# Study of the Fractional-Order HIV-1 Infection Model with Uncertainty in Initial Data

Authors: Yong Wu; Shabir Ahmad; Aman Ullah; Kamal Shah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Uncertainty always lives with us. We cannot take exact measurement of initial conditions or parameters values in a mathematical model. As humans, we are remaining alive in an environment where the uncertainties lie in the modelling of physical phenomena. There might be some incomplete information or estimation of the parameter or initial values. To handle uncertainty, we use fuzzy operators rather than classical operators. In this paper, we study a model of HIV-1 infection by taking uncertainty in the initial data under Caputo fractional operator. We explore the existence and uniqueness of the results through fixed-point theory. We study the Ulam–Hyres stability of the considered model. By using the fuzzy Laplace Adomian decomposition method, numerical results are obtained for specific fuzzy initial conditions. To better understand the behaviour of the fuzzy solution, we present the obtained numerical results graphically for various fractional orders where the uncertainty lies in [0, 1].


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## Body

## 1. Introduction

AIDS (acquired immune deficiency syndrome) is one of the most dangerous diseases caused by a pathogen virus called HIV (human immunodeficiency virus) and leads a person to death. In 1982, a patient in the USA was infected by HIV. At the end of 2019, about 38 million people, identified by the WHO (World Health Organization), were surviving with HIV. In 2019, 0.69 million people died from diseases associated with AIDS. To investigate how AIDS can spread or control, mathematical models are valuable instruments as they provide a short or a long-term prediction of the frequency of the disease. Many researchers have investigated the dynamics of HIV infection models [1–3]. A lot of methods have been used by researchers to find a solution to HIV infections [4–6]. In [7], the authors presented an HIV-1 infection model which contains five compartments, denoted by Mˇt (the uninfected CD4+ T cells), Mˇ∗t (the concentration of infected cells), Mˇ∗∗t (the concentration of double cells), and Vpt and Vrt (the densities of pathogen viruses), respectively. The integer-order model of HIV-1 infection is given by(1)ddtMˇt=λ−d1Mˇt−β1MˇtVpt,ddtMˇ∗t=β1MˇtVpt−d2ℳ∗t−ϵ1VrtMˇ∗t,ddtMˇ∗∗t=ϵ1VrtMˇ∗t−d3Mˇ∗∗t,ddtVpt=k′Mˇ∗t−d4Vpt,ddtVrt=c′Mˇ∗∗t−d5Vrt,where the host cells are produced at the rate λ. Moreover, d1,d2,d3,andd4 are the rates of death of host cells, infected cells, double infected cells, and pathogen virus, respectively. β1 is the rate of infection of healthy CD4+ T cells. The rate of production of HIV-1 by a cell is defined by k′, ϵ1 is infection rate of removal of HIV-1 infected cells, c′ is the rate of production of recombinant by a double-infected cell, and d5 is the removal rate of recombinant. The mathematician has taken an interest in fractional calculus in recent years as it has many applications in engineering and biological sciences [8–10]. In the last decade, the researcher converted many models from integer order to fractional order which provides best results as compared to integer-order derivatives. As fractional operator has a greater degree of freedom, different researchers studied the existence theory of the solution [11–13]. The fractional operators also have been used for modelling of different infectious diseases. Sweilam et al. investigated the optimal control for variable order fractional HIV/AIDS and malaria mathematical models with multitime delay [14]. The Legendre spectral-collocation method was used for solving fractional optimal control of HIV infection of CD4+T cells’ mathematical model in [15]. A complex fractional-order HIV infection model with drug resistance during therapy has been studied by Sweilam et al. [16]. Some other applications of the fractional-order operators in various field of sciences can be found in [17–19]. It is worthy to note here that the fractional derivative in the Caputo sense is the most commonly used definition among the definitions of the fractional derivative. The definition of Caputo is mathematically rigorous than the Riemann–Liouville definition. The Caputo derivative exists in the whole interval (0, 1). In addition, Caputo definition is very welcome in applied science and engineering. Furthermore, properties of the Caputo derivative are helpful in translating the higher fractional-order differential systems into lower ones. For a comparison between Caputo and Riemann–Liouville operators, the interested reader is referred to [20, 21]. Therefore, we will consider the fractional operator in Caputo sense.Confusion and uncertainty still live with us, and that is a reality. Many human beings are exposed to doubting anything surrounding them and questioning why for themselves or others. Because they are not clear, and their reports are incomplete or inaccurate. Now, suppose that we are in a situation where uncertainty occurs with all this inaccurate knowledge. It is a reality that, in many of our real questions where inaccurate data are involved, we do not know how to answer. For scientists, this mindset and attitude of uncertainty is very necessary. Our aim is to sort out how to understand it and operate through it rather than trying to fight ambiguity because the advancement, resources, and life that you want are uncertain. To overcome this situation, Zadeh introduced fuzzy set in 1965 [22]. The concept of fuzzy mapping and control has been addressed by Zadeh and Chang [23]. Furthermore, researchers extended the ordinary and fractional operators to fuzzy operators [24, 25]. These operators have been used by various authors to study different models [26–29]. If the information or data is imprecise, then fuzzy operators model physical phenomena easily and accurately. We are interested in investigating model (1) for fractional-order derivative under the uncertain initial conditions. For fractional order 0<γ≤1, consider(2)DtγMˇt,ϖ0=λ−d1Mˇt,ϖ0−β1Mˇt,ϖ0Vˇpt,ϖ0,DtγMˇ∗t,ϖ0=β1Mˇt,ϖ0Vˇpt,ϖ0−d2Mˇ∗t,ϖ0−ε1Vˇrt,ϖ0Mˇ∗t,ϖ0,DtγMˇ∗∗t,ϖ0=ε1Vˇrt,ϖ0Mˇ∗t,ϖ0−d3Mˇ∗∗t,ϖ0,DtγVˇpt,ϖ0=k′Mˇ∗t,ϖ0−d4Vˇpt,ϖ0,DtγVˇrt,ϖ0=c′Mˇ∗∗t,ϖ0−d5Vˇrt,ϖ0,along with fuzzy initial conditions,(3)Mˇ0,ϖ0=Mˇ¯0,ϖ0,Mˇ¯0,ϖ0,Mˇ∗0,ϖ0=Mˇ∗¯0,ϖ0,Mˇ∗¯0,ϖ0,Mˇ∗∗0,ϖ0=Mˇ∗∗¯0,ϖ0,Mˇ∗∗¯0,ϖ0,Vˇp0,ϖ0=Vˇp¯0,ϖ0,Vˇp¯0,ϖ0,Vˇr0,ϖ0=Vˇr¯0,ϖ0,Vˇr¯0,ϖ0,such that ϖ0∈0,1.


## 2. Preliminaries

This section provides the basic concepts of fuzzy sets and fuzzy fractional calculus. Let FFD, FFI, and LT denote the fuzzy fractional derivative, fuzzy fractional integral, and Laplace transform, respectively.Definition 1 (see [29, 30]).
A fuzzy setd˜:ℝ⟶0,1 is called a fuzzy number if(i)
d˜ is fuzzy convex.(ii)
d˜ is normal, i.e., for some z0∈ℝ;d˜z0=1.(iii)
d˜˙ is upper semicontinuous on ℝ.(iv)
The closure ofz∈ℝ,d˜z>0 is compact.Definition 2 (see [29, 30]).
The parametric form of a fuzzy numberd˜ is given by d˜¯u,ϖ0,d˜¯u,ϖ0, for ϖ0∈0,1, if and only if(i)
d˜¯u,ϖ0 is increasing, left on 0,1 and right continuous at 0, respectively.(ii)
d˜¯u,ϖ0 is decreasing, left on 0,1 and right continuous at 0, respectively.(iii)
d˜¯u,ϖ0≤d˜¯u,ϖ0.Definition 3 (see [31]).
Letℑ:E×E⟶ℝ be a function, and b=b¯ϖ0,b¯ϖ0 and c=c¯ϖ0,c¯ϖ0 are any two fuzzy numbers. Then, the H-distance between b and c is given by(4)ℑb,c=supϖ0∈0,1maxb¯ϖ0−c¯ϖ0,b¯ϖ0−c¯ϖ0.
InE, ℑ satisfies the properties given below:(i)
ℑb+υ,c+υ=ℑb,c∀b,υ,c∈E.(ii)
ℑb〉,c〉=〉ℑb,c∀〉∈ℝandb,c∈E.(iii)
ℑb+ξ,c+ς≤ℑb,c+ℑξ,ς∀b,c,ξ,ς∈E.(iv)
E,ℑ is a complete metric space.Definition 4 (see [31]).
LetΛ be a continuous fuzzy function on 0,d⊆ℝ, and a FFI is defined as(5)IκΛt=1Γγ∫0tt−ζγ−1Λζdζ.
Furthermore, ifCℛF0,b∩LℛF0,b, where CℛF0,b and LℛF0,b are the spaces of fuzzy continuous functions and fuzzy Lebesgue integrable functions, respectively, then the FFI can be written as(6)IγΛtϖ0=IγΛ¯ϖ0t,IγΛ¯ϖ0t,0≤ϖ0≤1,where(7)IγΛ¯ϖ0t=1Γγ∫0tt−ζγ−1Λ¯ϖ0tdζ,IγΛ¯ϖ0t=1Γγ∫0tt−ζγ−1Λ¯ϖ0tdζ.Definition 5 (see [31]).
A fuzzy functionΛ∈CℛF0,b∩LℛF0,b is such that Λ=Λ¯ϖ0t,Λ¯ϖ0t, t1∈0,b; then, the FFD in Caputo sense is given by(8)DγΛt0ϖ0=DγΛ¯ϖ0t0,DγΛ¯ϖ0t0,where(9)DγΛ¯ϖ0t0=1Γn−γ∫0tt−ζn−γ−1dndζnΛ¯ϖ0ζdζt=t0,DγΛ¯ϖ0t0=1Γn−γ∫0tt−ζn−γ−1dndζnΛ¯ϖ0ζdζt=t0,where n=γ.Definition 6 (see [32]).
LetV˜∈CℛF0,b∩LℛF0,b. Then, the fuzzy LT of Λ˜t is defined as(10)ℒΛ˜t=∫0∞Λ˜texp−stdt.
While, in the parametric form, the fuzzy LT is given by(11)∫0∞Λ˜texp−stdt=∫0∞Λ˜¯t,ϖ0exp−stdt,∫0∞Λ˜¯t,ϖ0exp−stdt.
Hence,(12)ℒΛ˜t=ℒΛ˜¯t,ϖ0,ℒΛ˜¯t,ϖ0.Theorem 1 (see [31]).
The LT of Caputo fractional derivative is defined as(13)LDγℑt=sγLℑt−sγ−1ℑ0.


## 3. Existence and Uniqueness Results

In this section, the existence and uniqueness of the solution of the fuzzy fractional model via fixed-point theory are discussed. Let us write the right-hand side of system (2) as(14)Ψt,Mˇt,ϖ0=λ−d1Mˇt,ϖ0−β1Mˇt,ϖ0Vˇpt,ϖ0,Ξt,Mˇ∗t,ϖ0=β1Mˇt,ϖ0Vˇpt,ϖ0−d2Mˇ∗t,ϖ0−ε1Vˇrt,ϖ0Mˇ∗t,ϖ0,ft,Mˇ∗∗t,ϖ0=ε1Vˇrt,ϖ0Mˇ∗t,ϖ0−d3Mˇ∗∗t,ϖ0,gt,Vˇpt,ϖ0=k′Mˇ∗t,ϖ0−d4Vˇpt,ϖ0,ht,Vˇrt,ϖ0=c′Mˇ∗∗t,ϖ0−d5Vˇrt,ϖ0.We can write the given system (2) as(15)DtγMˇt,ϖ0=Ψt,Mˇt,ϖ0,DtγMˇ∗t,ϖ0=Ξt,Mˇ∗t,ϖ0,DtγMˇ∗∗t,ϖ0=ft,Mˇ∗∗t,ϖ0,DtγVˇpt,ϖ0=gt,Vˇpt,ϖ0,DtγVˇrt,ϖ0=ht,Vˇrt,ϖ0.Now, using fractional integral, we obtain(16)t,ϖ0=Mˇ0,ϖ0+1Γγ∫0tt−sγ−1Ψs,Mˇs,ϖ0ds,Mˇ∗t,ϖ0=Mˇ∗0,ϖ0+1Γγ∫0tt−sγ−1Ξs,Mˇ∗s,ϖ0ds,Mˇ∗∗t,ϖ0=Mˇ∗∗0,ϖ0+1Γγ∫0tt−sγ−1fs,Mˇ∗∗s,ϖ0ds,Vˇpt,ϖ0=Vˇp0,ϖ0+1Γγ∫0tt−sγ−1gs,Vˇps,ϖ0ds,Vˇrt,ϖ0=Vˇr0,ϖ0+1Γγ∫0tt−sγ−1hs,Vˇrs,ϖ0ds.Now, consider a Banach space asB=B1×B2×B3×B4×B5, under the norm,(17)Mˇt,ϖ0,Mˇ∗t,ϖ0,Mˇ∗∗t,ϖ0,Vˇpt,ϖ0,Vˇrt,ϖ0=maxt∈0,TMˇt,ϖ0+Mˇ∗t,ϖ0+Mˇ∗∗t,ϖ0,+Vˇpt,ϖ0+Vˇrt,ϖ0.We can write equation (16) as(18)Πˇt,ϖ0=Πˇ0,ϖ0+1Γγ∫0tt−Ωγ−1ΘΛ,ΠˇΩdΩ,where Πˇt,ϖ0=Mˇt,ϖ0Mˇ∗t,ϖ0Mˇ∗∗t,ϖ0Vˇpt,ϖ0Vˇrt,ϖ0,Πˇ0,ϖ0=Xˇ0,ϖ0Mˇ∗0,ϖ0Mˇ∗∗0,ϖ0Vˇp0,ϖ0Vˇr0,ϖ0, and Θt,Πˇt,ϖ0=Ψt,Mˇt,ϖ0Ξt,Mˇ∗t,ϖ0ft,Mˇ∗∗t,ϖ0gt,Vˇpt,ϖ0ht,Vˇrt,ϖ0. We take some assumption on nonlinear function Θ:B⟶B as(1)
Lipschtiz condition: there exists a constantKΠˇ>0 such that, for each Πˇ1t,ϖ0,Πˇ2t,ϖ0∈B, (19)Θt,Πˇ1t,ϖ0−Θt,Πˇ2t,ϖ0≤KΠˇΠˇ1t,ϖ0−Πˇ2t,ϖ0.(2)
Growth condition: there exists constantsMΠˇ>0 and NΠˇ>0 such that(20)Θt,Πˇt,ϖ0≤MΠˇΠˇt,ϖ0+NΠˇ.Theorem 2.
If the growth condition is satisfied, then system (3) has at least one solution.Proof.
ConsiderA=Πˇt,ϖ0∈B:Πˇt,ϖ0≤r⊂B is closed and fuzzy convex, and a mapping ψ:A⟶A is defined by(21)ψΠˇt,ϖ0=Πˇ0,ϖ0+1Γγ∫0tt−Ωγ−1ΘΩ,Πˇt,ϖ0dΩ.
Then, for anyΠˇt,ϖ0∈A, we have(22)ψΠˇt,ϖ0=maxt∈0,TΠˇ0,ϖ0+1Γγ∫0tt−Ωγ−1ΘΩ,Πˇt,ϖ0dΩ,≤Πˇ0,ϖ0+1Γγ∫0tt−Ωγ−1ΘΩ,Πˇt,ϖ0dΩ,≤Πˇ0,ϖ0+1Γγ∫0tt−Ωγ−1MΠΠˇt,ϖ0+NΠdΩ,≤Πˇ0,ϖ0+τγΓγ+1MΠˇΠˇt,ϖ0+NΠˇ.
It follows thatψΠˇt,ϖ0≤r. Thus, ψA⊂A; it implies that ψ is bounded. To show ψ is equi-continuity, consider ϕ1,ϕ2∈0,T such that ϕ1<ϕ2. Then,(23)ψΠˇt,ϖ0ϕ2−ψΠˇt,ϖ0ϕ1=1Γγ∫0ϕ2ϕ2−Ωγ−1ΘΩ,Πˇt,ϖ0dΩ−γ∫0ϕ1ϕ1−Ωγ−1ΘΩ,Πˇt,ϖ0dΩ,≤ϕ2γ−ϕ1γMΠˇΠˇt,ϖ0+NΠˇΓγ+1.
Hence,(24)ψΠˇt,ϖ0ϕ2−ψΠˇt,ϖ0ϕ1⟶0asϕ2⟶ϕ1.
This show that the operatorψ is equi-continuous, which is completely continuous using Arzela–Ascoli theorem, as ψ is bounded. Therefore, system (3) has at least one solution using Schauder’s fixed-point theorem.Theorem 3.
Let the Lipschitz condition hold. If(25)τγKΠ<Γγ+1,then system (3) has a unique solution.Proof.
LetΠˇ1t,ϖ0,Πˇ2t,ϖ0∈B. Then,(26)ψΠˇ1t,ϖ0−ψΠˇ2t,ϖ0=maxt∈0,T1Γγ∫0tt−Ωγ−1ΘΩ,Πˇ1t,ϖ0dΩ−1Γγ∫0tt−Ωγ−1ΘΩ,Πˇ2t,ϖ0dΩ,≤τγΓγ+1KΠˇΠˇ1t,ϖ0−Πˇ2t,ϖ0,which shows that ψ is a contraction. Thus, system (3) has a unique solution by Banach contraction theorem.


## 4. Ulam–Hyres Stability

In this section, we show that the numerical results are stable. Take a small changeϕ∈C0,T, where ϕ0=0 depends on the solution of ψΠ˜t,ϖ0 as(i)
For any positiveε, ϕt≤ε(ii)
DtγψΠ˜t,ϖ0=Ψt,ψΠ˜t,ϖ0+ϕtLemma 1.
The above perturb problem has a solution,(27)DtγψΠ˜t,ϖ0=Ψt,ψΠ˜t,ϖ0+ϕt,ψΠ˜0,ϖ0=ψΠ˜0,ϖ0,if it satisfies(28)Π˜t,ϖ0−Π˜0t,ϖ0+1Γγ∫0tt−sγ−1Ψs,Π˜t,sds≤TγΓγ+1ε=ΩT,αε.Proof.
The proof is easy.Theorem 4.
Consider assumption (2) and relation (29) hold. Then, the numerical results of the considered model are Ulam–Hyres stable if Δ=Tγ/Γγ+1KΨ<1, by showing that equation (25) is Ulam–Hyres stable.Proof.
LetΠ˜t,ϖ0∈C be any solution and Π˜¯t,ϖ0∈C be at most one solution of equation (25); then,(29)Π˜t,ϖ0−Π˜¯t,ϖ0=Π˜t,ϖ0−Π˜0t,ϖ0+1Γγ∫0tt−sγ−1Ψs,Π˜¯s,ϖ0ds,≤Π˜t,ϖ0−Π˜0t,ϖ0+1Γγ∫0tt−sγ−1Ψs,Π˜¯s,ϖ0ds+1Γγ∫0tt−sγ−1Ψs,Π˜¯s,ϖ0ds−1Γγ∫0tt−sγ−1Ψs,Π˜¯s,ϖ0ds,≤ΩT,γε+Δ‖Π˜t,ϖ0−Π˜¯t,ϖ0‖.
From (29), we have(30)Π˜t,ϖ0−Π˜¯t,ϖ0≤ΩT,γ1−Δε.
Hence, from (30), it follows that system (6) is “UH stable.” Consequently, the considered model is UH stable.


### 4.1. Procedure for Solution

In this section, we deduce the solution of the given model by using LADM. Applying LT on (3), we obtain(31)LDtγMˇt,ϖ0=LΨt,Mˇt,ϖ0,LDtγMˇ∗t,ϖ0=LΞt,Mˇ∗t,ϖ0,LDtγMˇ∗∗t,ϖ0=Lft,Mˇ∗∗t,ϖ0,LDtγVˇpt,ϖ0=Lgt,Vˇpt,ϖ0,LDtγVˇrt,ϖ0=Lht,Vˇrt,ϖ0,(32)sγLMˇt,ϖ0=sγ−1Mˇ0,ϖ0+LΨt,Mˇt,ϖ0,sγLMˇ∗t,ϖ0=sγ−1Mˇ∗0,ϖ0+LΞt,Mˇ∗t,ϖ0,sγLMˇ∗∗t,ϖ0=sγ−1Mˇ∗∗0,ϖ0+Lft,Mˇ∗∗t,ϖ0,sγLVˇpt,ϖ0=sγ−1Vˇp0,ϖ0+Lgt,Vˇpt,ϖ0,sγLVˇrt,ϖ0=sγ−1Vˇr0,ϖ0+Lht,Vˇrt,ϖ0,(33)LMˇt,ϖ0=1sMˇ0,ϖ0+1sγLΨt,Mˇt,ϖ0,LMˇ∗t,ϖ0=1sMˇ∗0,ϖ0+1sγLΞt,Mˇ∗∗t,ϖ0,LMˇ∗∗t,ϖ0=1sMˇ∗∗0,ϖ0+1sγLft,Mˇ∗∗,LVˇpt,ϖ0=1sVˇp0,ϖ0+1sγLgt,Vˇpt,ϖ0,LVˇrt,ϖ0=1sVˇr0,ϖ0+1sγLht,Vˇrt,ϖ0.The infinite series solution is(34)Mˇt,ϖ0=∑m=0∞Mˇmt,ϖ0,Mˇ∗t,ϖ0=∑m=0∞Mˇm∗t,ϖ0,Mˇ∗∗t,ϖ0=∑m=0∞Mˇm∗∗t,ϖ0,Vˇpt,ϖ0=∑m=0∞Vˇpmt,ϖ0,Vˇrt,ϖ0=∑m=0∞Vˇrmt,ϖ0,(35)Mˇt,ϖ0Vˇpt,ϖ0=∑n=0∞Z1,nVˇrt,ϖ0Mˇ∗t,ϖ0∑n=0∞Z2,n,where Z1,nandZ2,n are Adomian polynomials and represent nonlinear terms.Taking inverse Laplace transform, we obtain(36)∑m=0∞Mˇmt,ϖ0=Mˇ0,ϖ0+L−11sγLΨt,Mˇt,ϖ0,∑m=0∞Mˇm∗t,ϖ0=Mˇ∗0,ϖ0+L−11sγLΞt,Mˇ∗∗t,ϖ0,∑m=0∞Mˇm∗∗t,ϖ0=Mˇ∗∗0,ϖ0+L−11sγLft,Mˇ∗∗,∑m=0∞Vˇpmt,ϖ0=Vˇp0,ϖ0+L−11sγLgt,Vˇpt,ϖ0,∑m=0∞Vˇrmt,ϖ0=Vˇr0,ϖ0+L−11sγLht,Vˇrt,ϖ0.Comparing the first two terms of the series in parametric form,(37)Mˇ0¯t,ϖ0=Mˇ¯0,ϖ0Mˇ0¯t,ϖ0=Mˇ¯0,ϖ0,Mˇ0∗¯t,ϖ0=Mˇ∗¯0,ϖ0Mˇ0∗¯t,ϖ0=Mˇ∗¯0,ϖ0,Mˇ0∗∗¯t,ϖ0=Mˇ∗∗¯0,ϖ0Mˇ0∗∗¯t,ϖ0=Mˇ∗∗¯0,ϖ0,Vˇp0¯t,ϖ0=Vˇp¯0,ϖ0Vˇp0¯t,ϖ0=Vˇp¯0,ϖ0,Vˇr0¯t,ϖ0=Vˇr¯0,ϖ0Vˇr0¯t,ϖ0=Vˇr¯0,ϖ0.(38)Mˇ1¯t,ϖ0=L−11sγLλ−d1Mˇ0¯t,ϖ0−β1Z1,0¯t,ϖ0,v1¯t,ϖ0=L−11sγLλ−d1Mˇ0¯t,ϖ0−β1Z1,0¯t,ϖ0,(39)Mˇ1∗¯t,ϖ0=L−11sγLβ1Z1,0¯t,ϖ0−d2Mˇ0∗¯−ε1Z2,0¯t,ϖ0,Mˇ1∗¯t,ϖ0=L−11sγLβ1Z1,0¯t,ϖ0−d2Mˇ0∗¯t,ϖ0−ε1Z2,0¯t,ϖ0,(40)Mˇ1∗∗¯t,ϖ0=L−11sγLε1Z2,0¯t,ϖ0−d3Mˇ0∗∗¯t,ϖ0,Mˇ1∗∗¯t,ϖ0=L−11sγLε1Z2,0¯t,ϖ0−d3Mˇ0∗∗¯t,ϖ0,(41)Vˇp1¯t,ϖ0=L−11sγLk′Mˇ0∗¯t,ϖ0−d4Vˇp0¯t,ϖ0,Vˇp1¯t,ϖ0=L−11sγLk′Mˇ0∗¯t,ϖ0−d4Vˇp0¯t,ϖ0,(42)Vˇr1¯t,ϖ0=L−11sγLc′Mˇ0∗∗¯t,ϖ0−d5Vˇr0¯t,ϖ0,Vˇp1¯t,ϖ0=L−11sγLc′Mˇ0∗∗¯t,ϖ0−d5Vˇr0¯t,ϖ0.We can find other terms by the similar way. Hence, the system has the series solution as follows:(43)Mˇ¯t,ϖ0=Mˇ0¯t,ϖ0+Mˇ1¯t,ϖ0+Mˇ2¯t,ϖ0+⋯Mˇ¯t,ϖ0=Mˇ0¯t,ϖ0+Mˇ1¯t,ϖ0+Mˇ2¯t,ϖ0+⋯Mˇ∗¯t,ϖ0=Mˇ0∗¯t,ϖ0+Mˇ1∗¯t,ϖ0+Mˇ2∗¯t,ϖ0+⋯Mˇ∗¯t,ϖ0=Mˇ0∗¯t,ϖ0+Mˇ1∗¯t,ϖ0+Mˇ2∗¯t,ϖ0+⋯Mˇ∗∗¯t,ϖ0=Mˇ0∗∗¯t,ϖ0+Mˇ1∗∗¯t,ϖ0+Mˇ2∗∗¯t,ϖ0+⋯Mˇ∗∗¯t,ϖ0=Mˇ0∗∗¯t,ϖ0+Mˇ1∗∗¯t,ϖ0+Mˇ2∗∗¯t,ϖ0+⋯Vˇp¯t,ϖ0=Vˇp0¯t,ϖ0+Vˇp1¯t,ϖ0+Vˇp2¯t,ϖ0+⋯Vˇp¯t,ϖ0=Vˇp0¯t,ϖ0+Vˇp1¯t,ϖ0+Vˇp2¯t,ϖ0+⋯Vˇr¯t,ϖ0=Vˇr0¯t,ϖ0+Vˇr1¯t,ϖ0+Vˇr2¯t,ϖ0+⋯Vˇr¯t,ϖ0=Vˇr0¯t,ϖ0+Vˇr1¯t,ϖ0+Vˇr2¯t,ϖ0+⋯Based on the procedure of the method, the following theorem discusses the convergence of the proposed method.Theorem 5.
LetH be a Banach space, ϑ˜pδ,t, ϑ˜nδ,t∈H, and 0<ξ<1. Let ϑ˜δ,t be an exact solution of the considered model. Then, the series solution ∑p=0∞ϑ˜pδ,t converges to the exact solution ϑ˜δ,t whenever ϑ˜pδ,t≤ξϑ˜p−1δ,t for all p>T; i.e., for any ω>0, there exists a positive number T such that ϑ˜p+nδ,t≤ω,∀m,n>T.Proof.
Let us define the sequence of∑p=0∞ϑ˜pδ,t as(44)S0δ,t=ϑ˜0δ,t,S1δ,t=ϑ˜0δ,t+ϑ˜1δ,t,S2δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t,S3δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+ϑ˜3δ,t,⋮Spδ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+⋯+ϑ˜pδ,t.
We need to show thatSpδ,t is a Cauchy sequence in H. For this, we consider(45)Sp+1δ,t−Spδ,t=ϑ˜p+1δ,t,≤ξϑ˜pδ,t≤ξ2ϑ˜p−1δ,t≤ξ3ϑ˜p−2δ,t⋮≤ξp+1ϑ˜0δ,t.
Now, for anyp,n∈ℕ, we have(46)Spδ,t−Snδ,t=ϑ˜p+nδ,t=Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+Sp−2δ,t−Sp−3δ,t+⋯+Sn+1δ,t−Snδ,t≤Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+⋯+Sn+1δ,t−Snδ,t.≤ξpϑ˜0δ,t+ξp−1ϑ˜0δ,t+⋯+ξp+1ϑ˜0δ,t=ϑ˜0δ,tξp+ξp−1+⋯+ξp+1=ϑ˜0δ,t1−ξp−n1−ξξn+1.
Sinceξ∈0,1 and ϑ˜0δ,t is bounded, letting ω=1−ξ/1−ξp−nξn+1ϑ˜0δ,t, we obtain(47)ϑ˜p+nδ,t≤ω,∀p,n>T.
Thus, the sequenceϑ˜pδ,tp=0∞ is a Cauchy sequence in H. This implies that ∃ϑ˜δ,t∈H such that limp⟶∞ϑ˜pδ,t=ϑ˜δ,t. Thus, the required solution converges to the exact solution.


## 4.1. Procedure for Solution

In this section, we deduce the solution of the given model by using LADM. Applying LT on (3), we obtain(31)LDtγMˇt,ϖ0=LΨt,Mˇt,ϖ0,LDtγMˇ∗t,ϖ0=LΞt,Mˇ∗t,ϖ0,LDtγMˇ∗∗t,ϖ0=Lft,Mˇ∗∗t,ϖ0,LDtγVˇpt,ϖ0=Lgt,Vˇpt,ϖ0,LDtγVˇrt,ϖ0=Lht,Vˇrt,ϖ0,(32)sγLMˇt,ϖ0=sγ−1Mˇ0,ϖ0+LΨt,Mˇt,ϖ0,sγLMˇ∗t,ϖ0=sγ−1Mˇ∗0,ϖ0+LΞt,Mˇ∗t,ϖ0,sγLMˇ∗∗t,ϖ0=sγ−1Mˇ∗∗0,ϖ0+Lft,Mˇ∗∗t,ϖ0,sγLVˇpt,ϖ0=sγ−1Vˇp0,ϖ0+Lgt,Vˇpt,ϖ0,sγLVˇrt,ϖ0=sγ−1Vˇr0,ϖ0+Lht,Vˇrt,ϖ0,(33)LMˇt,ϖ0=1sMˇ0,ϖ0+1sγLΨt,Mˇt,ϖ0,LMˇ∗t,ϖ0=1sMˇ∗0,ϖ0+1sγLΞt,Mˇ∗∗t,ϖ0,LMˇ∗∗t,ϖ0=1sMˇ∗∗0,ϖ0+1sγLft,Mˇ∗∗,LVˇpt,ϖ0=1sVˇp0,ϖ0+1sγLgt,Vˇpt,ϖ0,LVˇrt,ϖ0=1sVˇr0,ϖ0+1sγLht,Vˇrt,ϖ0.The infinite series solution is(34)Mˇt,ϖ0=∑m=0∞Mˇmt,ϖ0,Mˇ∗t,ϖ0=∑m=0∞Mˇm∗t,ϖ0,Mˇ∗∗t,ϖ0=∑m=0∞Mˇm∗∗t,ϖ0,Vˇpt,ϖ0=∑m=0∞Vˇpmt,ϖ0,Vˇrt,ϖ0=∑m=0∞Vˇrmt,ϖ0,(35)Mˇt,ϖ0Vˇpt,ϖ0=∑n=0∞Z1,nVˇrt,ϖ0Mˇ∗t,ϖ0∑n=0∞Z2,n,where Z1,nandZ2,n are Adomian polynomials and represent nonlinear terms.Taking inverse Laplace transform, we obtain(36)∑m=0∞Mˇmt,ϖ0=Mˇ0,ϖ0+L−11sγLΨt,Mˇt,ϖ0,∑m=0∞Mˇm∗t,ϖ0=Mˇ∗0,ϖ0+L−11sγLΞt,Mˇ∗∗t,ϖ0,∑m=0∞Mˇm∗∗t,ϖ0=Mˇ∗∗0,ϖ0+L−11sγLft,Mˇ∗∗,∑m=0∞Vˇpmt,ϖ0=Vˇp0,ϖ0+L−11sγLgt,Vˇpt,ϖ0,∑m=0∞Vˇrmt,ϖ0=Vˇr0,ϖ0+L−11sγLht,Vˇrt,ϖ0.Comparing the first two terms of the series in parametric form,(37)Mˇ0¯t,ϖ0=Mˇ¯0,ϖ0Mˇ0¯t,ϖ0=Mˇ¯0,ϖ0,Mˇ0∗¯t,ϖ0=Mˇ∗¯0,ϖ0Mˇ0∗¯t,ϖ0=Mˇ∗¯0,ϖ0,Mˇ0∗∗¯t,ϖ0=Mˇ∗∗¯0,ϖ0Mˇ0∗∗¯t,ϖ0=Mˇ∗∗¯0,ϖ0,Vˇp0¯t,ϖ0=Vˇp¯0,ϖ0Vˇp0¯t,ϖ0=Vˇp¯0,ϖ0,Vˇr0¯t,ϖ0=Vˇr¯0,ϖ0Vˇr0¯t,ϖ0=Vˇr¯0,ϖ0.(38)Mˇ1¯t,ϖ0=L−11sγLλ−d1Mˇ0¯t,ϖ0−β1Z1,0¯t,ϖ0,v1¯t,ϖ0=L−11sγLλ−d1Mˇ0¯t,ϖ0−β1Z1,0¯t,ϖ0,(39)Mˇ1∗¯t,ϖ0=L−11sγLβ1Z1,0¯t,ϖ0−d2Mˇ0∗¯−ε1Z2,0¯t,ϖ0,Mˇ1∗¯t,ϖ0=L−11sγLβ1Z1,0¯t,ϖ0−d2Mˇ0∗¯t,ϖ0−ε1Z2,0¯t,ϖ0,(40)Mˇ1∗∗¯t,ϖ0=L−11sγLε1Z2,0¯t,ϖ0−d3Mˇ0∗∗¯t,ϖ0,Mˇ1∗∗¯t,ϖ0=L−11sγLε1Z2,0¯t,ϖ0−d3Mˇ0∗∗¯t,ϖ0,(41)Vˇp1¯t,ϖ0=L−11sγLk′Mˇ0∗¯t,ϖ0−d4Vˇp0¯t,ϖ0,Vˇp1¯t,ϖ0=L−11sγLk′Mˇ0∗¯t,ϖ0−d4Vˇp0¯t,ϖ0,(42)Vˇr1¯t,ϖ0=L−11sγLc′Mˇ0∗∗¯t,ϖ0−d5Vˇr0¯t,ϖ0,Vˇp1¯t,ϖ0=L−11sγLc′Mˇ0∗∗¯t,ϖ0−d5Vˇr0¯t,ϖ0.We can find other terms by the similar way. Hence, the system has the series solution as follows:(43)Mˇ¯t,ϖ0=Mˇ0¯t,ϖ0+Mˇ1¯t,ϖ0+Mˇ2¯t,ϖ0+⋯Mˇ¯t,ϖ0=Mˇ0¯t,ϖ0+Mˇ1¯t,ϖ0+Mˇ2¯t,ϖ0+⋯Mˇ∗¯t,ϖ0=Mˇ0∗¯t,ϖ0+Mˇ1∗¯t,ϖ0+Mˇ2∗¯t,ϖ0+⋯Mˇ∗¯t,ϖ0=Mˇ0∗¯t,ϖ0+Mˇ1∗¯t,ϖ0+Mˇ2∗¯t,ϖ0+⋯Mˇ∗∗¯t,ϖ0=Mˇ0∗∗¯t,ϖ0+Mˇ1∗∗¯t,ϖ0+Mˇ2∗∗¯t,ϖ0+⋯Mˇ∗∗¯t,ϖ0=Mˇ0∗∗¯t,ϖ0+Mˇ1∗∗¯t,ϖ0+Mˇ2∗∗¯t,ϖ0+⋯Vˇp¯t,ϖ0=Vˇp0¯t,ϖ0+Vˇp1¯t,ϖ0+Vˇp2¯t,ϖ0+⋯Vˇp¯t,ϖ0=Vˇp0¯t,ϖ0+Vˇp1¯t,ϖ0+Vˇp2¯t,ϖ0+⋯Vˇr¯t,ϖ0=Vˇr0¯t,ϖ0+Vˇr1¯t,ϖ0+Vˇr2¯t,ϖ0+⋯Vˇr¯t,ϖ0=Vˇr0¯t,ϖ0+Vˇr1¯t,ϖ0+Vˇr2¯t,ϖ0+⋯Based on the procedure of the method, the following theorem discusses the convergence of the proposed method.Theorem 5.
LetH be a Banach space, ϑ˜pδ,t, ϑ˜nδ,t∈H, and 0<ξ<1. Let ϑ˜δ,t be an exact solution of the considered model. Then, the series solution ∑p=0∞ϑ˜pδ,t converges to the exact solution ϑ˜δ,t whenever ϑ˜pδ,t≤ξϑ˜p−1δ,t for all p>T; i.e., for any ω>0, there exists a positive number T such that ϑ˜p+nδ,t≤ω,∀m,n>T.Proof.
Let us define the sequence of∑p=0∞ϑ˜pδ,t as(44)S0δ,t=ϑ˜0δ,t,S1δ,t=ϑ˜0δ,t+ϑ˜1δ,t,S2δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t,S3δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+ϑ˜3δ,t,⋮Spδ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+⋯+ϑ˜pδ,t.
We need to show thatSpδ,t is a Cauchy sequence in H. For this, we consider(45)Sp+1δ,t−Spδ,t=ϑ˜p+1δ,t,≤ξϑ˜pδ,t≤ξ2ϑ˜p−1δ,t≤ξ3ϑ˜p−2δ,t⋮≤ξp+1ϑ˜0δ,t.
Now, for anyp,n∈ℕ, we have(46)Spδ,t−Snδ,t=ϑ˜p+nδ,t=Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+Sp−2δ,t−Sp−3δ,t+⋯+Sn+1δ,t−Snδ,t≤Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+⋯+Sn+1δ,t−Snδ,t.≤ξpϑ˜0δ,t+ξp−1ϑ˜0δ,t+⋯+ξp+1ϑ˜0δ,t=ϑ˜0δ,tξp+ξp−1+⋯+ξp+1=ϑ˜0δ,t1−ξp−n1−ξξn+1.
Sinceξ∈0,1 and ϑ˜0δ,t is bounded, letting ω=1−ξ/1−ξp−nξn+1ϑ˜0δ,t, we obtain(47)ϑ˜p+nδ,t≤ω,∀p,n>T.
Thus, the sequenceϑ˜pδ,tp=0∞ is a Cauchy sequence in H. This implies that ∃ϑ˜δ,t∈H such that limp⟶∞ϑ˜pδ,t=ϑ˜δ,t. Thus, the required solution converges to the exact solution.


## 5. Numerical Results and Discussion

In this section, we illustrate the obtained solution of the proposed model through MATLAB-17. We have taken the parameter values from [7]. The values of parameters are λ=2cell/mm3, d1=0.01/day, β1=0.004mm3/vir, d2=0.33/day, ϵ1=0.004mm3/vir, d3=2/day, k′=50vir/cell, d4=2/day, c′=2000vir/cell, and d5=2/day. Consider model (2) under initial conditions as(48)Mˇ0,ϖ0=ϖ0−1,1−ϖ0,Mˇ∗0,ϖ0=ϖ0−1,1−ϖ0,Mˇ∗∗0,ϖ0=ϖ0−1,1−ϖ0,Vˇp0,ϖ0=ϖ0−1,1−ϖ0,Vˇr0,ϖ0=ϖ0−1,1−ϖ0,where ϖ0∈0,1. Using the developed technique on model (2) by applying the initial conditions, we obtain(49)Mˇ0¯t,ϖ0=ϖ0−1,Mˇ0¯t,ϖ0=1−ϖ0,Mˇ0∗¯t,ϖ0=ϖ0−1,Mˇ0∗¯t,ϖ0=1−ϖ0,Mˇ0∗∗¯t,ϖ0=ϖ0−1,Mˇ0∗∗¯t,ϖ0=1−ϖ0,Vˇp0¯t,ϖ0=ϖ0−1,Vˇp0¯t,ϖ0=1−ϖ0,Vˇr0¯t,ϖ0=ϖ0−1,Vˇr0¯t,ϖ0=1−ϖ0,(50)Mˇ1¯t,ϖ0=λ−d1ϖ0−1+β1ϖ0−12tγΓγ+1,Mˇ1¯t,ϖ0=λ−d11,ϖ0+β11−ϖ02tγΓγ+1,(51)Mˇ1∗¯t,ϖ0=β1ϖ0−12−d2ϖ0−1−ε1ϖ0−12tγΓγ+1,Mˇ1∗¯t,ϖ0=β11−ϖ02−d21−ϖ0−ε11−ϖ02tγΓγ+1,(52)Mˇ1∗∗¯t,ϖ0=ε1ϖ0−12−d3ϖ0−1tγΓγ+1,Mˇ1∗∗¯t,ϖ0=ε11−ϖ02−d31−ϖ0tγΓγ+1,(53)Vˇp1¯t,ϖ0=k′ϖ0−1−d4ϖ0−1tγΓγ+1,Vˇp1¯t,ϖ0=k′1−ϖ0−d41−ϖ0tγΓγ+1,(54)Vˇr1¯t,ϖ0=c′ϖ0−1−d5ϖ0−1tγΓγ+1,Vˇr1¯t,ϖ0=c′1−ϖ0−d51−ϖ0tγΓγ+1.Other terms can be found out by the similar way.We have simulated the numerical results obtained for the different compartments via Matlab. From the literature, it is clear the fractional operators have a greater degree of freedom. Therefore, we have taken only a few orders for the graphical representation. Each class has a lower and upper solution and has presented at different fractional orders, where the uncertainty lies in [0, 1]. The odd-numbered Figures1–5 represent two-dimensional graphs for the various compartments at various fractional orders. While the even-numbered Figures 6–10 represent three-dimensional graphs of the different classes at various fractional orders and uncertainty belongs to [0, 1]. The fractional order has a great impact on the lower and upper solution of the considered model. As the fractional order increases from lower value to higher values, the increase in the uppercase and decrease in lowercase become faster. The solution curves of will converge to integer-order solution when the fractional order tends to unity. Thus, the considered model (2) is generalized than model (1).Figure 1
2D simulation ofMˇt,ϖ0 at different fractional orders and ϖ0=0.95. (a) Uppercase. (b) Lowercase.
(a)(b)Figure 2
2D simulation ofMˇ∗t,ϖ0 at different fractional orders and ϖ0=0.95. (a) Uppercase. (b) Lowercase.
(a)(b)Figure 3
2D simulation ofMˇ∗∗t,ϖ0 at different fractional orders and ϖ0=0.95. (a) Uppercase. (b) Lowercase.
(a)(b)Figure 4
2D simulation ofVpt,ϖ0 at different fractional orders and ϖ0=0.95. (a) Uppercase. (b) Lowercase.
(a)(b)Figure 5
2D simulation ofVrt,ϖ0 at different fractional orders and ϖ0=0.95. (a) Uppercase. (b) Lowercase.
(a)(b)Figure 6
3D simulation ofMˇt,ϖ0 at different fractional orders and ϖ0∈0,1.Figure 7
3D simulation ofMˇ∗t,ϖ0 at different fractional orders and ϖ0∈0,1.Figure 8
3D simulation ofMˇ∗∗t,ϖ0 at different fractional orders and ϖ0∈0,1.Figure 9
3D simulation ofVpt,ϖ0 at different fractional orders and ϖ0∈0,1.Figure 10
3D simulation ofVrt,ϖ0 at different fractional orders and ϖ0∈0,1.


## 6. Conclusion

We have successfully extended the fractional-order model of HIV to the fuzzy fractional model under the Caputo operator. We have derived the existence and uniqueness results through fixed-point theorems. We have also deduced the Ulam–Hyres stability of the proposed model through nonlinear analysis. We have presented a general method via the fuzzy LADM to obtain an approximate solution of the proposed model. We have achieved the numerical results for the specific fuzzy initial conditions through the developed scheme. We have provided the numerical simulations of the numerical results at different uncertainty values and different fractional-order. The simulations show that fuzziness with fractional operators provides global dynamics of a physical problem. Thus, these concepts open new doors for young researchers to model any physical problem with fuzzy fractional operators. The considered model can be studied under more generalized operators in the fuzzy case.



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