## 3 Numerical Results and Simulations  
In order to perform the numerical simulations, system (1) will be solved numerically using the fourth order Runge-Kutta iterative scheme. The parameter values or ranges used are presented in Table 1.  
Figure 2 shows the behavior of infection during the first 60 days. For the parameters used in this figure, the basic reproduction number is \(R_{0}=0.2209<1\); we clearly see that the solutions of the system converge to the disease-free equilibrium point \(E_{f}=(827.22,0,0,0,0)\). This numerical result is consistent with the theoretical result concerning the stability of \(E_{f}\). In addition, the plots showing the stability of this disease-free point in phase plane are given in Figure 3.  
Figure 4 shows the behavior of the infection for the first sixty days. The chosen parameters in this figure ensure that the basic reproduction number is greater than unity (\(R_{0}=11.049>1\)) and the immune response reproduction number is less than unity (\(R_{CTL}=3.596\times 10^{-1}<1\)). It is clearly seen that, in this case, all the solutions converge towards the CTL-free endemic equilibrium \(E_{1}=(19.96,5.98\times 10^{-1},1.14,199.78,0)\) that agrees with our theoretical finding concerning the stability of \(E_{1}\). Moreover, the plots showing the stability of this CTL-free endemic point in phase plane are given in Figure 5.  
Finally, Figure 6 illustrates the disease dynamics when both the basic reproduction number and the immune response reproduction number are greater than unity. Indeed, for the chosen parameters in this figure, we have \(R_{0}=11.049>1\) and \(R_{CTL}=4.13>1\). It can be seen that the infection persists and the convergence towards the infection steady state \(E_{3}=(285.12,6.55,3.33,555.55,660.86)\) is observed. This numerical result is in good agreement with the theoretical result concerning the stability of the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Meaning** & **Value** & **References** \\ \hline \(\lambda\) & Source rate of CD\({}^{+}\) T cells & \([0,10]\) & [16] \\ \(k_{1}\) & Average of infection & \([2.5\times 10^{-4},0.5]\) & [14] \\ \(d_{1}\) & Decay rate of healthy cells & \(0.0139\) & [14] \\ \(d_{2}\) & Death rate of exposed CD\({}^{+}\) T cells & \(0.0495\) & [14] \\ \(k_{2}\) & The rate that exposed become infected CD\({}^{4}\) T cells & \(1.1\) & [14] \\ \(d_{3}\) & Death rate of infected CD\({}^{4}\) T cells, not by CTL killing & \(0.5776\) & [14] \\ \(a\) & The rate of production the virus by infected CD\({}^{4}\) T cells & \([2,1250]\) & [14] \\ \(d_{4}\) & Clearance rate of virus & \([0.3466,2.4]\) & [14] \\ \(p\) & Clearance rate of infection & \(0.0024\) & [17] \\ \(c\) & Activation rate CTL cells & \(0.15\) & [17] \\ \(b\) & Death rate of CTL cells & \(0.5\) & [17] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters, their symbols and default values used in the suggested HIV model. last endemic equilibrium. The plots showing the stability of this final endemic point in phase plane are given in Figure 7. In these two last figures, we clearly observe the importance of CTL immune response in maximizing the healthy cells and reducing the viral load. It is also noteworthy that the CTLs do not vanish in this last case, which means that CTL cells will be always present when the infection becomes chronic. Figure 2: The behavior of the infection dynamics for \(\lambda=10\), \(d_{1}=0.0139\), \(k_{1}=0.04\), \(d_{2}=0.0495\), \(k_{2}=1.1\), \(d_{3}=0.5776\), \(a=2\), \(d_{4}=0.6\), \(p=0.0024\), \(c=0.15\), \(b=0.5\). For the parameters used in this figure, the basic reproduction number is \(R_{0}=0.2209<1\). Solutions of the system converge to the disease-free equilibrium point \(E_{f}=(827.22,0,0,0,0)\). Figure 4: _Cont._  
Figure 3: Selective phase portraits to illustrate the solution behavior of the infection dynamics when \(\lambda=10\), \(d_{1}=0.0139\), \(k_{1}=0.04\), \(d_{2}=0.0495\), \(k_{2}=1.1\), \(d_{3}=0.5776\), \(a=2\), \(d_{4}=0.6\), \(p=0.0024\), \(c=0.15\), \(b=0.5\).