# Dynamical Analysis of Approximate Solutions of HIV-1 Model with an Arbitrary Order

Authors: Asma; Nigar Ali; Gul Zaman; Anwar Zeb; Vedat Suat Erturk; Il Hyo Jung
Journal: Complexity
 (2019)

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## Abstract

This article studies the dynamical behavior of the analytical solutions of the system of fraction order model of HIV-1 infection. For this purpose, first, the proposed integer order model is converted into fractional order model. Then, Laplace-Adomian decomposition method (L-ADM) is applied to solve this fractional order HIV model. Moreover, the convergence of this method is also discussed. It can be observed from the numerical solution that (L-ADM) is very simple and accurate to solve fraction order HIV model.


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## Body

## 1. Introduction

AIDS (acquired immune deficiency syndrome) is a disease which is caused by the type of pathogen virus called human immune deficiency virus (HIV). This virus was introduced in 1981 in USA. AIDS is incurable disease that has high mortality rate (kills more than 25 million worldwide per year). Developing of AIDS takes about 6 months to 15 years. The virus attacks CD4+ T cells. The virus is transmitted through unprotected sexual contact and by sharing contaminated needle or transfusion of infected blood. It may also be transmitted from the mother to her child during pregnancy, lactation, or during birth.Applied mathematicians have a great interest to study the HIV/AIDS dynamics. This research helps the biologists to find the appropriate treatment for infected humans. Mathematical models are important tools in analyzing the spread and control of HIV/AIDS as they provide short- and long-term prediction of HIV and AIDS incidences.The dynamics of HIV epidemic models have been studied by several researchers [1–5]. For the numerical solutions of HIV-1 models, some methods have been introduced. Ghoreishi et al. [6] introduced homotopy analysis method for the solution of HIV-1 model. Further, Ongun [7] introduced the Laplace-Adomian decomposition method CLADMD for the solution of HIV-1 model. The HPM was introduced by Merdan in [8] to find the approximate solution of the model. But the most powerful method is LADM [7], which has gained much attention in the recent past.In this paper, we will generalize AIDS/HIV model to a fractional order system of orderζ in sense of Caputo definition because it is equivalent to ordinary differential equation when ζ=1. We will study the nonlinear model with a fractional order 0<ζ≤1. In recent years, the mathematician have taken interest in fractional calculus because it has many engineering and medical applications. Therefore, first, we consider the HIV infection integer order model which has been presented in [9]. This model consists of five compartments: xt which stands for the uninfected CD4+ T cells, x⋆t which represents the concentration of infected cells while the concentration of double cells is denoted by x⋆⋆, and the densities of pathogen viruses and recombinant viruses are denoted by vpt and vrt, respectively. This model can be written as
(1)ẋt=λ−d1xt−β1xtvpt,ẋ⋆t=β1xtvpt−d2x⋆t−α1vrtx⋆t,ẋ⋆⋆t=α1vrtx⋆t−d3x⋆⋆t,v̇pt=k'x⋆t−d4vpt,v̇rt=c'x⋆⋆t−d5vrt,with the initial condition x0=r1,x⋆o=r2,x⋆⋆o=r3,vp0=r4, and vr0=r5.The parameters used in the system (1) can be defined as follows: the healthy cells are produced at the rateλ. Moreover, d1, d2, d3, d4, and d5 are the rates of death of uninfected cells, infected cells, double infected cells, pathogen virus, and recombinant virus, respectively. β1 is the rate of infection of healthy CD4+ T cells. The rate of production of pathogen virus is defined by k'. α1 is infection rate of double infected cells. c' is the rate at which the recombinant viruses are produced.
(2)cDςxt=λ−d1xt−β1xtvpt,cDςx⋆t=β1xtvpt−d2yt−α1vrtx⋆t,cDςx⋆⋆t=α1vrtx⋆t−d3x⋆⋆t,cDςvpt=k'x⋆t−d4vpt,cDςvrt=c′x⋆⋆t−d5vrt,with the initial conditions x0=r1,x⋆0=r2,x⋆⋆0=r3,vp0=r4, and vr0=r5.In many branches of sciences, fractional differential equations (FDEs) have shown their importance as these equations have much application in different sciences such as Physics, Chemistry, Mechanics, and Engineering [10, 11]. Currently (FDEs), systems have gained much attention. In the recent past, various integer order models have been converted into fractional order models. Fractional order operators have greater degree of freedom, and hence, using these in mathematical models produce best results as compared to integer order derivatives.This research work will be organized as follows: the first section is related to some basic definitions from fractional calculus. In second section, Adomian decomposition method will be applied to the proposed system. Numerical simulation will be carried out in Section3. Section 4 is dedicated to convergence analysis of the proposed fractional order model. In the last section, conclusion will be drawn.


## 2. Preliminaries

We will discuss some basic definitions in this section. For this, we will use [12]. Throughout the paper, we have used Caputo fractional order derivative.Definition 1.
The fractional Riemann-Liouville type integral of orderκ∈0,1 of a function h∈L10,T~,ℝ is written as
(3)I0+κht=1Γκ∫0tt−sκ−1hsds,where cDκ is Caputo derivative and 0<κ<1. But κ is the order of fractional time derivative.Definition 2.
For the functionh, the Caputo derivative for fractional order on the interval 0,T~ can be defined as
(4)cD0+κht=1Γm−κ∫0tt−sm−κ−1hmsds.
Here,m=κ+1 and κ denote the integral part of m.Lemma 1.
The properties related to fractional order integral are satisfied.
(5)IκcDκgt=gt+u0+u1t+u2t2+⋯+um−1tm−1,for uj∈ℝ,j=0,1,2,…,m−1.Definition 3.
The Laplace transform of Caputo derivative is defined as follows:(6)LcDκyt=sκhs−∑k=0m−1sκ−k−1yk0,m−1<κ<m,m∈N,where m=κ+1 and κ denote the integer part of m.Theorem 1.
The initial value problem (2) with the given initial conditions (8) has unique solutions and will remain in R+5.Proof 1.
It can be proved that the solution of the initial value problem exists on (0,1) and is unique as derived in [13]. Moreover, we will show that R+5 is positively invariant region. From system (2), we can obtain the following results:
(7)dxdtx=0=λ≥0,dx⋆dtx⋆=0=βxvp≥0,dx⋆⋆dtx⋆⋆=0=αvrx⋆≥0,dvpdtv=0=kx⋆≥0,dvrdtvr=0=cx⋆⋆.
Hence, the solution will remain in the regionR+5.


## 3. The Laplace-Adomian Decomposition Method (L-ADM)

We will apply our considered method in the following steps. First, we will apply Laplace transform to both sides of (2) and obtain the equations as below
(8)sςℒxt−sς−1x0=ℒλ−d1xt−β1xtvpt,sςℒx⋆t−sς−1x⋆0=ℒβ1xtvpt−d2x⋆t−α1vrtx⋆t,sςℒx⋆⋆t−sς−1x⋆⋆0=ℒα1vrtx⋆t−d3zt,sςℒvpt−sς−1v0=ℒk'x⋆t−d4vpt,sςℒvrt−sς−1vr0=ℒc'x⋆⋆−qvr.Now, using the initial conditions and after some rearrangement, we obtain(9)ℒxt=r1s+1sςℒλ−d1xt−β1xtvptℒx⋆=r2s+1sςℒβ1xtvpt−d2x⋆t−α1vrtx⋆t,ℒx⋆⋆t=r3s+1sςℒα1x⋆tvrt−d3x⋆⋆tℒvpt=r4s+1sςℒk'x⋆t−d4vpt,ℒvrt=r5s+1sςℒc'x⋆⋆t−d5vrt.Assuming the solutions in the form of infinite series as(10)xt=∑m=0∞xm,x⋆t=∑m=0∞x⋆m,x⋆⋆t=∑m=0∞x⋆⋆m,vpt=∑m=0∞vpm,vrt=∑m=0∞vrm,and decomposing the nonlinear terms xtvpt and vrtx⋆ by using Adomian polynomial as
(11)xtvpt=∑m=0∞Kmt,vrtx⋆t=∑m=0∞Mmt,where Km and Mm can be defined by
(12)Kmt=1Γm+1dmdηm∑k=0mηkxk∑k=0mηkvpkη=0,Mmt=1Γm+1dmdηm∑k=0mηkvrk∑k=0mηkx∗kη=0.Here,Km and Mm are Adomian polynomials. The use of system (11) and system (13) in model (9) gives the following systems.
(13)ℒx0=r1s,ℒx⋆0=r2s,ℒx⋆⋆0=r3s,ℒvp0=r4s,ℒvr0=r5s,(14)ℒx1=1sςλ−d1x0−βK0,ℒx⋆1=1sςβ1K0−d1x⋆0−α1M0,ℒx⋆⋆1=1sςα1M0−d3x⋆⋆0,ℒv1=1sςk'x⋆0−d4vp0,ℒw1=1sςc'x⋆⋆0−d5vr0,and so on. The general terms for successive iterations taking m≥1 can be written as
(15)ℒxm+1=1sςλ−d1xm−β1Km,ℒx⋆m+1=1sςβ1Km−d2x⋆m−α1Mm,ℒx⋆⋆m+1=1sςα1Mm−d3x⋆⋆m,ℒvpm+1=1sςk'x⋆m−d4vpm,ℒvrm+1=1sςc'x⋆⋆m−d5vrm.To proceed further, the behaviors of the solutionsxt,x⋆t,x⋆⋆t,vpt, and vrt under different values of ζ will be discussed, and the differences of tendency of the behavior of the solutions between fractional order derivative and integer order derivative will be noted. To get the initial approximations x0,x⋆0,x⋆⋆0,vp0 and vr0, the inverse Laplace transform will be applied to the system (14). Then, using these values in the system (15), the second approximations, x1,x⋆1,x⋆⋆1,vp1, and vr1, are determined. Similarly, x2,x⋆2,x⋆⋆2,vp2, and vr2 can be calculated. We write these solutions as
(16)xt=x0+x1+x2+x3+⋯,x⋆t=x⋆0+x⋆1+x⋆2+x⋆3+⋯,x⋆⋆t=x⋆⋆0+x⋆⋆1+x⋆⋆2+x⋆⋆3+⋯,vpt=vp0+vp1+vp2+vp3+⋯,vrt=vr0+vr1+vr2+vr3+⋯.In the coming section, the numerical behavior of the proposed system will be presented.


## 4. Multistage Adomian Decomposition Method with Numerical

The following analytical approximate solution can be obtained by using L-ADM. These solutions are represented in terms of an infinite power series. Different values of the parameters used in the proposed system are taken from [9]. Therefore, we can write
(17)x0=3,x⋆0=6,x⋆⋆0=3,vp0=149,vr0=1.The first approximations under the given initial values can be found as follows:(18)x1=1.4936tζΓζ+1,x⋆1=−2.476tζΓζ+1,x⋆⋆1=−5.9928tζΓζ+1,vp1=299.89tζ5Γζ+1,vr1=117tζΓζ+1,Next, to calculate the second approximation, we will use the initial and first approximate values.Therefore, we get the following approximation:(19)x2=0.3t2ζ+1Γ2ζ+2−014336t2ζΓ2ζ+1−0.72238tζΓζ+1,x⋆2=3tζΓζ+1−0.3242tζ+1Γζ+2−0.508912t2ζ+1Γ2ζ+2,x⋆⋆2=0.0144tζ+1Γζ+2−5.41856t2ζ+1Γ2ζ+2,vp2=299.8212tζΓζ+1+296t2ζΓ2ζ+1,vr2=117tζΓζ+1−287.88t2ζΓ2ζ+1.The other approximation can be found in similar way. We get the following solutions after three terms by using L-ADM atζ=1 as
(20)xt=3+4.5136t−1.0728t2−0.5600t3,x⋆t=6+8.9928t−34.42t2−0.5089121t3,x⋆⋆t=3−5.9928t+0.0014t2−5.41856t3,vpt=149+598.8212t+296t2,vrt=1+234t−287.88t2.The system of series solution up to three terms forζ=0.95 is given below.
(21)xt=3+0.7870550345t0.95−0.7845218562t1.9+0.5661093291t2.90,x⋆t=6+1.773297592t0.95−0.7845218562t1.90+0.5661093291t2.90,x⋆⋆t=3−6.115846854t0.95−1.022499122t2.90+0.7536239617t1.95,vpt=149+612.0247390t0.95+162.4321420t2.90,vrt=1+238.8045928t0.95−157.5391685t1.90.Further, we obtain the following series after some simplification forζ=0.85 up to three terms as
(22)xt=3+0.8155783471t0.85−0.9280851532t1.70+0.7193120298t2.70,x⋆t=6+1.837562884t0.85−0.9280851532t1.70+0.7193120298t2.70,x⋆⋆t=3−6.337488549t0.85+0.8231484547t1.85−1.299211797t2.70,vpt=149+634.2048563t0.85+192.1563538t1.70,vrt=1+247.4590042t0.85−186.3679924t1.70.The solutions forζ=0.75 up to three terms is given by
(23)xt=3+0.8391376836t0.75−0.1078429583t1.50+0.9027033339et2.50,x⋆t=6+1.890643943t0.75−0.1078429583t1.50+0.9027033339t2.50,x⋆⋆t=3−6.520557442t0.75+0.8953222645t1.75−1.630450726t2.50,vpt=149+652.5249180t0.75+223.2845723t1.50,vrt=1+254.6072690t0.75−216.5585298t1.50.


## 5. Numerical Discussion

The numerical results shown in Figures1–5 imply the effectiveness of numerical methods discussed here. These methods give highly accurate results in very few iterations. Some simulations results (21), (22), and (23) are given as illustrations.Figure 1
The graph represents the solution behavior for the variablext versus time t under different values of the order of derivative. The solid line, dashed line, dot-dashed line, and dotted line represent solution behavior at ζ=1,ζ=0.95,ζ=0.85, and ζ=0.75, respectively.Figure 2
The graph represents the solution behavior for the variablex⋆t (concentration of infected cells) versus time t under different values of the order of derivative. The solid line, dashed line, dot-dashed line, and dotted line represent solution behavior at ζ=1,ζ=0.75,ζ=0.85, and ζ=0.95, respectively.Figure 3
The graph represents the solution behavior the variablex⋆⋆t (concentration of double infected cells) versus time t under different values of the order of derivative. The solid line, dashed line, dot-dashed line, and dotted line represent solution behavior at ζ=1,ζ=0.75,ζ=0.85, and ζ=0.95, respectively.Figure 4
The graph represents the solution behavior for the variablevpt (concentration of pathogen virus) versus time t under different values of the order of derivative. The solid line, dashed line, dot-dashed line, and dotted line represent solution behavior at ζ=1,ζ=0.75,ζ=0.85, and ζ=0.95, respectively.Figure 5
The graph represents the solution behavior for the variablevrt (concentration of recombinant virus) versus time t under different values of the order of the derivative. The solid line, dashed line, dot-dashed line, and dotted line represent solution behavior at ζ=1,ζ=0.95,ζ=0.85, and ζ=0.75, respectively.The figures show the behavior of the solutions of the variablesxt,x⋆t,x⋆t,vpt, and vrt. It is observed from the Figures (1–5) that the Laplace-Adomian decomposition method is more accurate. The presented method can predict the behavior of these variables accurately for the region under consideration. Figure 2 shows the simulation of approximate solutions by assuming ζ=1,0.95,0.85,0.75. It is clear from these figures that the behavior of the approximate solutions depends continuously on the time-fractional derivative. Moreover, we can see that the potential of this access can be rapidly increased if the step size is decreased.Remark 1.
Here, we assume thatw=x,x⋆,x⋆⋆,vp,vr,X=R+5.


## 6. Convergence Analysis

To check the convergence of the series solutions given in (19), we give the following result.Theorem 2 (see [14]).
LetT~:X→X be a nonlinear operator which is also contractive, where X is Banach space, that is, for all w,w¯∈X,T~w−T~w¯,0≤μw−w¯,0<μ<1. Then, in view of Banach contraction theorem, T~ has a unique point w∗ such that T~w∗=w∗. Further assume that the sequence of series generated by LADM can be written as the series (19) can be written as follows:
(24)wn=T~wn−1,wn−1=∑j=1m−1wj−1,n=1,2,3,…,and suppose that w0=w0∈Srw where Srw=w∗∈X:w∗−w<r, then,
(i)
wn∈Srw,
(ii)
limn→∞wn=w.Proof 2.
Letw=x,x⋆,x⋆⋆,vp,vr,X=R+5 be the exact solution of the model w=T~w. Also, wn=T~wn−1,w1=T~w0.
Now(25)w1−w=T~w0−T~w≤μ1w0−w.
Assume that(26)wn−1−w=T~w0−T~w≤μn−1w0−w.
We now prove forn=m,
(27)wn−w=T~wn−1−T~w≤μn−1T~wn−2−T~w=μwn−1=w/≤μμn−1w0−w≤μnw0−w,n→∞,μn→0.
Thus, we have(i)
wn∈Srw,
(ii)
limn→∞wn=w.


## 7. Conclusion

In this research work, the numerical solution of fractional order HIV-1 model is discussed by using the method known as L-ADM. For this purpose, different values are given toζ and numerical simulations for these values have been presented. It may be concluded from these figures that the concentration of virus decreases in the body as ζ increases from 0.75 to 0.85. But the viral load tends toward zero as ζ increases to 0.95 (see Figure 4). Moreover, the load of CD4+ T cells is increased by increasing the value of ζ (see Figure 1). One can conclude that increase in ζ can cause increase in the level of CD4+ T cells and decrease in viral level. Hence, when ζ→0, then the solution of the fractional order model is reduced to the solution of integer order model. Moreover, the convergence of the method is also discussed. Thus, it is observed that LADM can be used to increase the interval of convergence for the derived series solution. It is also shown that this method is very efficient and accurate method as compared with RK4 method and it is better to use this method for the solution of the other nonlinear systems.



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