8.53$
$0.5$	${\Pi }_2=(731.43,13.47,2.52,0.09,3.03,0.07,18.49,0.1,2,16.63)$	$5.67$	$4.14$
$1.0$	${\Pi }_2=(731.43,13.47,1.53,0.06,1.84,0.04,11.21,0.06,2,2.33)$	$2.39$	$1.74$
$1.2$	${\Pi }_1=(786.28,14.03,1,0.04,1.2,0.03,7.33,0.04,1.49,0.00)$	$1.72$	$1.25$
$1.3$	${\Pi }_1=(960.39,15.63,0.18,0.01,0.22,0.00,1.32,0.01,0.24,0.00)$	$1.1$	$1.46$
$1.5$	${\Pi }_0=\left(1003.3,16,0,0,0,0,0,0,0,0\right)$	$1.06$	$0.77$
2	${\Pi }_0=\left(1003.3,16,0,0,0,0,0,0,0,0\right)$	$0.50$	$0.37$

show the effect of the time delay parameter $\tau$ on the stability of ${\Pi }_0$, ${\Pi }_1$ and ${\Pi }_2$. It can be seen that, as $\tau$ is increased, the concentration of the uninfected target cells is increased, while the concentrations of short-lived, long-lived and latently infected cells, free HIV particles and B cells are decreased. From Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20, we can see that, in the case of smaller values of $\tau$, the trajectory of system will converge to ${\Pi }_2$. When the value of $\tau$ is increased, the trajectory will converge to ${\Pi }_1$ and finally approach ${\Pi }_0$. From a biological point of view, the intracellular time delay plays a similar role as the anti-HIV drugs in eliminating the virus. We observe that sufficiently large delay suppresses viral replication and clears the virus from the body. This gives us some suggestions on new drugs to prolong the increase in the intracellular delay period.

5. Discussion

All of the HIV infection mathematical models with three categories of infected cells, short-lived, long-lived and latent presented in the literature have studied the HIV infection and production in one class of target cells, CD4${}^{+}$ T cells. However, it has been reported in several papers that HIV can infect both CD4${}^{+}$ T cells and macrophages. In this paper, we have proposed an HIV infection model with three categories of infected cells and two classes of target cells, CD4${}^{+}$ T cells and macrophages. We have incorporated multiple distributed time delays to characterize the time between an HIV contacts an uninfected target cell and the creation of mature HIV particles. The effect of antibody immune response has been modeled. The production and removal rates of all compartments are represented by general nonlinear functions. The incidence rate of infection is also given by a general nonlinear function. The model can be seen as a generalization of several HIV dynamics models presented in the literature. We have shown that the solutions of the model are nonnegative and ultimately bounded, which ensures the well-posedness of the model. We have derived two threshold numbers $R_0$ (the basic reproduction number) and antibody immune response activation number $R_1$, which determine the stability of the three equilibria of the model. We have investigated the global stability of the equilibria of the model by using Lyapunov method and LaSalle’s invariance principle. We have proven that (i) if $R_0\le 1$, then the infection-free equilibrium ${\Pi }_0$ is globally asymptotically stable and the HIV is predicted to be completely cleared from the HIV infected patients, (ii) if the antibody-inactivated infection equilibrium ${\Pi }_1$ exists then it is globally asymptotically stable and a chronic HIV infection with inactive antibody immunity is attained, and (iii) if $R_1>1$, then the antibody-activated infection equilibrium ${\Pi }_2$ is globally asymptotically stable and a chronic HIV infection with active antibody immunity is attained. We have conducted numerical simulations and have shown that both the theoretical and numerical results are consistent.

Our analysis extends the results presented in [27], where the global stability was analyzed for a model with one target cell population. When we consider the HIV dynamics with only one class of target cells, CD4${}^{+}$ T cells, then model (10)–(15) leads to the following model:

$$\begin{array}{cc}\hfill {\dot{s}}_1\left(t\right)& ={\rho }_1-{\delta }_1{s}_1\left(t\right)+B{s}_1\left(t\right)\left(1-\frac{s_1\left(t\right)}{s_{max}}\right)-(1-{\epsilon }_1){\overline{\lambda }}_1{s}_1\left(t\right)p\left(t\right),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\dot{w}}_1\left(t\right)& =(1-{\epsilon }_1)\overline{\lambda }{}_{11}{e}^{-{\mu }_{11}{\tau }_{11}}{s}_1\left(t-{\tau }_{11}\right)p\left(t-{\tau }_{11}\right)-({\alpha }_1+{\beta }_1)w_1\left(t\right),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\dot{y}}_1\left(t\right)& =(1-{\epsilon }_1)\overline{\lambda }{}_{21}{e}^{-{\mu }_{21}{\tau }_{21}}{s}_1\left(t-{\tau }_{21}\right)p\left(t-{\tau }_{21}\right)+{\alpha }_1{w}_1\left(t\right)-{\eta }_1{y}_1\left(t\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\dot{u}}_1\left(t\right)& =(1-{\epsilon }_1)\overline{\lambda }{}_{31}{e}^{-{\mu }_{31}{\tau }_{31}}{s}_1\left(t-{\tau }_{31}\right)p\left(t-{\tau }_{31}\right)-{\nu }_1{u}_1\left(t\right),\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill \dot{p}\left(t\right)& =(1-{\epsilon }_2)\overline{N_1}{\eta }_1{e}^{-{\mu }_{41}{\tau }_{41}}{y}_1\left(t-{\tau }_{41}\right)+(1-{\epsilon }_2)\overline{M_1}{\nu }_1{e}^{-{\mu }_{51}{\tau }_{51}}{u}_1\left(t-{\tau }_{51}\right)\hfill \\ & -gp\left(t\right)-\mu p\left(t\right)x\left(t\right),\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \dot{x}\left(t\right)& =rp\left(t\right)x\left(t\right)-\omega x\left(t\right).\hfill \end{array}$$

(42)

Let us define the overall HAART effect as ${\epsilon }_e={\epsilon }_1+{\epsilon }_2-{\epsilon }_1{\epsilon }_2$ [39]. If ${\epsilon }_e=0$,then the HAART has no effect, if ${\epsilon }_e=1$, the HIV-1 growth is completely halted. Consequently, the basic reproduction number for system (37)–(42) is given by

$$\begin{array}{cc}\hfill R_0^C\left({\epsilon }_e\right)& =\frac{(1-{\epsilon }_e)\left[{\overline{N}}_1{\overline{A}}_1{e}^{-{\mu }_{41}{\tau }_{41}}+{\overline{M}}_1{\overline{\lambda }}_{31}({\alpha }_1+{\beta }_1)e^{-{\mu }_{31}{\tau }_{31}-{\mu }_{51}{\tau }_{51}}\right]}{g({\alpha }_1+{\beta }_1)}{s}_1^0,\hfill \\ \hfill {\overline{A}}_1& ={\alpha }_1{\overline{\lambda }}_{11}{e}^{-{\mu }_{11}{\tau }_{11}}+({\alpha }_1+{\beta }_1){\overline{\lambda }}_{21}{e}^{-{\mu }_{21}{\tau }_{21}}.\hfill \end{array}$$

For comparison purposes, we assume that $h=f=1$; then, the basic reproduction number for system (27)–(36) can be written as: $R_0=R_0^{CM}\left({\epsilon }_e\right)=R_0^C\left({\epsilon }_e\right)+R_0^M\left({\epsilon }_e\right)$ where:

$$\begin{array}{cc}\hfill R_0^M\left({\epsilon }_e\right)& =\frac{(1-{\epsilon }_e)\left[{\overline{N}}_2{\overline{A}}_2{e}^{-{\mu }_{42}{\tau }_{42}}+{\overline{M}}_2{\overline{\lambda }}_{32}({\alpha }_2+{\beta }_2)e^{-{\mu }_{32}{\tau }_{32}-{\mu }_{52}{\tau }_{52}}\right]}{g({\alpha }_2+{\beta }_2)}{s}_2^0,\hfill \\ \hfill {\overline{A}}_2& ={\alpha }_2{\overline{\lambda }}_{12}{e}^{-{\mu }_{12}{\tau }_{12}}+({\alpha }_2+{\beta }_2){\overline{\lambda }}_{22}{e}^{-{\mu }_{22}{\tau }_{22}},\hfill \end{array}$$

where $R_0^M\left({\epsilon }_e\right)$ is the basic reproduction number of a model that describes the HIV dynamics with only macrophages and neglecting the CD4${}^{+}$ T cells. For system (37)–(42), one can determine drug efficacy ${\epsilon }_e^C$ such that $R_0^C\left({\epsilon }_e^C\right)=1$ as:

$${\epsilon }_e^C=max\left\{\frac{R_0^C\left(0\right)-1}{R_0^C\left(0\right)},0\right\}.$$

Therefore, if ${\epsilon }_e^C\le {\epsilon }_e\le 1$, then $R_0^C\left({\epsilon }_e^C\right)\le 1$. For system (27)–(36), one can also determine the drug efficacy ${\epsilon }_e^{CM}$ such that $R_0^{CM}\left({\epsilon }_e^{CM}\right)=1$ as:

$${\epsilon }_e^{CM}=max\left\{\frac{R_0^{CM}\left(0\right)-1}{R_0^{CM}\left(0\right)},0\right\}.$$

Therefore, if ${\epsilon }_e^{CM}\le {\epsilon }_e\le 1$, then $R_0^{CM}\left({\epsilon }_e^{CM}\right)\le 1$.

Assume that $R_0^C\left(0\right)>1$, then ${\Pi }_0$ of system (37)–(42) is unstable in the absence of treatment. Since $R_0^C\left(0\right)<R_0^{CM}\left(0\right),$ then ${\epsilon }_e^C<{\epsilon }_e^{CM}$. Therefore, if we apply drugs with ${\epsilon }_e$ such that ${\epsilon }_e^C\le {\epsilon }_e<{\epsilon }_e^{CM}$, this guarantees that $R_0^C\left({\epsilon }_e\right)\le 1$ and the system (37)–(42) can be stabilized around ${\Pi }_0$; however, $R_0^{CM}\left({\epsilon }_e\right)>1$ and then ${\Pi }_0$ of (27)–(36) is unstable. Therefore, more accurate drug efficacy ${\epsilon }_e$ is determined when using the model with two classes of target cells. This shows the importance of considering the effect of the macrophages in the HIV dynamics.