Here, our primary objective is to illustrate numerical method to demonstrate the solution pathways of the recommended system. These numerical schemes will be utilized to visualize the dynamics of the system and to illustrate the chaotic phenomena of the system.Solution through Caputo-FabrizioThe solution analysis of a fractional system plays a pivotal role in enhancing our understanding of system behavior, validating models, making predictions, optimizing performance, and informing decision-making across diverse fields of study. There are many numerical methods available, but we will use the method developed in37 to analyze our recommended fractional system (4) for HIV infection. We start by looking at the first equation of our system:$$\begin{aligned} \mathcal {J}_1(t)-\mathcal {J}_1(0)= \frac{1-\upsilon }{\mathcal {W}(\upsilon )} \textrm{V}_1 (t,\mathcal {J}_1)+\frac{\upsilon }{\mathcal {W}(\upsilon )} \int _0^t \textrm{V}_1 (\chi , \mathcal {J}_1) d \chi . \end{aligned}$$
(13)
Let \(t=t_{\ell +1}, \ell =0,1, \dots ,\) so we obtain$$\begin{aligned} \mathcal {J}_1(t_{\ell +1})-\mathcal {J}_{1}(0)= \frac{1-\upsilon }{\mathcal {W}(\upsilon )} \textrm{V}_1 (t_\ell ,\mathcal {J}_1(t_\ell ))+\frac{\upsilon }{\mathcal {W}(\upsilon )} \int _0^{t_{\ell +1}} \textrm{V}_1 (t, \mathcal {J}_1) d t, \end{aligned}$$
(14)
and$$\begin{aligned} \mathcal {J}_1(t_{\ell })-\mathcal {J}_{1}(0)= \frac{1-\upsilon }{\mathcal {W}(\upsilon )} \textrm{V}_1 (t_{\ell -1},\mathcal {J}_1(t_{\ell -1}))+\frac{\upsilon }{\mathcal {W}(\upsilon )} \int _0^{t_{\ell }} \textrm{V}_1 (t, \mathcal {J}_1) d t. \end{aligned}$$
(15)
The successive terms difference is stated below$$\begin{aligned} \mathcal {J}_{1_{\ell +1}}-\mathcal {J}_{1_\ell }= \frac{1-\upsilon }{\mathcal {W}(\upsilon )} \bigg (\textrm{V}_1 (t_\ell ,\mathcal {J}_{1_\ell })-\textrm{V}_1 (t_{\ell -1},\mathcal {J}_{1_{\ell -1}}) \bigg )+\frac{\upsilon }{\mathcal {W}(\upsilon )} \int _\ell ^{t_{\ell +1}} \textrm{V}_1 (t, \mathcal {J}_1) d t. \end{aligned}$$
(16)
Now we approximate the function \(\textrm{V}_1 (t, \mathcal {J}_1)\) in the time interval \([t_\kappa ,t_{\kappa +1}]\) by utilizing interpolation polynomial and obtain$$\begin{aligned} \mathcal {P}_\kappa (t)\cong \frac{\textrm{V}_1 (t_\kappa , \mathcal {J}_\kappa )}{q} (t-t_{\kappa -1})- \frac{\textrm{V}_1 (t_{\kappa -1}, \mathcal {J}_{\kappa -1})}{q} (t-t_{\kappa }), \end{aligned}$$
(17)
where q represent the time spent and \(q=t_\ell -t_{\ell -1}\). The above stated \(\mathcal {P}_\kappa (t)\) is utilized to obtain$$\begin{aligned} \int _\ell ^{t_{\ell +1}} \textrm{V}_1 (t, \mathcal {J}_1) d t&= \int _\ell ^{t_{\ell +1}} \bigg ( \frac{\textrm{V}_1 (t_\ell , \mathcal {J}_{1_\ell })}{q} (t-t_{\ell -1})-\frac{\textrm{V}_1 (t_{\ell -1}, \mathcal {J}_{1_{\ell -1}})}{q} (t-t_{\ell }) \bigg ) dt,\nonumber \\&=\frac{3q}{2} \textrm{V}_1 (t_\ell , \mathcal {J}_{1_\ell })-\frac{q}{2} \textrm{V}_1 (t_{\ell -1}, \mathcal {J}_{1_{\ell -1}}). \end{aligned}$$
(18)
Here, putting Eq. (29) in Eq. (16), yield the following result$$\begin{aligned} \mathcal {J}_{1_{\ell +1}}&= \mathcal {J}_{1_\ell }+\bigg ( \frac{1-\upsilon }{\mathcal {W}(\upsilon )}+\frac{3\upsilon q}{2 \mathcal {W} (\upsilon )} \bigg ) \textrm{V}_1 (t_\ell , \mathcal {J}_{1_\ell }) \nonumber \\&\quad – \bigg ( \frac{1-\upsilon }{\mathcal {W}(\upsilon )}+\frac{\upsilon q}{2 \mathcal {W} (\upsilon )} \bigg )\textrm{V}_1 (t_{\ell -1}, \mathcal {J}_{1_{\ell -1}}), \end{aligned}$$
(19)
which is required solution for the first equation of the system. In the same way, we can determine for the second and third equation of (4) given by$$\begin{aligned} \mathcal {J}_{2_{\ell +1}}&= \mathcal {J}_{2_\ell }+\bigg ( \frac{1-\upsilon }{\mathcal {W}(\upsilon )}+\frac{3\upsilon q}{2 \mathcal {W} (\upsilon )} \bigg ) \textrm{V}_2 (t_\ell , \mathcal {J}_{2_\ell }) \nonumber \\&\quad – \bigg ( \frac{1-\upsilon }{\mathcal {W}(\upsilon )}+\frac{\upsilon q}{2 \mathcal {W} (\upsilon )} \bigg )\textrm{V}_2 (t_{\ell -1}, \mathcal {J}_{2_{\ell -1}}), \end{aligned}$$
(20)
and$$\begin{aligned} \mathcal {J}_{3_{\ell +1}}&= \mathcal {J}_{3_\ell }+\bigg ( \frac{1-\upsilon }{\mathcal {W}(\upsilon )}+\frac{3\upsilon q}{2 \mathcal {W} (\upsilon )} \bigg ) \textrm{V}_3 (t_\ell , \mathcal {J}_{3_\ell }) \nonumber \\&\quad – \bigg ( \frac{1-\upsilon }{\mathcal {W}(\upsilon )}+\frac{\upsilon q}{2 \mathcal {W} (\upsilon )} \bigg )\textrm{V}_3 (t_{\ell -1}, \mathcal {J}_{3_{\ell -1}}). \end{aligned}$$
(21)
The two-step Adams-Bashforth approach (ABA) used in this method for the CF takes into consideration the nonlinearity of the kernel as well as the exponential decay rule for the CF.Solution through Atangana-BaleanuHere, we will represent the solution pathways of our system (5) of HIV infection. We initially adopt the following fractional system to develop the necessary numerical method for our fractional model as follows:$$\begin{aligned} ^{{ABC}}_{0}D^{\upsilon }_{t}\mathcal {J}(t)=f(t,\mathcal {J}(t)), \end{aligned}$$using the theory of fractional calculus, we attain$$\begin{aligned} \mathcal {J}(t)-\mathcal {J}(0)= \frac{1-\upsilon }{ABC(\upsilon )} f (t,\mathcal {J}(t))+\frac{\upsilon }{ABC(\upsilon ) \gimel (\upsilon )} \int _0^t (t-\varkappa )^{\upsilon -1} f (\varkappa , \mathcal {J}(\varkappa )) d \varkappa . \end{aligned}$$
(22)
Take \(t=t_\zeta\), then we obtain$$\begin{aligned} \mathcal {J}(t_\zeta )-\mathcal {J}(0)= \frac{1-\upsilon }{ABC(\upsilon )} f (t_{\zeta -1},\mathcal {J}(t_{\zeta -1}))+\frac{\upsilon }{ABC(\upsilon ) \gimel (\upsilon )} \int _0^{t_\zeta } (t_\zeta -\varkappa )^{\upsilon -1} f (\varkappa , \mathcal {J}(\varkappa )) d \varkappa . \end{aligned}$$
(23)
and for \(t_{\zeta +1}\), we get$$\begin{aligned} \mathcal {J}(t_{\zeta +1})-\mathcal {J}(0)= \frac{1-\upsilon }{ABC(\upsilon )} f (t_{\zeta +1},\mathcal {J}(t_{\zeta +1}))+\frac{\upsilon }{ABC(\upsilon ) \gimel (\upsilon )} \int _0^{t_{\zeta +1}} (t_{\zeta +1}-\varkappa )^{\upsilon -1} f (\varkappa , \mathcal {J}(\varkappa )) d \varkappa . \end{aligned}$$
(24)
We get the difference for above equation as$$\begin{aligned} \mathcal {J}(t_{\zeta +1})-\mathcal {J}(t_\zeta )&= \frac{1-\upsilon }{ABC(\upsilon )} \bigg [ f (t_{\zeta },\mathcal {J}(t_{\zeta }))-f (t_{\zeta -1},\mathcal {J}(t_{\zeta -1})) \bigg ]+ \frac{\upsilon }{ABC(\upsilon ) \gimel (\upsilon )} \nonumber \\&\int _0^{t_{\zeta +1}} (t_{\zeta +1}-\varkappa )^{\upsilon -1} f (\varkappa , \mathcal {J}(\varkappa )) d \varkappa -\int _0^{t_{\zeta }} (t_{\zeta }-\varkappa )^{\upsilon -1} f (\varkappa , \mathcal {J}(\varkappa )) d \varkappa . \end{aligned}$$
(25)
$$\begin{aligned} \mathcal {J}(t_{\zeta +1})-\mathcal {J}(t_\zeta )&= \frac{1-\upsilon }{ABC(\upsilon )} \bigg [ f (t_{\zeta },\mathcal {J}(t_{\zeta }))-f (t_{\zeta -1},\mathcal {J}(t_{\zeta -1})) \bigg ]+ B_{\upsilon ,1}-B_{\upsilon , 2}. \end{aligned}$$
(26)
where$$\begin{aligned} B_{\upsilon ,1}= \frac{\upsilon }{ABC(\upsilon ) \gimel (\upsilon )} \int _0^{t_{\zeta +1}} (t_{\zeta +1}-\varkappa )^{\upsilon -1} f (\varkappa , \mathcal {J}(\varkappa )) d \varkappa . \end{aligned}$$Using approximation we obtain$$\begin{aligned} P(t)\cong \frac{f (t_\zeta , \mathcal {J}_\zeta )}{q} (t-t_{\zeta -1})- \frac{f (t_{\zeta -1}, \mathcal {J}_{\zeta -1})}{q} (t-t_{\zeta }), \end{aligned}$$
(27)
where \(q=t_\ell -t_{\ell -1}\). Then, we get the following$$\begin{aligned} B_{\upsilon ,1}&= \frac{\upsilon }{ABC(\upsilon ) \gimel (\upsilon )} \int _0^{t_{\zeta +1}} (t_{\zeta +1}-\varkappa )^{\upsilon -1} \bigg [ \frac{f (t_\zeta , \mathcal {J}_\zeta )}{q} (t-t_{\zeta -1})- \frac{f (t_{\zeta -1}, \mathcal {J}_{\zeta -1})}{q} (t-t_{\zeta }) \bigg ] d \varkappa \nonumber \\&= \frac{\upsilon }{ABC(\upsilon )\gimel (\upsilon )} \int _0^{t_{\zeta +1}} (t_{\zeta +1}-\varkappa )^{\upsilon -1} \bigg [ \frac{f (t_\zeta , \mathcal {J}_\zeta )}{q} (t-t_{\zeta -1})- \frac{f (t_{\zeta -1}, \mathcal {J}_{\zeta -1})}{q} (t-t_{\zeta }) \bigg ] d \varkappa , \end{aligned}$$
(28)
$$\begin{aligned} B_{\upsilon ,1}&= \frac{\upsilon f(t_\zeta ,\mathcal {J}_\zeta )}{ABC(\upsilon )\gimel (\upsilon )q} \bigg [ \frac{2qt_{\zeta +1}^{\upsilon }}{\upsilon }-\frac{t_{\zeta +1}^{\upsilon +1}}{\upsilon +1} \bigg ]-\frac{\upsilon f(t_{\zeta -1},\mathcal {J}_{\zeta -1})}{ABC(\upsilon )\gimel (\upsilon )q} \bigg [ \frac{qt_{\zeta +1}^{\upsilon }}{\upsilon }-\frac{t_{\zeta +1}^{\upsilon +1}}{\upsilon +1} \bigg ]. \end{aligned}$$
(29)
Similarly, we obtain$$\begin{aligned} B_{\upsilon ,2}=\frac{\upsilon f(t_\zeta ,\mathcal {J}_\zeta )}{ABC(\upsilon )\gimel (\upsilon )q} \bigg [ \frac{q t^\upsilon _\zeta }{\upsilon }-\frac{t_\zeta ^{\upsilon +1}}{\upsilon +1} \bigg ]- \frac{ f(t_{\zeta -1},\mathcal {J}_{\zeta -1})}{ABC(\upsilon )\gimel (\upsilon )q}, \end{aligned}$$so, we get the following result$$\begin{aligned} \mathcal {J}(t_{\zeta +1})-\mathcal {J}(t_\zeta )&= \frac{1-\upsilon }{ABC(\upsilon )} \bigg [ f (t_{\zeta },\mathcal {J}(t_{\zeta }))-f (t_{\zeta -1},\mathcal {J}(t_{\zeta -1})) \bigg ]+ \frac{\upsilon f(t_\zeta ,\mathcal {J}_\zeta )}{ABC(\upsilon )\gimel (\upsilon )q} \bigg [ \frac{2qt_{\zeta +1}^{\upsilon }}{\upsilon }-\frac{t_{\zeta +1}^{\upsilon +1}}{\upsilon +1} \bigg ] \nonumber \\&\quad -\frac{\upsilon f(t_{\zeta -1},\mathcal {J}_{\zeta -1})}{ABC(\upsilon )\gimel (\upsilon )q} \bigg [ \frac{qt_{\zeta +1}^{\upsilon }}{\upsilon }-\frac{t_{\zeta +1}^{\upsilon +1}}{\upsilon +1} \bigg ]-\frac{\upsilon f(t_\zeta ,\mathcal {J}_\zeta )}{ABC(\upsilon )\gimel (\upsilon )q} \bigg [ \frac{q t^\upsilon _\zeta }{\upsilon }-\frac{t_\zeta ^{\upsilon +1}}{\upsilon +1} \bigg ]\nonumber \\&\quad +\frac{ f(t_{\zeta -1},\mathcal {J}_{\zeta -1})}{ABC(\upsilon )\gimel (\upsilon )q}. \end{aligned}$$
(30)
The above yields that$$\begin{aligned} \mathcal {J}(t_{\zeta +1})&= \mathcal {J}(t_\zeta )+ f(t_\zeta ,y_\zeta ) \bigg [ \frac{1-\upsilon }{ABC(\upsilon )}+ \frac{\upsilon }{ABC(\upsilon )q} \bigg \{ \frac{2qt^{\upsilon }_{\zeta +1}}{\upsilon }-\frac{t^{\upsilon +1}_{\zeta +1}}{\upsilon +1} \bigg \} \nonumber \\&\quad – \frac{\upsilon }{ABC(\upsilon )\gimel (\upsilon )q} \bigg \{ \frac{qt_{\zeta }^{\upsilon }}{\upsilon }-\frac{t_{\zeta }^{\upsilon +1}}{\upsilon +1} \bigg \} \bigg ] +f(t_{\zeta -1},y_{\zeta -1}) \times \nonumber \\&\quad – \bigg [\frac{\upsilon -1}{ABC(\upsilon )}-\frac{\upsilon }{ABC(\upsilon )\gimel (\upsilon )q} \bigg \{ \frac{qt_{\zeta +1}^{\upsilon }}{\upsilon }-\frac{t_{\zeta +1}^{\upsilon +1}}{\upsilon +1} +\frac{ t^{\upsilon +1}}{q ABC(\upsilon )\gimel (\upsilon )} \bigg \} \bigg ]. \end{aligned}$$
(31)
The method described above is for the ABC fractional derivative. Here, we run different simulations to see how different factors affect the relationship between HIV and T-cells. The settings we use for the parameters in this section are meant to highlight the chaotic and up-and-down behavior of the system (5) through simulations.
Exploring the dynamics of HIV and CD4+ T-cells with non-integer derivatives involving nonsingular and nonlocal kernel
