Stability of a delay-distributed HIV infection model with silent infected cell-to-cell spread and CTL-mediated immunity

Abstract

In this paper, we formulate a mathematical model to investigate a within-host HIV dynamics under the effect of cytotoxic T lymphocytes immune response. The model incorporates two modes of transmission, virus-to-cell (VTC) and cell-to-cell (CTC). The CTC infection is due to the contact of healthy CD$4^{+}$ T cells with (i) silent HIV-infected cells, and (ii) active HIV-infected cells. The model integrates three types of distributed time delays. We show that the model is well posed and it has three equilibria. The existence and stability of equilibria are governed by two threshold parameters. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle’s invariance principle. We have presented numerical simulations to illustrate the theoretical results. We have studied the effects of CTC transmission and time delays on the dynamical behavior of the system. We have shown that inclusion of time delay can significantly increase the concentration of healthy CD4$^{+}$ T cells and reduce the concentrations of infected cells and free HIV particles. While the inclusion of CTC transmission decreases the concentration of healthy CD4$^{+}$ T cells and increases the concentrations of infected cells and free HIV particles.

1 Introduction

Acquired immunodeficiency syndrome (AIDS) is one of the fatal human diseases which is caused by human immunodeficiency virus (HIV). HIV is a retrovirus that mainly infects the healthy (uninfected) CD$4^{+}$ T cells and other immune cells such as macrophages and dendritic cells. For long period up to 10 years [1], the HIV infection can be controlled by two main components (i) cytotoxic T lymphocytes (CTLs) that kill the HIV-infected cells, and (ii) B cells that generate specific antibodies which in turn neutralize the viruses. However, during this period of time the concentration of healthy CD$4^{+}$ T cells declines. When the concentration of the CD$4^{+}$ T cells reaches below 200 cells/mm$^{3}$, the patient is said to have progressed to AIDS. During the last decades, mathematical modeling of a within-host HIV infection has witnessed a significant development. Moreover, mathematical analysis of the HIV dynamics models has also become one of the most important and fundamental methods for understanding the within-host HIV dynamics [2,3,4,5,6,7,8]. Nowak and Bangham  [6] have introduced an HIV infection model which describes the interaction between healthy CD$4^{+}$ T cells (W), active HIV-infected cells (M), free HIV particles (N) and HIV-specific CTLs (P):

$$\begin{aligned} \left\{ \begin{array}[c]{l} \dot{W}(t)=\rho -\alpha W(t)-\eta _{1}W(t)N(t),\\ \dot{M}(t)=\eta _{1}W(t)N(t)-aM(t)-\mu P(t)M(t),\\ \dot{N}(t)=bM(t)-\varepsilon N(t),\\ \dot{P}(t)=\sigma P(t)M(t)-\pi P(t). \end{array} \right. \end{aligned}$$

(1)

The healthy CD$4^{+}$ T cells are generated at specific constant rate $\rho$ and die at rate $\alpha W$. The term $\eta _{1}WN$ refers to the rate at which new infectious appears by virus-to-cell (VTC) transmission between free HIV particles and healthy CD$4^{+}$ T cells. The active HIV-infected cells die at rate aM. The term $\mu PM$ is the killing rate of active HIV-infected cells due to their HIV-specific CTL-mediated immunity. The free HIV particles are generated at rate bM and cleared from the plasma at rate $\varepsilon N$. The proliferation rate of the effective HIV-specific CTLs is given by $\sigma PM$. The term $\pi P$ represents the decay rate of the CTLs. HIV infection models with CTL-mediated immune response have been investigated in many papers (see, e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]).

Model (1) is formulated on the assumption that HIV can only spread by VTC transmission. However, several works have reported that there is another mode of transmission called cell-to-cell (CTC) where HIV can be transmitted directly from an infected cell to a healthy CD$4^{+}$ T cell through the formation of virological synapses (see, e.g., [17,18,19,20]). Sourisseau et al.  [21] have shown that CTC transmission plays an efficient role in the HIV replication. Sigal et al. [22] have demonstrated the importance of CTC transmission in the HIV infection process during the antiviral treatment. Iwami et al.  [19] have shown that more than 50% of HIV infections are due to CTC transmission. The effects of both VTC and CTC transmissions on the virus dynamics have been addressed in several works (see, e.g., [23,24,25,26,27,28,29]), although the CTL-mediated immunity has not been considered. Virus dynamics models with both VTC and CTC transmissions and CTL-mediated immunity have been investigated in [30,31,32]. Wang et al.  [31] have proposed the following virus dynamics model with CTL-mediated immunity, both VTC and CTC transmissions and a distributed time delay for production of active infected cells:

$$\begin{aligned} \left\{ \begin{array} [c]{l} \dot{W}(t)=\rho -\alpha W(t)-\eta _{1}W(t)N(t)-\eta _{2}W(t)U(t),\\ \dot{M}(t)=\int \limits _{0}^{\infty }\Lambda (\varphi )e^{-\hslash \varphi }W(t-\varphi )\left[ \eta _{1}N(t-\varphi )+\eta _{2}U(t-\varphi )\right] \hbox {d}\varphi -aM(t)-\mu P(t)M(t),\\ \dot{N}(t)=bM(t)-\varepsilon N(t),\\ \dot{P}(t)=\sigma P(t)M(t)-\pi P(t). \end{array} \right. \nonumber \\ \end{aligned}$$

(2)

The healthy CD$4^{+}$ T cells are contacted with active CD$4^{+}$ T infected cells and become infected due to CTC transmission at rate $\eta _{2}WU$. The factor $\Lambda (\varphi )e^{-\hslash \varphi }$ represents the probability that healthy CD$4^{+}$ T cells contacted by HIV particles or active HIV-infected cells at time $t-\varphi$ survived $\varphi$ time units and become active infected at time t. The delay parameter $\varphi$ is random taken from a probability distribution function $\Lambda (\varphi )$ over the time interval $(0,\infty )$.

It is known that current anti-retroviral drugs can suppress HIV replication to a low level but cannot enucleate HIV from the body. One of the main reasons of this fact is the presence of silent (latent) CD$4^{+}$ T infected cells where the HIV provirus can reside [33, 34]. Silent HIV-infected cells live long, but they can be activated to produce new HIV particles. Silent HIV-infected cells have been included in the virus dynamics models with both VTC and CTC transmissions in [35,36,37,38,39,40]. In a very recent work [41], it has been shown that both silent and active HIV-infected cells can infect the healthy CD$4^{+}$ T cells through CTC mechanism. In the literature, all viral infection models with CTC transmission and silent infected cells have assumed that the CTC transmission only occurs due to the active infected cells.

In the present paper, we extend model (2) by considering in the dynamics (i) both silent and active HIV-infected cells, (ii) three types of distributed time delays and (iii) three types of infection modes, VTC, silent HIV-infected CTC and active HIV-infected CTC transmissions. The well-posedness of the model is investigated. We deduce two threshold parameters which govern the existence and stability of the three equilibria. Global stability of all equilibria is proven by formulating Lyapunov functions and utilizing LaSalle’s invariance principle. We perform some numerical simulations to illustrate the strength of our theoretical results.

2 Model formulation

We formulate a distributed delay HIV infection model with CTL-mediated immunity. We assume that the HIV virions can replicate by two mechanisms VTC and CTC transmissions. The CTC infection has two sources, (i) the contact between healthy CD$4^{+}$ T cells and silent HIV-infected cells, and (ii) the contact between healthy CD$4^{+}$ T cells and active HIV-infected cells. Under these assumptions, we propose the following model:

$$\begin{aligned} \left\{ \begin{array} [c]{l} \dot{W}(t)=\rho -\alpha W(t)-\eta _{1}W(t)N(t)-\eta _{2}W(t)U(t)-\eta _{3}W(t)M(t),\\ \dot{U}(t)=\int \limits _{0}^{\kappa _{1}}\Lambda _{1}(\varphi )e^{-\hslash _{1}\varphi }W(t-\varphi )\left[ \eta _{1}N(t-\varphi )+\eta _{2}U(t-\varphi )+\eta _{3}M(t-\varphi )\right] \hbox {d}\varphi -\left( \lambda +\gamma \right) U(t),\\ \dot{M}(t)=\lambda \int \limits _{0}^{\kappa _{2}}\Lambda _{2}(\varphi )e^{-\hslash _{2}\varphi }U(t-\varphi )\hbox {d}\varphi -aM(t)-\mu P(t)M(t),\\ \dot{N}(t)=b\int \limits _{0}^{\kappa _{3}}\Lambda _{3}(\varphi )e^{-\hslash _{3}\varphi }M(t-\varphi )\hbox {d}\varphi -\varepsilon N(t),\\ \dot{P}(t)=\sigma P(t)M(t)-\pi P(t), \end{array} \right. \end{aligned}$$

(3)

where U(t) is the concentration of silent HIV-infected cells. The healthy CD$4^{+}$ T cells are contacted with silent HIV-infected cells and become infected due to CTC transmission at rate $\eta _{3}WM$. The term $\lambda U$ is the rate of silent HIV-infected cells that become active HIV-infected cells. $\gamma U$ represents the death rate of the silent HIV-infected cells. The factor $\Lambda _{1}(\varphi )e^{-\hslash _{1}\varphi }$ represents the probability that healthy CD$4^{+}$ T cells contacted by HIV particles or HIV-infected cells at time $t-\varphi$ survived $\varphi$ time units and become silent infected at time t. The term $\Lambda _{2} (\varphi )e^{-\hslash _{2}\varphi }$ is the probability that silent HIV-infected cells survived $\varphi$ time units before transmitted to be active at time t. Moreover, the factor $\Lambda _{3}(\varphi )e^{-\hslash _{3}\varphi }$ demonstrates the probability of new immature HIV particles at time $t-\varphi$ lost $\varphi$ time units and become mature at time t. Here, $\hslash _{i},$ $i=1,2,3$ are positive constants. The delay parameter $\varphi$ is random taken from a probability distribution function $\Lambda _{i}(\varphi )$ over the time interval $\left[ 0,\kappa _{i}\right] ,$ $i=1,2,3$, where $\kappa _{i}$ is the limit superior of this delay period. The function $\Lambda _{i} (\varphi ),$ $i=1,2,3$ satisfies $\Lambda _{i}(\varphi )>0$ and

$$\begin{aligned} \int \limits _{0}^{\kappa _{i}}\Lambda _{i}(\varphi )\hbox {d}\varphi =1\text { and } \int \limits _{0}^{\kappa _{i}}\Lambda _{i}(\varphi )e^{-u\varphi }\hbox {d}\varphi <\infty , \end{aligned}$$

where $u>0$. Let us denote

$$\begin{aligned} \mathcal {\bar{H}}_{i}(\varphi )=\Lambda _{i}(\varphi )e^{-\hslash _{i}\varphi }\text { and }\mathcal {H}_{i}=\int \limits _{0}^{\kappa _{i}}\mathcal {\bar{H}} _{i}(\varphi )\hbox {d}\varphi , \end{aligned}$$

where $i=1,2,3.$ Thus, $0<\mathcal {H}_{i}\le 1,$ $i=1,2,3$. The initial conditions of system (3) are given by:

$$\begin{aligned} W(x)&=\epsilon _{1}(x), U(x)=\epsilon _{2}(x), M(x)=\epsilon _{3}(x), N(x)=\epsilon _{4}(x), P(x)=\epsilon _{5}(x),\nonumber \\ \epsilon _{j}(x)&\ge 0,{\ }x\in [-\kappa ,0],\ j=1,2,\ldots ,5, \, \, \kappa =\max \{\kappa _{1},\kappa _{2},\kappa _{3}\}, \end{aligned}$$

(4)

where $\epsilon _{j}(x)\in \mathcal {C}([-\kappa ,0],\mathbb {R}_{\ge 0})$, $j=1,2,\ldots ,5$ and $\mathcal {C}=\mathcal {C}([-\kappa ,0],\mathbb {R}_{\ge 0})$ is the Banach space of continuous functions with norm $\Vert \epsilon _{j} \Vert =\sup \limits _{-\kappa \le m\le 0}\vert \epsilon _{j}(m)\vert$ for $\epsilon _{j}\in \mathcal {C}$. Therefore, system (3) with initial conditions (4) has a unique solution by using the standard theory of functional differential equations [42, 43]. All remaining variables and parameters have the same biological meaning as explained in Sect. 1. Table 1 summarizes all parameters and their definitions.

Table 1 Parameters of model (3) and their interpretations

3 Well-posedness of solutions

Proposition 1

All solutions of system (3) with initial conditions (4) are nonnegative and ultimately bounded.

Proof

First, we show the nonnegativity of solutions. From the first equation of system (3), we have $\dot{W}\mid _{W=0}=\rho >0,$ then $W(t)>0$ for all $t\ge 0$. Moreover, the rest equations of system (3) give us the following:

$$\begin{aligned} U(t)&=\epsilon _{2}(0)e^{-\left( \lambda +\gamma \right) t}\\&\quad +\,\int \limits _{0}^{t}e^{-\left( \lambda +\gamma \right) (t-\varkappa )}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )W(\varkappa -\varphi )\left[ \eta _{1}N(\varkappa -\varphi )+\eta _{2} U(\varkappa -\varphi )\right. \\&\quad \quad \left. +\,\eta _{3}M(\varkappa -\varphi )\right] \hbox {d}\varphi \hbox {d}\varkappa \ge 0, \\ M(t)&=\epsilon _{3}(0)e^{-\int _{0}^{t}\left( a+\mu P(y)\right) \hbox {d}y}+\lambda \int \limits _{0}^{t}e^{-\int _{\varkappa }^{t}\left( a+\mu P(y)\right) \hbox {d}y}\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2} (\varphi )U(\varkappa -\varphi )\hbox {d}\varphi \hbox {d}\varkappa \ge 0,\\ N(t)&=\epsilon _{4}(0)e^{-\varepsilon t}+b\int \limits _{0}^{t} e^{-\varepsilon (t-\varkappa )}\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}} _{3}(\varphi )M(\varkappa -\varphi )\hbox {d}\varphi \hbox {d}\varkappa \ge 0,\\ P(t)&=\epsilon _{5}(0)e^{-\int _{0}^{t}\left( \pi -\sigma M(y)\right) \hbox {d}y}\ge 0, \end{aligned}$$

for all $t\in \left[ 0,\kappa \right]$. Thus, by a recursive argument, we get $W(t),U(t),M(t),N(t),P(t)\ge 0$ for all $t\ge 0$. Hence, the solutions of system (3) satisfy $(W(t),U(t),M(t),N(t),P(t))\in \mathbb {R}_{\ge 0} ^{5}$ for all $t\ge 0$.

Now, we investigate the boundedness. From the first equation of system (3), we obtain $\lim \sup _{t\rightarrow \infty }W(t)\le \frac{\rho }{\alpha }$. Let $\Psi _{1}(t)=\int \nolimits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1} (\varphi )W(t-\varphi )\hbox {d}\varphi +U(t),$ then

$$\begin{aligned} \dot{\Psi }_{1}(t)&=\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}} _{1}(\varphi )\left[ \rho -\alpha W(t-\varphi )\right] \hbox {d}\varphi -\left( \lambda +\gamma \right) U(t)\\&=\rho \mathcal {H}_{1}-\alpha \int \limits _{0} ^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )W(t-\varphi )\hbox {d}\varphi -\left( \lambda +\gamma \right) U(t)\\&\le \,\rho -\phi _{1}\left[ \int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}} _{1}(\varphi )W(t-\varphi )\hbox {d}\varphi +U(t)\right] =\rho -\phi _{1}\Psi _{1}(t), \end{aligned}$$

where $\phi _{1}=\min \{\alpha ,\lambda +\gamma \}$. It follows that, $\lim \sup _{t\rightarrow \infty }\Psi _{1}(t)\le \Omega _{1}$, where $\Omega _{1} =\frac{\rho }{\phi _{1}}$. Since $\int \nolimits _{0}^{\kappa _{1}}\mathcal {\bar{H} }_{1}(\varphi )W(t-\varphi )\hbox {d}\varphi$ and U(t) are nonnegative, then $\lim \sup _{t\rightarrow \infty }U(t)\le \Omega _{1}$. Moreover, we let $\Psi _{2}(t)=M(t)+\frac{\mu }{\sigma }P(t)$, then

$$\begin{aligned} \dot{\Psi }_{2}(t)&=\lambda \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H} }_{2}(\varphi )U(t-\varphi )\hbox {d}\varphi -aM(t)-\frac{\mu \pi }{\sigma }P(t)\le \lambda \mathcal {H}_{2}\Omega _{1}-aM(t)-\frac{\mu \pi }{\sigma }P(t)\\&\le \lambda \Omega _{1}-aM(t)-\frac{\mu \pi }{\sigma }P(t)\le \lambda \Omega _{1}-\phi _{2}\left( M(t)+\frac{\mu }{\sigma }P(t)\right) =\lambda \Omega _{1}-\phi _{2}\Psi _{2}(t), \end{aligned}$$

where $\phi _{2}=\min \{a,\pi \}$. It follows that, $\lim \sup _{t\rightarrow \infty }\Psi _{2}(t)\le \Omega _{2}$, where $\Omega _{2}=\dfrac{\lambda \Omega _{1} }{\phi _{2}}$. Since $M(t)\ge 0$ and $P(t)\ge 0$, then $\lim \sup _{t\rightarrow \infty }M(t)\le \Omega _{2}$ and $\lim \sup _{t\rightarrow \infty }P(t)\le \Omega _{3}$, where $\Omega _{3}=\frac{\sigma }{\mu }\Omega _{2}$. Finally, from the fourth equation of system (3), we have

$$\begin{aligned} \dot{N}(t)=b\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )M(t-\varphi )\hbox {d}\varphi -\varepsilon N(t)\le b\mathcal {H}_{3}\Omega _{2}-\varepsilon N(t)\le b\Omega _{2}-\varepsilon N(t). \end{aligned}$$

Thus, $\lim \sup _{t\rightarrow \infty }N(t)\le \Omega _{4}$, where $\Omega _{4}=\dfrac{b\Omega _{2}}{\varepsilon }$. $\square$

According to Proposition 1, we can show that the region

$$\begin{aligned} \Xi =\left. \left\{ (W,U,M,N,P)\in \mathcal {C}_{\ge 0}^{5}:\left\| W\right\| \le \Omega _{1},\left\| U\right\| \le \Omega _{1},\left\| M\right\| \le \Omega _{2},\left\| P\right\| \le \Omega _{3},\left\| N\right\| \le \Omega _{4}\right\} \right. \end{aligned}$$

is positively invariant with respect to system (3). $\square$

4 Equilibria

Let (W, U, M, N, P) be any equilibrium of system (3) satisfying the following equations:

$$\begin{aligned} 0&=\rho -\alpha W-\eta _{1}WN-\eta _{2}WU-\eta _{3}WM, \end{aligned}$$

(5)

$$\begin{aligned} 0&=\mathcal {H}_{1}\left( \eta _{1}WN+\eta _{2}WU+\eta _{3}WM\right) -\left( \lambda +\gamma \right) U, \end{aligned}$$

(6)

$$\begin{aligned} 0&=\lambda \mathcal {H}_{2}U-aM-\mu PM, \end{aligned}$$

(7)

$$\begin{aligned} 0&=b\mathcal {H}_{3}M-\varepsilon N, \end{aligned}$$

(8)

$$\begin{aligned} 0&=\left( \sigma M-\pi \right) P. \end{aligned}$$

(9)

The straightforward calculation finds that system (3) admits five equilibria.

(i) It is obvious that system (3) has an infection-free equilibrium, Đ$_{0}=(W_{0},0,0,0,0)$, where $W_{0}=\rho /\alpha$. This case describes the situation of healthy state where the HIV infection is absent.

(ii) When $P_{1}=0$, we have a chronic HIV infection equilibrium with inactive CTL-mediated immune response, Đ $_{1}=(W_{1},U_{1},M_{1},N_{1},0),$ where

$$\begin{aligned} W_{1}&=\dfrac{a\varepsilon \left( \gamma +\lambda \right) }{\mathcal {H} _{1}\left[ a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right] },\\ U_{1}&=\frac{a\varepsilon \alpha }{a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }\left[ \frac{W_{0}\mathcal {H}_{1}\left\{ a\varepsilon \eta _{2} +\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right\} }{a\varepsilon \left( \gamma +\lambda \right) }-1\right] ,\\ M_{1}&=\frac{\varepsilon \alpha \lambda \mathcal {H}_{2}}{a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }\left[ \frac{W_{0}\mathcal {H}_{1}\left\{ a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right\} }{a\varepsilon \left( \gamma +\lambda \right) }-1\right] ,\\ N_{1}&=\frac{\alpha b\lambda \mathcal {H}_{2}\mathcal {H}_{3}}{a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }\left[ \frac{W_{0}\mathcal {H}_{1}\left\{ a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right\} }{a\varepsilon \left( \gamma +\lambda \right) }-1\right] . \end{aligned}$$

Therefore, Đ$_{1}$ exists when

$$\begin{aligned} \frac{W_{0}\mathcal {H}_{1}\left[ a\varepsilon \eta _{2}+\lambda \mathcal {H} _{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right] }{a\varepsilon \left( \gamma +\lambda \right) }>1. \end{aligned}$$

At the equilibrium Đ$_{1}$, the chronic HIV infection persists while the CTL-mediated immune response is unstimulated. In order to state the threshold dynamics of infection-free equilibrium, it is necessary to define the basic HIV reproduction number $\mathfrak {R}_{0}$ of the model. If the antiviral drugs are taken into account in that HIV dynamics model, then $\mathfrak {R}_{0}$ can be used to determine the minimum drug efficacy which stabilizes the system around the infection-free equilibrium and clears the viruses from the body. The basic HIV reproduction number of model (3) can be calculated by different methods such as (i) the next-generation matrix method of van den Driessche and Watmough [44], (ii) local stability of the infection-free equilibrium, and (iii) the existence of the chronic HIV infection equilibrium with inactive CTL-mediated immune response. In the present paper, we derive $\mathfrak {R}_{0}$ by method (iii) as:

$$\begin{aligned} \mathfrak {R}_{0}=\frac{W_{0}\mathcal {H}_{1}\left[ a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right] }{a\varepsilon \left( \gamma +\lambda \right) }=\mathfrak {R}_{01}+\mathfrak {R}_{02} +\mathfrak {R}_{03}, \end{aligned}$$

where

$$\begin{aligned} \mathfrak {R}_{01}=\frac{W_{0}\lambda b\eta _{1}\mathcal {H}_{1}\mathcal {H}_{2} \mathcal {H}_{3}}{a\varepsilon \left( \gamma +\lambda \right) },\,\, \mathfrak {R}_{02}=\frac{W_{0}\eta _{2}\mathcal {H}_{1}}{\gamma +\lambda },\,\, \mathfrak {R}_{03}=\frac{W_{0}\lambda \eta _{3}\mathcal {H}_{1}\mathcal {H}_{2} }{a\left( \gamma +\lambda \right) }. \end{aligned}$$

The parameter $\mathfrak {R}_{0}$ determines whether or not the infection will be chronic in the absence of the immune response. In fact, $\mathfrak {R}_{01}$ determines the average number of secondary HIV-infected cells caused by an existing free HIV particle due to VTC transmission, while $\mathfrak {R}_{02}$ and $\mathfrak {R}_{03}$ determine the average numbers of secondary HIV-infected cells caused by living silent and active HIV-infected cell, respectively, due to CTC transmission. In terms of $\mathfrak {R}_{0}$, we can write

$$\begin{aligned} W_{1}&=\dfrac{W_{0}}{\mathfrak {R}_{0}}, U_{1}=\frac{a\varepsilon \alpha }{a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) }\left( \mathfrak {R}_{0}-1\right) ,\nonumber \\ M_{1}&=\frac{\varepsilon \alpha \lambda \mathcal {H}_{2}}{a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }\left( \mathfrak {R}_{0}-1\right) , N_{1}=\frac{\alpha b\lambda \mathcal {H}_{2}\mathcal {H}_{3}}{a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }\left( \mathfrak {R}_{0}-1\right) . \end{aligned}$$

(10)

(iii) When $P\ne 0$, we have $M_{2}=\dfrac{\pi }{\sigma }$. In this case, we consider a chronic HIV infection equilibrium with active CTL-mediated immune response, Đ $_{2}=(W_{2},U_{2},M_{2},N_{2},P_{2})$, where

$$\begin{aligned} W_{2}=\frac{\rho \varepsilon \sigma }{b\pi \eta _{1}\mathcal {H}_{3}+\varepsilon \left( \pi \eta _{3}+\alpha \sigma +\sigma \eta _{2}U_{2}\right) }, N_{2}=\frac{b\pi \mathcal {H}_{3}}{\varepsilon \sigma }, P_{2}=\frac{a}{\mu }\left( \frac{\lambda \sigma \mathcal {H}_{2}U_{2}}{a\pi }-1\right) , \end{aligned}$$

(11)

and $U_{2}$ satisfies the quadratic equation

$$\begin{aligned} \tilde{A}U_{2}^{2}+\tilde{B}U_{2}+\tilde{C}=0, \end{aligned}$$

(12)

where

$$\begin{aligned} \tilde{A}&=\varepsilon \eta _{2}{\sigma }\left( \gamma +\lambda \right) , \nonumber \\ \tilde{B}&=\pi \left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \left( \gamma +\lambda \right) +\varepsilon {\sigma }\left[ \alpha \left( \gamma +\lambda \right) -\eta _{2}\rho \mathcal {H}_{1}\right] \, , \, \nonumber \\ \tilde{C}&=-\pi \rho \mathcal {H}_{1}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) . \end{aligned}$$

(13)

Since $\tilde{A}>0$ and $\tilde{C}<0$, then $\tilde{B}^{2}-4\tilde{A} \tilde{C}>0$ and there are two distinct real roots of Eq. (12). The positive root is given by

$$\begin{aligned} U_{2}=\frac{-\tilde{B}+\sqrt{\tilde{B}^{2}-4\tilde{A}\tilde{C}}}{2\tilde{A}}. \end{aligned}$$

It follows that $W_{2}>0$ and $P_{2}>0$ only when $\dfrac{\lambda \sigma \mathcal {H}_{2}U_{2}}{a\pi }>1$. We define the HIV specific CTL-mediated immunity reproduction number as follows:

$$\begin{aligned} \mathfrak {R}_{1}=\frac{\lambda \sigma \mathcal {H}_{2}U_{2}}{a\pi }. \end{aligned}$$

Thus, $P_{2}=\dfrac{a}{\mu }\left( \mathfrak {R}_{1}-1\right)$. Therefore, Đ$_{2}$ exists when $\mathfrak {R}_{1}>1$. The parameter $\mathfrak {R}_{1}$ determines whether or not the HIV-specific CTL-mediated immune response is stimulated.

The threshold parameters are given as follows:

$$\begin{aligned} \mathfrak {R}_{0}=\frac{W_{0}\mathcal {H}_{1}\left[ a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right] }{a\varepsilon \left( \gamma +\lambda \right) }, \,\,\mathfrak {R}_{1}=\frac{\lambda \sigma \mathcal {H}_{2}U_{2}}{a\pi }. \end{aligned}$$

$\square$

5 Global stability analysis

In this section, we prove the global asymptotic stability of all equilibria by constructing Lyapunov functional following the method presented in [45,46,47,48,49]. Define $\digamma (x)=x-1-\ln x$. Denote $(W,U,M,N,P)=(W(t),U(t),M(t),N(t),P(t))$ and $(W_{\varphi },U_{\varphi },M_{\varphi },N_{\varphi })=(W(t-\varphi ),U(t-\varphi ),M(t-\varphi ),N(t-\varphi )).$

Theorem 1

If $\mathfrak {R}_{0}\le 1$, then Đ$_{0}$ is globally asymptotically stable (G.A.S).

Proof

Construct a Lyapunov functional candidate $\Theta _{0}(W,U,M,N,P)$ as follows:

$$\begin{aligned} \Theta _{0}&=W_{0}\digamma \left( \frac{W}{W_{0}}\right) +\frac{1}{\mathcal {H}_{1}}U+\frac{W_{0}\left( b\eta _{1}\mathcal {H}_{3} +\varepsilon \eta _{3}\right) }{a\varepsilon }M+\frac{\eta _{1}W_{0}}{\varepsilon }N+\frac{\mu W_{0}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }P\\&\quad +\frac{1}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}} _{1}(\varphi )\int \limits _{t-\varphi }^{t}W(\varkappa )\left[ \eta _{1}N(\varkappa )+\eta _{2}U(\varkappa )+\eta _{3}M(\varkappa )\right] \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{\lambda W_{0}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}} _{2}(\varphi )\int \limits _{t-\varphi }^{t}U(\varkappa )\hbox {d}\varkappa \hbox {d}\varphi +\frac{b\eta _{1}W_{0}}{\varepsilon }\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\int \limits _{t-\varphi }^{t}M(\varkappa )\hbox {d}\varkappa \hbox {d}\varphi . \end{aligned}$$

It is seen that, $\Theta _{0}(W,U,M,N,P)>0$ for all $W,U,M,N,P>0,$ and $\Theta _{0}$ has a global minimum at Đ$_{0}$. We calculate $\frac{\hbox {d}\Theta _{0}}{\hbox {d}t}$ along the solutions of model (3) as:

$$\begin{aligned} \frac{\hbox {d}\Theta _{0}}{\hbox {d}t}&=\left( 1-\frac{W_{0}}{W}\right) \left( \rho -\alpha W-\eta _{1}WN-\eta _{2}WU-\eta _{3}WM\right) \nonumber \\&\quad +\frac{1}{\mathcal {H}_{1}}\left[ \int \limits _{0}^{\kappa _{1}} \mathcal {\bar{H}}_{1}(\varphi )W_{\varphi }\left( \eta _{1}N_{\varphi }+\eta _{2}U_{\varphi }+\eta _{3}M_{\varphi }\right) \hbox {d}\varphi -\left( \lambda +\gamma \right) U\right] \nonumber \\&\quad +\frac{W_{0}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }\left[ \lambda \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H} }_{2}(\varphi )U_{\varphi }\hbox {d}\varphi -aM-\mu PM\right] \nonumber \\&\quad +\frac{\eta _{1}W_{0}}{\varepsilon }\left[ b\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )M_{\varphi }\hbox {d}\varphi -\varepsilon N\right] +\frac{\mu W_{0}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }\left( \sigma PM-\pi P\right) \nonumber \\&\quad +\frac{1}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}} _{1}(\varphi )\left[ W\left( \eta _{1}N+\eta _{2}U+\eta _{3}M\right) -W_{\varphi }\left( \eta _{1}N_{\varphi }+\eta _{2}U_{\varphi }+\eta _{3} M_{\varphi }\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\lambda W_{0}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}} _{2}(\varphi )\left( U-U_{\varphi }\right) \hbox {d}\varphi +\frac{b\eta _{1}W_{0} }{\varepsilon }\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\left( M-M_{\varphi }\right) \hbox {d}\varphi \end{aligned}$$

(14)

Collecting terms of Eq. (14), we get

$$\begin{aligned} \frac{\hbox {d}\Theta _{0}}{\hbox {d}t}&=\left( 1-\frac{W_{0}}{W}\right) \left( \rho -\alpha W\right) +\eta _{2}W_{0}U-\frac{\lambda +\gamma }{\mathcal {H}_{1} }U\\&\quad +\frac{\lambda W_{0}\mathcal {H}_{2}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }U-\frac{\mu \pi W_{0}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }P. \end{aligned}$$

Using $W_{0}=\rho /\alpha$, we obtain

$$\begin{aligned} \frac{\hbox {d}\Theta _{0}}{\hbox {d}t}&=-\alpha \frac{(W-W_{0})^{2}}{W}+\frac{\lambda +\gamma }{\mathcal {H}_{1}}\left[ \frac{W_{0}\mathcal {H}_{1}\left\{ a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) \right\} }{a\varepsilon \left( \lambda +\gamma \right) }-1\right] U\\&\quad -\frac{\mu \pi W_{0}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }P\\&=-\alpha \frac{(W-W_{0})^{2}}{W}+\frac{\lambda +\gamma }{\mathcal {H}_{1} }\left( \mathfrak {R}_{0}-1\right) U-\frac{\mu \pi W_{0}\left( b\eta _{1} \mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }P. \end{aligned}$$

Therefore, $\frac{\hbox {d}\Theta _{0}}{\hbox {d}t}\le 0$ for all $W,U,M,N,P>0$ with equality holding when $W=W_{0}$ and $U=P=0.$ Let $\Upsilon _{0}=\left\{ (W,U,M,N,P):\frac{\hbox {d}\Theta _{0}}{\hbox {d}t}=0\right\}$ and $\Upsilon _{0}^{^{\prime }}$ be the largest invariant subset of $\Upsilon _{0}$. The solutions of system (3) converge to $\Upsilon _{0}^{^{\prime }}$ [42]. The set $\Upsilon _{0}^{^{\prime }}$ is invariant and contains elements which satisfy $W(t)=W_{0}$ and $U(t)=P(t)=0$. According to the LaSalle’s invariance principle, $\lim \nolimits _{t\rightarrow \infty }$ $W(t)=W_{0}$ and $\lim \nolimits _{t\rightarrow \infty }$ $U(t)=\lim \nolimits _{t\rightarrow \infty }P(t)=0$. Then, $\dot{W}(t)=0$ and $\dot{U}(t)=\dot{P}(t)=0.$ From the third and fourth equations of system (3), we have

$$\begin{aligned} \dot{M}(t)&=-\,aM(t), \end{aligned}$$

(15)

$$\begin{aligned} \dot{N}(t)&=b\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3} (\varphi )M_{\varphi }\hbox {d}\varphi -\varepsilon N(t). \end{aligned}$$

(16)

We define a Lyapunov function:

$$\begin{aligned} \tilde{\Theta }_{0}=M(t)+\frac{a}{2b\mathcal {H}_{3}}N(t)+\frac{a}{2\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3} (\varphi )\int \limits _{t-\varphi }^{t}M(\varkappa )\hbox {d}\varkappa \hbox {d}\varphi . \end{aligned}$$

Therefore, the time derivative of $\tilde{\Theta }_{0}$ along the solutions of system (15)–(16) can be calculated as follows:

$$\begin{aligned} \frac{\hbox {d}\tilde{\Theta }_{0}}{\hbox {d}t}=-\frac{a}{2}\left( M(t)+\frac{\varepsilon }{b\mathcal {H}_{3}}N(t)\right) \le 0. \end{aligned}$$

Clearly, $\frac{\hbox {d}\tilde{\Theta }_{0}}{\hbox {d}t}=0$ if and only if $M(t)=N(t)=0$ for all t. Let $\Upsilon _{0}^{^{^{\prime \prime }}}=\left\{ (W,U,M,N,P)\in \Upsilon _{0}^{^{\prime }}\right. :$ $\left. \frac{\hbox {d}\tilde{\Theta }_{0}}{\hbox {d}t}=0\right\} .$ It follows that $\Upsilon _{0}^{^{^{\prime \prime }}}=\left\{ (W,U,M,N,P)\in \Upsilon _{0}^{^{\prime }}:W=W_{0},U=M=N=P=0\right\} =\left\{ {">} _{0}\right\}$. Hence, all solution trajectories approach Đ$_{0}$ and this means that Đ$_{0}$ is G.A.S [42, 43]. $\square$

Lemma 1

If $\mathfrak {R}_{1}\le 1$, then $M_{1}\le M_{2}$.

Proof

Let $\mathfrak {R}_{1}\le 1$, then $\frac{\lambda \sigma \mathcal {H} _{2}U_{2}}{a\pi }\le 1$ and hence

$$\begin{aligned}&\left. U_{2}\le \frac{a\pi }{\lambda \sigma \mathcal {H}_{2}}\Longrightarrow \frac{-\tilde{B}+\sqrt{\tilde{B}^{2}-4\tilde{A}\tilde{C}}}{2\tilde{A}} \le \frac{a\pi }{\lambda \sigma \mathcal {H}_{2}}\right. \\&\quad \Longrightarrow \sqrt{\tilde{B}^{2}-4\tilde{A}\tilde{C}}\le \frac{2\tilde{A}a\pi +\lambda {\sigma }\mathcal {H}_{2}\tilde{B}}{\lambda \sigma \mathcal {H}_{2}}\\&\quad \Longrightarrow \left( \frac{2\tilde{A}a\pi +\lambda \sigma \mathcal {H} _{2}\tilde{B}}{\lambda \sigma \mathcal {H}_{2}}\right) ^{2}+4\tilde{A}\tilde{C}-\tilde{B}^{2}\ge 0. \end{aligned}$$

Using Eqs. (10) and (13), we obtain

$$\begin{aligned} \frac{4a\pi \varepsilon {\eta }_{2}\sigma (\gamma +\lambda )^{2}\left[ a\varepsilon \eta _{2}+\lambda \mathcal {H}_{2}\left( b \eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) \right] }{\lambda ^{2}\mathcal {H}_{2}^{2}}\left( M_{2}-M_{1}\right) \ge 0. \end{aligned}$$

Hence, $M_{1}\le M_{2}$.$\square$

We consider the following equalities to be used in the proceeding theorems:

$$\begin{aligned}&\left. \ln \left( \frac{W_{\varphi }N_{\varphi }}{WN}\right) =\ln \left( \frac{W_{\varphi }N_{\varphi }U_{n}}{W_{n}N_{n}U}\right) +\ln \left( \frac{W_{n}}{W}\right) +\ln \left( \frac{N_{n}U}{NU_{n}}\right) ,\right. \nonumber \\&\left. \ln \left( \frac{W_{\varphi }U_{\varphi }}{WU}\right) =\ln \left( \frac{W_{\varphi }U_{\varphi }}{W_{n}U}\right) +\ln \left( \frac{W_{n}}{W}\right) ,\right. \nonumber \\&\left. \ln \left( \frac{W_{\varphi }M_{\varphi }}{WM}\right) =\ln \left( \frac{W_{\varphi }M_{\varphi }U_{n}}{W_{n}M_{n}U}\right) +\ln \left( \frac{W_{n}}{W}\right) +\ln \left( \frac{M_{n}U}{MU_{n}}\right) ,\right. \nonumber \\&\left. \ln \left( \frac{U_{\varphi }}{U}\right) =\ln \left( \frac{U_{\varphi }M_{n}}{U_{n}M}\right) +\ln \left( \frac{U_{n}M}{UM_{n}}\right) ,\right. \nonumber \\&\left. \ln \left( \frac{M_{\varphi }}{M}\right) =\ln \left( \frac{M_{\varphi }N_{n}}{M_{n}N}\right) +\ln \left( \frac{M_{n}N}{MN_{n}}\right) , \,\,n=1,2.\right. \end{aligned}$$

(17)

$\square$

Theorem 2

Suppose that $\mathfrak {R}_{1}\le 1<\mathfrak {R}_{0}$, then Đ$_{1}$ is G.A.S.

Proof

We define a functional $\Theta _{1}(W,U,M,N,P)$ as:

$$\begin{aligned} \Theta _{1}&=W_{1}\digamma \left( \frac{W}{W_{1}}\right) +\frac{1}{\mathcal {H}_{1}}U_{1}\digamma \left( \frac{U}{U_{1}}\right) \\&\quad +\frac{W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }M_{1}\digamma \left( \frac{M}{M_{1}}\right) +\frac{\eta _{1}W_{1}}{\varepsilon }N_{1}\digamma \left( \frac{N}{N_{1}}\right) \\&\quad +\frac{\mu W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }P+\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H} _{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{W(\varkappa )N(\varkappa )}{W_{1}N_{1}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{\eta _{2}W_{1}U_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{W(\varkappa )U(\varkappa )}{W_{1}U_{1}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{\eta _{3}W_{1}M_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{W(\varkappa )M(\varkappa )}{W_{1}M_{1}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{\lambda W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) U_{1}}{a\varepsilon }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H} }_{2}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{U(\varkappa )}{U_{1}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{b\eta _{1}W_{1}M_{1}}{\varepsilon }\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{M(\varkappa )}{M_{1}}\right) \hbox {d}\varkappa \hbox {d}\varphi . \end{aligned}$$

Calculating $\frac{\hbox {d}\Theta _{1}}{\hbox {d}t}$ as:

$$\begin{aligned} \frac{\hbox {d}\Theta _{1}}{\hbox {d}t}&=\left( 1-\frac{W_{1}}{W}\right) \left( \rho -\alpha W-\eta _{1}WN-\eta _{2}WU-\eta _{3}WM\right) \nonumber \\&\quad +\frac{1}{\mathcal {H}_{1}}\left( 1-\frac{U_{1}}{U}\right) \left[ \int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )W_{\varphi }\left( \eta _{1}N_{\varphi }+\eta _{2}U_{\varphi }+\eta _{3}M_{\varphi }\right) \hbox {d}\varphi -\left( \lambda +\gamma \right) U\right] \nonumber \\&\quad +\frac{W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }\left( 1-\frac{M_{1}}{M}\right) \left[ \lambda \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )U_{\varphi } \hbox {d}\varphi -aM-\mu PM\right] \nonumber \\&\quad +\frac{\eta _{1}W_{1}}{\varepsilon }\left( 1-\frac{N_{1}}{N}\right) \left[ b\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )M_{\varphi } \hbox {d}\varphi -\varepsilon N\right] +\frac{\mu W_{1}\left( b\eta _{1} \mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }\nonumber \\&\quad \times \left( \sigma PM-\pi P\right) +\frac{\eta _{1}W_{1}N_{1} }{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1} (\varphi )\left[ \frac{WN}{W_{1}N_{1}}-\frac{W_{\varphi }N_{\varphi }}{W_{1}N_{1}}+\ln \left( \frac{W_{\varphi }N_{\varphi }}{WN}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{2}W_{1}U_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{WU}{W_{1}U_{1}}-\frac{W_{\varphi }U_{\varphi }}{W_{1}U_{1}}+\ln \left( \frac{W_{\varphi }U_{\varphi }}{WU}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{3}W_{1}M_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{WM}{W_{1}M_{1}}-\frac{W_{\varphi }M_{\varphi }}{W_{1}M_{1}}+\ln \left( \frac{W_{\varphi }M_{\varphi }}{WM}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\lambda W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) U_{1}}{a\varepsilon }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H} }_{2}(\varphi )\left[ \frac{U}{U_{1}}-\frac{U_{\varphi }}{U_{1}}+\ln \left( \frac{U_{\varphi }}{U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{b\eta _{1}W_{1}M_{1}}{\varepsilon }\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\left[ \frac{M}{M_{1}}-\frac{M_{\varphi } }{M_{1}}+\ln \left( \frac{M_{\varphi }}{M}\right) \right] \hbox {d}\varphi . \end{aligned}$$

(18)

Collecting terms of Eq. (18), we derive

$$\begin{aligned} \frac{\hbox {d}\Theta _{1}}{\hbox {d}t}&=\left( 1-\frac{W_{1}}{W}\right) \left( \rho -\alpha W\right) +\eta _{2}W_{1}U-\frac{\lambda +\gamma }{\mathcal {H}_{1} }U-\frac{\eta _{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }N_{\varphi }U_{1}}{U}\hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}} \mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }U_{\varphi }U_{1}}{U} \hbox {d}\varphi -\frac{\eta _{3}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }M_{\varphi }U_{1}}{U} \hbox {d}\varphi +\frac{\lambda +\gamma }{\mathcal {H}_{1}}U_{1}\nonumber \\&\quad -\frac{\lambda W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}} _{2}(\varphi )\frac{U_{\varphi }M_{1}}{M}\hbox {d}\varphi +\frac{W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon }M_{1}\nonumber \\&\quad +\frac{\mu W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }PM_{1}-\frac{b\eta _{1}W_{1}}{\varepsilon } \int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\frac{M_{\varphi }N_{1}}{N}\hbox {d}\varphi +\eta _{1}W_{1}N_{1}\nonumber \\&\quad -\frac{\mu {\pi } W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }P+\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H} _{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }N_{\varphi }}{WN}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{2}W_{1}U_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }U_{\varphi }}{WU}\right) \hbox {d}\varphi +\frac{\eta _{3}W_{1}M_{1}}{\mathcal {H}_{1}} \int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }M_{\varphi }}{WM}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{\lambda W_{1}\mathcal {H}_{2}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }U+\frac{\lambda W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) U_{1}}{a\varepsilon } \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\ln \left( \frac{U_{\varphi }}{U}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{b\eta _{1}W_{1}M_{1}}{\varepsilon }\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\ln \left( \frac{M_{\varphi }}{M}\right) \hbox {d}\varphi . \end{aligned}$$

(19)

Using the equilibrium conditions for Đ$_{1}$, we get

$$\begin{aligned}&\left. \rho =\alpha W_{1}+\eta _{1}W_{1}N_{1}+\eta _{2}W_{1}U_{1}+\eta _{3}W_{1}M_{1}{,}\right. \\&\left. \eta _{1}W_{1}N_{1}+\eta _{2}W_{1}U_{1}+\eta _{3}W_{1}M_{1} =\frac{\lambda +\gamma }{\mathcal {H}_{1}}U_{1},\right. \\&\left. \frac{\lambda \mathcal {H}_{2}U_{1}}{a}=M_{1},\,\, N_{1}=\frac{b\mathcal {H}_{3}M_{1}}{\varepsilon }.\right. \end{aligned}$$

In addition,

$$\begin{aligned} \eta _{1}W_{1}N_{1}+\eta _{3}W_{1}M_{1}=\frac{W_{1}\left( b\eta _{1} \mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon }M_{1}=\frac{\lambda W_{1}\mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }U_{1}. \end{aligned}$$

Then, we obtain

$$\begin{aligned} \frac{\hbox {d}\Theta _{1}}{\hbox {d}t}&=\left( 1-\frac{W_{1}}{W}\right) \left( \alpha W_{1}-\alpha W\right) +\left( \eta _{1}W_{1}N_{1}+\eta _{2}W_{1}U_{1}+\eta _{3}W_{1}M_{1}\right) \left( 1-\frac{W_{1}}{W}\right) \\&\quad -\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }N_{\varphi }U_{1}}{W_{1}N_{1} U}\hbox {d}\varphi -\frac{\eta _{2}W_{1}U_{1}}{\mathcal {H}_{1}}\int \limits _{0} ^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }U_{\varphi }}{W_{1}U}\hbox {d}\varphi \\&\quad -\frac{\eta _{3}W_{1}M_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }M_{\varphi }U_{1}}{W_{1}M_{1} U}\hbox {d}\varphi +\eta _{1}W_{1}N_{1}+\eta _{2}W_{1}U_{1}+\eta _{3}W_{1}M_{1}\\&\quad -\frac{\eta _{1}W_{1}N_{1}+\eta _{3}W_{1}M_{1}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\frac{U_{\varphi }M_{1}}{U_{1}M}\hbox {d}\varphi +\eta _{1}W_{1}N_{1}+\eta _{3}W_{1}M_{1}\\&\quad +\frac{\mu W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }PM_{1}-\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H}_{3} }\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\frac{M_{\varphi }N_{1}}{M_{1}N}\hbox {d}\varphi +\eta _{1}W_{1}N_{1}\\&\quad -\frac{\mu \pi W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma a\varepsilon }P+\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H} _{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }N_{\varphi }}{WN}\right) \hbox {d}\varphi \\&\quad +\frac{\eta _{2}W_{1}U_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }U_{\varphi }}{WU}\right) \hbox {d}\varphi +\frac{\eta _{3}W_{1}M_{1}}{\mathcal {H}_{1}} \int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }M_{\varphi }}{WM}\right) \hbox {d}\varphi \\&\quad +\frac{\eta _{1}W_{1}N_{1}+\eta _{3}W_{1}M_{1}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\ln \left( \frac{U_{\varphi }}{U}\right) \hbox {d}\varphi +\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\ln \left( \frac{M_{\varphi }}{M}\right) \hbox {d}\varphi . \end{aligned}$$

Using the equalities given by (17) in case of $n=1$, we get

$$\begin{aligned} \frac{\hbox {d}\Theta _{1}}{\hbox {d}t}&=-\alpha \frac{(W-W_{1})^{2}}{W}-\left( \eta _{1}W_{1}N_{1}+\eta _{2}W_{1}U_{1}+\eta _{3}W_{1}M_{1}\right) \left[ \frac{W_{1}}{W}-1-\ln \left( \frac{W_{1}}{W}\right) \right] \nonumber \\&\quad -\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{W_{\varphi }N_{\varphi }U_{1} }{W_{1}N_{1}U}-1-\ln \left( \frac{W_{\varphi }N_{\varphi }U_{1}}{W_{1}N_{1} U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{2}W_{1}U_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{W_{\varphi }U_{\varphi }}{W_{1} U}-1-\ln \left( \frac{W_{\varphi }U_{\varphi }}{W_{1}U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{3}W_{1}M_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{W_{\varphi }M_{\varphi }U_{1} }{W_{1}M_{1}U}-1-\ln \left( \frac{W_{\varphi }M_{\varphi }U_{1}}{W_{1}M_{1} U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{1}W_{1}N_{1}+\eta _{3}W_{1}M_{1}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\left[ \frac{U_{\varphi }M_{1}}{U_{1}M}-1-\ln \left( \frac{U_{\varphi }M_{1}}{U_{1} M}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{1}W_{1}N_{1}}{\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\left[ \frac{M_{\varphi }N_{1}}{M_{1}N} -1-\ln \left( \frac{M_{\varphi }N_{1}}{M_{1}N}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\mu W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }\left( M_{1}-\frac{\pi }{\sigma }\right) P. \end{aligned}$$

(20)

Therefore, Eq. (20) becomes

$$\begin{aligned} \frac{\hbox {d}\Theta _{1}}{\hbox {d}t}&=-\alpha \frac{(W-W_{1})^{2}}{W}-\frac{\eta _{1} W_{1}N_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}} _{1}(\varphi )\left[ \digamma \left( \frac{W_{\varphi }N_{\varphi }U_{1}}{W_{1}N_{1}U}\right) +\digamma \left( \frac{W_{1}}{W}\right) \right] \hbox {d}\varphi \\&\quad -\frac{\eta _{2}W_{1}U_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \digamma \left( \frac{W_{\varphi }U_{\varphi }}{W_{1}U}\right) +\digamma \left( \frac{W_{1}}{W}\right) \right] \hbox {d}\varphi \\&\quad -\frac{\eta _{3}W_{1}M_{1}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \digamma \left( \frac{W_{\varphi }M_{\varphi }U_{1}}{W_{1}M_{1}U}\right) +\digamma \left( \frac{W_{1}}{W}\right) \right] \hbox {d}\varphi \\&\quad -\frac{\eta _{1}W_{1}N_{1}+\eta _{3}W_{1}M_{1}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\digamma \left( \frac{U_{\varphi }M_{1}}{U_{1}M}\right) \hbox {d}\varphi \\&\quad -\frac{\eta _{1}W_{1}N_{1} }{\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3} (\varphi )\digamma \left( \frac{M_{\varphi }N_{1}}{M_{1}N}\right) \hbox {d}\varphi \\&\quad +\frac{\mu W_{1}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{a\varepsilon }\left( M_{1}-M_{2}\right) P. \end{aligned}$$

Using Lemma 1 and since $\mathfrak {R}_{1}\le 1$ then $M_{1}\le M_{2}$ and $\frac{\hbox {d}\Theta _{1}}{\hbox {d}t}\le 0$ for all $W,U,M,N,P>0$ with equality holding when $W=W_{1}$, $P=0$ and $\digamma =0.$ Let $\Upsilon _{1}^{\prime }$ be the largest invariant subset of $\Upsilon _{1}=\left\{ (W,U,M,N,P):\frac{\hbox {d}\Theta _{1}}{\hbox {d}t}=0\right\}$. The trajectories of system (3) converge to $\Upsilon _{1}^{^{\prime }}$. The set $\Upsilon _{1}^{^{\prime }}$ is invariant and contains elements with $W(t)=W_{1}$ and $\digamma =0,$ i.e.,

$$\begin{aligned} \frac{W_{\varphi }N_{\varphi }U_{1}}{W_{1}N_{1}U}=\frac{W_{\varphi }U_{\varphi } }{W_{1}U}=\frac{W_{\varphi }M_{\varphi }U_{1}}{W_{1}M_{1}U}=\frac{U_{\varphi }M_{1}}{U_{1}M}=\frac{M_{\varphi }N_{1}}{M_{1}N}=1, \end{aligned}$$

(21)

for all $t\in [0,\kappa ].$ If $W(t)=W_{1}$, then from Eq. (21), we get $U(t)=U_{1}$, $M(t)=M_{1}$ and $N(t)=N_{1}$. Then, $\Upsilon _{1}^{\prime }=\left\{ {">} _{1}\right\}$ and Đ$_{1}$ is G.A.S using LaSalle’s invariance principle.$\square$

Theorem 3

Suppose that $\mathfrak {R}_{1}>1$, then Đ$_{2}$ is G.A.S.

Proof

Define a function $\Theta _{2}(W,U,M,N,P)$ as:

$$\begin{aligned} \Theta _{2}&=W_{2}\digamma \left( \frac{W}{W_{2}}\right) +\frac{1}{\mathcal {H}_{1}}U_{2}\digamma \left( \frac{U}{U_{2}}\right) +\frac{W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }M_{2}\digamma \left( \frac{M}{M_{2} }\right) \\&\quad +\frac{\eta _{1}W_{2}}{\varepsilon }N_{2}\digamma \left( \frac{N}{N_{2}}\right) +\frac{\mu W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma \varepsilon \left( a+\mu P_{2}\right) }P_{2} \digamma \left( \frac{P}{P_{2}}\right) \\&\quad +\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1} (\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{W(\varkappa )N(\varkappa )}{W_{2}N_{2}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{\eta _{2}W_{2}U_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{W(\varkappa )U(\varkappa )}{W_{2}U_{2}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{\eta _{3}W_{2}M_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{W(\varkappa )M(\varkappa )}{W_{2}M_{2}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{\lambda W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) U_{2}}{\varepsilon \left( a+\mu P_{2}\right) }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{U(\varkappa )}{U_{2}}\right) \hbox {d}\varkappa \hbox {d}\varphi \\&\quad +\frac{b\eta _{1}W_{2}M_{2}}{\varepsilon }\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\int \limits _{t-\varphi }^{t}\digamma \left( \frac{M(\varkappa )}{M_{2}}\right) \hbox {d}\varkappa \hbox {d}\varphi . \end{aligned}$$

We calculate $\frac{\hbox {d}\Theta _{2}}{\hbox {d}t}$ as:

$$\begin{aligned} \frac{\hbox {d}\Theta _{2}}{\hbox {d}t}&=\left( 1-\frac{W_{2}}{W}\right) \left( \rho -\alpha W-\eta _{1}WN-\eta _{2}WU-\eta _{3}WM\right) \nonumber \\&\quad +\frac{1}{\mathcal {H}_{1}}\left( 1-\frac{U_{2}}{U}\right) \left[ \int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )W_{\varphi }\left( \eta _{1}N_{\varphi }+\eta _{2}U_{\varphi }+\eta _{3}M_{\varphi }\right) \hbox {d}\varphi -\left( \lambda +\gamma \right) U\right] \nonumber \\&\quad +\frac{W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }\left( 1-\frac{M_{2}}{M}\right) \left[ \lambda \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2} (\varphi )U_{\varphi }\hbox {d}\varphi -aM-\mu PM\right] \nonumber \\&\quad +\frac{\eta _{1}W_{2}}{\varepsilon }\left( 1-\frac{N_{2}}{N}\right) \left[ b\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )M_{\varphi } \hbox {d}\varphi -\varepsilon N\right] +\frac{\mu W_{2}\left( b\eta _{1} \mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma \varepsilon \left( a+\mu P_{2}\right) }\left( 1-\frac{P_{2}}{P}\right) \nonumber \\&\quad \times \left( \sigma PM-\pi P\right) +\frac{\eta _{1}W_{2}N_{2} }{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1} (\varphi )\left[ \frac{WN}{W_{2}N_{2}}-\frac{W_{\varphi }N_{\varphi }}{W_{2}N_{2}}+\ln \left( \frac{W_{\varphi }N_{\varphi }}{WN}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{2}W_{2}U_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{WU}{W_{2}U_{2}}-\frac{W_{\varphi }U_{\varphi }}{W_{2}U_{2}}+\ln \left( \frac{W_{\varphi }U_{\varphi }}{WU}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{3}W_{2}M_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{WM}{W_{2}M_{2}}-\frac{W_{\varphi }M_{\varphi }}{W_{2}M_{2}}+\ln \left( \frac{W_{\varphi }M_{\varphi }}{WM}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{\lambda W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) U_{2}}{\varepsilon \left( a+\mu P_{2}\right) }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\left[ \frac{U}{U_{2} }-\frac{U_{\varphi }}{U_{2}}+\ln \left( \frac{U_{\varphi }}{U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad +\frac{b\eta _{1}W_{2}M_{2}}{\varepsilon }\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\left[ \frac{M}{M_{2}}-\frac{M_{\varphi } }{M_{2}}+\ln \left( \frac{M_{\varphi }}{M}\right) \right] \hbox {d}\varphi . \end{aligned}$$

(22)

Collecting terms of Eq. (22), we derive

$$\begin{aligned} \frac{\hbox {d}\Theta _{2}}{\hbox {d}t}&=\left( 1-\frac{W_{2}}{W}\right) \left( \rho -\alpha W\right) +\eta _{2}W_{2}U+\eta _{3}W_{2}M-\frac{\lambda +\gamma }{\mathcal {H}_{1}}U-\frac{\eta _{1}}{\mathcal {H}_{1}}\int \limits _{0} ^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }N_{\varphi }U_{2} }{U}\hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}} \mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }U_{\varphi }U_{2}}{U} \hbox {d}\varphi -\frac{\eta _{3}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }M_{\varphi }U_{2}}{U} \hbox {d}\varphi \nonumber \\&\quad +\frac{\lambda +\gamma }{\mathcal {H}_{1}}U_{2}-\frac{aW_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }M-\frac{\lambda W_{2}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\frac{U_{\varphi }M_{2}}{M}\hbox {d}\varphi \nonumber \\&\quad +\frac{aW_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }M_{2}+\frac{\mu W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }PM_{2}-\frac{b\eta _{1}W_{2}}{\varepsilon }\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\frac{M_{\varphi }N_{2} }{N}\hbox {d}\varphi \nonumber \\&\quad +\eta _{1}W_{2}N_{2}-\frac{\mu \pi W_{2}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) }{\sigma \varepsilon \left( a+\mu P_{2}\right) }P-\frac{\mu W_{2}\left( b\eta _{1}\mathcal {H}_{3} +\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) } P_{2}M\nonumber \\&\quad +\frac{\mu \pi W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\sigma \varepsilon \left( a+\mu P_{2}\right) }P_{2}+\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}} \mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }N_{\varphi }}{WN}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{2}W_{2}U_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }U_{\varphi }}{WU}\right) \hbox {d}\varphi +\frac{\eta _{3}W_{2}M_{2}}{\mathcal {H}_{1}} \int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }M_{\varphi }}{WM}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{\lambda W_{2}\mathcal {H}_{2}\left( b\eta _{1}\mathcal {H} _{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }U+\frac{\lambda W_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) U_{2}}{\varepsilon \left( a+\mu P_{2}\right) }\int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\ln \left( \frac{U_{\varphi }}{U}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{b\mathcal {H}_{3}\eta _{1}W_{2}}{\varepsilon }M+\frac{b\eta _{1} W_{2}M_{2}}{\varepsilon }\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}} _{3}(\varphi )\ln \left( \frac{M_{\varphi }}{M}\right) \hbox {d}\varphi . \end{aligned}$$

(23)

Using the equilibrium conditions for Đ $_{2}$:

$$\begin{aligned}&\left. \rho =\alpha W_{2}+\eta _{1}W_{2}N_{2}+\eta _{2}W_{2}U_{2}+\eta _{3}W_{2}M_{2}{,}\right. \\&\left. \eta _{1}W_{2}N_{2}+\eta _{2}W_{2}U_{2}+\eta _{3}W_{2}M_{2} =\frac{\lambda +\gamma }{\mathcal {H}_{1}}U_{2},\right. \\&\left. \lambda \mathcal {H}_{2}U_{2}=\left( a+\mu P_{2}\right) M_{2}, \,\,M_{2}=\frac{\pi }{\sigma }, \,\,N_{2} =\frac{b\mathcal {H}_{3}}{\varepsilon }M_{2}.\right. \end{aligned}$$

Further,

$$\begin{aligned} \eta _{1}W_{2}N_{2}+\eta _{3}W_{2}M_{2}=\frac{W_{2}\left( b\eta _{1} \mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon }M_{2}=\frac{\lambda W_{2}\mathcal {H}_{2}\left( b\eta _{1}\mathcal {H}_{3}+\varepsilon \eta _{3}\right) }{\varepsilon \left( a+\mu P_{2}\right) }U_{2}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \frac{\hbox {d}\Theta _{2}}{\hbox {d}t}&=\left( 1-\frac{W_{2}}{W}\right) \left( \alpha W_{2}-\alpha W\right) +\left( \eta _{1}W_{2}N_{2}+\eta _{2}W_{2}U_{2}+\eta _{3}W_{2}M_{2}\right) \left( 1-\frac{W_{2}}{W}\right) \nonumber \\&\quad -\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }N_{\varphi }U_{2}}{W_{2}N_{2} U}\hbox {d}\varphi -\frac{\eta _{2}W_{2}U_{2}}{\mathcal {H}_{1}}\int \limits _{0} ^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }U_{\varphi }}{W_{2}U}\hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{3}W_{2}M_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\frac{W_{\varphi }M_{\varphi }U_{2}}{W_{2}M_{2} U}\hbox {d}\varphi +\eta _{1}W_{2}N_{2}+\eta _{2}W_{2}U_{2}+\eta _{3}W_{2}M_{2}\nonumber \\&\quad -\frac{\eta _{1}W_{2}N_{2}+\eta _{3}W_{2}M_{2}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\frac{U_{\varphi }M_{2}}{U_{2}M}\hbox {d}\varphi +\eta _{1}W_{2}N_{2}+\eta _{3}W_{2}M_{2}\nonumber \\&\quad -\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\frac{M_{\varphi }N_{2}}{M_{2}N}\hbox {d}\varphi +\eta _{1}W_{2}N_{2}+\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }N_{\varphi }}{WN}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{2}W_{2}U_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }U_{\varphi }}{WU}\right) \hbox {d}\varphi +\frac{\eta _{3}W_{2}M_{2}}{\mathcal {H}_{1}} \int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}}_{1}(\varphi )\ln \left( \frac{W_{\varphi }M_{\varphi }}{WM}\right) \hbox {d}\varphi \nonumber \\&\quad +\frac{\eta _{1}W_{2}N_{2}+\eta _{3}W_{2}M_{2}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\ln \left( \frac{U_{\varphi }}{U}\right) \hbox {d}\varphi +\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3}}\mathcal {\bar{H}}_{3}(\varphi )\ln \left( \frac{M_{\varphi }}{M}\right) \hbox {d}\varphi . \end{aligned}$$

(24)

Using the equalities given by (17) in case of $n=2$, we get

$$\begin{aligned} \frac{\hbox {d}\Theta _{2}}{\hbox {d}t}&=-\alpha \frac{(W-W_{2})^{2}}{W}-\left( \eta _{1}W_{2}N_{2}+\eta _{2}W_{2}U_{2}+\eta _{3}W_{2}M_{2}\right) \left[ \frac{W_{2}}{W}-1-\ln \left( \frac{W_{2}}{W}\right) \right] \nonumber \\&\quad -\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{W_{\varphi }N_{\varphi }U_{2} }{W_{2}N_{2}U}-1-\ln \left( \frac{W_{\varphi }N_{\varphi }U_{2}}{W_{2}N_{2} U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{2}W_{2}U_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{W_{\varphi }U_{\varphi }}{W_{2} U}-1-\ln \left( \frac{W_{\varphi }U_{\varphi }}{W_{2}U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{3}W_{2}M_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \frac{W_{\varphi }M_{\varphi }U_{2} }{W_{2}M_{2}U}-1-\ln \left( \frac{W_{\varphi }M_{\varphi }U_{2}}{W_{2}M_{2} U}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{1}W_{2}N_{2}+\eta _{3}W_{2}M_{2}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\left[ \frac{U_{\varphi }M_{2}}{U_{2}M}-1-\ln \left( \frac{U_{\varphi }M_{2}}{U_{2} M}\right) \right] \hbox {d}\varphi \nonumber \\&\quad -\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\left[ \frac{M_{\varphi }N_{2}}{M_{2}N} -1-\ln \left( \frac{M_{\varphi }N_{2}}{M_{2}N}\right) \right] \hbox {d}\varphi . \end{aligned}$$

(25)

Eq. (25) can be rewritten as follows:

$$\begin{aligned} \frac{\hbox {d}\Theta _{2}}{\hbox {d}t}&=-\alpha \frac{(W-W_{2})^{2}}{W}-\frac{\eta _{1} W_{2}N_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1}}\mathcal {\bar{H}} _{1}(\varphi ))\left[ \digamma \left( \frac{W_{\varphi }N_{\varphi }U_{2}}{W_{2}N_{2}U}\right) +\digamma \left( \frac{W_{2}}{W}\right) \right] \hbox {d}\varphi \\&\quad -\frac{\eta _{2}W_{2}U_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \digamma \left( \frac{W_{\varphi }U_{\varphi }}{W_{2}U}\right) +\digamma \left( \frac{W_{2}}{W}\right) \right] \hbox {d}\varphi \\&\quad -\frac{\eta _{3}W_{2}M_{2}}{\mathcal {H}_{1}}\int \limits _{0}^{\kappa _{1} }\mathcal {\bar{H}}_{1}(\varphi )\left[ \digamma \left( \frac{W_{\varphi }M_{\varphi }U_{2}}{W_{2}M_{2}U}\right) +\digamma \left( \frac{W_{2}}{W}\right) \right] \hbox {d}\varphi \\&\quad -\frac{\eta _{1}W_{2}N_{2}+\eta _{3}W_{2}M_{2}}{\mathcal {H}_{2}} \int \limits _{0}^{\kappa _{2}}\mathcal {\bar{H}}_{2}(\varphi )\digamma \left( \frac{U_{\varphi }M_{2}}{U_{2}M}\right) \hbox {d}\varphi \\&\quad -\frac{\eta _{1}W_{2}N_{2}}{\mathcal {H}_{3}}\int \limits _{0}^{\kappa _{3} }\mathcal {\bar{H}}_{3}(\varphi )\digamma \left( \frac{M_{\varphi }N_{2}}{M_{2} N}\right) \hbox {d}\varphi . \end{aligned}$$

Hence, if $\mathfrak {R}_{1}>1$, then $\frac{\hbox {d}\Theta _{2}}{\hbox {d}t}\le 0$ for all $W,U,M,N,P>0$ with equality holding when $W=W_{2}$ and $\digamma =0$. The solutions of system (3) tend to $\Upsilon _{2}^{^{\prime }}$ the largest invariant subset of $\Upsilon _{2}=\left\{ (W,U,M,N,P):\frac{\hbox {d}\Theta _{2}}{\hbox {d}t}=0\right\}$. The set $\Upsilon _{2}^{^{\prime }}$ contains elements with $W(t)=W_{2}$ and $\digamma =0$, i.e.,

$$\begin{aligned} \frac{W_{\varphi }N_{\varphi }U_{2}}{W_{2}N_{2}U}=\frac{W_{\varphi }U_{\varphi } }{W_{2}U}=\frac{W_{\varphi }M_{\varphi }U_{2}}{W_{2}M_{2}U}=\frac{U_{\varphi }M_{2}}{U_{2}M}=\frac{M_{\varphi }N_{2}}{M_{2}N}=1, \end{aligned}$$

(26)

for all $t\in [0,\kappa ].$ If $W(t)=W_{2}$, then from Eq. (26), we get $U(t)=U_{2},$ $M(t)=M_{2}$ and $N(t)=N_{2}$. Substituting in the third equation of system (3), we obtain

$$\begin{aligned} 0=\dot{M}(t)=\lambda {U}_{2}-aM_{2}-\mu P(t)M_{2} , \end{aligned}$$

which yields $P(t)=P_{2}$ for all t. Therefore, $\Upsilon _{2}^{^{\prime } }=\left\{ ">_{2}\right\}$. LaSalle’s invariance principle implies that Đ$_{2}$ is G.A.S.

$\square$

6 Numerical results

In this section, we illustrate the results of Theorems 1-3 by performing numerical simulations. We address the influence of antiviral drugs efficacy, CTC transmission and time delays on the dynamical behavior of the system. For numerical purposes, we transform the distributed-time delay model (3) to a discrete-time delay one by choosing a Dirac delta function D(.) as a special form of the kernel $\Lambda _{i}(.)$ as [9]:

$$\begin{aligned} \Lambda _{i}(x)=D\left( x-\varphi _{i}\right) , \,\,\varphi _{i} \in \left[ 0,\kappa _{i}\right] {, \ }i=1,2,3. \end{aligned}$$

Let $\kappa _{i}$ tends to $\infty$, then the properties of D(.) imply that:

$$\begin{aligned} \int \limits _{0}^{\infty }\Lambda _{j}(\varsigma )\hbox {d}\varsigma =1, \,\,\mathcal {H}_{j}=\int \limits _{0}^{\infty }D\left( \varsigma -\varphi _{j}\right) e^{-\hslash _{j}\varsigma }\hbox {d}\varsigma =e^{-\hslash _{j}\varphi _{j} }, \,\,j=1,2,3. \end{aligned}$$

Thus, model (3) becomes:

$$\begin{aligned} \left\{ \begin{array} [c]{l} \dot{W}=\rho -\alpha W-\eta _{1}WN-\eta _{2}WU-\eta _{3}WM,\\ \dot{U}=e^{-\hslash _{1}\varphi _{1}}W_{\varphi _{1}}\left( \eta _{1} N_{\varphi _{1}}+\eta _{2}U_{\varphi _{1}}+\eta _{3}M_{\varphi _{1}}\right) -\left( \lambda +\gamma \right) U,\\ \dot{M}=\lambda e^{-\hslash _{2}\varphi _{2}}U_{\varphi _{2}}-aM-\mu PM,\\ \dot{N}=be^{-\hslash _{3}\varphi _{3}}M_{\varphi _{3}}-\varepsilon N,\\ \dot{P}=\sigma PM-\pi P. \end{array} \right. \end{aligned}$$

(27)

For model (27), the threshold parameters are given by:

$$\begin{aligned} \mathfrak {R}_{0}=\frac{W_{0}e^{-\hslash _{1}\varphi _{1}}\left[ a\varepsilon \eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left( b\eta _{1}e^{-\hslash _{3}\varphi _{3}}+\varepsilon \eta _{3}\right) \right] }{a\varepsilon \left( \gamma +\lambda \right) }, \,\,\mathfrak {R}_{1}=\frac{\lambda \sigma e^{-\hslash _{2}\varphi _{2}}U_{2}}{a\pi }. \end{aligned}$$

(28)

Table 2 Some values of the parameters of model (27)

To solve system (27) numerically, we fix the values of some parameters (see Table 2) and the others will be varied. We have chosen the values of parameters of the model just to perform the numerical simulations. This is indeed because 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 can be established. We note that the stability of the three equilibria is controlled by two threshold parameters $\mathfrak {R}_{0}\$and $\mathfrak {R}_{1}$. We can see that $\mathfrak {R}_{0}$ depends on the incidence rate constants $\eta _{1}$, $\eta _{2}$ and $\eta _{3}$. These parameters can significantly affected by the antiviral drug therapy as will be shown below. On the other hand, the proliferation rate constant for effective HIV-specific CTLs $\sigma$ can significantly change the parameter $\mathfrak {R}_{1}$. Therefore, to verify the results of Theorems 1-3 we vary the parameters $\eta _{1}$, $\eta _{2}$, $\eta _{3}$ and $\sigma$. In addition, to address the effect of the time delays on the HIV dynamics we vary the delays parameters $\varphi _{1},$ $\varphi _{2}$ and $\varphi _{3}$.

Fig. 1

The behavior of solution trajectories of system (27) in case of $\mathfrak {R}_{0}\le 1$

Fig. 2

The behavior of solution trajectories of system (27) in case of $\mathfrak {R}_{1}\le 1<\mathfrak {R}_{0}$

Fig. 3

The behavior of solution trajectories of system (27) in case of $\mathfrak {R}_{1}>1$

Fig. 4

The influence of antiviral drug efficacy on the behavior of solution trajectories of system (29)

Fig. 5

The influence of time delay parameters on the behavior of solution trajectories of system (29)

6.1 Stability of the equilibria

We consider the values $\varphi _{1}=3,$ $\varphi _{2}=2,$ $\varphi _{3}=1\$and choose the following three different initial conditions for model (27):

IV-1 : $(W(\varphi ),U(\varphi ),M(\varphi ),N(\varphi ),P(\varphi ))=(500,5,0.8,0.8,3)$, (Solid lines in the figures),

IV-2: $(W(\varphi ),U(\varphi ),M(\varphi ),N(\varphi ),P(\varphi ))=(650,4,0.6,0.6,2),$ (Dashed lines in the figures),

IV-3: $(W(\varphi ),U(\varphi ),M(\varphi ),N(\varphi ),P(\varphi ))=(800,3,0.4,0.4,1)$, (Dotted lines in the figures), where $\varphi \in [-3,0].$ Choosing selected values of $\eta _{1}$ , $\eta _{2}$, $\eta _{3}$ and $\sigma$ under the mentioned initial conditions leads to the following cases:

Stability of Đ$_{0}$. $\eta _{1}=0.0003,$ $\eta _{2}=0.00001$, $\eta _{3}=0.0001$ and $\sigma =0.002$. With these parameters, we have $\mathfrak {R}_{0}=0.34<1$. Figure 1 displays that the trajectories initiating with IV-1, IV2 and IV-3 reach the equilibrium Đ$_{0}=(1000,0,0,0,0)$. This shows that Đ$_{0}$ is G.A.S according to Theorem 1. In this case, the HIV particles will be cleared from the body.

Stability of Đ$_{1}$. $\eta _{1}=0.003,$ $\eta _{2}=0.00002$, $\eta _{3}=0.001$ and $\sigma =0.002$. With such a choice, we get $\mathfrak {R}_{1}=0.10<1<3.29=\mathfrak {R}_{0}$. It is clear that the equilibrium point Đ$_{1}$ exists with Đ$_{1}=\left( 303.7,12.90,3.46,6.40,0\right)$. Figure 2 displays that the trajectories initiating with IV-1, IV2 and IV-3 tend to Đ $_{1}$. Therefore, the numerical results support Theorem 2. Hence, a chronic HIV infection with inactivated CTL-mediated immune response is attained.

Stability of Đ$_{2}$. $\eta _{1}=0.003,$ $\eta _{2}=0.00002$, $\eta _{3}=0.001$ and $\sigma =0.2$. Then, we calculate $\mathfrak {R}_{1}=2.50>1$. In Fig. 3, we show that Đ$_{2} =\left( 747.86,4.67,0.5,0.93,3.76\right)$ exists and it is G.A.S and this agrees with Theorem 3. Hence, a chronic HIV infection with CTL-mediated immune response is attained.

6.2 Effect of antiviral drugs

To study the effect of antiviral drugs on the HIV dynamics, we incorporate three types of antiviral drugs into model (27) as:

$$\begin{aligned} \left\{ \begin{array} [c]{l} \dot{W}=\rho -\alpha W-(1-\xi _{1})\eta _{1}WN-(1-\xi _{2})\eta _{2}WU-(1-\xi _{3})\eta _{3}WM,\\ \dot{U}=e^{-\hslash _{1}\varphi _{1}}W_{\varphi _{1}}\left[ (1-\xi _{1})\eta _{1}N_{\varphi _{1}}+(1-\xi _{2})\eta _{2}U_{\varphi _{1}}+(1-\xi _{3})\eta _{3}M_{\varphi _{1}}\right] -\left( \lambda +\gamma \right) U,\\ \dot{M}=\lambda e^{-\hslash _{2}\varphi _{2}}U_{\varphi _{2}}-aM-\mu PM,\\ \dot{N}=be^{-\hslash _{3}\varphi _{3}}M_{\varphi _{3}}-\varepsilon N,\\ \dot{P}=\sigma PM-\pi P, \end{array} \right. \end{aligned}$$

(29)

where $\xi _{1}\in [0,1]$ is the efficacy of antiviral therapy in blocking infection by VTC mechanism. Moreover, $\xi _{2}\in [0,1]$ and $\xi _{3} \in [0,1]$ are efficacies of therapy in blocking infection by silent HIV-infected CTC and active HIV-infected CTC mechanisms, respectively [50].

For model (29), the basic HIV reproduction number is given by:

$$\begin{aligned}&\mathfrak {R}_{0(29)}(\xi _{1},\xi _{2},\xi _{3})\\&\quad =\frac{W_{0}e^{-\hslash _{1}\varphi _{1}}\left[ a\varepsilon (1-\xi _{2})\eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left\{ b\eta _{1}(1-\xi _{1})e^{-\hslash _{3}\varphi _{3} }+\varepsilon \eta _{3}(1-\xi _{3})\right\} \right] }{a\varepsilon \left( \gamma +\lambda \right) }. \end{aligned}$$

We let $\xi =\xi _{1}=\xi _{2}=\xi _{3},$ then

$$\begin{aligned} \mathfrak {R}_{0(29)}(\xi )=(1-\xi )\frac{W_{0}e^{-\hslash _{1}\varphi _{1} }\left[ a\varepsilon \eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left( b\eta _{1}e^{-\hslash _{3}\varphi _{3}}+\varepsilon \eta _{3}\right) \right] }{a\varepsilon \left( \gamma +\lambda \right) }. \end{aligned}$$

(30)

Now, we want to determine the minimum drug efficacy that stabilizes the system around the infection-free equilibrium and consequently the viruses will be cleared from the body. Let $\mathfrak {R}_{0(29)}(\xi )\le 1$, then we obtain

$$\begin{aligned}&1-\xi \le \frac{a\varepsilon \left( \gamma +\lambda \right) }{W_{0} e^{-\hslash _{1}\varphi _{1}}\left[ a\varepsilon \eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left( b\eta _{1}e^{-\hslash _{3}\varphi _{3}}+\varepsilon \eta _{3}\right) \right] }\\&\quad \Longrightarrow \xi \ge 1-\frac{a\varepsilon \left( \gamma +\lambda \right) }{W_{0}e^{-\hslash _{1}\varphi _{1}}\left[ a\varepsilon \eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left( b\eta _{1}e^{-\hslash _{3}\varphi _{3} }+\varepsilon \eta _{3}\right) \right] }. \end{aligned}$$

Since $0\le \xi \le 1$, then for $\xi _{(29)}^{\min }<\xi \le 1$, the infection-free equilibrium Đ$_{0}$ of model (29) is G.A.S, where

$$\begin{aligned} \xi _{(29)}^{\min }=\max \left\{ 0,1-\frac{a\varepsilon \left( \gamma +\lambda \right) }{W_{0}e^{-\hslash _{1}\varphi _{1}}\left[ a\varepsilon \eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left( b\eta _{1}e^{-\hslash _{3}\varphi _{3}}+\varepsilon \eta _{3}\right) \right] }\right\} . \end{aligned}$$

(31)

Using the values of the parameters given in Table 2 and choosing the parameters $\varphi _{1}=3,$ $\varphi _{2}=2,$ $\varphi _{3}=1,$ $\eta _{1}=0.003,$ $\eta _{2}=0.00002$, $\eta _{3}=0.001$ and  $\sigma =0.2$. We compute $\xi _{(29)}^{\min }$ as $\xi _{(29)}^{\min }=0.6963$ and then we have the following scenarios:

(i) If $0.6963\le \xi \le 1$, then $\mathfrak {R}_{0(29)}(\xi )\le 1$ and hence Đ$_{0}$ is G.A.S. In this scenario, the virus particles will be removed from the body due to the strength of antiviral treatment.

(ii) If $0\le \xi <0.6963$, then $\mathfrak {R}_{0(29)}(\xi )>1$ and thus one of the other equilibria is G.A.S. This means that the HIV particles cannot be eradicated from the body due to less efficient treatment.

To study the influence of drug efficacies on the HIV dynamics model (29), we consider different values of $\xi$ and solve system (29) under the following initial condition:

$$\begin{aligned} (W(\varphi ),U(\varphi ),M(\varphi ),N(\varphi ),P(\varphi ))=(600,2.5,0.5,4,0.5),\text { where }\varphi \in [-3,0]. \end{aligned}$$

The plots given in Fig. 4 show the solution trajectories of the system with different drug efficacies. We observe that as the drug efficacy is increased, the concentration of healthy CD${{4}^{+}}$ T cells is increased, while the concentrations of silent/active HIV-infected cells, free HIV particles, and HIV-specific CTLs are decreased due to the strength of the usage drugs.

6.3 Effect of time delays on the HIV dynamics

In this part, we vary the delay parameters $\varphi _{1},$ $\varphi _{2},$ $\varphi _{3}$ and fix the parameters $\eta _{1}=0.003,$ $\eta _{2} =0.00002$, $\eta _{3}=0.001,$ $\sigma =0.2$ and $\xi =0$. Since $\mathfrak {R}_{0}$ given by Eq. (30) depends on $\varphi _{1},$ $\varphi _{2}$ and $\varphi _{3}$, then changing the parameters $\varphi _{1},$ $\varphi _{2}$ and $\varphi _{3}$ will change the stability of equilibria. Let us take the following values:

$$\begin{aligned} \begin{array}{ll} \mathbf{(I) } &{} \varphi _{1}=\varphi _{2}=\varphi _{3}=0, \\ \mathbf{(II) } &{} \varphi _{1}=4,\varphi _{2}=3 \, \hbox {and} \, \varphi _{3}=2, \\ \mathbf{(III) } &{} \varphi _{1}=5,\varphi _{2}=4 \,\hbox {and} \, \varphi _{3}=3, \\ \mathbf{(V) } &{} \varphi _{1}=7,\varphi _{2}=6 \, \hbox {and} \, \varphi _{3}=5. \end{array} \end{aligned}$$

With these values, we solve system (29) under initial condition IV-3. The numerical solutions are displayed in Fig. 5. We observe that inclusion of time delays can significantly increase the concentration of healthy CD4$^{+}$ T cells and reduce the concentrations of other compartments.

In Table 3, we present the values $\mathfrak {R}_{0(29)}$ for selected values of $\varphi _{1},$ $\varphi _{2}$ and $\varphi _{3}$. It is clear that, $\mathfrak {R}_{0(29)}$ is decreased when $\varphi _{1},$ $\varphi _{2}$ and $\varphi _{3}$ are increased and the stability of Đ$_{0}$ will be changed. Now, we want to calculate the critical value of the time delay that changes the stability of Đ$_{0}$. To do so, we fix the parameters $\varphi _{2}$ and $\varphi _{3}$ and write $\mathfrak {R}_{0(29)}$ as a function of $\varphi _{1}$ as:

$$\begin{aligned} \mathfrak {R}_{0(29)}(\varphi _{1})=\frac{W_{0}e^{-\hslash _{1}\varphi _{1} }\left[ a\varepsilon \eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left( b\eta _{1}e^{-\hslash _{3}\varphi _{3}}+\varepsilon \eta _{3}\right) \right] }{a\varepsilon \left( \gamma +\lambda \right) }. \end{aligned}$$

When $\mathfrak {R}_{0(29)}\left( \varphi _{1}\right) \le 1$, we obtain

$$\begin{aligned} \varphi _{1}\ge \varphi _{1}^{\min },\text { where }\varphi _{1}^{\min } =\max \left\{ 0,\frac{1}{\hslash _{1}}\ln \frac{W_{0}\left\{ a\varepsilon \eta _{2}+\lambda e^{-\hslash _{2}\varphi _{2}}\left( b\eta _{1}e^{-\hslash _{3}\varphi _{3}}+\varepsilon \eta _{3}\right) \right\} }{a\varepsilon \left( \gamma +\lambda \right) }\right\} . \end{aligned}$$

Therefore, if $\varphi _{1}\ge \varphi _{1}^{\min }$, then Đ$_{0}$ is G.A.S. Let $\varphi _{2}=5$ and $\varphi _{3}=4$ and compute $\varphi _{1}^{\min }$ as $\varphi _{1}^{\min }=2.22266$. It follows that:

• (i) If $\varphi _{1}\ge 2.22266$, then $\mathfrak {R}_{0(29)}(\varphi _{1})\le 1$ and Đ$_{0}$ is G.A.S.

• (ii) If $\varphi _{1}<2.22266$, then $\mathfrak {R}_{0(29)}(\varphi _{1})>1$ and Đ$_{0}$ will lose its stability and one of the other equilibria will be G.A.S.

We observe that the time delay has a similar effect as the drug efficacy. This gives some impression to develop a new class of treatment which causes an increase in the delay period and then suppress the HIV replication.

Table 3 The variation of $\mathfrak {R}_{{0}(29)}$ with respect to the delay parameters

Fig. 6

The evolution of HIV dynamics (29) under different modes of transmission

6.4 Effects of CTC transmission

Here, we investigate the influence of different modes of transmission on the HIV dynamics (29). We use the parameters given in Table 2 and choose the values $\sigma =0.05$, $\varphi _{1}=3,$ $\varphi _{2}=2$, $\varphi _{3}=1$ and $\xi =0$ with the following initial condition:

IV-4. $(W(\varphi ),U(\varphi ),M(\varphi ),N(\varphi ),P(\varphi ))=(600,10,2,0.5,1)$, where $\varphi \in [-3,0].$

We choose three sets of parameters $\eta _{1},$ $\eta _{2}$, $\eta _{3}$ and investigate the following illustrative cases:

Case 1: HIV dynamics with VTC, silent HIV-infected CTC and active HIV-infected CTC transmissions: Here, we consider the parameters $\eta _{1}=0.005,$ $\eta _{2}=0.002$ and $\eta _{3}=0.003$. Figure 6 shows that the solutions of the system approach the equilibrium Đ$_{4}=\left( 151.64,15.71,2,3.70,2.77\right)$.

Case 2: HIV dynamics with VTC, silent HIV-infected CTC and active HIV-infected CTC transmissions: In this case, we choose the parameters $\eta _{1}=0.004,$ $\eta _{2}=0.001$ and $\eta _{3}=0.002$. We can see from Fig. 6 that the trajectories of the system tend to the equilibrium Đ$_{4}=\left( 232.38,14.22,2,3.70,2.26\right)$.

Case 3: HIV dynamics with both VTC and active HIV-infected CTC transmissions: In this case, we select the values $\eta _{1}=0.003,$ $\eta _{2}=0$ and $\eta _{3}=0.001$. From Fig. 6, we observe that the solution trajectories converge to the equilibrium Đ$_{3}=\left( 432.67,10.51,2,3.70,1.02\right)$.

Case 4: HIV dynamics with only VTC transmission: Here, we consider the values $\eta _{1}=0.002$ and $\eta _{2}=\eta _{3}=0$. Figure 6 displays that the solution trajectories approach the equilibrium Đ$_{3}=\left( 574.44,7.88,2,3.70,0.14\right)$.

Case 5: HIV dynamics with only VTC transmission: In this situation, we pick the parameters $\eta _{1}=0.001$ and $\eta _{2}=\eta _{3}=0$. It is clear from Fig. 6 that the solution trajectories reach the equilibrium Đ$_{0}=\left( 1000,0,0,0,0\right)$.

From the above discussion, we note that the presence of silent HIV-infected CTC and/or active HIV-infected CTC transmissions increases the infection rate. As a result, the concentration of healthy CD${{4}^{+}}$ T cells is decreased, while the concentrations of silent/active HIV-infected cells, free HIV particles and HIV-specific CTLs are increased as shown in Fig. 6.

7 Conclusion and discussion

In this paper, we formulated and analyzed an HIV dynamics model with CTL-mediated immunity. We incorporated both VTC and CTC transmissions. We assumed that the CTC infection has two sources, (i) the contact between healthy CD${{4}^{+}}$ T cells and silent HIV-infected cells, and (ii) the contact between healthy CD${{4}^{+}}$ T cells and active HIV-infected cells. We incorporated three types of distributed-time delays into the model. We established that the model is well posed by proving that the solutions of the model are nonnegative and bounded. We calculated the three possible equilibria of the model, the infection-free equilibrium, Đ$_{0}$, the chronic HIV infection equilibrium with inactive CTL-mediated immune response, Đ$_{1}$, and the chronic HIV infection equilibrium with active CTL-mediated immune response, Đ$_{2}$. The existence and global stability of the three equilibria are governed by two threshold parameters, $\mathfrak {R}_{0}$ (the basic HIV reproduction number) and $\mathfrak {R}_{1}$ (the HIV specific CTL-mediated immunity reproduction number). The global asymptotic stability of the three equilibria Đ$_{0}$, Đ$_{1}$ and Đ$_{2}$ was investigated by constructing Lyapunov functionals and utilizing LaSalle’s invariance principle. To illustrate the theoretical results, we performed some numerical simulations. We developed the model to take into account three types of antiviral drugs. We showed that the inclusion of CTC transmission decreases the concentration of healthy CD4$^{+}$ T cells and increases the concentrations of infected cells and free HIV particles. On the other side, the inclusion of time delay can significantly suppress the HIV replication and increase the concentration of healthy CD4$^{+}$ T cells.

We observe that the inclusion of silent HIV-infected CTC and active HIV-infected CTC transmissions into the HIV infection model increases the basic HIV reproduction number $\mathfrak {R}_{0(29)}$, since $\mathfrak {R}_{0(29)}=\mathfrak {R}_{01(29)}+\mathfrak {R}_{02(29)} +\mathfrak {R}_{03(29)}>\mathfrak {R}_{01(29)}$. Therefore, neglecting the CTC transmission will lead to under-evaluated basic reproduction number. When the CTC is neglected then model (29) leads to the following form:

$$\begin{aligned} \left\{ \begin{array} [c]{l} \dot{W}=\rho -\alpha W-(1-\xi )\eta _{1}WN,\\ \dot{U}=(1-\xi )e^{-\hslash _{1}\varphi _{1}}\eta _{1}W_{\varphi _{1}} N_{\varphi _{1}}-\left( \lambda +\gamma \right) U,\\ \dot{M}=\lambda e^{-\hslash _{2}\varphi _{2}}U_{\varphi _{2}}-aM-\mu PM,\\ \dot{N}=be^{-\hslash _{3}\varphi _{3}}M_{\varphi _{3}}-\varepsilon N,\\ \dot{P}=\sigma PM-\pi P, \end{array} \right. \end{aligned}$$

(32)

and the basic HIV reproduction number is given by

$$\begin{aligned} \mathfrak {R}_{0(32)}(\xi )=(1-\xi )\frac{W_{0}\lambda b\eta _{1}e^{-(\hslash _{1}\varphi _{1}+\hslash _{2}\varphi _{2}+\hslash _{3}\varphi _{3})}}{a\varepsilon \left( \gamma +\lambda \right) }. \end{aligned}$$

Then, the infection-free equilibrium Đ$_{0}$ can be stabilized for $\xi _{(32)}^{\min }<\xi \le 1$, where

$$\begin{aligned} \xi _{(32)}^{\min }=\max \left\{ 0,1-\frac{a\varepsilon \left( \gamma +\lambda \right) }{W_{0}\lambda b\eta _{1}e^{-(\hslash _{1}\varphi _{1}+\hslash _{2}\varphi _{2}+\hslash _{3}\varphi _{3})}}\right\} . \end{aligned}$$

(33)

Comparing Eqs. (31) and (33), we get that $\xi _{(32)}^{\min }\le \xi _{(29)}^{\min }$. Therefore, if we apply drugs with $\xi$ such that $\xi _{(32)}^{\min }\le \xi <\xi _{(29)}^{\min }$, this guarantees that $\mathfrak {R}_{0(32)}(\xi )\le 1$ and then Đ$_{0}$ of system (32) is G.A.S, however, $\mathfrak {R}_{0(29)}>1$ and then Đ$_{0}$ of system (29) is unstable. Therefore, more accurate drug efficacy which is required to clear the HIV from the body is calculated by using our proposed model. This shows the importance of considering the effect of CTC transmission in the HIV dynamics. Consequently, this observation sheds a light on the great importance of considering the influence of CTC transmission in the HIV dynamics. Our proposed model (3) can be extended by incorporating age structure of the infected cells or diffusion [51,52,53,54]. Looking ahead to further developments an interesting perspective would be introducing a stochastic internal variable, as in [55], to account for virus mutations which generally develops by a stochastic dynamics. Moreover, since the exact analytical solution of our proposed HIV dynamics model is not known, therefore, we can only obtain an approximate solution of the model. Therefore, the discrete-time version of the model needs to be investigated (see, e.g., [56,57,58]). These extensions, indeed, require more investigations; therefore, we leave it for future works.