## 1 Introduction  
Human immunodeficiency virus (HIV) is known as a pathogen causing the acquired immunodeficiency syndrome (AIDS), which is the end-stage of the infection. After that, the immune system fails to play its life-sustaining role [1; 2]. Indeed, it is well known that the cellular immune responses play an indispensable role during the HIV viral infection, especially in people known as elite controllers who are infected by HIV but maintain a normal CD4 count for many years and remain asymptomatic or have a very delayed disease progression over the course of their HIV infection [3; 4].  
The basic viral infection model with CTL's immune was first studied in [5] and its global stability was studied in [6; 7]. A prevailing theory suggests that CTL cells are responsible for the sudden decrease HIV virus load and the subsequent quasi-steady state level of the viral load after a sharp increase during the primary infection phase [8; 9; 10; 11]. In [12], the authors consider the interaction between CTL, viruses and macrophages and study the global stability in a model with a mass action infection term. With a saturated infection term, the same mathematical question was tackled in [13] with a model consisting of four ordinary differential equations with a Beddington-DeAngelis infection term.  
More recently, the model describing the interaction between the HIV viruses, CD4\({}^{+}\) T cells, infected cells with a standard incidence rate function describing saturated viral infection is formulated and studied in [14]. The authors study the local and global stability of the endemic states and illustrate that their model can fit well with some clinical HIV infection data sets. In this paper, we extend their work by incorporating the cytotoxic T lymphocytes (CTL) immune response. The dynamics of HIV infection with CTL response that we consider is given by the following nonlinear system of differential equations:  
\[\left\{\begin{aligned} \dot{x}&=\lambda-d_{1}x-\frac{k_{1}xv}{x+v}, \\ \dot{s}&=\frac{k_{1}xv}{x+v}-d_{2}s-k_{2}s,\\ \dot{y}&=k_{2}s-d_{3}y-pyz,\\ \dot{v}&=ay-d_{4}v,\\ \dot{z}&=cyz-bz,\end{aligned}\right. \tag{1}\]  
with the initial conditions \(x(0)=x_{0}\), \(s(0)=s_{0}\), \(y(0)=y_{0}\), \(v(0)=v_{0}\) and \(z(0)=z_{0}\).  
In this model, \(x\), \(y\), \(s\), \(v\) and \(z\) denote the concentration of uninfected cells, infected cells, exposed cells, free virus and CTL cells, respectively. Susceptible host cells CD4\({}^{+}\) T cells are produced at a rate \(\lambda\), die at a rate \(d_{1}x\) and become infected by virus at a rate \(\frac{k_{1}xv}{x+v}\). Exposed cells die at a rate \(d_{2}s\) and become infected at a rate of \(k_{2}s\). Infected cells increase at rate \(k_{2}s\) and die at rate \(d_{3}y\) and are killed by the CTL response at a rate \(pyz\). Free virus is produced by infected cells at a rate \(ay\) and decays at a rate \(d_{4}v\). Finally, CTLs expand in response to viral antigen derived from infected cells at a rate \(cyz\) and decay in the absence of antigenic stimulation at a rate \(bz\). Note that this model (1) employs the more realistic standard incidence rate function \(\frac{k_{1}xv}{x+v}\) as in [14; 15], which better describes the rate of viral infection. The schematic representing the viral dynamics of the problem under consideration is illustrated in Figure 1.  
The rest of the paper is organized as follows. In Section 2, we study the positivity and boundedness of solutions. We present the steady states and give a local stability result in Section 3. Section 4 is dedicated to the global stability results. Numerical simulations are performed in Section 5 and we conclude the paper in the last section.