# The Fractional Differential Model of HIV-1 Infection of CD4+ T-Cells with Description of the Effect of Antiviral Drug Treatment

Authors: Bijan Hasani Lichae; Jafar Biazar; Zainab Ayati
Journal: Computational and Mathematical Methods in Medicine
 (2019)

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## Abstract

In this paper, the fractional-order differential model of HIV-1 infection of CD4+ T-cells with the effect of drug therapy has been introduced. There are three components: uninfected CD4+ T-cells, x, infected CD4+ T-cells, y, and density of virions in plasma, z. The aim is to gain numerical solution of this fractional-order HIV-1 model by Laplace Adomian decomposition method (LADM). The solution of the proposed model has been achieved in a series form. Moreover, to illustrate the ability and efficiency of the proposed approach, the solution will be compared with the solutions of some other numerical methods. The Caputo sense has been used for fractional derivatives.


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## Body

## 1. Introduction

Human immunodeficiency virus (HIV) is a retrovirus that causes acquired immunodeficiency syndrome (AIDS) [1]. HIV infects, damages, and reduces CD4+ T-cells. Therefore, it causes to decrease the resistance of immune system [2]. The body becomes more gradually sensitive to infections and loses its safety. AIDS is one the most important and dangerous diseases in our time. According to UNAIDS 2017 annual report, “36.7 million people globally were living with HIV and 1.8 million people became newly infected with HIV and 1 million people died from AIDS-related illnesses in 2016.” In spite of the great progress in controlling the disease, no vaccine has been yet discovered for HIV. In the last two decades, a lot of efforts have been made to design and solve mathematical models that have essential rule in analyzing to control and prevent the spread of HIV-related diseases [3–13]. Usually almost all of these mathematical models explain the relation between HIV viruses and uninfected CD4+ cells and the effect of drug therapy to infected cells. Bonhoeffer et al. [4] presented a model for virus dynamics with two components x and y, where x denotes the density of infected cells and y shows the density of virus-producing cells.The proposed model is as follows:(1)dxdt=c−βx−γxy,dydt=γxy−dy,where c is the rate of production of infected cells, β is the natural death rate of infected cells, d is the rate of virus-producing cells’ death, and γ is the rate of infection of uninfected cells. This model and many such models were inspired from Anderson’s model [14, 15]. Anderson’s model is one of the first and the most important models of infectious diseases. Tuckwell and Wan [16] introduced a modified model of equation (1) with three components: uninfected, infected CD4+ T-cells, and density of virions in plasma (x, y, and z, respectively). The presented model with three equations is as follows:(2)dxdt=s′−μx−βxz,dydt=βxz−εy,dzdt=cy−γz,with the initial conditions x0=k1,y0=k2, and z0=k3 where s′, μ, β, ε, c, and γ are constant coefficients, s′ is the rate of creation or production of CD4+ T-cells, μ is the natural death rate, β is the rate of infected CD4+ cells from uninfected CD4+ cells, ε is the rate of virus-producing cells’ death, c is the rate of creation of virions viruses by infected cells, and γ is the rate of virus particle death. For the sake of comparison and showing the ability of the proposed approach, we use the parameter values reported in references [6, 16]. The parameter values are as follows: s′=0.272 (day/mm3), μ=0.00136 (day/mm3), β=0.00027 (day/virion/mm3), ε=0.33 (day/mm3), c=50 (virion/CLM/day), and γ=2.0 (day). The rate of some coefficients will change if drug therapy is not 100% successful. When the drug treatment begins, infected cells which create virus components are affected. If the drug therapy is not effective, a part of infected cells will improve and remaining cells will begin to produce virus [8].Mathematical modeling of many problems in biology and other branches of sciences appears as differential equations in fractional order. Because the fractional-order differential equations save memory on themselves and are related to fractals [8, 17–19], we prefer to use the fractional-order form of the model (2) as follows:(3)Dα1x=s′−μx−βxz,Dα2y=βxz−εy,Dα3z=cy−γz,with the same initial conditions, where 0<αi≤1, i=1,2,3. There are some numerical approaches for solving these types of mathematical models. Some of these methods are as follows: homotopy analysis, optimal homotopy asymptotic, homotopy perturbation, Adomian decomposition, and variational iteration [20–30]. In [8], system of fractional equation (3) has been solved by homotopy analysis method (HAM) and generalized Euler method (GEM). In [31], equation (3) has also been solved by homotopy perturbation method (HPM). Adomian [32], introduced a decomposition method (ADM) which is a powerful method to get analytic approximate solution of differential equations. Using Laplace transform method with couple of ADM (LADM) to solve systems of differential equations leads to an effective method that finds many applications in applied mathematics. In this paper, we will solve equation (3) by LADM and will compare the results with the results achieved by generalized Euler, homotopy analysis, homotopy perturbation, and Runge–Kutta methods. The structure of the paper is as follows: in Section 2, we will present a brief review of fractional calculus. In Sections 3 and 4, we will solve the fractional-order HIV-1 model by LADM. In Section 5, the convergence of the method will be discussed. In the last section, we present the conclusion.


## 2. Fractional Calculus

The purpose of this section is to recall a few preliminaries about what appears in this research.Definition 2.1.
The Riemann–Liouville fractional integral of orderα for a function X:0,∞⟶R is defined as(4)JαXs=1Γα∫0ss−tα−1Xtdt,where αε0,∞ (see [33]).Definition 2.2.
The Caputo fractional derivative for a functionX:0,∞⟶R on the closed interval 0,S is defined as(5)DαXs=1Γm−α∫0ss−tm−α−1Xmtdt,m=α+1,where α is the integer part of α. Another presentation of the Caputo fractional derivative can be shown as follows (see [33]):(6)DαXs=Jm−αDmXs.Lemma 2.1.
Ifαε0,∞ and m=α+1, then the following result holds for fractional calculus:(7)JαDαXs=Xs+∑j=0m−1Xj0j!sj.Proof.
See [33, 34].Definition 2.3.
The Laplace transform of Caputo fractional derivative is defined as follows:(8)ℒDαXt=sαYs−∑j=0m−1sα−k−1Xj0,m−1<α<m,m∈ℕ.


## 3. Solution of Model (3)

In this section, LADM has been implemented to solve system of fractional equation (3) with the initial conditions.We apply Laplace transform on both sides of each equation of equation (3):(9)LDα1x=Ls′−μx−βxz,LDα2y=Lβxz−εy,LDα3z=Lcy−γz,which implies that(10)sα1Lx−sα1−1x0=Ls′−μx−βxz,sα2Ly−sα2−1y0=Lβxz−εy,sα3Lz−sα3−1z0=Lcy−γz.Substitution of the initial conditions in equation (10) and applying inverse Laplace transform results in(11)x=k1+ℒ−11sα1ℒs′−μx−βxz,y=k2+ℒ−11sα2ℒβxz−εy,z=k3+ℒ−11sα3ℒcy−γz.To apply ADM, let us considerx, y, and z as the following series:(12)x=∑i=0∞xi,y=∑i=0∞yi,z=∑i=0∞zi.To decompose the nonlinear termxz, let us follow an alternate algorithm [35] to get,(13)xz=∑i=0∞pi,where pi is as the following equation:(14)pi=∑k=0ixkzi−k,substituting equations (12)–(14) into (11) reads(15)ℒx0=k1s,ℒy0=k2s,ℒz0=k3s,ℒx1=s′sα1+1−μsα1ℒx0−βsα1ℒp0,ℒy1=βsα2ℒp0−εsα2ℒy0,ℒz1=csα3ℒy0−γsα3ℒz0,ℒx2=−μsα1ℒx1−βsα1ℒp1,ℒy2=βsα2ℒp1−εsα2ℒy1,ℒz2=csα3ℒy1−γsα3ℒz1,⋮ℒxn+1=−μsα1ℒxn−βsα1ℒpn,ℒyn+1=βsα2ℒpn−εsα2ℒyn,ℒzn+2=csα3ℒyn−γsα3ℒzn.We take inverse Laplace transform on both sides of each equation of equation (15):(16)x0=k1,y0=k2,z0=k3,x1=s′−μx0−βx0z0tα1Γα1+1,y1=βx0z0−εy0tα2Γα2+1,z1=cy0−γz0tα3Γα3+1,x2=−μ+βz0s′−μx0−βx0z0t2α2Γ2α2+1−βx0cy0−γz0tα1+α3Γα1+α3+1,y2=βx0cy0−γz0tα2+α3Γα2+α3+1+βz0s′−μx0−βx0z0tα1+α2Γα1+α2+1−εβx0z0−εy0t2α2Γ2α2+1,z2=cβx0z0−εy0tα2+α3Γα2+α3+1−γcy0−γz0t2α3Γ2α3+1,x3=βx0cβx0z0−εy0tα1+α2+α3Γα1+α2+α3+1+βx0γcy0−γz0tα1+2α3Γα1+2α3+1+β2x0z0cy0−γz0t2α1+α2Γ2α1+α2+1+s′−μx0−βx0z0βz0μ+z0+μμ+βz0t3α1Γ3α1+1+βcy0−γz0μx0−s′−μx0−βx0z0Γα1+α3+1Γα1+1Γα3+1t2α1+α3Γ2α1+α3+1,y3=βx0cβx0z0−εy0−εcy0−γz0t2α2+α3Γ2α2+α3+1−βx0γcy0−γz0tα2+2α3Γα2+2α3+1−βz0μ+βz0s′−μx0−βx0z0t2α1+α2Γ2α1+α2+1+βs′−μx0−βx0z0Γα1+α3+1Γα1+1Γα3+1tα1+α2+α3Γα1+α2+α3+1−β2x0z0cy0−γz0+βz0εs′−μx0−βx0z0tα1+2α2Γα1+2α2+1+ε2βx0z0−εy0t3α2Γ3α2+1,z3=βx0ccy0−γz0−γcβx0z0−εy0tα2+2α3Γα2+2α3+1+γ2cy0−γz0t3α3Γ3α3+1+βz0cs′−μx0−βx0z0tα1+α2+α3Γα1+α2+α3+1−εcβx0z0−εy0t2α2+α3Γ2α2+α3+1.We have calculated four terms of the infinite series ofx, y, and z as an approximate solution. To get any desired accuracy, one is able to proceed the process and obtain more terms. Finally, the solution of mathematical model can be obtained as follows:(17)xt=∑i=0∞xit≈x0t+x1t+x2t+x3t,yt=∑i=0∞yit≈y0t+y1t+y2t+y3t,zt=∑i=0∞zit≈z0t+z1t+z2t+z3t.


## 4. Numerical Simulation

In this section, constants and initial values are substituted in equation (16) to obtain an approximate solution.Substituting the following values:s′  = 0.272 (day/mm3), μ = 0.00136 (day/mm3), β = 0.00027 (day/virion/mm3), ε = 0.33 (day/mm3), c = 50 (virion/CLM/day), and γ = 2.0 (day) and the initial conditions x0=100,y0=0, and z0=1 in equation (16), we get,(18)x0=100,y0=0,z0=1,x1=0.1090tα1Γα1+1,y1=0.0270tα2Γα2+1,z1=−2.0000tα3Γα3+1,x2=−0.0001776700t2α2Γ2α2+1+0.05400tα1+α3Γα1+α3+1,y2=−0.05400tα2+α3Γα2+α3+1+0.0000294300tα1+α2Γα1+α2+1−0.0089100t2α2Γ2α2+1,z2=1.35000tα2+α3Γα2+α3+1+4.0000t2α3Γ2α3+1,x3=−0.0364500tα1+α2+α3Γα1+α2+α3+1−0.10800tα1+2α3Γα1+2α3+1−0.0000145800t2α1+2α2Γ2α1+α2+1+0.000029711656t3α1Γ3α1+1−0.00054Γα1+α3+1Γα1+1Γα3+1t2α1+α3Γ2α1+α3+1,y3=0.054270000t2α2+α3Γ2α2+α3+1+0.10800tα2+2α3Γα2+2α3+1−0.000000047971t2α1+α2Γ2α1+α2+1+0.0000294300Γα1+α3+1Γα1+1Γα3+1tα1+α2+α3Γα1+α2+α3+1+0.000004868100tα1+2α3Γα1+2α3+1+0.0002940300t3α2Γ3α2+1,z3=0.054270000tα2+2α3Γα2+2α3+1−8.0000t3α3Γ3α3+1+0.0014715000tα1+α2+α3Γα1+α2+α3+1−0.4455000t2α2+α3Γ2α2+α3+1.Three terms approximations can be written as the following form:(19)xt=100+0.1090tα1Γα1+1−0.0001776700t2α2Γ2α2+1+0.05400tα1+α3Γα1+α3+1−0.0364500tα1+α2+α3Γα1+α2+α3+1−0.10800tα1+2α3Γα1+2α3+1−0.0000145800t2α1+α2Γ2α1+α2+1+0.000029711656t3α1Γ3α1+1−0.00054Γα1+α3+1Γα1+1Γα3+1t2α1+α3Γ2α1+α3+1,yt=0.0270tα2Γα2+1−0.05400tα2+α3Γα2+α3+1+0.0000294300tα1+α2Γα1+α2+1−0.0089100t2α2Γ2α2+1+0.054270000t2α2+α3Γ2α2+α3+1+0.10800tα2+2α3Γα2+2α3+1−0.000000047971t2α1+α2Γ2α1+α2+1+0.0000294300Γα1+α3+1Γα1+1Γα3+1tα1+α2+α3Γα1+α2+α3+1+0.000004868100tα1+2α2Γα1+2α2+1+0.0002940300t3α2Γ3α2+1,zt=1−2.0000tα3Γα3+1+1.35000tα2+α3Γα2+α3+1+4.0000t2α3Γ2α3+1+0.054270000tα2+2α3Γα2+2α3+1−8.0000t3α3Γ3α3+1+0.0014715000tα1+α2+α3Γα1+α2+α3+1−0.4455000t2α2+α3Γ2α2+α3+1.Let us takeα1, α2, and α3 equal to α, so the solution of fractional-order of model (3) is obtained as follows:(20)xt=100+0.1090tαΓα+1+0.0538223300t2αΓ2α+1−0.1444348683t3αΓ3α+1−0.00054Γ2α+1t3αΓ3α+1,yt=0.0270tαΓα+1−0.0628805700t2αΓ2α+1+0.1625688501t3αΓ3α+1+0.0000294300Γ2α+1t3αΓ3α+1,zt=1−2.0000tαΓα+1+5.35000t2αΓ2α+1−8.389758500t3αΓ3α+1.Forα1=α2=α3=1, the solution of equation (3) will be as follows:(21)xt=100+0.1090t+0.02691116500t2−0.02425247806t3,yt=0.0270t−0.03144028500t2+0.02710461836t3,zt=1−2.0000t+2.6750000t2−1.398293083t3.In Tables1–3, one can compare the approximate solution of fractional-order of model (3) with the results of GEM, HAM, RK4 in [8], and HPM in [31] using traditional order α=1. The results of LADM are more accurate than the results obtained by other methods.Table 1
Numerical results ofxt (uninfected CD4+ T-cells).
t
LADM
GEM
HPM
HAM
RK4
0
100
100
100
100
100
0.2
100.023
100.023
100.023
100.023
100.023
0.4
100.046
100.047
100.047
100.047
100.047
0.6
100.070
100.071
100.071
100.071
100.071
0.8
100.092
100.097
100.097
100.096
100.097
1.0
100.112
100.122
100.123
100.122
100.122Table 2
Numerical results ofyt (infected CD4+ T-cells).
t
LADM
GEM
HPM
HAM
RK4
0
0
0
0
0
0
0.2
0.00436
0.00434
0.00434
0.00434
0.00434
0.4
0.00750
0.00715
0.00721
0.00714
0.00715
0.6
0.01074
0.00908
0.00934
0.00909
0.00908
0.8
0.01336
0.01049
0.01117
0.01063
0.01049
1.0
0.01866
0.01161
0.01276
0.01194
0.01161Table 3
Numerical results ofzt (density of virions in plasma).
t
LADM
GEM
HPM
HAM
RK4
0
1
1
1
1
1
0.2
0.69581
0.69030
0.69071
0.69059
0.69070
0.4
0.53851
0.51152
0.51208
0.51237
0.51190
0.6
0.46097
0.41069
0.41394
0.40994
0.41103
0.8
0.39607
0.35656
0.37749
0.35148
0.35684
1.0
0.27671
0.33053
0.42419
0.32869
0.33073Figures1–3 show the results for different values of α, and the results can be compared.Figure 1
Dynamics of uninfected CD4+ T-cells for various values of α.Figure 2
Dynamics of infected CD4+ T-cells for various values of α.Figure 3
Dynamics of density of virions in plasma for various values ofα.


## 5. Convergence Analysis of the Method

In this section, convergence of the proposed method, using the idea presented in [36], is studied.Consider the following functional equation:(22)Fvt=gt,where F is a functional operator and can be decomposed as F=Dα+R+N and g is a known function.D
α is a Caputo fractional derivative operator, R is a linear operator, and N is a nonlinear analytic operator, respectively. So equation (22) can be written as follows:(23)Dαvt=gt−Rvt−Nvt.The goal is to find a functionvt satisfying equation (22). By applying the Laplace transform on both sides of equation (23) reads(24)ℒDαvt=ℒgt−Rvt−Nvt.By using definition (2.3), equation (23) can be written as follows:(25)ℒvt=v0s+ℒgtsα−ℒRvtsα−ℒNvtsα,by considering v0=v0 and using inverse of Laplace transform on both sides of equation (25) results in(26)ℒ−1ℒvt=ℒ−1v0s+ℒgtsα−ℒRvtsα−ℒNvtsα,which implies that(27)vt=v0+ℒ−1ℒgtsα−ℒ−1ℒRvtsα−ℒ−1ℒNvtsα.By implementing ADM and assuming the solutionvt as an infinite series say, vt=∑n=0∞vnt, and writing the nonlinear term based on Adomian polynomials such as(28)Nvt=∑n=0∞pnv0t,v1t,…,vnt,where(29)pnv0t,v1t,…,vnt=1Γn+1dndλnNλ∑i=0nvitλiλ=0.Equation (27) can be written as the following form:(30)∑n=0∞vnt=v0+ℒ−1ℒgtsα−ℒ−1ℒR∑n=0∞vntsα−ℒ−1ℒ∑n=0∞pnv0t,v1t,…,vntsα.So, we have(31)∑n=0∞vnt=v0+ℒ−1ℒgtsα−∑n=0∞ℒ−1ℒRvntsα−∑n=0∞ℒ−1ℒpnv0t,v1t,…,vntsα.From which we can define(32)v0t=v0,v1t=ℒ−1ℒgtsα−ℒ−1ℒRv0tsα−ℒ−1ℒp0v0tsα,v2t=−ℒ−1ℒRv1tsα−ℒ−1ℒp1v0t,v1tsα,⋮vn+1t=−ℒ−1ℒRvntsα−ℒ−1ℒpnv0t,v1t,…,vntsα.Theorem 5.1.
LADM for equation (21) with the solution (19) is equivalent to(33)snt=v0t+v1t+…+vnt,s0t=v0t.By using the following iterative scheme:(34)sn+1t=v0+ℒ−1ℒgtsα−ℒ−1ℒRsntsα−ℒ−1ℒNsntsα,where(35)N∑i=0nvit=∑i=0npiv0t,v1t,…,vit,n=0,1,2,….Proof.
Forn=0, from equation (33):(36)s1t=v0+ℒ−1ℒgtsα−ℒ−1ℒRs0tsα−ℒ−1ℒNs0tsα=v0+ℒ−1ℒgtsα−ℒ−1ℒRv0tsα−ℒ−1ℒp0v0tsα.
Then, by assumption of equation (33), we have(37)v1t=ℒ−1ℒgtsα−ℒ−1ℒRv0tsα−ℒ−1ℒp0v0tsα,for n=1:(38)s2t=v0+ℒ−1ℒgtsα−ℒ−1ℒRs1tsα−ℒ−1ℒNs1tsα=v0+ℒ−1ℒgtsα−ℒ−1ℒRv0t+v1tsα−ℒ−1ℒp0v0t+p1v0t,v1tsα=v0t+v1t−ℒ−1ℒRv1tsα−ℒ−1ℒp1v0t,v1tsα.
We know thats2t=v0t+v1t+v2t, so we obtain(39)v2t=−ℒ−1ℒRv1tsα−ℒ−1ℒp1v0t,v1tsα.
By strong induction, let us have(40)vk+1t=−ℒ−1ℒRvktsα−ℒ−1ℒpkv0t,v1t,…,vktsα,k=1,2,…,n−1,and prove the following for k=n,(41)sn+1t=v0+ℒ−1ℒgtsα−ℒ−1ℒRsntsα−ℒ−1ℒNsntsα=v0+ℒ−1ℒgtsα−ℒ−1ℒR∑k=0nvktsα−ℒ−1ℒ∑k=0npkv0t,v1t,…,vktsα=v0+ℒ−1ℒgtsα−∑k=0nℒ−1ℒRvktsα−∑k=0nℒ−1ℒpkv0t,v1t,…,vktsα=v0t+v1t+…vnt−ℒ−1ℒRvntsα−ℒ−1ℒpnv0t,v1t,…,vntsα.
Then, from equation (33), we derive(42)vn+1t=−ℒ−1ℒRvntsα−ℒ−1ℒpnv0t,v1t,…,vntsα.
This entails the statement is true and the theorem is proved.Theorem 5.2.
LetX be a Banach space.(i)
∑i=0∞vit resulted from equation (31), convergence to s∈X, if ∃c∈0,1,s.t∀n∈ℕ⟹vn+1≤cvn,(ii)
st=∑i=0∞vit satisfies in(43)st=v0+ℒ−1ℒgtsα−ℒ−1ℒRstsα−ℒ−1ℒNstsα.Proof.
(44)
s
n
+
1
−
s
n
=
v
n
+
1
≤
c
v
n
≤
c
2
v
n
−
1
≤
…
≤
c
n
+
1
v
0
.
∀
n
,
m
∈
ℕ
,
n
≥
m, we have(45)sn−sm=sn−sn−1+sn−1−sn−2+…+sm+1−sm≤sn−sn−1+sn−1−sn−2+…+sm+1−sm≤cnv0+cn−1v0+…+cm+1v0≤cn+cn−1+…+cm+1v0≤cm+11+c+c2+…+cn+…≤cm+11−cv0.
This means thatlimn,m⟶∞sn−sm=0, therefore, sn is a Cauchy sequence in the Banach space of X and is convergent. So, ∃sϵX,s.tlimn⟶∞sn=s.
From equation (33), we derive(46)limsntn⟶∞=v0+ℒ−1ℒgtsα−ℒ−1ℒlimn⟶∞Rsntsα−ℒ−1ℒlimn⟶∞Nsntsα=v0+ℒ−1ℒgtsα−ℒ−1ℒlimn⟶∞R∑k=0nvktsα−ℒ−1ℒlimn⟶∞N∑k=0nvktsα=v0+ℒ−1ℒgtsα−ℒ−1ℒlimn⟶∞∑k=0nRvktsα−ℒ−1ℒlimn⟶∞∑k=0npkv0t,v1t,…,vktsα=v0+ℒ−1ℒgtsα−ℒ−1ℒ∑k=0∞Rvktsα−ℒ−1ℒ∑k=0∞pkv0t,v1t,…,vktsα,
From equation (35), we have(47)N∑i=0∞vit=∑i=0∞piv0t,v1t,…,vit,n=0,1,2.….
So,(48)st=v0+ℒ−1ℒgtsα−ℒ−1ℒR∑k=0∞vktsα−ℒ−1ℒN∑k=0∞vktsα=v0+ℒ−1ℒgtsα−ℒ−1ℒRstsα−ℒ−1ℒNstsα.Lemma 5.1.
Equation (43) is equivalent to equation (22).Proof.
By using Laplace transform on both sides of equation (43) reads to(49)ℒst=ℒv0+ℒgtsα−ℒRstsα−ℒNstsα=v0s+ℒgtsα−ℒRstsα−ℒNstsα=sα−1v0sα+ℒgtsα−ℒRstsα−ℒNstsα,so we can write(50)sαℒst−sα−1v0=ℒgt−ℒRst−ℒNst.
In virtue of definition 2.3 and linearity of the Laplace transform, equation (50) can be written as the follows:(51)ℒDαst=ℒgt−Rst−Nst.
By applying the inverse of Laplace transform on both sides of equation (51), we derive(52)Dαst=gt−Rst−Nst.
Consideringvt=st, one gets equation (22). So, the solution of equation (43) is the same as the solution of equation (22).


## 6. Conclusion

In this paper, a fractional-order model of HIV-1 with three components has been introduced. Whenα⟶1, then Dαxt⟶Dxt; therefore, the fractional-order of presented model reduces to traditional model. By applying Laplace transform and Adomian decomposition method, or LADM for short, which is a strong approach to compute numerical solution of fractional differential equations, we gain an approximate solution of the proposed model. The accuracy of the proposed approach has made it a reliable method. We have calculated four terms of the infinite series of x, y, and z as an approximate solution. The result of LADM has been compared with the results of some other methods such as GEM, HAM, RK4 [8], and HPM [31]. The results are presented in Tables 1–3. Figures 1–3, show that the uninfected CD4+ T-cells, x, infected CD4+ T-cells, y, and density of virions in plasma, z, depend on the various values of α, the various values of the parameters, and the time fractional derivative. A comparison of the approximate solutions shows that LADM can work more accurate than other methods. Convergence of the proposed method is studied. Because of the fact that obtaining the exact solution for system of fractional equation is difficult or impossible, we would like to suggest such an easy and reliable approach for further research, in the future.



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