# A Modified Iterative Algorithm for Numerical Investigation of HIV Infection Dynamics
## 4 Results and Discussions  
We take a scenario of [PERSON] et al. [29], where the initial conditions are \(x_{0}=0.1\), \(y_{0}=0\), \(z_{0}=0.1\), with parameter values \(p=0.1\), \(\alpha=0.02\), \(\beta=0.3\), \(r=3\), \(q=2.4\), \(k=1500\), \(k=0.0027\), \(N=10\). All our calculations as well as our graphs are carried out by Maple 2020. The Maple built-in programme has been used for RK4 solutions with step size \(\Delta t=0.001\). The new iterative method (NIM) and modified new iterative method's (MNIM) simulation results are illustrated graphically and numerically. We displayed the graphical outcomes of both NIM and MNIM strategies related to the RK4 method in Figure 1. The figure depicts that the 5-iterations of NIM solutions fail to provide suitable accuracy as time increases and solutions are only valid till \(t\leq 1\), whereas the 2-iterations of modified NIM precisions agree well with those of RK4 precisions. Additionally, we contrasted the absolute errors of the proposed MNIM scheme to those of other conventional methods, such as cGP(2) and LWCM [29], GA-IPA and GA-ASA [28], LADM-Pade [21], MVIM [26], HPM [25], Bessel collocation [23] and NIM relative to RK4 method, as shown in Tables 2-4 for \(X(t),\ Y(t),Z(t)\), respectively. The recent results via cGP(2) and LWCM [29] match to at least four decimal places with RK4. Further, the numeric solutions via NIM matched at least one decimal place with RK4. In contrast, the results show that the solutions by 2-iterations of MNIM match with those of RK4 at least eight decimal places. The comparisons with other methods used for the model clearly indicate that the MNIM scheme outperformed all the pre-existing solutions. Therefore, it suggests the high accuracy and validity of the proposed MNIM scheme with fewer iterative steps. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**LWCM**} & \multicolumn{2}{c}{**c} & \multicolumn{2}{c}{**c} & \multicolumn{2}{c}{**LADM-Pad\(\acute{\acute{e}}\)**} & \multicolumn{2}{c}{**Bessel**} & \multicolumn{2}{c}{**NIM**} & \multicolumn{2}{c}{**MNMIM**} \\ \(t\) & [29] & [29] & **GA-IPA [28]** & **GA-ASA [28]** & [21] & **MVIM [26]** & **HPM [25]** & **Bessel** & **NIM** & **HPM [25]** \\ \hline
0.2 & \(7.50\times 10^{-6}\) & \(5.81\times 10^{-6}\) & \(1.32\times 10^{-3}\) & \(1.32\times 10^{-3}\) & \(7.77\times 10^{-5}\) & \(7.85\times 10^{-5}\) & \(7.78\times 10^{-5}\) & \(4.87\times 10^{-3}\) & \(6.979\times 10^{-6}\) & \(1\times 10^{-10}\) \\
0.4 & \(2.70\times 10^{-5}\) & \(2.71\times 10^{-5}\) & \(1.06\times 10^{-5}\) & \(1.06\times 10^{-5}\) & \(1.65\times 10^{-5}\) & \(3.00\times 10^{-4}\) & \(1.95\times 10^{-4}\) & \(2.56\times 10^{-2}\) & \(0.0000535\) & \(7\times 10^{-10}\) \\
0.6 & \(7.34\times 10^{-5}\) & \(5.74\times 10^{-5}\) & \(5.69\times 10^{-5}\) & \(1.69\times 10^{-3}\) & \(2.43\times 10^{-3}\) & \(8.49\times 10^{-4}\) & \(1.10\times 10^{-3}\) & \(6.81\times 10^{-2}\) & \(0.0006497\) & \(5.03\times 10^{-9}\) \\
0.8 & \(1.77\times 10^{-4}\) & \(1.38\times 10^{-4}\) & \(3.83\times 10^{-3}\) & \(3.83\times 10^{-3}\) & \(3.46\times 10^{-2}\) & \(2.14\times 10^{-3}\) & \(1.29\times 10^{-2}\) & \(1.36\times 10^{-1}\) & \(0.01457\) & \(2.14\times 10^{-8}\) \\
1 & \(3.98\times 10^{-4}\) & \(3.98\times 10^{-4}\) & \(4.19\times 10^{-3}\) & \(4.18\times 10^{-3}\) & \(2.58\times 10^{-1}\) & \(5.14\times 10^{-3}\) & \(7.89\times 10^{-2}\) & \(2.04\times 10^{-1}\) & \(0.1816\) & \(7.92\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of Absolute Errors for \(X(t)\) of MNIM Scheme and LWCM, cGT(P2), GA-IPA, GA-ASA, LADM-Pad\(\acute{e}\), MVIM, HPM, Bessel collocation, NMIM relative to RX4 Method for HIV Infection Model.