A Study on Dynamics of CD4 + T-Cells under the Effect of HIV-1 Infection Based on a Mathematical Fractal-Fractional Model via the Adams-Bashforth Scheme and Newton Polynomials

: In recent decades, AIDS has been one of the main challenges facing the medical community around the world. Due to the large human deaths of this disease, researchers have tried to study the dynamic behaviors of the infectious factor of this disease in the form of mathematical models in addition to clinical trials. In this paper, we study a new mathematical model in which the dynamics of CD4 + T-cells under the effect of HIV-1 infection are investigated in the context of a generalized fractal-fractional structure for the first time. The kernel of these new fractal-fractional operators is of the generalized Mittag-Leffler type. From an analytical point of view, we first derive some results on the existence theory and then the uniqueness criterion. After that, the stability of the given fractal-fractional system is reviewed under four different cases. Next, from a numerical point of view, we obtain two numerical algorithms for approximating the solutions of the system via the Adams-Bashforth method and Newton polynomials method. We simulate our results via these two algorithms and compare both of them. The numerical results reveal some stability and a situation of lacking a visible order in the early days of the disease dynamics when one uses the Newton polynomial. derivative; HIV-1 infection; Newton polynomial; Adams-Bashforth MSC: 34A08; 65P99; 49J15 graphical results showed that these two numerical algorithms give the same outcomes and differences are small. Also, we investigated the effect of frcatal dimensions and fractional orders on these simulations. Also, the effect of different values for the average number of infected particles and the supply rate of new T-cells were simulated in some graphs under the Adams-Bashforth method. This stude showed that we can predict the next behavior of the fractal-fractional CD4 + -HIV-1-model via the two mentioned numerical methods and their results are more accurate and identical. This shows the power of simulation of the frcatal-fractional models in comparison to the fractional models. In the next researches, we can develope our numerical methods on different fractal-fractional models of diseases.


Introduction
According to medical definitions and clinical findings and virology, human immunodeficiency virus (HIV) is a type of retrovirus that leads to acquired immunodeficiency syndrome (AIDS) in humans [1]. In fact, CD4 + T-lymphocytes are the largest number of white blood cells in the human immune system that are attacked by HIV viruses, which attack CD4 + T-cells and infect them, reducing their number and efficiency. They disrupt cells. Therefore, this process reduces the resistance of the immune system in the human body and weakens it [1]. Although the HIV virus infects other cells, it causes the most damage to T cells by causing the degradation and destruction of CD4 + T-cells. As a result, the affected person's body gradually becomes sensitive to various types of infections and exponential law and generalized Mittag-Leffler law with fractal derivatives. There exist two components for such fractal-fractional operators: the fractional order and fractal dimension (order). Fractal-fractional differential equations transfer the order and dimension of every dynamical system into a rational order one. In fact, we are able to generalize each standard differential equation to the generalized systems having arbitrary order and dimension of derivatives. The main goal of such a combination is to analyze a vast range of nonlocal BVPs or IVPs that possess fractal behaviors. In this direction, a limited researchers obtained some results in which we see that the generalized fractal-fractional operators give accurate and more exact simulations for describing mathematical models of real-world phenomena. Some of new works in this regard are [37][38][39][40][41].
Due to the novelty and efficacy of these new fractal-fractional operators, in this paper, we aim to design a mathematical model of CD4 + T-cells under the effect of HIV-1 infection in which derivatives are fractal-fractional operators in the sense of Atangana-Baleanu. It is notable that in 2021, Ahmad et al. [42] studied the dynamics of HIV primary infection in the context of a model designed by the fractal-fractional operators. The main contribution and novelty of our work in comparison to their paper [42] is that we analyze all qualitative behaviors of such a fractal-fractional system. In other words, we first review the existence and uniqueness of solutions and further, we complete our study by giving new results about the stability (Ulam-Hyers-Rassias) of solutions. Also, for the first time, in this paper, we derive numerical schemes for the fractal-fractional CD4 + -HIV-1 model with the help of the Newton polynomials and by applying some real data, we compare our results with the Adams-Bashforth simulations. In this direction, we can see some dynamical behaviors of the solutions in our simulations.
The arrangement of the paper is as follows: we introduce our fractal-fractional model in Section 2 and describe its parameters and coefficients. The existence results are given in Section 3 and further, in Section 4, we investigate the uniqueness. The stability criterion are implemented in Section 5. In the sequel, we derive two numerical schemes. In other words, in Section 6, the Adams-Bashforth method are done and then, we derive another algorithm in Section 7 by using the Newton polynomials. We present some simulations and discussions about both numerical methods in Section 8. We end the paper by giving conclusions in Section 9.

The Structure of the Model for CD4 + T-Cells and HIV-1
In 2006, Wang and Li [1] formulated an integer-order classical mathematical structure of dynamics of CD4 + T-cells under the HIV-1 infection in three-compartmental model as via the initial values T(0) = T 0 , U(0) = U 0 , and V(0) = V 0 , and also the state functions T(s), U(s), V(s) are an amount of susceptible CD4 + T-cells, an amount of infectious CD4 + T-cells, and the free particles of the infection of the HIV virus in the blood at the time s ∈ J := [0, S], (S > 0), respectively. Moreover, the parameter N stands for the average number of infected particles by an existing infected cell, ϑ is the natural rate of death for the virus, ζ is the return rate of infected cells to susceptible compartment, κ is the rate of death for infected T-cells, q is the rate of infection T-cells, ρ is the natural rate of death, and θ shows the supply rate for new T-cells. They considered all parameters as positive values and T 0 , U 0 , V 0 ≥ 0.
To upgrade and improve the exact results, inspired by the standard model (1), we present a mathematical fractal-fractional model on dynamics of CD4 + T-cells under the effect of HIV-1 infection via the generalized Mittag-Leffler-type kernel (fractal-fractional CD4 + -HIV-1-model) as where all assumptions and parameters are similar to above classical model (1). Also, FFML D (δ,σ) 0,s is the (δ, σ)-fractal-fractional derivative with the fractional order δ ∈ (0, 1] and the fractal order σ ∈ (0, 1] via the Mittag-Leffler-type kernel. In other words, let a continuous map Ψ : (a, b) → [0, ∞) be fractal differentiable of dimension σ. In this case, the (δ, σ)-fractal-fractional derivative of Ψ of the generalized Mittag-Leffler-type kernel of order δ in the Riemann-Liouville sense is defined as is the fractal derivative and AB(δ) and AB(0) = AB(1) = 1 [36].

Existence Property
In this section, the existence property is investigated based on fixed point theory. For the qualitative analysis, make the Banach space X = M 3 , where M = C(J, R) with K X = T, U, V X = max |W(s)| : s ∈ J , for |W| := |T| + |U| + |V|. We reformulate the R.H.S. of the fractal-fractional CD4 + -HIV-1model (2) as: In this case, the fractal-fractional CD4 + -HIV-1-model (2) is transformed into the following system In view of (6), we rewrite the developed tripled-system with the compact IVP which takes the form    ABR D δ 0,s K(s) = σs σ−1 Q s, K(s) , where and By the definition and by (7), we have In the sequel, operating the fractal-fractional Atangana-Baleanu integral on (10), we get . (11) Due to the above compact form of the fractal-fractional integral equation, the extended representation of it is illustrated as We consider a self-map to derive a fixed-point problem, by defining F : X → X as The Leray-Schauder theorem is utilized to prove the existence property in relation to the fractal-fractional CD4 + -HIV-1-model (2). Theorem 1. [43] Let X be a Banach space, E ⊂ X a convex closed bounded set, O ⊂ E an open set, and 0 ∈ O. Then for the continuous and compact map F :Ō → E, either: (HY1)∃ y ∈Ō s.t. y = F(y), or (HY2)∃ y ∈ ∂O and 0 < µ < 1 s.t. y = µF(y).
Proof. First, consider F : X → X which is defined by (13) and assume N r = K ∈ X : K X ≤ r , for some r > 0. Clearly, as Q is continuous, thus F is so. From (P1), we get for K ∈ N r . Hence Thus F is uniformly bounded on X. Now, take s, v ∈ [0, S] s.t. s < v and K ∈ N r . By denoting sup We see that (16) approaches to 0 independent of K, as v → s. Consequently when v → s. This gives the equicontinuity of F, and accordingly the compactness of F on N r by the Arzelá-Ascoli thoerem. Since all conditions of Theorem 1 are fulfilled on F, we have one of (HY1) or (HY2). From (P2), we set for some ω > 0 s.t.

Uniqueness Property
Here, on the fractal-fractional CD4 + -HIV-1-model (2), we investigate the Lipschitz property in the first step and further, the uniqueness property.
Then the kernels Q 1 , Q 2 , Q 3 defined in (5) are fulfilled the Lipschitz property with constants α 1 , α 2 , α 3 > 0 w.r.t. the relevant components, where Proof. For Q 1 , we take T, T * ∈ M := C(J, R) arbitrarily, and we have From the above, we find out that Q 1 is Lipschitz w.r.t. T under the constant α 1 > 0. For Q 2 , we choose two arbitrary elements U, U * ∈ M := C(J, R), and estimate The above inequality means that Q 2 is Lipschitz w.r.t. U under the constant α 2 > 0. Finally, for both arbitrary elemets V, V * ∈ M := C(J, R), we have This confirms that Q 3 is Lipschitz w.r.t. V under the constant α 3 > 0. Therefore kernel functions Q 1 , Q 2 , Q 3 are Lipschitze w.r.t. under the constants α 1 , α 2 , α 3 > 0, respectively. Now, by invoking the above lemma, we are able to prove the uniqueness property for solutions of the fractal-fractional system (2).

Proof.
To prove the desired result, we consider the contrary of the conclusion of theorem. That is, consider the existence of another solution for the fractal-fractional CD4 From (12), we have In this case, we estimate and so From (19), we know that the above inequality holds if T − T * = 0, or T = T * . In the similar manner, from The latter equality confirms that the fractal-fractional CD4 + -HIV-1-model (2) possesses a unique solution.

Ulam-Hyers-Rassias Stability
In this part, the stability result of solutions in the context of four types of the Ulam-Hyers, Ulam-Hyers-Rassias and their generalizations are proved on the tripled system of the fractal-fractional CD4 + -HIV-1-model (2).
If we takeh  (s) = 1, in this case Definition 3 yields the Ulam-Hyers criterion for the stability of solutions.
The following lemmas are useful for our main theorems. (20). Then the functions T * , U * , V * ∈ M fulfill the inequalities and and Proof. Let r 1 > 0 be arbitrary. Since T * ∈ M satisfies so, by Remark 1, we are allowed to take a function z 1 (s) s.t.
In this case, we estimate This states that the inequality (22) is fulfilled. Similarly, we can obtain the inequalities (23) and (24).
Then, we estimate Similarly, we can obtain the remaining inequalities.
Proof. Let r 1 > 0 and T * ∈ M be an arbitrary solution of (20). Also, from Theorem 3, we assume T ∈ M as a unique solution of the fractal-fractional CD4 + -HIV-1-model (2). Then T(s) is defined as Therefore, by Lemma 2 and with the help of the triangle inequality, we estimate Hence, Hence, the Ulam-Hyers stability of the fractal-fractional CD4 + -HIV-1-model (2) is fulfilled. On the other hand, if we take then a Q  (0) = 0, and the generalized Ulam-Hyers stability is simply proved.

Numerical Scheme via Adams-Bashforth Method
In the present section of the paper, we aim to derive a numerical algorithm for the fractal-fractional CD4 + -HIV-1-model (2). To do this, we apply a technique based on twostep Lagrange polynomials entitled fractional Adams-Bashforth method. We re-define fractal-fractional integral equations (12) at s k+1 . In fact, we discretize these integral equations (12) for s = s k+1 as follows The approximation of above integrals are formulated by Next, we approximate three functions w σ−1 Q  (w, T(w), U(w), V(w)),  = 1, 2, 3, on the interval [s , s +1 ] by applying two-step Lagrange interpolation polynomials under the step size h = s − s −1 . By straighforward computations, we obtain algorithms which yield the numerical solutions to the fractal-fractional CD4 + -HIV-1-model (2) as whereŶ

Numerical Scheme via Newton Polynomials Method
In this section, we produce a new numerical scheme for solutions of our fractalfractional CD4 + -HIV-1-model (2) which was introduced by Atangana and Araz in the book [44] in 2021. To do this, we again use the compact form of IVP (7) with the conditions (8) and (9). In this case, we have Set Q * (s, K(s)) = σs σ−1 Q(s, K(s)). Then By discretizing the above equation at s = s k+1 = (k + 1)h, we get If we approximate the above integral, then it becomes In this step, the function Q * (s, K(s)) is approximated by the Newton polynomial as Substitute (33) into (32): We simplify the above relations, and we get In consequence, On the other hand, we compute above three integrals separately, and we get and By putting (35)-(37) in (34), we obtain Lastly, we replace Q * (s, K(s)) = σs σ−1 Q(s, K(s)) in (38), and we get Based on the numerical scheme obtained in (39), numerical solutions of the fractalfractional CD4 + -HIV-1-model (2) are given by and and where the constantsΨ  (k, , δ) are introduced in (40) for  = 1, 2, 3.
In the provided figures, we show the behaviors of three state functions T, U, V under the effect of different values for the fractal-fractional orders δ = σ = 1.00, 0.98, 0.96, 0.94, 0.92, 0.90 simultaneously. We also compare the results in the graphs with respect to two numerical algorithms given by (29)- (31) and (41)- (43) as shown in Tables 1-3.
In these graphs, the quantity of the time s interprets the number of days. Note that Figures 1 and 2 demonstrate the dynamics of the CD4 + T-cells via the Adams-Bashforth method (Section 6). Figure 1a indicates that when the fractal dimensions and fractional orders decrease, the number of susceptible CD4 + T-cells steadily increases from the 42nd day to the 64th day, and steadily decreases from the 65th day to the 102nd day. Further it increases again from the 103rd day and finally converges to the integer-order at the 250th day to the end of the simulation period. Figure 1b,c show that when the fractal dimensions and fractional orders move from the integer order, the peak of the amount of infectious CD4 + T-cells and the free particles of the infection of the HIV-1 virus in the blood decrease respectively and also give slight differences in their asymptotic stabilities. In Figure 2a,c, we show a 0.01-variation in fractal-fractional order. In this case, the numerical trajectories show that a slight change in the fractal-fractional order produces a slight changes in the asymptotic behavior of the HIV-1 virus on CD4 + T-cells.  Figure 3a also indicates that when the fractal dimensions and fractional orders decrease, the number of susceptible CD4 + T-cells steadily increases from the 42nd day to the 64th day, and steadily decreases from the 65th day to the 102nd day. Also it increases again from the 103rd day and finally converges to the integer-order at the 250th day to the end of the simulation period, but it shows a disorganized behavior from the 11th day to the 24th day, which indicates the early stages of HIV in the CD4 + T-cells. Figure 3b,c show that when the fractal dimensions and fractional orders move from the integer order, the peak of the amount of infectious CD4 + T-cells and the free particles of the infection of the HIV-1 virus in the blood decrease, respectively and also give slight differences in their asymptotic stabilities, but again, they show a disorganized behavior from the 11th day to the 24th day. In Figure 4a,c, we show a 0.01-variation in fractal-fractional orders, and the numerical trajectories show that a slight change in the fractal-fractional orders produce slight changes in the asymptotic behavior of the HIV-1 virus on CD4 + T-cells.  In Figure 5, based on the Adams-Bashforth method, we see the numerical trajectories of three state functions by varying the average number of infected particles N for the values N = 500, 600, 700, 800, 900, 1000 under the fractal-fractional order δ = σ = 0.95. Thus, in Figure 5a, it shows that the susceptible CD4 + T-cells increase as the average number of infected particles by an existing infected cell reduces by 10%, and that of the peak of the amount of infectious CD4 + T-cells and the free particles of the infection of the HIV-1 virus in the blood decrease, respectively in Figure 5b,c. In Figure 6, based on the Adams-Bashforth method, we see the numerical trajectories by varying the supply rate for new T-cells for θ = 10, 15, 20, 25, 30, 35 under the fractal order and fractional order δ = σ = 0.95. Thus, in Figure 6a, it shows that the susceptible CD4 + T-cells increase for a period of time and gradually decrease as the number of the supply rate θ increases before converging at the 150th day. In Figure 6b,c, we noticed that the amount of infectious CD4 + T-cells and the free particles of the infection of the HIV-1 virus in the blood increase, respectively. In Tables 1-3, we see some results of the two numerical schemes (the Adams-Bashforth method and Newton polynomials method) for all of three state functions under the fractal dimension and fractional order δ = σ = 0.95 with step size h = 0.1.

Conclusions
In this paper, we designed a fractal-fractional CD4 + -HIV-1-model and analyzed the dynamics of CD4 + T-cells under the infection of HIV-1 virus. We considered three compartments for this model by defining three state functions T, U, and V for the amount of susceptible CD4 + T-cells, amount of infectious CD4 + T-cells, and the free particles of the infection of the HIV virus in the blood. We derived three fractal-fractional integral equations and proved that their kernels are Lipschitz. In this direction, we could prove the existence and uniqueness criteria for solutions of the fractal-fractional CD4 + -HIV-1model. In the sequel, we investigated four stability results with the help of two auxiliary inequality. We extracted two algorithms via the Adams-Bashforth method and also via the Newton polynomials and simulated our real data in relation to the given fractal-fractional CD4 + -HIV-1-model. The numerical and graphical results showed that these two numerical algorithms give the same outcomes and differences are small. Also, we investigated the effect of frcatal dimensions and fractional orders on these simulations. Also, the effect of different values for the average number of infected particles and the supply rate of new T-cells were simulated in some graphs under the Adams-Bashforth method. This stude showed that we can predict the next behavior of the fractal-fractional CD4 + -HIV-1-model via the two mentioned numerical methods and their results are more accurate and identical. This shows the power of simulation of the frcatal-fractional models in comparison to the fractional models. In the next researches, we can develope our numerical methods on different fractal-fractional models of diseases. Data Availability Statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.