# Numerical Solution of Fractional-Order HIV Model Using Homotopy Method

Authors: Sami Aljhani; Mohd Salmi Md Noorani; A. K. Alomari
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this study, we construct a convergent algorithm for generating an approximate analytic solution for the fractional HIV infection ofCD4+ T cells with Atangana–Baleanu fractional derivatives in the Caputo sense. We compute the solution by utilizing the fractional homotopy analysis transform method (FHATM) and achieved a convergence region of the solution by employing an auxiliary parameter. Moreover, we apply a numerical scheme proposed by Toufik and Atangana for solving this kind of problem and compared with our results. A good agreement between the new algorithm and the numerical scheme is remarkable. The solution via the present algorithm can be obtained without any linearization or discretization which makes it reliable and easy to apply.


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## Body

## 1. Introduction

Fractional calculus has played a significant role within the field of science and engineering, and many mathematicians and scientists have been working in this field lately. In recent decades, fractional calculus has been used in several areas of physics, biology, engineering, and others. Further details about fractional calculus and its applications can be found in the literature [1–9].Because most nonlinear fractional differential equations cannot be solved exactly, it is necessary to use approximate and numerical methods. Various powerful mathematical techniques such as the Adomian decomposition method (ADM) [10, 11], homotopy analysis method (HAM) [12–15], optimal homotopy asymptotic method (OHAM) [16], homotopy perturbation method (HPM) [17], and variational iterative method (VIM) [18, 19] have been used to obtain an exact and approximate analytical solution.The HAM was first introduced and employed in 1992 by Liao [20], after which many researchers successfully applied this method to solve linear and nonlinear differential equations. In recent years, many researchers have devoted their attention to obtaining a solution of linear and nonlinear differential equations using a variety of methods based on Laplace transform such as the Laplace decomposition method (LDM) [21] and the homotopy perturbation transform method (HPTM) [22]. Khan et al. [23] and Kumar et al. [24–26] coupled the HAM with Laplace transform to solve a nonlinear differential equation and Volterra integral equation. The homotopy analysis transform method (HATM) is a combination of HAM with Laplace transformation. The main advantage of this method is its ability to combine two powerful methods to obtain a rapid convergent series for fractional differential equations. This method provides us with a convenient way to control the convergence of the series solution.Recently, Toufik and Atangana [27] developed a numerical scheme to solve a nonlinear fractional differential equation considering the Atangana–Baleanu fractional derivative. This method is a combination of the fundamental fractional calculus theorem with two-step Lagrange polynomial which is successfully used to solve many real-world problems [28–30].Since the early 1980s, researchers have made an enormous effort to mathematically model the human immunodeficiency virus (HIV), the virus responsible for causing acquired immune deficiency syndrome (AIDS). In 1989, Perelson [31] considered the interaction of uninfected (T) and infected CD4+ (I) T cells, and free virus molecules (F) in his model, following which Perelson et al. [32] extended the original model [31]. Culshaw and Ruan [33] reduced the model discussed in [32] as follows:(1)dTdt=p−μTT+kT1−T+ITmax−k1FT,(2)dIdt=k1′FT−μII,(3)dFdt=MμbI−k1FT−μFF,where Tt, It, and Ft represent the concentration of healthy CD4+ T cells at time t, infected CD4+ T cells, and the free HI virus at time t, respectively. Table 1 summarizes the meanings of functions and parameters. Equation (1) describes the rate of change in the uninfected population of CD4+ T cells. The first term is the constant rate at which the body produces CD4+ T cells from precursors in the bone marrow. Because the virus can infect both thymocytes and T cells, as for all cells in the body, these cells have a finite lifetime; thus, the second term describes the decreasing source. The third term describes the logistic growth of the healthy CD4+ T cells, and the proliferation of infected CD4+ T cells is neglected. The last term models the rate at which the free virus infects a CD4+ T cell. Once a T cell has been infected, it becomes an infected cell; therefore, k1 FT is subtracted from equations (1) and (3) and added to equation (2). Hence, F and T decrease concurrently.Table 1
List of parameters and functions.
Parameters and functions
Description
Values
T
t
Concentration of uninfectedCD4+ T cells
T
0
=
1000
I
t
Concentration of infectedCD4+ T cells
I
0
=
0
F
t
Concentration of HIV RNA
F
0
=
0.001
μ
T
Natural death rate ofCD4+T cells (concentration)
0.02
μ
I
Blanket death rate of infectedCD4+ T cells
0.26
μ
b
Lytic death rate of infected cells
0.24
μ
F
Death rate of free virus
2.4
k
1
Rate at whichCD4+ T cells become infected with the virus
2.4
×
10
−
5
k
1
′
Rate at which infected cells become active
2
×
10
−
5
k
Growth rate of concentration ofCD4+ T cells
0.03
M
Number of virion produced by infectedCD4+ T cells
500
T
max
Maximal concentration ofCD4+ T cells
1500
p
Source term for uninfectedCD4+ T cells
10Equation (2) describes the rate of change in the infected population of actively infected T cells. The first term represents the rate of infection of CD4+ T cells by the virus. The second term represents the rate of disappearance of infected cells.The three terms in equation (3) refer to the rate of production and destruction of the free infection virus. An actively infected CD4+ T cell produces M virus particles; thus, the rate at which the virus is produced is set equal to M times the lytic death rate for the infected cell. A free virus is lost as a result of binding to an uninfected CD4+ T cell at k1 FT. The third term accounts for the loss of viral infectivity, viral death, and/or clearance from the body.In this study, our approach to solving the fractional HIV model is to determine the order in which the fractional derivative changes by extending the classical HIV model (1)–(3) to the following set of fractional ordinary differential equations of the order α, β, and γ:(4)DtαaABCTt=p−μTT+kT1−T+ITmax−k1FT,0<α<1,(5)DtβaABCFt=k1′FT−μII,0<β<1,(6)Ft=MμbI−k1FT−μFF,0<γ<1,with initial conditions(7)T0=T00,I0=I00,F0=F00,where DtαaABC, DtβaABC, and DtγaABC are the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). To the best of our knowledge, this is the first work that solves fractional HIV infection of the CD4+ T cells model in ABC sense analytically and numerically. To indicate the strength of our proposed method, we compare our findings with the algorithm of Toufik and Atangana [27].


## 2. Preliminaries and Notations

The Atangana–Baleanu fractional derivative in the Caputo sense (ABC) is defined as [6, 34](8)DtαaABCft=Mα1−α∫atddtfsEα−α1−αt−sαds,n−1<α≤n,where α∈ℝ, Mα>0 is a normalization function satisfying(9)Mα=1−α+αΓα,with M0=M1=1, Eα⋅ denotes the Mittag-Leffler function, defined by(10)Eαz=∑k=0∞zkΓαk+1,and Γ⋅ denotes Euler’s gamma function defined as(11)Γz=∫0∞tz−1e−tdt,ℜz.The fractional integral for the ABC, which is newly defined with a nonlocal kernel and does not have singularities att=s, is defined as follows [6]:(12)IaABCtαft=1−αMαft+αMαΓα∫atfst−sα−1ds.Here, when α equals zero, the initial function is recovered, and when α equals unity, the classical ordinary integral is obtained.The Laplace transform of the fractional definitions with ABC is given as follows [6]:(13)ℒDtαaABCfts=Mα1−αsαℒfts−sα−1fasα+α/1−α.


## 3. Homotopy and Laplace Transform for FHATM

Applying the Laplace transform to equations (4)–(6) and using the formula for the Laplace transform of the ABC and then simplifying these equations, we find that(14)ℒTt;q=T0s+pssα1−α+αsαMα+sα1−α+αsαMαℒk−μTTt−kTmaxTt2−kTmaxTtIt−k1FtTt,ℒIt;q=I0s+sβ1−β+βsβMβℒk1′FtTt−μIIt,ℒFt;q=F0s+sγ1−γ+γsγMγℒMμbTt−k1FtTt−μFFt.Next, defining the nonlinear operators as(15)NTφ1t;q,φ2t;q,φ3t;q=ℒφ1t;q−T0s−pssα1−α−αsαMα−sα1−α+αsαMαℒk−μTφ1t;q;−kTmaxφ1t;q2−kTmaxφ1t;qφ2t;q−k1φ3t;qφ1t;q,NIφ1t;q,φ2t;q,φ3t;q=ℒφ2t;q−I0s−sβ1−β+βsβMβℒk1′φ3t;qφ1t;q−μIφ2t;q,NFφ1t;q,φ2t;q,φ3t;q=ℒφ3t;q−F0s−sγ1−γ+γsγMγℒMμbφ1t;q−k1φ3t;qφ1t;q−μFφ3t;q,where NT, NI, and NF are the nonlinear operators. Let ℏ be a nonzero auxiliary parameter. Using the embedding parameter q∈0,1, we construct the so-called zeroth-order deformation equation:(16)1−qℒφ1t;q−T0t=qℏN1φ1t;q,φ2t;q,φ3t;q,(17)1−qℒφ2t;q−I0t=qℏN2φ1t;q,φ2t;q,φ3t;q,(18)1−qℒφ3t;q−F0t=qℏN3φ1t;q,φ2t;q,φ3t;q,where ℒ is the Laplace operator, subject to the initial conditions(19)φ10;q=T00,φ20;q=I00,φ30;q=F00.Clearly if q=0 and q=1 we obtain(20)φ1t;0=T0t,φ1t;1=Tt,φ2t;0=I0t,φ2t;1=It,φ3t;0=F0t,φ3t;1=Ft,when q varies from zero to unity, the solution of the model (4)–(6) will vary from the initial guesses T0t, I0t, and F0t to the exact solution Tt, It, and Ft of the model (4)–(6). Expanding φit;q, i=1,2,3 by the Taylor series with respect to the embedding parameter q yields(21)φ1t;q=T0t+∑m=1∞Tmtqm,(22)φ2t;q=I0t+∑m=1∞Imtqm,(23)φ3t;q=F0t+∑m=1∞Fmtqm,where(24)Tmt=1m!∂mφ1t;q∂qm∣q=0,Imt=1m!∂mφ2t;q∂qm∣q=0,Fmt=1m!∂mφ3t;q∂qm∣q=0.The convergence of equations (21)–(23) depends on the nonzero auxiliary parameters ℏ [20]. Moreover, if the initial values guessed for T0t, I0t, and F0t and the auxiliary parameter ℏ are appropriately selected, then at q=1, series (21)–(23) converges(25)φ1t;1=T0t+∑m=1∞Tmti.eTt=T0t+∑m=1∞Tmt,(26)φ2t;1=I0t+∑m=1∞Imti.eIt=I0t+∑m=1∞Imt,(27)φ3t;1=F0t+∑m=1∞Fmti.eFt=F0t+∑m=1∞Fmt,which must be one of the solution of model (4)–(6), as proved by [20]. The equations governing the unknown functions can be deduced from the zeroth-deformation equations (16)–(18). Define the vectors(28)T⟶mt=T0t,T1t,…,Tmt,m=1,2,…,n,I⟶mt=I0t,I1t,…,Imt,m=1,2,…,n,F⟶mt=F0t,F1t,…,Fmt,m=1,2,…,n.Differentiating the zeroth-deformation equations (16)–(18) m-times with respect to the embedding parameter q, then setting q=0, and finally dividing them by m!, enables the mth-order deformation equations to be obtained:(29)ℒTmt−χmTm−1t=ℏRm,TT⟶m−1,I⟶m−1,F⟶m−1,m=1,2,…,n,(30)ℒImt−χmIm−1t=ℏRm,IT⟶m−1,I⟶m−1,F⟶m−1,m=1,2,…,n,(31)ℒFmt−χmFm−1t=ℏRm,FT⟶m−1,I⟶m−1,F⟶m−1,m=1,2,…,n,with initial conditions(32)Tm0=0,Im0=0,Fm0=0.It should be emphasized that themth-order deformation equations (29)–(31) are linear; hence, they can be solved by Mathematica or MATLAB. For simplicity, we can specify the auxiliary functions to be equal to unity. Applying the inverse Laplace transform to equations (29)–(31), we obtain(33)Tmt=χmTm−1t+ℏℒ−1Rm,TT⟶m−1,I⟶m−1,F⟶m−1,m=1,2,…,n,Imt=χmIm−1t+ℏℒ−1Rm,IT⟶m−1,I⟶m−1,F⟶m−1,m=1,2,…,n,Fmt=χmFm−1t+ℏℒ−1Rm,FT⟶m−1,I⟶m−1,F⟶m−1,m=1,2,…,n,where(34)Rm,Tt=ℒTm−1t−1−χmT0+sαtαMαΓα+1+s1−αMα−sα1−α+αsαMαℒμT−kTm−1t+kTmax∑i=0m−1TitTm−1−it,+kTmax∑i=0m−1TitIm−1−it+k1∑i=0m−1TitFm−1−it,Rm,It=ℒIm−1t−1−χmI0−sβ1−β+βsβMβℒk1′∑i=0m−1FitTm−1−it−μIIm−1ti,Rm,Ft=ℒFm−1t−1−χmF0−sγ1−γ+γsγMγℒk1∑i=0m−1FitTm−1−it+μFFm−1t−MμbIm−1t,χm=0,m≤1,1,m>1.Next, the solution of themth-order deformation equations (29)–(31) is given as(35)Tmt=χm+ℏTm−1t−ℏ1−χmT0+sαtαMαΓα+1+s1−αMα−ℏℒ−1sα1−α+αsαMαℒμT−kTm−1t+kTmax∑i=0m−1TitTm−1−it+kTmax∑i=0m−1TitIm−1−it+k1∑i=0m−1TitFm−1−it,(36)Imt=χm+ℏIm−1t−ℏ1−χmI0−ℏℒ−1sβ1−β+βsβMβℒk1′∑i=0m−1FitTm−1−it−μIIm−1t,(37)Fmt=χm+ℏFm−1t−ℏ1−χmF0−ℏℒ−1sγ1−γ+γsγMγℒk1∑i=0m−1FitTm−1−it+μFFm−1t−MμbIm−1t.Taking initial conditions and equations (35)–(37), we obtain(38)T1=−ℏMαp+k−μTT0+kT01−T0+I0Tmax−k1T0F01−α+αtαΓ1+α,I1=−ℏk1′T0F0−μII0Mβ1−β+βtβΓ1+β,F1=−ℏMμbI0−k1T0F0−μFF0Mγ1−γ+γtγΓ1+γ,T2=1+ℏT1+ℏ2m1Mα2k−μT−kTmax2T0+I0−k1F0p+k−μTT0+kT01−T0+I0Tmax−k1T0F0−ℏ2kT0m2MαMβk1′T0F0−μII0Tmax−ℏ2k1T0m3MαMγMμbI0−k1T0F0−μFF0,I2=1+ℏI1−ℏ2μIm4Mβ2k1′T0F0−μII0+ℏ2k1′F0m2MαMβp+k−μTT0+kT01−T0+I0Tmax−k1T0F0+ℏ2k1′T0m5MγMβMμbI0−k1T0F0−μFF0,F2=1+ℏF1−ℏ2k1T0+μFm6Mγ2MμbI0−k1T0F0−μFF0−ℏ2k1F0m3MαMγp+k−μTT0+kT01−T0+I0Tmax−k1T0F0+ℏ2Mμbm5MγMβk1′T0F0−μII0,where mi,i=1,2,…,6, are given by(39)m1=1−α2+21−ααtαΓ1+α+αtα2Γ2α+1,m4=1−β2+21−ββtβΓ1+β+βtβ2Γ2β+1,m6=1−γ2+21−γγtγΓ1+γ+γtγ2Γ2γ+1,m2=1−α1−β+1−αβtβΓ1+β+1−βαtαΓ1+α+αβtα+βΓα+β+1,m3=1−α1−γ+1−αγtγΓ1+γ+1−γαtαΓ1+α+αγtα+γΓα+γ+1,m5=1−β1−γ+1−βγtγΓ1+γ+1−γβtβΓ1+β+βγtα+γΓβ+γ+1.In a similar way,Tm, Im, and Fm, for m≥3 can be obtained. Finally, the solution of model (2) is given by(40)Tt=∑m=0n−1Tmt,(41)It=∑m=0n−1Imt,(42)Ft=∑m=0n−1Fmt,and by choosing a suitable value for ℏ for the convergence of the series according to [20]. The analysis of the convergence of the HATM can found in the literature [35].


## 4. Numerical Illustration

According to [36], it should be noted that the solution of the series contain the auxiliary parameter ℏ, which offers an easy way to control the convergence of the solution of the series. Because it is essential to assure that the series equations (25)–(27) is convergent, we plotted the ℏ curve of 6 terms of the FHATM solution for the fractional-time ABC equations in Figures 1–3. Using these ℏ curves, we note that the straight line that parallels the ℏ axis provides the region of convergence. These valid regions are listed in Table 2.Figure 1
ℏ curves obtained by the (a) 6th-order and (b) 5th-order approximation of the FHATM.
(a)
(b)Figure 2
ℏ curves obtained by the (a) 6th-order and (b) 5th-order approximation of the FHATM.
(a)
(b)Figure 3
ℏ curves obtained by the (a) 6th-order and (b) 5th-order approximation of the FHATM.
(a)
(b)Table 2
Regions of convergence, optimal values ofℏ, and minimum values when t=0.0001.
ℏ
ℏ
∗
Minimum values
−
1.32
≤
ℏ
≤
−
0.62
−
0.804051
1
.
84022
×
10
−
11
−
1.35
≤
ℏ
≤
−
0.65
−
0.805328
1
.
77401
×
10
−
11
−
1.35
≤
ℏ
≤
−
0.65
−
0.808061
5
.
58464
×
10
−
7Furthermore, ifℏ is appropriately chosen, equations (25)–(27) may converge fast. To this end, we have to compute the optimal values of the convergence control parameters from the minimum of the averaged residual errors.Niu and Chun [37] introduced several methods to obtain the optimal value of ℏ. The optimal value of the convergence control parameter is defined by using the concept of the square residual error. An error analysis is presented to determine the optimal values of ℏ. We substitute equations (47)–(49) into equations (4)–(6) and obtain the residual functions as follows:(43)Em,Tt;ℏ1=DtαaABCψTt;ℏ1−p+μTψTt;ℏ1−kψTt;ℏ11−ψTt;ℏ1+ψIt;ℏ1Tmax+k1ψFt;ℏ1ψTt;ℏ1,(44)Em,It;ℏ2=DtβaABCψIt;ℏ2−k1′ψFt;ℏ2ψTt;ℏ2+μIψIt;ℏ2,(45)Em,Ft;ℏ3=DtγaABCψFt;ℏ3−MμbψIt;ℏ3+k1ψFt;ℏ3ψTt;ℏ3+μFψFt;ℏ3.Then, the square residual error for the sixth-order approximation is defined as(46)SEm,Tℏ1=1N+1∑l=0NEm,T∑z=1mTlΔt,SEm,Iℏ2=1N+1∑l=0NEm,I∑z=1mIlΔt,SEm,Fℏ3=1N+1∑l=0NEm,F∑z=1mFlΔt.The use of the first derivative test enables us to determine the values of the auxiliary parameters ℏ1, ℏ2, and ℏ3 for which SEm,Tℏ1, SEm,Iℏ2, and SEm,Fℏ3 are minimized. It should be emphasized that the approximation procedures that are used to select the optimal value of ℏ in FHATM are similar to those of HAM [38].In Table3, the minimum values of the square residual error are given for the optimal values of ℏ1, ℏ2, and ℏ3 when α=β=γ=0.99.Table 3
Residual errorsEm,T, Em,I, and Em,F for the optimal ℏ.
t
E
m
,
T
t
;
ℏ
1
∗
E
m
,
I
t
;
ℏ
2
∗
E
m
,
F
t
;
ℏ
3
∗
0
0.0000234312
0.0000194777
0.00234336
0.2
1
.
47222
×
10
−
8
1
.
17419
×
10
−
8
1
.
24526
×
10
−
6
0.4
1
.
21518
×
10
−
8
8
.
98671
×
10
−
9
6
.
94956
×
10
−
7
0.6
1
.
72222
×
10
−
8
1
.
12419
×
10
−
8
2
.
78878
×
10
−
7
0.8
2
.
67042
×
10
−
8
3
.
1236
×
10
−
8
6
.
87927
×
10
−
6
1
1
.
09289
×
10
−
8
1
.
12475
×
10
−
8
2
.
43936
×
10
−
6The absolute residual errors that were calculated for varioust∈0,1 are listed in Table 2. These results show that the FHATM obtains an accurate approximate solution for the fractional HIV model (4)–(6). The residual errors are plotted in Figure 4 for t∈0,1 and various values of ℏ. The square residual errors are plotted in Figure 5, and Figure 6 shows the absolute residual functions for the optimal ℏ. As these figures show, the solution obtained by using FHATM provides us with a sufficiently accurate analytical solution that only requires a few iterative steps. Mathematica software was used to calculate the six-term approximations for T, I, and F, respectively.(47)ϒTt;ℏ=∑m=06Tmt=1000+1.44837×10−6ℏ+3.70969×10−6ℏ2+5.06867×10−6ℏ3+3.89645×10−6ℏ4+1.59786×10−6ℏ5+2.73079×10−7ℏ6+0.000143992ℏt0.99+0.000377629ℏ2t0.99+0.000528248ℏ3t0.99+0.000415689ℏ4t0.99+0.000174472ℏ5t0.99+0.0000305132ℏ6t0.99+⋯,(48)ϒIt;ℏ=∑m=06Imt=−1.20698×10−6ℏ−3.0988×10−6ℏ2−4.24412×10−6ℏ3−3.27012×10−6ℏ4−1.34403×10−6ℏ5−2.30202×10−7ℏ6−0.000119993ℏt0.99−0.000316179ℏ2t0.99−0.000444257ℏ3t0.99−0.000351059ℏ4t0.99−0.000147925ℏ5t0.99−0.0000259663ℏ6t0.99−⋯,(49)ϒFt;ℏ=∑m=06Fmt=0.001+0.000146285ℏ+0.000378272ℏ2+0.000521647ℏ3+0.000404615ℏ4+0.000167371ℏ5+0.0000288456ℏ6+0.0145431ℏt0.99+0.0388549ℏ2t0.99;+0.0552967ℏ3t0.99+0.0442161ℏ4t0.99+0.0188364ℏ5t0.99+0.00334019ℏ6t0.99+⋯.Figure 4
Errors of residual functions equations (39)–(41) using the sixth order of the approximation solution for various values of ℏ. (a) Comparison errors of residual function ET for various ℏ, (b) comparison errors of residual function EI for various ℏ, and (c) comparison errors of residual function EF for various ℏ.
(a)
(b)
(c)Figure 5
Square residual function equations (42)–(44) using the sixth order of the approximation solution for ℏ∈−0.9,−0.5. (a) Square residual errors of T versus ℏ, (b) square residual errors of I versus ℏ, and (c) square residual errors of F versus ℏ.
(a)
(b)
(c)Figure 6
Absolute residual function equations (39)–(41) using the sixth order of the approximation solution for ℏ∗. (a) Absolute residual error functions for T and the optimal ℏ, (b) absolute residual error functions for I and the optimal ℏ, and (c) absolute residual error functions for F and the optimal ℏ.
(a)
(b)
(c)Note that, if we setα=β=γ=1, then the FHATM solution is the same as that obtained with the HAM in [13]. The numerical results are plotted in Figure 7.Figure 7
Numerical simulation of the concentrations of uninfectedCD4+ T cells T (t), infected CD4+ T cells I (t), and HIV RNA F (t) for different values of α, β, and γ and the optimal values of ℏ∗. (a) Approximate solut[[parms resize(1),pos(50,50),size(200,200),bgcol(156)]], and (c) approximate solutions of F (t).
(a)
(b)
(c)


## 5. Numerical Scheme

In this section, we solve fractional a HIV model numerically using the numerical scheme introduced by Toufik and Atangana [27]. The numerical solution and FHATM solution are compared in Figure 8.Figure 8
Comparison of the numerical solution with that obtained by FHATM forℏ=0.01. (a) Comparison of the numerical solution and FHATM solution for α = 0.99, (b) comparison of the numerical solution and FHATM solution for β = 0.99, and (c) comparison of the numerical solution and FHATM solution for γ = 0.99.
(a)
(b)
(c)


## 6. Conclusion and Further Work

In this study, we successfully solved the fractional HIV infection by using theCD4+ T cells model numerically and analytically, which includes an operator of the type of the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). Analytically approximate solution was obtained for this derivative by incorporating the FHATM in the model of the fractional HIV infection of CD4+ T cells. The solution includes the auxiliary parameter ℏ, which provides an easy way to control the convergence region of the resulting infinite series. The results we obtained show that the FHATM is a successful technique for obtaining an approximate solution of the fractional HIV infection of CD4+ T cells. Moreover, our result agrees strongly with the computation of Toufik and Atangana [27]. Studying the dynamics and stability of the system based on the ABC fractional definition and the FHATM algorithm is an interesting idea for the researchers.



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*Source: 2149037-2020-04-15.xml*