# Numerical Solution of Fractional Model of HIV-1 Infection in Framework of Different Fractional Derivatives

Authors: Sara Salem Alzaid; Badr Saad T. Alkahtani; Shivani Sharma; Ravi Shanker Dubey
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we have extended the model of HIV-1 infection to the fractional mathematical model using Caputo-Fabrizio and Atangana-Baleanu fractional derivative operators. A detailed proof for the existence and the uniqueness of the solution of fractional mathematical model of HIV-1 infection in Atangana-Baleanu sense is presented. Numerical approach is used to find and study the behavior of the solution of the stated model using different derivative operators, and the graphical comparison between the solutions obtained for the Caputo-Fabrizio and the Atangana-Baleanu operator is presented to see which fractional derivative operator is more efficient.


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## Body

## 1. Introduction

HIV stands for the Human Immunodeficiency Virus. This virus attacks a person’s immune system. The immune system of a person’s body works as a defence mechanism against infections such as bacteria and viruses.All viruses proliferate by infecting the living cells of the body. HIV targets such immune system cells which are meant to defend the body. These cells are known as CD4+ T cells. CD4+ T cells are basically white blood cells that fight infection. CD4+ T cell count is a measure of immune function in a patient with HIV. It is an imperative determinant for the need for opportunistic infection (OI) prophylaxis. CD4+ T cell count is determined from blood as a part of laboratory monitoring for HIV infection.When HIV takes over these cells, it transforms a cell into a virus factory. It makes the cell to produce thousands of copies of the virus, and these copies then infect the other CD4+ T cells. The infected cells do not function desirably and die early. The paucity of CD4+ T cells deteriorates the immune system and makes it challenging for the body to stay healthy.CD4+ T cell count is generally measured after being diagnosed with HIV (at baseline). It is measured in every three to six months during the first two years until these cell counts in a patient’s body increase above 300 cells/mm3; else, it is measured in every twelve months. Most HIV patients can expect an average increase of about 50-100 cells/mm3. Patients who indulge in therapy with low CD4+ T cell count or at an older age may not experience the same increase in their CD4+ T cell count despite virologic suppression.There are a lot of factors that affect these cell counts. Medication is one of the options to keep these T cell counts high.The characteristics of Acquired Immunodeficiency Syndrome (AIDS) pathogenesis are progressive depletion of CD4+ T cell population in close association with progressive impairment of cellular immunity and increased susceptibility to opportunistic infections (OI). HIV was originally hypothesized to be a consequence of sluggish CD4+ T cell destruction. However, massive CD4+ memory T cell destruction now occurs at a primitive level of infection. In most individuals, the initial destruction is countered by CD4+ memory T cell regeneration that protects these T cell numbers and functions above the threshold associated with overt immunodeficiency (see [1, 2]).The mathematical model of HIV-1 infection of CD4+ T cells is given as follows
(1)dUdt=ε−dU−αUV,U0=γ1,dIdt=αUV−sI,I0=γ2,dVdt=cI−δV,V0=γ3,here, Ut represents the population with uninfected CD4+ T cells, It represents the population with infected CD4+ T cells, and Vt represents the density of virions in plasma at time “t”, respectively. Further, ε is the rate of generation of CD4+ T cells, d is the natural death rate, α is the infected death rate of CD4+ T cells, s is the rate by which viruses make the cells dead, c is the rate of formation of virions viruses due to infected CD4+ T cells, and δ gives the death rate of viruses (see [3]). In the current work, we will be concerned with the fractional mathematical model of HIV-1 infection. Most biological systems have aftereffects or memory, so the modeling of such biological systems using fractional order derivatives have many advantages in which the effects like memory are neglected. It has been concluded that there is fractional order electrical conductance in the cell membrane of many biological organisms, and they are classified in groups of noninteger models. Thus, fractional derivatives, see also ([4–12]) study the behavior of these biological models more efficiently. Fractional calculus have been successfully growing and giving fruitful and countless developments in fields of chemistry, physics, biochemistry, medicine, biology, etc. (see [13–16]). In this work, we will consider the fractional mathematical model of HIV-1 infection in sense of Caputo-Fabrizio derivative operator [17] and Atangana-Baleanu derivative operator [18]. A detailed proof for the existence and the uniqueness of the solution is presented. The numerical solutions are presented for the fractional mathematical model in sense of the Caputo-Fabrizio and the Atangana-Baleanu derivative operator. Further, the results obtained are compared graphically to see which fractional derivative operator gives the better result.Definition 1 (see [17]).
Letf be an integrable function on ℝ, t>0, 0<β<1, the Caputo-Fabrizio fractional derivative of order β is defined as
(2)ζtβ0CFft=Nβ1−β∫0texp−βt−τ1−βf′τdτ,where ζtβ0CF represents the Caputo-Fabrizio fractional derivative of order β and Nβ is a normalization function and the following holds N0=N1=1.Definition 2 (see [19]).
Letf be an integrable function on ℝ, t>0, 0<β<1, the Caputo-Fabrizio time fractional integral of order β is given as
(3)ICFtβft=21−β2−βNβft+2β2−βNβ∫0tfτdτ,where Nβ is the normalization function and the following holds N0=N1=1.Definition 3 (see [18]).
Letf be an integrable function on ℝ, let 0<β<1, then the Atangana-Baleanu fractional derivative is given as
(4)ζtβ0ABCft=Nβ1−β∫0tf′τEβ−βt−τβ1−βdτ,where ζtβ0ABC is the Atangana-Baleanu fractional derivative of order β in Caputo sense, Eβ is the Mittag-Leffler function, and Nβ is the normalization function such that N0=N1=1.Definition 4 (see [18]).
Letf be an integrable function on ℝ, the fractional integral of Atangana-Baleanu fractional derivative of order β is given as
(5)IABtβft=1−βNβft+βNβΓβ∫0tfτt−τβ−1dτ.Theorem 5 (see [18]).
The fractional differential equation(6)IABtβft=vt,possesses a solution which is unique given as
(7)ft=1−βMβvt+βMβΓβ∫0tvτt−τβ−1dτ.


## 2. Fractional HIV-1 Model in Caputo-Fabrizio Sense

The fractional mathematical model of HIV-1 infection in Caputo-Fabrizio sense is given as follows.(8)ζtβ0CFUt=ε−dU−αUV,U0=γ1,ζtβ0CFIt=αUV−sI,I0=γ2,ζtβ0CFVt=cI−δV,V0=γ3,here, 0CFζtβ denotes the Caputo-Fabrizio fractional derivative of order β.


### 2.1. Derivation of Numerical Scheme in Sense of Caputo-Fabrizio Derivative

To illustrate the method ([20, 21]), we contemplate the following general equation:
(9)ζtβ0CFηt=gt,ηt,t≥0,η0=η0,where ηt stands for Ut,It,Vt and gt,ηt is an integrable function that stands for ε−dU−αUV, αUV−sI, and cI−δV, respectively. Using the fundamental theorem, the above equation can be written as
(10)ηt−η0=1−βNβgt,ηt+βNβ∫0tgτ,ητdτ.At pointt=tk+1, for k=0,1,2⋯, equation (10) becomes
(11)ηtk+1−η0=1−βNβgtk,ηtk+βNβ∫0tk+1gτ,ητdτ.At pointt=tk, for k=0,1,2⋯, equation (10) becomes
(12)ηtk−η0=1−βNβgtk−1,ηtk−1+βNβ∫0tkgτ,ητdτ.From the above two equations, we get(13)ηtk+1−ηtk=1−βNβgtk,ηtk−gtk−1,ηtk−1+βNβ∫tktk+1gτ,ητdτ.Considering g(τ, ητ) through Lagrange polynomial interpolation,
(14)qn=gτ,ητ=τ−ti−1ti−ti−1gti,ηti+τ−titi−1−tigti−1,ηti−1,where ηti is a function at time ti and ηti−1 is a function at time ti−1. Substituting the value of gτ,ητ in equation (13), we get
(15)ηk+1−ηk=1−βNβgtk,ηtk−gtk−1,ηtk−1+βNβ∫tktk+1gtk,ηkhτ−tk−1−gtk−1,ηk−1hτ−tkdτ.Substitutingh=ti−ti−1 and on solving, we get
(16)ηk+1=η0+1−βNβ+3h2Nβgtk,ηtk−1−βNβ+βh2Nβgtk−1,ηtk−1.Using the aforementioned concept of numerical scheme, the numerical scheme for the fractional model of HIV-1 infection in framework of Caputo-Fabrizio derivative operator is given as(17)Uk+1=U0+1−βNβ+3h2NβL1tk,ηtk−1−βNβ+βh2NβL1tk−1,ηtk−1,where
(18)L1t,U=ε−dU−αUV,Ik+1=I0+1−βNβ+3h2NβL2tk,ηtk−1−βNβ+βh2NβL2tk−1,ηtk−1,where
(19)L2t,I=αUV−sI,Vk+1=V0+1−βNβ+3h2NβL3tk,ηtk−1−βNβ+βh2NβL3tk−1,ηtk−1,where
(20)L3t,V=cI−δV.


## 2.1. Derivation of Numerical Scheme in Sense of Caputo-Fabrizio Derivative

To illustrate the method ([20, 21]), we contemplate the following general equation:
(9)ζtβ0CFηt=gt,ηt,t≥0,η0=η0,where ηt stands for Ut,It,Vt and gt,ηt is an integrable function that stands for ε−dU−αUV, αUV−sI, and cI−δV, respectively. Using the fundamental theorem, the above equation can be written as
(10)ηt−η0=1−βNβgt,ηt+βNβ∫0tgτ,ητdτ.At pointt=tk+1, for k=0,1,2⋯, equation (10) becomes
(11)ηtk+1−η0=1−βNβgtk,ηtk+βNβ∫0tk+1gτ,ητdτ.At pointt=tk, for k=0,1,2⋯, equation (10) becomes
(12)ηtk−η0=1−βNβgtk−1,ηtk−1+βNβ∫0tkgτ,ητdτ.From the above two equations, we get(13)ηtk+1−ηtk=1−βNβgtk,ηtk−gtk−1,ηtk−1+βNβ∫tktk+1gτ,ητdτ.Considering g(τ, ητ) through Lagrange polynomial interpolation,
(14)qn=gτ,ητ=τ−ti−1ti−ti−1gti,ηti+τ−titi−1−tigti−1,ηti−1,where ηti is a function at time ti and ηti−1 is a function at time ti−1. Substituting the value of gτ,ητ in equation (13), we get
(15)ηk+1−ηk=1−βNβgtk,ηtk−gtk−1,ηtk−1+βNβ∫tktk+1gtk,ηkhτ−tk−1−gtk−1,ηk−1hτ−tkdτ.Substitutingh=ti−ti−1 and on solving, we get
(16)ηk+1=η0+1−βNβ+3h2Nβgtk,ηtk−1−βNβ+βh2Nβgtk−1,ηtk−1.Using the aforementioned concept of numerical scheme, the numerical scheme for the fractional model of HIV-1 infection in framework of Caputo-Fabrizio derivative operator is given as(17)Uk+1=U0+1−βNβ+3h2NβL1tk,ηtk−1−βNβ+βh2NβL1tk−1,ηtk−1,where
(18)L1t,U=ε−dU−αUV,Ik+1=I0+1−βNβ+3h2NβL2tk,ηtk−1−βNβ+βh2NβL2tk−1,ηtk−1,where
(19)L2t,I=αUV−sI,Vk+1=V0+1−βNβ+3h2NβL3tk,ηtk−1−βNβ+βh2NβL3tk−1,ηtk−1,where
(20)L3t,V=cI−δV.


## 3. Fractional HIV-1 Model in Atangana-Baleanu Sense

The fractional mathematical model of HIV-1 infection in Atangana-Baleanu sense is given as follows.(21)ζtβ0ABCUt=ε−dU−αUV,U0=γ1,ζtβ0ABCIt=αUV−sI,I0=γ2,ζtβ0ABCVt=cI−δV,V0=γ3,here, ζtβ0ABC is the Atangana-Baleanu fractional derivative of order β.


### 3.1. Existence and Uniqueness of the Solution

Theorem 6.
The kernels(22)L1t,U=ε−dU−αUV,L2t,I=αUV−sI,L3t,V=cI−δV,satisfy the Lipschitz condition and contractions if following hold:
(23)0<k1<1,0<k2<1,0<k3<1,where k1=d+n1, k2=s, and k3=δ.Proof.
Consider the kernel(24)L1t,U=ε−dU−αUV.
LetU and U1 be two functions, then
(25)L1t,U−L1t,U1=ε−dU−αUV−ε−dU1−αU1V≤dU−U1+αVU−U1.
Letn1=maxtVt, then
(26)L1t,U−L1t,U1≤k1Ut−U1t,where
(27)k1=d+n1α.
Hence, this proves Lipschitz’s condition forL1t,U, and if 0<k1<1, then this proves contraction for L1t,U. Similarly, we can prove the result for the kernels L2t,I and L3t,V.Theorem 7.
The time fractional HIV-1 infection model(28)ζtβ0ABCUt=ε−dU−αUV,U0=γ1,(29)ζtβ0ABCIt=αUV−sI,I0=γ2,(30)ζtβ0ABCVt=cI−δV,V0=γ3,possesses a unique solution under the conditions that we are able to search tmax which satisfies
(31)1−βNβki+tmaxβNβΓβki<1,fori=1,2,3,where k1=d+n1, k2=s, and k3=δ.Proof.
We prove the result for equation (28).
Consider equation (28)
(32)ζtβ0ABCUt=ε−dU−αUV,U0=γ1.
LetL1t,U=ε−dU−αUV.
Then, equation (32) can be written as
(33)ζtβ0ABCUt=L1t,U.
Using Theorem5, we get
(34)Ut=U0+1−βNβL1t,Ut+βNβΓβ∫0tt−τβ−1L1τ,Uτdτ.
LetJ=0,T and define an operator Z:CJ,ℝ3⟶CJ,ℝ3 such that
(35)ZUt=U0+1−βNβL1t,Ut+βNβΓβ∫0tt−τβ−1L1τ,Uτdτ.
So equation (34) can be seen as ZUt=Ut. Define the supremum norm on J as U=Supt∈JUt. Then, CJ,ℝ3 and . define a Banach Space. Finally, consider
(36)ZU1t−ZU2t=1−βNβL1t,U1t−L2t,U2t+βNβΓβ∫0tt−τβ−1L1τ,U1τ−L1τ,U1τdτ.
Taking modulus on both sides of equation (36) and using triangle inequality, we get
(37)ZU1t−ZU2t≤1−βNβL1t,U1t−L2t,U2t+βNβΓβ∫0tt−τβ−1L1τ,U1τ−L1τ,U1τdτ.
Lastly, using the fact that the kernelL1t,Ut satisfies Lipschitz condition, we get
(38)ZU1−ZU2≤1−βNβk1+tmaxβNβΓβk1U1−U2.
Equation (38) is a contraction if
(39)1−βNβk1+tmaxβNβΓβk1<1.
Hence, using the Banach Fixed Point theorem, we govern the existence of a unique solution for the fractional model of HIV-1 infection in framework of Atangana-Baleanu derivative operator.


### 3.2. Derivation of Numerical Scheme in Sense of ABC Derivative

Toufik and Atangana ([21, 22]) introduced a numerical scheme for solving fractional derivatives having nonsingular and nonlocal kernel. To illustrate the method, contemplate the following fractional differential equation:
(40)ζtβ0ABCηt=gt,ηt,t≥0,η0=η0.Using Theorem5, the above equation can be written as
(41)ηt−η0=1−βNβgt,ηt+βNβΓβ∫0tt−τβ−1gτ,ητdτ.At pointt=tk+1, for k=0,1,2⋯, equation (41) becomes
(42)ηtn+1−η0=1−βNβgtk,ηtk+βNβΓβ∫0tk+1tk+1−τβ−1gτ,ητdτ,(43)ηk+1=ηtk+1=η0+1−βNβgtk,ηtk+βNβΓβ∑j=0k∫tjtj+1tk+1−τβ−1gτ,ητdτ.Consideringgτ,ητ through Lagrange polynomial interpolation,
(44)qn=gτ,ητ=τ−ti−1ti−ti−1gti,ηti+τ−titi−1−tigti−1,ηti−1.Substituting the value ofgτ,ητ in equation (43), we get
(45)ηk+1=η0+1−βNβgtk,ηtk+βNβΓβ∑i=0kgti,ηtih∫titi+1t−ti−1tk+1−tβ−1dt−gti−1,ηti−1h∫titi+1t−ti−1tk+1−tβ−1dt.Substitutingh=ti−ti−1 and on solving, we get
(46)ηk+1=η0+1−βNβgtk,ηtk+βNβ∑i=0khβgti,ηtiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβgti−1,ηti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β.Using the aforementioned concept of numerical scheme, the numerical scheme for the fractional model of HIV-1 infection in framework of Atangana-Baleanu derivative operator is given as(47)Uk+1=U0+1−βNβL1tk,Utk+βNβ∑i=0khβL1ti,UtiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβL1ti−1,Uti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β,where
(48)L1t,U=ε−dU−αUV,Ik+1=I0+1−βNβL2tk,Itk+βNβ∑i=0khβL2ti,ItiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβL2ti−1,Iti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β,where,
(49)L2t,I=αUV−sI,Vk+1=V0+1−βNβL3tk,Vtk+βNβ∑i=0khβL3ti,VtiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβL3ti−1,Vti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β,where,
(50)L3t,V=cI−δV.


## 3.1. Existence and Uniqueness of the Solution

Theorem 6.
The kernels(22)L1t,U=ε−dU−αUV,L2t,I=αUV−sI,L3t,V=cI−δV,satisfy the Lipschitz condition and contractions if following hold:
(23)0<k1<1,0<k2<1,0<k3<1,where k1=d+n1, k2=s, and k3=δ.Proof.
Consider the kernel(24)L1t,U=ε−dU−αUV.
LetU and U1 be two functions, then
(25)L1t,U−L1t,U1=ε−dU−αUV−ε−dU1−αU1V≤dU−U1+αVU−U1.
Letn1=maxtVt, then
(26)L1t,U−L1t,U1≤k1Ut−U1t,where
(27)k1=d+n1α.
Hence, this proves Lipschitz’s condition forL1t,U, and if 0<k1<1, then this proves contraction for L1t,U. Similarly, we can prove the result for the kernels L2t,I and L3t,V.Theorem 7.
The time fractional HIV-1 infection model(28)ζtβ0ABCUt=ε−dU−αUV,U0=γ1,(29)ζtβ0ABCIt=αUV−sI,I0=γ2,(30)ζtβ0ABCVt=cI−δV,V0=γ3,possesses a unique solution under the conditions that we are able to search tmax which satisfies
(31)1−βNβki+tmaxβNβΓβki<1,fori=1,2,3,where k1=d+n1, k2=s, and k3=δ.Proof.
We prove the result for equation (28).
Consider equation (28)
(32)ζtβ0ABCUt=ε−dU−αUV,U0=γ1.
LetL1t,U=ε−dU−αUV.
Then, equation (32) can be written as
(33)ζtβ0ABCUt=L1t,U.
Using Theorem5, we get
(34)Ut=U0+1−βNβL1t,Ut+βNβΓβ∫0tt−τβ−1L1τ,Uτdτ.
LetJ=0,T and define an operator Z:CJ,ℝ3⟶CJ,ℝ3 such that
(35)ZUt=U0+1−βNβL1t,Ut+βNβΓβ∫0tt−τβ−1L1τ,Uτdτ.
So equation (34) can be seen as ZUt=Ut. Define the supremum norm on J as U=Supt∈JUt. Then, CJ,ℝ3 and . define a Banach Space. Finally, consider
(36)ZU1t−ZU2t=1−βNβL1t,U1t−L2t,U2t+βNβΓβ∫0tt−τβ−1L1τ,U1τ−L1τ,U1τdτ.
Taking modulus on both sides of equation (36) and using triangle inequality, we get
(37)ZU1t−ZU2t≤1−βNβL1t,U1t−L2t,U2t+βNβΓβ∫0tt−τβ−1L1τ,U1τ−L1τ,U1τdτ.
Lastly, using the fact that the kernelL1t,Ut satisfies Lipschitz condition, we get
(38)ZU1−ZU2≤1−βNβk1+tmaxβNβΓβk1U1−U2.
Equation (38) is a contraction if
(39)1−βNβk1+tmaxβNβΓβk1<1.
Hence, using the Banach Fixed Point theorem, we govern the existence of a unique solution for the fractional model of HIV-1 infection in framework of Atangana-Baleanu derivative operator.


## 3.2. Derivation of Numerical Scheme in Sense of ABC Derivative

Toufik and Atangana ([21, 22]) introduced a numerical scheme for solving fractional derivatives having nonsingular and nonlocal kernel. To illustrate the method, contemplate the following fractional differential equation:
(40)ζtβ0ABCηt=gt,ηt,t≥0,η0=η0.Using Theorem5, the above equation can be written as
(41)ηt−η0=1−βNβgt,ηt+βNβΓβ∫0tt−τβ−1gτ,ητdτ.At pointt=tk+1, for k=0,1,2⋯, equation (41) becomes
(42)ηtn+1−η0=1−βNβgtk,ηtk+βNβΓβ∫0tk+1tk+1−τβ−1gτ,ητdτ,(43)ηk+1=ηtk+1=η0+1−βNβgtk,ηtk+βNβΓβ∑j=0k∫tjtj+1tk+1−τβ−1gτ,ητdτ.Consideringgτ,ητ through Lagrange polynomial interpolation,
(44)qn=gτ,ητ=τ−ti−1ti−ti−1gti,ηti+τ−titi−1−tigti−1,ηti−1.Substituting the value ofgτ,ητ in equation (43), we get
(45)ηk+1=η0+1−βNβgtk,ηtk+βNβΓβ∑i=0kgti,ηtih∫titi+1t−ti−1tk+1−tβ−1dt−gti−1,ηti−1h∫titi+1t−ti−1tk+1−tβ−1dt.Substitutingh=ti−ti−1 and on solving, we get
(46)ηk+1=η0+1−βNβgtk,ηtk+βNβ∑i=0khβgti,ηtiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβgti−1,ηti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β.Using the aforementioned concept of numerical scheme, the numerical scheme for the fractional model of HIV-1 infection in framework of Atangana-Baleanu derivative operator is given as(47)Uk+1=U0+1−βNβL1tk,Utk+βNβ∑i=0khβL1ti,UtiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβL1ti−1,Uti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β,where
(48)L1t,U=ε−dU−αUV,Ik+1=I0+1−βNβL2tk,Itk+βNβ∑i=0khβL2ti,ItiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβL2ti−1,Iti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β,where,
(49)L2t,I=αUV−sI,Vk+1=V0+1−βNβL3tk,Vtk+βNβ∑i=0khβL3ti,VtiΓβ+2k+1−iβk−i+2+β−k−iβk−i+2+2β−hβL3ti−1,Vti−1Γβ+2k+1−iβ+1−k−iβk−i+1+β,where,
(50)L3t,V=cI−δV.


## 4. Conclusion

The graphical representation shows the change in the number of uninfected CD4+ T cells, infected CD4+ T cells and the density of virions in plasma with respect to time. We have also inferred the effect of different fractional order derivatives to see the change in the number of uninfected CD4+ T cells, infected CD4+ T cells and the density of virions in plasma with respect to time. From Figures 1 and 2, we see that on increasing the value of β (β close to 1), the number of uninfected CD4+ T cells increases well while the number of infected CD4+ T cells and the density of virions in plasma decreases. In Figures 3–5, we have shown the graphical comparison between the Caputo-Fabrizio operator and the Atanagana-Baleanu derivative operator. We have seen that on decreasing the value of β, the number of uninfected CD4+ T cells decreases well for the Atangana-Baleanu (ABC) operator as compared to the Caputo-Fabrizio (CF) operator while the number of infected CD4+ T cells and the density of virions in plasma increases well for the ABC operator than the CF operator. This shows that ABC derivative operator provides better results than the CF derivative operator.Figure 1
The graphical results to show the (a) uninfected CD4+ T cells, (b) infected CD4+ T cells, and (c) density of virions in plasma for different order fractional derivatives in sense of Caputo-Fabrizio operator, where blue represents graph for β=0.3, yellow for β=0.5, green for β=0.7, and red for β=0.9, respectively.
(a)(b)(c)Figure 2
The graphical results to show the (a) uninfected CD4+ T cells, (b) infected CD4+ T cells, and (c) density of virions in plasma for different order fractional derivatives in sense of ABC operator, where blue represents graph for β=0.3, yellow for β=0.5, green for β=0.7, and red for β=0.9, respectively.
(a)(b)(c)Figure 3
The comparison of the behavior of uninfected CD4+ T cells in Caputo-Fabrizio and Atangana Baleanu sense for different order fractional derivatives, where blue colour represents graph for ABC operator and orange represents graph for CF operator.
(a)
β=0.3(b)
β=0.5(c)
β=0.7(d)
β=0.9Figure 4
The comparison of the behavior of infected CD4+ T cells in Caputo-Fabrizio and Atangana Baleanu sense for different order fractional derivatives, where blue colour represents graph for ABC operator and orange represents graph for CF operator.
(a)
β=0.3(b)
β=0.5(c)
β=0.7(d)
β=0.9Figure 5
The comparison of the behavior of density of virions in plasma in Caputo-Fabrizio and Atangana Baleanu sense for different order fractional derivatives, where blue colour represents graph for ABC operator and orange represents graph for CF operator.
(a)
β=0.3(b)
β=0.5(c)
β=0.7(d)
β=0.9



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*Source: 6642957-2021-03-22.xml*