Conformable Fractional-Order Modeling and Analysis of HIV/AIDS Transmission Dynamics

Te mathematical model of the dynamics of HIV/AIDS infection transmission is developed by adding the set of infected but noninfectious persons, using a conformable fractional derivative in the Liouville–Caputo sense. Some fxed point theorems are applied to this model to investigate the existence and uniqueness of the solutions. It is determined what the system’s fundamental reproduction number R 0 is. Te disease-free equilibrium displays the model’s stability and the local stability around the equilibrium. Te study also examined the efects of diferent biological features on the system through numerical simulations using the Adams–Moulton approach. Additionally, varied values of fractional orders are simulated numerically, demonstrating that the results generated by the conformable fractional derivative-based model are more physiologically plausible than integer-order derivatives.


Introduction
HIV/AIDS has been a global epidemic since the late 20th century, infecting millions of people around the world [1].HIV is an infectious disease that targets the body's defenses, specifcally targeting CD + 4 T-cells, which act as a bodyguard against any viral infection [2].Body fuids are the primary vector for HIV transmission.Also, unprotected sexual contact, sharing of needles and syringes among drug injectors, and mother-to-child transmission during delivery or nursing are common routes of transmission.Terefore, if we cannot manage HIV efectively, it can gradually destroy the body's immunity, making it difcult for the body to resist infections and diseases.As this virus has negative impacts on public health, healthcare systems, and communities, and with advances in medical research and treatment, a large number of researchers have sought and put in much efort to manage HIV and limit its spread and allowing individuals infected with HIV to live a better life and be healthier and reduce the risk of infecting others.Classical diferential equations use integer-order derivatives to explain population dynamics.However, the fractional diferential equations (FDEs) extend this idea to fractional-order derivatives, allowing a more accurate depiction of complex systems as the propagation of infectious illnesses.In recent decades, FDEs have gained much attention due to their use in modeling a wide range of phenomena in various disciplines, including biology, chemistry, economics, engineering, physics, and others [3].FDEs are among the most important tools used to build these mathematical models.It has been found that they give more accurate results than those that can be obtained using diferential equations in the integer order.Te researchers point out in [4] that the model can predict the course of infection and the efectiveness of diferent treatment strategies.It can also be used to understand the disease's mechanisms and identify potential targets for new therapies [5,6].Many fractional diferential operators have been used in so-called mathematical epidemiology to model many infectious diseases as mathematical models of infectious diseases are important sources for understanding and interpreting the dynamics of those diseases such as coronaries, polycystic ovarian syndrome, tuberculosis, immunodefciency, infuenza, cancer, and hepatitis [7][8][9][10][11][12][13][14][15][16][17].FDEs are crucial in understanding the dynamics of HIV/AIDS transmission.Many mathematical models consisting of FDEs and fractional diferential operators have emerged to describe the dynamics of HIV/AIDS transmission, control its spread, and understand its transmission mechanisms between people (see [18]).Mustapha et al. [19] established a stochastic HIV/AIDS model with protection awareness.Teir paper revealed the impact of protection awareness on the control of AIDS.In [20], the authors studied how the human immunodefciency virus (HIV) and acquired immune defciency syndrome (AIDS) spread.Hassani et al. [21], investigated the fractional mathematical model of the HIV/AIDS spread optimization technique by using generalized shifted Jacobi polynomials.In [22], the authors presented a new mathematical model that examines the interaction between the immune system and cancer cells using Caputo and Caputo-Fabrizio derivatives.Te model shows a reduction in the number of cancer cells for both derivatives, with a higher reduction observed for the Caputo-Fabrizio derivative.In [23], the chaotic behavior of a fractional-order HIV-1 model involving interactions between cancer cells, healthy CD4+T lymphocytes, and virus-infected CD + 4 − T lymphocytes was investigated.Here, we discuss some literature about the study's subject.In [24], the authors discussed a modeling approach to study the role of prostitution in HIV transmission.It considers a fractional-order HIV epidemic model and analyzes the spread of HIV through prostitution.An HIV infection model of CD + 4 − T cells was discussed using the Atangana-Baleanu fractional derivative, and the existence and uniqueness of solutions for the model were analyzed in [25].Javidi et al. [26] investigated a fractionalorder HIV/AIDS (FOHA) epidemic system with treatment and solved the system obtained by using the fourth-order Runge-Kutta method.Ding et al. [27] discussed the conditions that must be met for a general fractional optimal control to be optimal control of the HIV-immune system with memory whose FD is described in the Caputo sense.Carla et al. [28] studied the FD model for HIV and TB coinfection, and they considered vertical HIV transmission and treatment for both ailments.Tey also analyzed the numerical results of the suggested model for various FD order values.Also, Carla et al. [29] studied the FD order model for the three phases of HIV epidemics with drug resistance.Te model includes CD + 4 − T cells, CTLs, macrophages, and pathogen populations, and they simulated the model for diferent values of the FD.A fractional-order nonlinear mathematical model is presented for analyzing and monitoring the propagation of HIV/AIDS by Zahra et al. [30].A population fractional model for transmitting human immunodefciency virus (HIV) was presented in [31]; the authors assumed a homogenous mixing population and considered the availability of anti-HIV preventive vaccines.Hikal and Zahra in [32] investigated the fractional model of HIV/AIDS that includes treatment and time delay.Tey also presented the numerical simulations of the model by a fnite diference method for a fractional system.Liu et al. [33] introduced the fractional-order HIV/AIDS infection model with the Beddington-DeAngelis functional response rate.Te fractional-order HIV/AIDS model of analyzing and monitoring the propagation was considered by Zafar et al. [34].Tey also investigated the numerical simulations and the impact of the system parameter on the spread of the disease using the Adams-Bashforth-Moulton algorithm and Grunwald-Letnikov method.FDs were used by Arshad et al. [35] to analyze the HIV infection model.Teir analysis focused primarily on the degree of T-cell depletion from viral cytopathicity.Numerical simulations employing the Caputo derivative's fnite diference approximation were used to demonstrate the analytical conclusions.Cristiana et al. [36] proposed a Caputo fractional-order SICA epidemiological model for HIV/AIDS transmission.Te model extended an integer-order HIV/AIDS model, incorporating memory efects, long-range interactions, and hereditary properties through fractional diferentiation.Te local and uniform asymptotic stability of the disease-free and endemic equilibrium points was analyzed.A mathematical model for the transmission dynamics of HIV/AIDS with fractional order and fractal-fractional order has been studied in [1].Te model was evaluated for its basic properties, equilibria, and reproductive number.
A fractional conformable derivative (FCD) shares many characteristics with a classical derivative.It has been used to model several physical and biological problems.Te author in [37] used a mathematical operator called the conformable derivative in the sense of the Liouville-Caputo derivative (LC) to investigate measles infection.
Motivated by the works mentioned above, we expand the HIV transmission model proposed in [36,38] by adding the set of individuals who are infected but not infectious under FCD in the LC sense.We frst verify the existence and uniqueness of the model solution.Ten, we discuss the fundamental characteristics of the model, such as the diseasefree equilibrium and basic reproduction number.Additionally, the model is numerically solved using the Adams-Moulton technique.Lastly, the graphical efects of diferent parameters are shown for various values of fractional order.Tis study aims to investigate the behavior of the HIV/AIDS virus infection model under diferent 9 values and to analyze the efect of various factors on the disease dynamics.
Tis work used fractional-order conformable derivatives in the Liouville-Caputo sense to suggest a novel model of HIV/AIDS transmission.Tis allowed the model to represent the intricate dynamics involved in developing the disease.Te fractional-order method enabled more physiologically accurate simulations than the conventional integer-order models.We conducted mathematical analysis and obtained analytical formulas describing the basic reproduction number R 0 to ensure the model is well-posed.Tis research paper is presented as follows.Section 2 provides some fundamental topics related to fractional calculus.Te model structure and its associated results are given in Section 3. In Section 4, we prove the existence and uniqueness of the results by using fxed point theory and nonlinear analysis.In Section 5, we calculate disease-free equilibrium and R 0 .Section 6 shows active numerical sketches for the HIV model solution by asymptomatic transporters.

Preliminaries
Here are some fundamental defnitions that will be used in this study.
Remark 4 (see [40]).Te most crucial feature of the FCD that links it with classical derivatives is Defnition 5 (see [41]).Let Ten, the FCD in the sense of LC is defned as ( Defnition 6 (see [41]).Let Ten, the fractional conformable integral in the sense of LC is defned as

The Model
Te FCD model is frst formulated using integer-order derivatives.To further explore the model at hand, we will examine the scenario in which the recruitment rate remains constant, but the people size varies.1.
In light of the preceding discussion and hypotheses, the proposed compartmental model for HIV/AIDS can be expressed by the following diferential equation system: Te HIV/AIDS model ( 7) is now transformed into fractional order.Fractional-order diferential systems are preferred over conventional ones because they can help reduce mistakes caused by overlooked parameters in realworld application modeling.Te model (7) with FCD of order q in the sense of LC is where S(t 0 ) � S 0 , E(t 0 ) � E 0 , I(t 0 ) � I 0 , C(t 0 ) � C 0 , and A (t 0 ) � A 0 are initial conditions.

Existence and Uniqueness Results
For model (8), it will be shown here that the solution exists and is unique with FCD of order q in the LC sense.At frst, we put Using the fractional integral operator on both sides of model ( 8), we get 4 International Journal of Diferential Equations which implies and Te symbols ξ 1 , ξ 2 , ξ 3 , ξ 4 , and ξ 5 have to hold for the Lipschitz condition only if S, E, I, C, and A possess an upper bound.Suppose that S and S * are couple functions, and we have Treatment failure rate 0.090 [36] International Journal of Diferential Equations Take ω 1 as Proceeding in the same way as before, we obtain where . Tis means that the Lipschitz condition has been done for all four functions.Let us now take the expressions iteratively.Indeed, (10) yields with S(t 0 ) � S 0 , E(t 0 ) � E 0 , I(t 0 ) � I 0 , C(t 0 ) � C 0 , and A (t 0 ) � A 0 .
When the diference between the following terms is taken, we obtain 6 International Journal of Diferential Equations . Also, by using ( 15)-( 17) and fact that Te following theorem will prove the existence of the solution and its uniqueness.

Disease-Free Equilibrium (DFE)
To calculate the equilibrium point of model (8), we make the model's left side (8) equal to zero as follows: By solving the above HIV/AIDS model regarding no disease condition, we fnd the DFE of the model as follows: E, I, C, A) � (ρ/d, 0,0,0,0).

Theorem .
Te DFE E 0 ought to achieve Re(Ψ j ) < 0, j � 1, . . ., 5 for being locally asymptotically stable (LAS), when Ψ the eigenvalue the Jacobian matrix calculated at each equilibrium point.Proof.To prove the LAS of the DFE point in our model (8), it is necessary to establish that all eigenvalues of the Jacobian matrix J, which is derived from model ( 8) of equations and evaluated at the DFE J(E 0 ) point, satisfy a specifc requirement.
Te next analysis aims to determine the J matrix at the DFE for model (3).Terefore, the Jacobian matrix is denoted by J(E 0 ): where To simplify, we will symbolize the following: Te eigenvalues of J(E 0 ) are the roots of P(Ψ).Hence, we put It follows that Ψ � − d and Let us make To prove that all roots of the polynomial (27) satisfy |arg(Ψ i )| > cπ/2, we use the fractional Ruth-Hurwitz criterion.Te discriminant D(p) of the polynomial P(Ψ) is given as If the subsequent conditions are satisfed, the solution to polynomial (28) the solution to polynomial (28) does not satisfy then the solution to polynomial (28) satisfes |arg(Ψ i )| > cπ/2, for c ∈ (0,1).Te necessary condition that the solution to polynomial (28) satisfes . R 0 is the number of infected cases due to the transmission of infection from a previously injured person.It can be calculated as explained by Driessche in [43].By using the relation R 0 � λ(FV − 1 ), where λ the spectral radius of the second-generation operator, F and V are the matrices for the new disease class and for the rest of the transitional terms, respectively.Te matrices F and V connected to model ( 8) are given by

International Journal of Diferential Equations
Set and and we obtain R 0 � λ(FV − 1 ) � ρι(μσ ), which is greater than one; by using (Teorem 2, [43]), we can say that the disease can infect the population.

Discussion and Numerical Results
Tis section discusses the numerical simulations for model (8) utilizing the Adams-Moulton approach, an iterative method proposed in [44][45][46] that yields approximate solutions of fractional-order ordinary diferential equations to provide results of simulations for state variables in the model (8).Te Cauchy ODE by the LC of the order 9 has been considered as the following: where j � 0,1,2, . . ., 9 − 1. Te mentioned Cauchy initial value problem can be turned into a Volterra integral equation of the second kind as follows: To get the repeated approach, we suppose the constant time step size ∆t � B/N, t j � j∆t, j � 0, 1, 2, . . ., N where N is the number of times of integration in the interval [0, B].Estimating the preceding equation in terms of fractions by taking the diferential operator into account the conformable derivative of order q, we get the Adams-Moulton technique [37] for the fractional conformable derivative of order q with LC of order 9: By using the fractional conformable derivative of order q, we get Applying the iterative process described in (35), the HIV model (8) with the fractional conformable derivative of order q with (LC) operator of order β can be expressed as follows: (n + 1 − j) 9 − (n − j) 9  D q 0,t f 2 t j , S j , E j , I j , C j , A j  , International Journal of Diferential Equations where Troughout simulations, ∆t is the magnitude of the time step which is equal to 10 − 3 .Te time interval is taken from 0 to 100 and the initial conditions are assumed as 0.6256, 0.03, 0.0222, 0, 0.0522, while the parameter values are given as shown in Table 1.
Our simulations are based on continuous model tracking, where the orders of the derivative are considered at diferent values.
Figure 1 describes how the susceptible population changes over time; we note that the susceptible cases continue to increase in the frst days until 14.88 when 9 � 1, 14.08 when 9 � 0.95, 13.6 when 9 � 0.90, 12.68 when 9 � 0.85, and 11.6 when 9 � 0.80 (FCD) while keeping the q value constant.After ffteen days, the susceptible cases begin to decrease, and as a result of this decrease, there is an increase in E, I, A as in Figures 3-6.Similarly, in Figure 2, the susceptible cases continue to increase          8) for a fractional order q � 0.95 of FCD and diferent fractional orders of LC. 14 International Journal of Diferential Equations in the frst days until 14 for all values of (q) while keeping 9 � 0.95.Following that, it decreases, and, as a result of this there is an increase in E, I, A as we can see in Figures 7-10.We note that the approximate method gives us the same results with little diference that is barely noticeable when we use diferent values for a local derivative q ∈ (0, 1].In contrast, we note that the increase and decrease in the curves of the categories representing the model come according to the increase or decrease in values of (iterated or fractionalizing index) 9 ∈ (0,1].Finally, we note that the values of q do not signifcantly afect, unlike those caused by the values of 9.

Conclusion
In this study, the fractional conformable derivative (FCD) of order q in the fractional Liouville-Caputo (LC) sense of order 9 is applied to develop the model of the transmission dynamics of HIV/AIDS infection.A new fractional HIV/ AIDS infection model is presented, with people divided into fve classes.Fixed point theorems have been used to investigate the existence and uniqueness of the solutions for the proposed model.Te model's basic reproduction number R 0 has been determined.Te stability of the model and local stability around the equilibrium in the disease-free case were presented.Using numerical simulations with the aid of the Adams-Moulton approach, the study also investigated the efects of numerous biological characteristics on the system.Numerical simulations for various fractional order values have been carried out, showing that results produced with the FCD-based model are more physiologically reasonable than with models of integer-order cases.Further research can explore applying the conformable fractional derivative-based model to other infectious diseases beyond HIV/AIDS to assess its efectiveness in capturing the dynamics of diferent diseases.Future studies can focus on refning the model by incorporating more detailed biological characteristics and considering the interactions between diferent subpopulations, such as high-risk groups or individuals with varying immunity levels.

Figure 1 :
Figure 1: Te approximate solution of S of the considered model (8) for a fractional order q � 0.95 of FCD and diferent fractional orders of LC.

Figure 2 :
Figure 2: Te approximate solution of S of the considered model (8) for diferent fractional orders of FCD and a fractional order 9 � 0.95 of LC.

Figure 3 :Figure 4 :
Figure 3: Te approximate solution of E of the considered model (8) for diferent fractional orders of FCD and a fractional order 9 � 0.95 of LC.

Figure 5 :
Figure 5: Te approximate solution of C of the considered model (8) for diferent fractional orders of FCD and a fractional order 9 � 0.95 of LC.

Figure 6 :
Figure 6: Te approximate solution of A of the considered model (8) for diferent fractional orders of FCD and a fractional order 9 � 0.95 of LC.

Figure 7 :
Figure7: Te approximate solution of E of the considered model (8) for a fractional order q � 0.95 of FCD and diferent fractional orders of LC.

Figure 8 :
Figure 8: Te approximate solution of I of the considered model (8) for a fractional order q � 0.95 of FCD and diferent fractional orders of LC.

Figure 9 :Figure 10 :
Figure 9: Te approximate solution of C of the considered model (8) for a fractional order q � 0.95 of FCD and diferent fractional orders of LC.