## 2 Mathematical Analysis of the HIV Model  
### Existence and Local Stability  
For the problems deal with cell population evolution, the cell densities should remain non-negative and bounded. In this section, we will establish the positivity and boundedness of solutions of the  
Figure 1: Schematic of the model under consideration. model (1). First of all, for biological reasons, the parameters \(x_{0}\), \(s_{0}\), \(y_{0}\), \(v_{0}\) and \(z_{0}\) must be larger than or equal to 0. Hence, we have the following result:  
**Proposition 1**. : _For any non-negative initial conditions \((x_{0},s_{0},y_{0},v_{0},z_{0})\), system (1) has a unique solution. Moreover, this solution is non-negative and bounded for all \(t\geq 0\)._  
Proof. : The proof is given in Appendix A. Moreover, system (1) has an infection-free equilibrium \(E_{f}=(\dfrac{\lambda}{d_{1}},0,0,0,0)\), corresponding to the maximal level of healthy CD4\({}^{+}\) T-cells. The basic reproduction number of our problem is given as follows:  
\[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]  
where \(\dfrac{k_{2}}{d_{2}+k_{2}}\) is the proportion of exposed cells to become productively infected cells, \(\dfrac{a}{d_{3}}\) is the number of free virus production by an infected cell and \(\dfrac{1}{d_{4}}\) is the average life of virus. From a biological point of view, \(R_{0}\) stands for the average number of secondary infections generated by one infected cell when all cells are susceptibles. Depending on the value of this basic reproduction number \(R_{0}\); in other words, depending on these three biological proportions, we will study the stability of the free-endemic and the endemic equilibria. In addition to the disease free equilibrium, system (1) has three endemic equilibria. The first of them is \(E_{1}=(x_{1},s_{1},y_{1},v_{1},z_{1})\), where  
\[x_{1} =\dfrac{\lambda}{d_{1}+k_{1}(1-\frac{1}{R_{0}})},\] \[s_{1} =\dfrac{k_{1}\lambda R_{0}(1-\frac{1}{R_{0}})}{(d_{2}+k_{2})(d_{1 }+k_{1}(1-\frac{1}{R_{0}}))((1-\frac{1}{R_{0}})R_{0}+1)},\] \[y_{1} =\dfrac{d_{4}\lambda R_{0}(1-\frac{1}{R_{0}})}{ad_{1}+ak_{1}(1- \frac{1}{R_{0}})},\] \[v_{1} =\dfrac{\lambda R_{0}(1-\frac{1}{R_{0}})}{d_{1}+k_{1}(1-\frac{1}{ R_{0}})},\] \[z_{1} =0.\]  
The second endemic steady state is \(E_{2}=(x_{2},s_{2},y_{2},v_{2},z_{2})\), where  
\[x_{2} =\dfrac{-abd_{1}-abk_{1}+\lambda cd_{4}+\sqrt{A}}{2 cd_{1}d_{4}},\] \[s_{2} =\dfrac{d_{3}R_{0}}{k_{2}}\dfrac{b(-abd_{1}-abk_{1}+\lambda cd_{ 4}+\sqrt{A})}{c(abd_{1}-abk_{1}+\lambda cd_{4}+\sqrt{A})},\] \[y_{2} =\dfrac{b}{c},\] \[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]0 \[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]1The third endemic steady state is \(E_{3}=(x_{3},y_{3},v_{3},z_{3})\), where  
\[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]2 \[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]3 \[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]4 \[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]5 \[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]6  
with \(A=(abk_{1}-\lambda cd_{4})^{2}+a^{2}b^{2}d_{1}^{2}+2a^{2}b^{2}d_{1}k_{1}+2 \lambda abcd_{1}d_{4}\). The last endemic steady-state \(E_{3}\) will not be taken under consideration since \(x_{3}<0\), which is not possible biologically. On the other hand, from the components of \(E_{1}\), it is clear that, when \(R_{0}>1\), this endemic point exists. In what follows, we will make use of the following CTL immune response reproduction number \(R_{CTL}\) to classify the model system dynamics  
\[R_{0}=k_{1}\dfrac{k_{2}}{d_{2}+k_{2}}\dfrac{a}{d_{3}}\dfrac{1}{d_{4}}, \tag{2}\]7  
depending on the value of this CTL immune reproduction number \(R_{CTL}\); in other words, depending on \(R_{0}\) and the parameters describing the effectiveness of CTL like \(c\) and \(b\), we will study the stability of the endemic equilibria.