Analysis of the Fractional HIV Model under Proportional Hadamard-Caputo Operators

Abstract

Modeling human immunodeficiency virus (HIV) via fractional operators has several benefits over the classical integer-order HIV model. The reason is that the fractional HIV model relies not only on the recent status but also on the former conduct of the model. Thus, we are motivated to introduce and analyze a new fractional HIV model. This article focuses on a novel fractional HIV model under the proportional Hadamard-Caputo fractional operators. The study of this model involves the existence and uniqueness (EU) of its solution and the stability examination. We employ Leray–Schauder nonlinear alternative (L-SNLA) and Banach’s fixed point theorems to analyze the EU results. In addition, for this provided model, we develop several forms of Ulam’s stability findings. As a special case of our results, we give and analyze a new fractional HIV model with Hadamard-Caputo operators. Moreover, by appropriate choice of the fractional parameters, the obtained outcomes are valid for analysis of the fractional HIV models formed by several fractional operators defined in the past literature.

1. Introduction

In recent years, mathematical modeling and simulations have been employed as critical tools for predicting the occurrence and severity of infections, as well as for gaining insight into the infection’s behavior, see [1]. These models are salutary instruments that help us understand the dynamics of the immune response to many pathogens. Human immunodeficiency virus (HIV) infection of helper T-cells (HTCs) with therapy is one of the most important models. A retrovirus, the human immunodeficiency virus (HIV) is the cause of acquired immunodeficiency syndrome (AIDS) [2]. HIV targets HTCs, which are the immune system’s most abundant white blood cells and perform a crucial part in safeguarding the human body from illness. If HIV infects a person, the virus assaults HTCs and attempts to multiply. The HIV life cycle [3] is divided into seven stages:

• Binding: HTCs have receptors on their surfaces that the virus binds to.
• Fusion: HIV initiates the fusion of its envelope with the helper T-cell (HTC) membrane once it connects to receptors on HTCs. This stage allows the virus to enter the cell.
• Reverse transcription: The process of transferring genetic information in the form of RNA into DNA through the use of reverse transcription enzyme is known as reverse transcription. HIV can enter the nucleus of HTCs at this stage.
• Integration: HIV then releases another enzyme called integrase into the nucleus of the HTC after the reverse transcription process. This enzyme is used by the virus to join its DNA with the HTC’s DNA. The virus is still considered inactive at this time, and even sensitive laboratory tests have difficulty detecting it.
• Replication: HIV may now use the machinery of HTCs to produce viral proteins since it has been integrated into their DNA. It can also manufacture more of its genetic material (RNA) during this time. These two factors make it possible for it to produce additional virus particles.
• Assembly: Fresh HIV RNA and proteins are sent to the edge of the HTC during the assembly stage, where they mature into immature HIV. In their current state, these viruses are not infectious.
• Budding: Immature viruses push out of HTCs during the budding stage. They next release a protease that changes the virus’ proteins and turns them into a mature and infectious form.

Antiretroviral therapy (ART) is defined as the use of HIV medications to remedy HIV infection and safeguard the immune system by preventing the virus from growing at specific phases of HIV’s cycle [4].

On the other hand, researchers have used differential equations to solve real-world problems in a variety of fields. For applied researchers, the differential equations theory is well understood. It is well-known in a variety of sectors and is applied in various areas. Various researchers have produced fractional-order differential equations throughout the previous three centuries. They have been widely used to solve real-world problems in a variety of fields; see [5]. Existence theory, stability analysis, numerical and optimization approaches are some of the well-known features that have lately been established. The fractional derivatives presented in fractional calculus were introduced by mathematician researchers. Utilizing fractional operators and applying them to the HIV paradigm has resulted in a variety of research findings. Using numerical simulations, Ding and Ye [6] investigated the stability of balance for a fractional model for HIV infection of HTCs in 2009. The proposed numerical system is intended to be a rough approximation.

Arafa et al. [7] constructed a fractional HIV model and studied the dynamics of HTCs and HIV during primary infection in 2012. This model was solved numerically using the generalized Euler method (GEM). Arafa et al. [8] used the GEM to study and assess the solution attitude for a fractional model of HIV with HTCs in the Caputo style. Based on a Caputo fractional derivative, Arshad et al. [9] investigated the impact of an HIV model on the HTC population in 2017. The numerical approach has been proposed using a finite difference approximation. Khan et al. [10] established the fractional HIV/AIDS model’s stability results and numerical solutions in terms of an Atangana–Baleanu derivative operator. A Caputo fractional derivative HIV model was introduced by Ferrari et al. [11]; it indicated the presence of a reverse transcriptase inhibitor. They proved the model’s EU, and its stability. Using the Mittag–Leffler kernel, Khan et al. [12] examined and evaluated the existence, numerical solutions, and stability of the HIV-TB fractional model. Nazir et al. [13] investigated a model of HIV using the Caputo–Fabrizio fractional operator. Using fixed-point theory, they were able to determine the existence conditions for the solutions. In Hyers–Ulam style, the stability of the relevant solution is likewise demonstrated. The Adomian decomposition method and the Laplace transform approach are used to confirm the model’s approximate solution.

Kongson et al. [14] recently used prominent fixed-point theorems, including Banach and Larey’s nonlinear Schauder’s alternative to investigate the EU of solutions for the fractional HIV model. Furthermore, stability analysis is explored in the perspective of distinct Ulam’s stability. Eventually, they employed a predictor-corrector approach given in [15] to discover approximate fractional HIV model solutions for various fractional derivative orders. Moreover, Albalawi et al. [16] looked at Banach fixed-point theory to analyze EU for the solution of a time-fractional Emden–Fowler-type model and approximation of symmetric solutions utilizing the rational homotopy perturbation method were examined. On the other hand, a numerical investigation of the path-tracking damped oscillatory behavior of a model for the HIV infection of CD4+ T cells was introduced by Shah et al. [17]. A further numerical solution for the HIV model is provided [18]. Readers can view examples of numerical approaches for more information see [19,20].

The motivation of the existing work is to offer and investigate a novel fractional HIV model under the proportional Hadamard-Caputo fractional operators. The EU characteristics for the solution of this model are shown by Banach and Leray’s nonlinear Schauder’s alternative fixed-point forms. Further, different kinds of Ulam’s stability are investigated for the suggested HIV fractional model. As a particular case of our results, we provide and analyze a new fractional HIV model with Hadamard-Caputo operators. Due to the appropriate choice of the fractional parameters, the obtained results are applicable for analyzing some fractional HIV models created by a number of fractional operators in earlier literature.

Our study is divided into different categories, the first of which deals with the introduction of fundamental ideas, notations, and the fractional HIV model associated with the generalized proportional Hadamard-Caputo fractional operator. The second is concerned with examining the resultant model’s EU. The third section defines a few distinct sorts of stability assumptions and uses these assumptions to examine the stability of our model. The final portion concludes final observations and recommendations.

2. Fundamental Instruments

This section covers the basic concepts, essential facts, and notations for generalized fractional derivative and integral operators that will be useful during the rest of our study.

The space of all absolutely continuous functions $\zeta$ is referred to as ${\mathfrak{C}}^s[1,T]$. Such that $(s-1)$-derivative of $\zeta$ is absolutely continuous on [1,T].

Definition 1

([5]). If $\zeta :[1,\infty )\to \mathbb{R}$ is a continuous and integrable function and for each $k>0$, then the one-side fractional integral of Hadamard of $kth$ order is obtained by

$${\gimel }^k\zeta \left(\varrho \right)=\frac{1}{\mathsf{\Gamma }\left(k\right)}{\int }_1^{\varrho }{(ln\varrho -lnv)}^{k-1}\frac{\zeta \left(v\right)}{v}dv,k\in \mathbb{R},k>0.$$

(1)

Definition 2

([5]). The Hadamard fractional derivative of $kth$ order is defined by

$${\mathfrak{D}}^k\zeta \left(\varrho \right)=\frac{\left(\varrho \frac{d}{d\varrho }\right)}{\mathsf{\Gamma }(n-k)}{\int }_1^{\varrho }{(ln\varrho -lnv)}^{n-k-1}\frac{\zeta \left(v\right)}{v}dv,$$

(2)

where $n=\left[k\right]+1$ and $k>0.$

Definition 3

([5]). The fractional HCD of order k is obtained by

$${\overline{\mathfrak{D}}}^k\zeta \left(\varrho \right)=\frac{1}{\mathsf{\Gamma }(n-k)}{\int }_1^{\varrho }{(ln\varrho -lnv)}^{n-k-1}[v\frac{d}{dv}\zeta \left(v\right)]\frac{dv}{v},$$

(3)

Recently, Barakat et al. [21] employed some highly fascinating fractional derivative and integral fractional operators known as generalized proportional Hadamard-Caputo derivative and integral, respectively, and described by the following definitions.

Definition 4

([21]). The proportional Hadamard-Caputo fractional integral of k order is defined by

$${}_{PHC}{\mathfrak{J}}^{k,\sigma }\zeta \left(\varrho \right)=\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^{\varrho }{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{\varrho }{v}\right)}{(ln\varrho -lnv)}^{k-1}\frac{\zeta \left(v\right)}{v}dv,$$

(4)

where $k>0$ and $\sigma \in (0,1].$

Definition 5

([21]). The one-side fractional derivative of order k of the proportional Hadamard-Caputo fractional derivative is given by

$${}_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(\varrho \right)=\frac{1}{{\sigma }^{l-k}\mathsf{\Gamma }(l-k)}{\int }_1^{\varrho }{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{\varrho }{v}\right)}{(ln\varrho -lnv)}^{l-k-1}\left[{\mathfrak{D}}^{l,\sigma }\zeta \left(v\right)\right]\frac{dv}{v},$$

(5)

where $k>0,l=\left[k\right]+1,$$\sigma \in (0,1].$

On the other hand, the following important lemmas for our anticipated consequences were recently demonstrated in [21,22].

Lemma 1

([22]). Let $r,k\in \mathbb{C}$ such that $Re\left(r\right)>0$ and $Re\left(k\right)>0.$ Then, for any $\sigma \in (0,1]$ we have

• $\left({}_{PHC}{\mathfrak{J}}^{k,\sigma }{e}^{\frac{\sigma -1}{\sigma }lnu}{(lnu)}^{r-1}\right)\left(\varrho \right)={\displaystyle \frac{\mathsf{\Gamma }\left(r\right)}{{\sigma }^k\mathsf{\Gamma }(r+k)}}{e}^{\frac{\sigma -1}{\sigma }ln\varrho }{(ln\varrho )}^{r+k-1}.$
• $\left({}_{PHC}{\mathfrak{D}}^{k,\sigma }{e}^{\frac{\sigma -1}{\sigma }lnu}{(lnu)}^{r-1}\right)\left(\varrho \right)={\displaystyle \frac{{\sigma }^k\mathsf{\Gamma }\left(r\right)}{\mathsf{\Gamma }(r-k)}}{e}^{\frac{\sigma -1}{\sigma }ln\varrho }{(ln\varrho )}^{r-k-1}.$

Lemma 2

([21]). Assume that all absolutely continuous functions ζ whose derivative of order $(l-1)$ is absolutely continuous on [1,T] are referred to as the set ${\mathfrak{AC}}^l[1,T]$. Let k be a complex number with $Re\left(k\right)>0,$$\sigma \in (0,1],$ $l=\left[Re\right(k\left)\right]+1,$$\zeta \in L_1[1,T]$ and ${(}_{PHC}{\mathfrak{J}}^{k,\sigma }\zeta )\left(\varrho \right)$$\in {\mathfrak{AC}}^l[1,T].$ Then

$${}_{PHC}{\mathfrak{J}}^{k,\sigma }{}_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(\varrho \right)=\zeta \left(\varrho \right)-\sum _{j=1}^l\frac{{\mathfrak{D}}^{k-j,\sigma }\zeta \left(1\right)}{\mathsf{\Gamma }(k-j+1){\sigma }^{k-j}}{e}^{\frac{\sigma -1}{\sigma }ln\varrho }{(ln\varrho )}^{k-j}.$$

Also, the fractional differential equation

$${}_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(\varrho \right)=0,$$

(6)

has a solution

$$\zeta \left(\varrho \right)=e^{\frac{\sigma -1}{\sigma }ln\varrho }\sum _{s=0}^{l-1}{M}_s{(ln\varrho )}^s,$$

(7)

where $M_s=\frac{{\mathfrak{D}}^{s,\sigma }\zeta \left(1\right)}{\mathsf{\Gamma }(s+1){\sigma }^s}.$

Model Description

Reverse transcription (RT) inhibition occurs prior to the infected cell producing virus, according to a recent model described by Kongson et al. [14]. They looked at the dynamics of the viral population, infected HTCs, and uninfected HTCs. They also divided the infected HTCs into two classes: pre-RT, the stage of affected cells without RT, and post-RT, the stage of affected cells with RT. The fractional HIV model under generalized Caputo derivative is shown as follows

$$\left\{\begin{array}{c}\left({}_C{\mathfrak{D}}^{k,\sigma }V\right)\left(t\right)=\mathsf{\Omega }-\theta M\left(t\right)V\left(t\right)-{\nu }_{1}V\left(t\right)+(\beta \gamma +h)P\left(t\right),\hfill \\ \left({}_C{\mathfrak{D}}^{k,\sigma }P\right)\left(t\right)=\theta M\left(t\right)V\left(t\right)-({\nu }_2+\gamma +h)P\left(t\right),\hfill \\ \left({}_C{\mathfrak{D}}^{k,\sigma }C\right)\left(t\right)=(1-\beta )\gamma P\left(t\right)-{\nu }_{3}C\left(t\right),\hfill \\ \left({}_C{\mathfrak{D}}^{k,\sigma }M\right)\left(t\right)=T{\nu }_{3}C\left(t\right)-\varphi M\left(t\right),\hfill \end{array}\right.$$

(8)

where ${}_C{\mathfrak{D}}^{k,\sigma }(.)$ represents the generalized Caputo fractional derivative, $V\left(t\right)$ is the number of HTCs that are susceptible, $D\left(t\right)$ denotes the number of viral HTCs before RT, $C\left(t\right)$ represents the concentration of HTCs that are infected and have finished RT, making them able to generate virus, and $M\left(t\right)$ is the number of viruses. Moreover, consideration was given to the next positive numbers: $\mathsf{\Omega }$ is the influx rate of HTCs, $\theta$ is the infection rate caused by HTC interactions, ${\nu }_1$ is the proportion of HTCs that naturally die, $\beta$ is the RT inhibitor’s effectiveness $(\beta \in (0,1\left)\right)$, $\gamma$ is the typical rate of HTC conversion from pre-RT to post-RT, h is the rate of reverse transcription failure-induced uninfected cell reversion in infected cells, ${\nu }_2$ is the infection-related HTC mortality rate, ${\nu }_3$ is the percentage of HTCs that die after becoming infected, T is the overall quantity of viral particles generated by infected HTCs, and $\varphi$ represents the virus’s elimination rate.

In this paper, we look at the preceding HIV model (8) in terms of the generalized proportional Hadamard-Caputo fractional derivative.

$$\left\{\begin{array}{c}\left({}_{PHC}{\mathfrak{D}}^{k,\sigma }V\right)\left(t\right)={\mathsf{\Lambda }}_1(t,V,P,C,M),\hfill \\ \left({}_{PHC}{\mathfrak{D}}^{k,\sigma }P\right)\left(t\right)={\mathsf{\Lambda }}_2(t,V,P,C,M),\hfill \\ \left({}_{PHC}{\mathfrak{D}}^{k,\sigma }C\right)\left(t\right)={\mathsf{\Lambda }}_3(t,V,P,C,M),\hfill \\ \left({}_{PHC}{\mathfrak{D}}^{k,\sigma }M\right)\left(t\right)={\mathsf{\Lambda }}_4(t,V,P,C,M).\hfill \end{array}\right.$$

(9)

Such that for each $t\in [1,T]$ the functions ${\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2,{\mathsf{\Lambda }}_3,{\mathsf{\Lambda }}_4$ are nonlinearly defined as follows:

$$\left\{\begin{array}{c}{\mathsf{\Lambda }}_1(t,V,P,C,M)=\mathsf{\Omega }-\theta M\left(t\right)V\left(t\right)-{\nu }_{1}V\left(t\right)+(\beta \gamma +h)P\left(t\right),\hfill \\ {\mathsf{\Lambda }}_2(t,V,P,C,M)=\theta M\left(t\right)V\left(t\right)-({\nu }_2+\gamma +h)P\left(t\right),\hfill \\ {\mathsf{\Lambda }}_3(t,V,P,C,M)=(1-\beta )\gamma P\left(t\right)-{\nu }_{3}C\left(t\right),\hfill \\ {\mathsf{\Lambda }}_4(t,V,P,C,M)=T{\nu }_{3}C\left(t\right)-\varphi M\left(t\right).\hfill \end{array}\right.$$

(10)

Having the initial conditions $V\left(1\right)=V_0,P\left(1\right)=P_0,C\left(1\right)=C_0,M\left(1\right)=M_0.$

3. Existence and Uniqueness

It is critical that we analyze the model to ensure the existence of its solution. Various ideas and tools can be used to derive this concept. Fixed-point theory is a strong tool for studying the aforementioned requirement. As a result, we use the fixed point theorem to prove the existence of at least one solution and its uniqueness for model (9) in this section of the manuscript.

Take $\mathbb{A}={\mathfrak{C}}^k([1,T],\mathbb{R})$ be the Banach space of absolutely continuous functions $\zeta$ from $[1,T]\mathrm{into}\mathbb{R}$ with the norm:

$$\left|\right|\zeta \left|\right|=\underset{t\in [1,T]}{sup}\left|\zeta \left(t\right)\right|,\left|\zeta \left(t\right)\right|=\left|V\left(t\right)\right|+\left|P\left(t\right)\right|+\left|C\left(t\right)\right|+\left|M\left(t\right)\right|.$$

Now, we modify the HIV model (9) to include the following assumptions:

$$\begin{array}{ccc}\zeta \left(t\right)=\left\{\begin{array}{c}V\left(t\right),\hfill \\ P\left(t\right),\hfill \\ C\left(t\right),\hfill \\ M\left(t\right).\hfill \end{array}\right.& {\zeta }_0=\left\{\begin{array}{c}{V}_0,\hfill \\ P_0,\hfill \\ C_0,\hfill \\ M_0.\hfill \end{array}\right.& \mathsf{\Lambda }(t,\zeta \left(t\right))=\left\{\begin{array}{c}{\mathsf{\Lambda }}_1(t,V,P,C,M),\hfill \\ {\mathsf{\Lambda }}_2(t,V,P,C,M),\hfill \\ {\mathsf{\Lambda }}_3(t,V,P,C,M),\hfill \\ {\mathsf{\Lambda }}_4(t,V,P,C,M),\hfill \end{array}\right.\end{array}$$

(11)

and

$$\begin{array}{c}\mathsf{\Lambda }(1,\zeta \left(1\right))=\left\{\begin{array}{c}{\mathsf{\Lambda }}_1(1,V\left(1\right),P\left(1\right),C\left(1\right),M\left(1\right)),\hfill \\ {\mathsf{\Lambda }}_2(1,V\left(1\right),P\left(1\right),C\left(1\right),M\left(1\right)),\hfill \\ {\mathsf{\Lambda }}_3(1,V\left(1\right),P\left(1\right),C\left(1\right),M\left(1\right)),\hfill \\ {\mathsf{\Lambda }}_4(1,V\left(1\right),P\left(1\right),C\left(1\right),M\left(1\right)).\hfill \end{array}\right.\end{array}$$

(12)

By using Lemma 2, (11) and (12). The HIV model (9) can be expressed by the next system

$$\begin{array}{c}\left\{\begin{array}{c}\left({}_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \right)\left(t\right)=\mathsf{\Lambda }(t,\zeta \left(t\right)),\hfill \\ \zeta \left(1\right)={\zeta }_0.\hfill \end{array}\right.\end{array}$$

(13)

Which is the same as the integral operator that follows

$$\zeta \left(t\right)={\zeta }_0+\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv.$$

(14)

Define the operator $\mathfrak{T}:\mathbb{A}\to \mathbb{A}$ such that

$$\left(\mathfrak{T}\zeta \right)\left(t\right)={\zeta }_0+\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv.$$

(15)

The operator $\mathfrak{T}$ has fixed points if and only if the problem (13), which is analogous to the model (9), has solutions.

For the purposes of further investigation, we assert that the following assumptions are true:

$\left({\mathcal{G}}_1\right)$

There is a constant ${\delta }_{\mathsf{\Lambda }}>0,$ such that

$$\left|\right|\mathsf{\Lambda }(t,{\zeta }_1\left(t\right))-\mathsf{\Lambda }(t,{\zeta }_2\left(t\right))\left|\right|\le {\delta }_{\mathsf{\Lambda }}\left|\right|{\zeta }_1\left(t\right)-{\zeta }_2\left(t\right)\left|\right|$$

$\left({\mathcal{G}}_2\right)$

There is a nondecreasing continuous function $\mathbb{X}:[0,\infty )\to [0,\infty )$ which satisfies each $\mu \ge 1$$\mathbb{X}\left(\mu \zeta \right)\le \mu \mathbb{X}\left(\zeta \right),$ and a function $\psi \in \mathbb{A}$ such that

$$\left|\right|\mathsf{\Lambda }(t,\zeta (t\left)\right)\left|\right|\le \psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(t\right)\left|\right).$$

$\left({\mathcal{G}}_3\right)$

There are some constants $ϵ,B>0$ such that

$$\frac{B}{\left|\right|{\zeta }_0\left|\right|+\frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(ϵ\right){(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}}>1.$$

The following theorem gives the existence of the solution for the HIV model (9) by using the nonlinear Leray–Schauder alternative fixed-point theory.

Theorem 1.

If assumptions $\left({\mathcal{G}}_2\right)$ and $\left({\mathcal{G}}_3\right)$ are fulfilled. Then the HIV model (9) has at least one solution on $[1,T].$

Proof. 

Chose $ϵ>0$ such that ${\left[\mathfrak{B}\right]}_r=\{\zeta \in \mathbb{A};|\left|\zeta \right||\le ϵ\}.$ Utilising $\left({\mathcal{G}}_2\right)$ for each $t\in [1,T]$ we get

$$\begin{array}{ccc}\hfill \left|\right(\mathfrak{T}\zeta \left)\right(t\left)\right|& & \le \left|\right|{\zeta }_0\left|\right|+\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(t,\zeta (t\left)\right)\left|\right|}{v}dv\hfill \\ & & \le \left|\right|{\zeta }_0\left|\right|+\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(t\right)\left|\right)}{v}dv\hfill \\ & & \le \left|\right|{\zeta }_0\left|\right|+\frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(ϵ\right)\left|\right)}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{dv}{v}.\hfill \end{array}$$

Making use of the fact that $e^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}\le 1,$ since $\sigma \in (0,1],$ and $1\le v<t\le T$ we have

$$\left|\right|\mathfrak{T}\zeta \left|\right|\le \left|\right|{\zeta }_0\left|\right|+\frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(ϵ\right)\left|\right)}{{\sigma }^k\mathsf{\Gamma }(k+1)}{(lnT)}^k.$$

Hence, bounded balls are mapped into bounded sets in $\mathbb{A}$ via the operator $\mathfrak{T}$.

On the other hand, we show that $\mathfrak{T}$ is completely continuous. To do that, choose $\zeta \in \mathbb{A}$ and $1\le v_1<v_2.$ Such that

$$\begin{array}{ll}\hfill |\left(\mathfrak{T}\zeta \right)& \left(v_2\right)-\left(\mathfrak{T}\zeta \right)\left(v_1\right)|\le |\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^{v_2}{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{v_2}{v}\right)}{(ln{v}_2-lnv)}^{k-1}\frac{\mathsf{\Lambda }(v,\zeta (v\left)\right)}{v}dv\\ & -\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^{v_1}{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{v_1}{v}\right)}{(ln{v}_1-lnv)}^{k-1}\frac{\mathsf{\Lambda }(v,\zeta (v\left)\right)}{v}dv|\\ & \le \frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}[{\int }_1^{v_1}{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{v_2}{v}\right)}{(ln{v}_2-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(v,\zeta (v\left)\right)\left|\right|}{v}dv\\ & +{\int }_{v_1}^{v_2}{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{v_2}{v}\right)}{(ln{v}_2-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(v,\zeta (v\left)\right)\left|\right|}{v}dv\\ & -\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^{v_1}{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{v_1}{v}\right)}{(ln{v}_1-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(v,\zeta (v\left)\right)\left|\right|}{v}dv]\\ & \le \frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(ϵ\right)\left|\right)}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^{v_1}\left[e^{\frac{\sigma -1}{\sigma }ln\left(\frac{v_2}{v}\right)}{(ln{v}_2-lnv)}^{k-1}-e^{\frac{\sigma -1}{\sigma }ln\left(\frac{v_1}{v}\right)}{(ln{v}_1-lnv)}^{k-1}\right]\frac{dv}{v}\\ & +\frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(ϵ\right)\left|\right)}{{\sigma }^k\mathsf{\Gamma }(k+1)}{(ln{v}_2-ln{v}_1)}^k\to 0\mathrm{as}{v}_2\to v_1.\end{array}$$

Therefore, the operator $\mathfrak{T}$ is completely continuous, according to the Arzel’a–Ascoli theorem. Finally, we prove that the collection of all solutions that satisfy $\zeta =\rho \mathfrak{T}\zeta$ for every $0<\rho <1$ is bounded. Let $\zeta \in \mathbb{A}$ be a solution, then for each $1<t<T$ we have

$$\left|\zeta \left(T\right)\right|=\rho \left(\mathfrak{T}\zeta \right)\left(t\right)\le \left|\right|{\zeta }_0\left|\right|+\frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(ϵ\right)\left|\right)}{{\sigma }^k\mathsf{\Gamma }(k+1)}{(lnT)}^k.$$

Hence,

$$\left|\right|\zeta \left|\right|\le \left|\right|{\zeta }_0\left|\right|+\frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(\right|\zeta \left(ϵ\right)\left|\right)}{{\sigma }^k\mathsf{\Gamma }(k+1)}{(lnT)}^k.$$

By using the assumption ${\mathcal{G}}_3,\exists B>0$ with $\left|\right|\zeta \left|\right|\ne B.$ Set $\mathcal{Q}:=\{\zeta \in \mathbb{A};|\left|\zeta \right||<B\}.$ It is worth noting that $\mathfrak{T}:\overline{\mathcal{Q}}\to \mathbb{A}$ is completely continuous. By the definition of $\overline{\mathcal{Q}}$ no $\zeta \in \partial \overline{\mathcal{Q}}$ satisfies $\left|\zeta \right(T\left)\right|=\rho \mathfrak{T}\zeta$. As a result of the (L-SNLA), we can deduce that (9) has a solution on $[1,T].$ □

We will now show how the Banach fixed-point principle can be used to prove EU of the solution for the HIV model (9).

Theorem 2.

If assumption $\left({\mathcal{G}}_1\right)$ is fulfilled for each ${\zeta }_1,{\zeta }_2\in \mathbb{A},t\in [1,T]$ and if

$$\begin{array}{c}\hfill \frac{{\delta }_{\mathsf{\Lambda }}{(lnT)}^{k\sigma }}{{\sigma }^k\mathsf{\Gamma }(k+1)}<1.\end{array}$$

(16)

Then, for the HIV model (9) there is a unique solution on $[1,T].$

Proof. 

Let ${sup}_{v\in [1,T]}\parallel \mathsf{\Lambda }(v,0)\parallel =\overline{P}<\infty$. Choose ${\mathbb{B}}_{{\rho }_1}=\left\{\zeta \in \mathbb{A}:\parallel \zeta \parallel \le {\rho }_1\right\}$, where

$$\parallel {\zeta }_0\parallel +\frac{\overline{P}{(lnT)}^{k\sigma }}{{\sigma }^k\mathsf{\Gamma }(k+1)}\le {\rho }_1\left(1-\frac{\overline{P}{(lnT)}^{k\sigma }}{{\sigma }^k\mathsf{\Gamma }(k+1)}\right).$$

(17)

we conclude that ${\mathbb{B}}_{{\rho }_1}$ is closed, bounded, and convexly set in $\mathbb{A}$.

First, we will show that $\mathfrak{T}{\mathbb{B}}_{{\rho }_1}\subset {\mathbb{B}}_{{\rho }_1}$. Letting $\zeta \in {\mathbb{B}}_{{\rho }_1}$, we get

$$\begin{array}{ccc}\hfill \left|\right(\mathfrak{T}\zeta \left)\right(v\left)\right|& \le & \parallel {\zeta }_0\parallel +\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(t,\zeta (t\left)\right)\left|\right|}{v}dv\hfill \\ & \le & \parallel {\zeta }_0\parallel +\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(v,\zeta (v\left)\right)-\mathsf{\Lambda }(v,0)\left|\right|+\left|\right|\mathsf{\Lambda }(v,0)\left|\right|}{v}dv\hfill \\ & \le & \parallel {\zeta }_0\parallel +\frac{{\rho }_1{\delta }_{\mathsf{\Lambda }}+\overline{P}}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{dv}{v}\hfill \\ & \le & \parallel {\zeta }_0\parallel +\frac{{\rho }_1{\delta }_{\mathsf{\Lambda }}+\overline{P}}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{(lnt-lnv)}^{k-1}\frac{dv}{v}\hfill \\ & \le & \parallel {\zeta }_0\parallel +\frac{({\rho }_1{\delta }_{\mathsf{\Lambda }}+\overline{P}){(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}\le {\rho }_1.\hfill \end{array}$$

(18)

Hence, $\mathfrak{T}{\mathbb{B}}_{{\rho }_1}\subset {\mathbb{B}}_{{\rho }_1}$.

Secondly, we shall prove that $\mathfrak{T}$ is a contraction operator. For each ${\zeta }_1,{\zeta }_2\in \mathbb{A},t\in [1,T],$ we get

$$\begin{array}{ccc}\hfill \left|\left(\mathfrak{T}{\zeta }_1\right)\left(t\right)-\left(\mathfrak{T}{\zeta }_2\right)\left(t\right)\right|& \le & \frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(v,{\zeta }_1\left(v\right))-\mathsf{\Lambda }(v,{\zeta }_2\left(v\right))\left|\right|}{v}dv\hfill \\ & \le & \frac{{\delta }_{\mathsf{\Lambda }}}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}∥{\zeta }_1\left(t\right)-{\zeta }_2\left(t\right)∥\frac{du}{v}\hfill \\ & \le & \frac{{\delta }_{\mathsf{\Lambda }}{(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}∥{\zeta }_1-{\zeta }_2∥.\hfill \end{array}$$

(19)

Since $\frac{{\delta }_{\mathsf{\Lambda }}{(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}<1$, there exist $\zeta \in {\mathbb{B}}_{{\rho }_1}$ such that $\zeta =\mathfrak{T}\zeta$. Hence, the problem (13), which is analogous to the model (9), has solutions.

Finally, we will demonstrate the fixed point’s uniqueness. Assume that ${\zeta }_1,{\zeta }_2$ are two separate fixed points and from (19) we get

$$\left|\right|{\zeta }_1-{\zeta }_2\left|\right|=\left|\left(\mathfrak{T}{\zeta }_1\right)\left(t\right)-\left(\mathfrak{T}{\zeta }_2\right)\left(t\right)\right|<\left|\right|{\zeta }_1-{\zeta }_2\left|\right|.$$

This is contradictory. Hence, a fixed point $\zeta$ is a unique solution for problem (15). Accordingly, the model (9) has a unique solution on $[1,T]$. □

Remark 1.

Hadmard-Caputo integral and differential operators can be obtained by setting $\sigma =1$ in (4) and (5), respectively. These operators are applied to model (9) to produce the Hadamard-Caputo HIV model.

The EU of the Hadamard-Caputo HIV model are given by the following corollaries, which use the nonlinear Leray–Schauder alternative fixed-point theory and the Banach contraction principle, respectively.

Corollary 1.

If assumptions $\left({\mathcal{G}}_2\right)$ and $\left({\mathcal{G}}_3^{\prime }\right)$ are fulfilled. Then, the Hadamard-Caputo HIV model has at least one solution on $[1,T],$ where $\left({\mathcal{G}}_3^{\prime }\right)$ is defined as if there are some constants $ϵ,B>0$, such that

$$\frac{B}{\left|\right|{\zeta }_0\left|\right|+\frac{{sup}_{t\in [1,T]}\psi \left(t\right)\mathbb{X}\left(ϵ\right){(lnT)}^k}{\mathsf{\Gamma }(k+1)}}>1.$$

Corollary 2.

If assumption $\left({\mathcal{G}}_1\right)$ is fulfilled for each ${\zeta }_1,{\zeta }_2\in \mathbb{A},t\in [1,T]$ and if

$$\begin{array}{c}\hfill \frac{{\delta }_{\mathsf{\Lambda }}{(lnT)}^k}{\mathsf{\Gamma }(k+1)}<1.\end{array}$$

Then, for the Hadamard-Caputo HIV model there is a unique solution on $[1,T].$

4. Results on Ulam–Hyers Stability (UHS)

In this section, we provide some necessary requirements for model (9) to satisfy the assumptions of different types of stability. UHS, extended UHS, Ulam–Hyers–Rassias stability (UHRS), and extended UHRS are examples of these types of stability. Before we show the stability theorems, we need to define a few terms.

Let us take a real positive constant $\lambda$ and assume that ${\mathsf{\Psi }}_{\mathsf{\Lambda }}:[1,T]\to {\mathbb{R}}_{+}$ is a continuous function. The inequalities that will be used to define stability are as follows

$${\parallel }_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(t\right)-\mathsf{\Lambda }(t,\zeta \left(t\right))\parallel \le \lambda ,\forall t\in [1,T],$$

(20)

$${\parallel }_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(t\right)-\mathsf{\Lambda }(t,\zeta \left(t\right))\parallel \le \lambda {\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right),\forall t\in [1,T],$$

(21)

$${\parallel }_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(t\right)-\mathsf{\Lambda }(t,\zeta \left(t\right))\parallel \le {\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right),\forall t\in [1,T].$$

(22)

Definition 6

([23]). The Equation (13) is seen as being stable under the UH condition if ∃ a solution $\zeta \in \mathbb{A}$ for the Equation (13) moreover, there are $C_{\mathsf{\Lambda }}>0,$ such that for each $x\in \mathbb{A}$ fulfilling the inequality (20) and for all $\lambda >0$ we have

$$\parallel x\left(t\right)-\zeta \left(t\right)\parallel \le C_{\mathsf{\Lambda }}\lambda ,t\in [1,T],$$

(23)

where $C_{\mathsf{\Lambda }}=max\left(C_{{\mathsf{\Lambda }}_1},C_{{\mathsf{\Lambda }}_2},C_{{\mathsf{\Lambda }}_3},C_{{\mathsf{\Lambda }}_4}\right).$

Definition 7

([23]). If there exists ${\mathsf{\Psi }}_{\mathsf{\Lambda }}\in \mathcal{C}[1,T]$, with ${\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(1\right)=0$, and for each $\zeta \in \mathbb{A}$ fulfilling the inequality (21), then ∃ a solution $x\in \mathbb{A}$ for (13) with

$$\parallel \zeta \left(t\right)-x\left(t\right)\parallel \le {\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(\lambda \right),t\in [1,T],$$

(24)

where ${\mathsf{\Psi }}_{\mathsf{\Lambda }}=max\left\{{\mathsf{\Psi }}_{{\mathsf{\Lambda }}_1},{\mathsf{\Psi }}_{{\mathsf{\Lambda }}_2},{\mathsf{\Psi }}_{{\mathsf{\Lambda }}_3},{\mathsf{\Psi }}_{{\mathsf{\Lambda }}_4}\right\}.$ Then, the extended UH condition on the problem (13) is said to be stable.

A fundamental attribute that can be used to achieve UHS and extended UHS is now provided.

Lemma 3.

Suppose that $k>0$ and $\sigma \in (0,1]$. The subsequent inequality

$$∥\zeta \left(t\right)-{\zeta }_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv∥\le \frac{\lambda {(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}.$$

(25)

is satisfied if $\zeta \in \mathbb{A}$ is a solution of the inequality (20).

Proof. 

Since $\zeta$ satisfies (20), then ∃ a function $y\in \mathbb{A}$ (dependent on $\zeta$) satisfies

$$\parallel y\left(t\right)\parallel \le \lambda ,y=max(y_1,y_2,y_3,y_4),\forall t\in [1,T].$$

(26)

And

$$\left\{\begin{array}{cc}{}_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(t\right)=\mathsf{\Lambda }(t,\zeta \left(t\right))+y\left(t\right),\hfill & t\in [1,T],\hfill \\ \zeta \left(1\right)={\zeta }_0\ge 0.\hfill \end{array}\right.$$

(27)

Lemma 2 which states that the solution to problem (27) can be expressed as

$$\begin{array}{ccc}\hfill \zeta \left(t\right)& & ={\zeta }_0+\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv\hfill \\ & & +\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{y\left(t\right)}{v}dv.\hfill \end{array}$$

(28)

By using both Equations (28) and (26), we have

$$\begin{array}{ccc}\hfill & & ∥\zeta \left(t\right)-{\zeta }_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv∥\hfill \\ & & \le \frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\left|\right|y\left(t\right)\left|\right|}{v}dv\le \frac{\lambda {(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}.\hfill \end{array}$$

(29)

Thus, inequality (25) is established. □

We are now prepared to demonstrate the UHS plus extended UHS.

Theorem 3.

Consider $\mathsf{\Lambda }(t,\zeta \left(t\right))\in \mathcal{C}([1,T],{\mathbf{R}}^{+})$ for each $\zeta \in \mathbb{A}$. Model (9) or problem (13) is stable under UHS and the extended UHS conditions if both $\left({\mathcal{G}}_3\right)$ and Equation (16) are all met.

Proof. 

Consider that $x\in \mathbb{A}$ is a unique solution of the problem (13). Further, consider that $\lambda >0$ and $\zeta \in \mathbb{A}$ satisfies the inequality (20). Utilizing Lemma 3 and Equation (13), we obtain

$$\begin{array}{ccc}\hfill \parallel \zeta \left(t\right)-x\left(t\right)\parallel & \le & ∥\zeta \left(t\right)-x_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,x(t\left)\right)}{v}dv∥\hfill \\ & \le & ∥\zeta \left(t\right)-{\zeta }_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv∥\hfill \\ & +& \frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(t,\zeta (t\left)\right)-\mathsf{\Lambda }(t,x(t\left)\right)\left|\right|}{v}dv\hfill \\ & \le & ∥\zeta \left(t\right)-{\zeta }_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv∥\hfill \\ & +& \frac{{\delta }_{\mathsf{\Lambda }}}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{(lnt-lnv)}^{k-1}\frac{\left|\right|\zeta \left(t\right)-x\left(t\right)\left|\right|}{v}dv\hfill \\ & \le & \frac{\lambda {(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}+\frac{{\delta }_{\mathsf{\Lambda }}{(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}\parallel \zeta \left(t\right)-x\left(t\right)\parallel .\hfill \end{array}$$

(30)

Hence, $\parallel \zeta \left(t\right)-x\left(t\right)\parallel \le \lambda C_{\mathsf{\Lambda }}$, where

$$C_{\mathsf{\Lambda }}=\frac{{(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)-{\delta }_{\mathsf{\Lambda }}{(lnT)}^k}.$$

(31)

Therefore, model (9) is (UH)-stable. Additionally, it is implied that model (9) is extended (UH)-stable by setting $\lambda C_{\mathsf{\Lambda }}={\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(\lambda \right)$ such that ${\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(1\right)=0.$ The proof is now complete. □

The following significant result, which provides UHS for the Hadamard-Caputo HIV model, is obtained by using Corollary 1 and Theorem 3.

Corollary 3.

Consider $\mathsf{\Lambda }(t,\zeta \left(t\right))\in \mathcal{C}([1,T],{\mathbf{R}}^{+})$ for each $\zeta \in \mathbb{A}$. Then, the Hadamard-Caputo HIV model is stable under (UH) and the extended (UH) conditions if both $\left({\mathcal{G}}_3^{\prime }\right)$ and Equation (16) with respect to fractional Hadamard-Caputo operators are all met.

Finally, the (UHRS) and extended UHRS assumptions can be realized by model (9) if certain requirements are met. The following characteristics of these stability categories can be seen as follows.

Definition 8

([23]). The Equation (13) is seen as being stable under UHR condition with respect to ${\mathsf{\Psi }}_{\mathsf{\Lambda }},$ if there exists a solution $\zeta \in \mathbb{A}$ for the Equation (13). Moreover, there are ${\mathcal{M}}_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}>0,$ such that for each $x\in \mathbb{A}$ fulfilling the inequality (20) and for all $\lambda >0$ we have

$$\parallel x\left(t\right)-\zeta \left(t\right)\parallel \le \lambda {\mathcal{M}}_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right),t\in [1,T].$$

(32)

Definition 9

([23]). The Equation (13) is seen as being stable under extended UHR condition with respect to ${\mathsf{\Psi }}_{\mathsf{\Lambda }},$ if there exists a solution $\zeta \in \mathbb{A}$ for the Equation (13). Moreover, there are ${\mathcal{M}}_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}>0,$ such that for each $x\in \mathbb{A}$ fulfilling the inequality (20) we have

$$\parallel x\left(t\right)-\zeta \left(t\right)\parallel \le {\mathcal{M}}_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}.{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right),t\in [1,T].$$

(33)

A fundamental lemma that will be explained will be used to demonstrate the UHRS and generalized UHRS result.

Lemma 4.

Consider the following condition:

$\left({\mathcal{G}}_4\right)$

 ∃ a non-decreasing operator ${\mathsf{\Psi }}_{\mathsf{\Lambda }}\in \mathbb{A}$ and ∃ ${\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}>0$, such that the following inequality holds:

$${}_{PHC}{\mathfrak{J}}^{k,\sigma }{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right)\le {\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right)t\in [1,T].$$

(34)

If $k>0$ and $\sigma \in (0,1]$, and the subsequent inequality

$$∥\zeta \left(t\right)-{\zeta }_0{-}_{PHC}{\mathfrak{J}}^{k,\sigma }\mathsf{\Lambda }(t,\zeta \left(t\right))∥\le \lambda {\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right)$$

(35)

is satisfied for each $\zeta \in \mathbb{A}$ which is a solution of inequality (22).

Proof. 

Since $\zeta$ fulfills (21), then ∃ an operator $V\in \mathbb{A}$ (dependent on $\zeta$) which satisfies

$$\parallel V\left(t\right)\parallel \le \lambda {\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right),V=max(V_1,V_2,V_3,V_4),\forall t\in [1,T].$$

(36)

And

$$\left\{\begin{array}{cc}{}_{PHC}{\mathfrak{D}}^{k,\sigma }\zeta \left(t\right)=\mathsf{\Lambda }(t,\zeta \left(t\right))+V\left(t\right),\hfill & t\in [1,T],\hfill \\ \zeta \left(1\right)={\zeta }_0\ge 0.\hfill \end{array}\right.$$

(37)

Hence, the solution to problem (37) can be expressed as

$$\begin{array}{ccc}\hfill \zeta \left(t\right)& & ={\zeta }_0{+}_{PHC}{\mathfrak{J}}^{k,\sigma }\mathsf{\Lambda }(t,\zeta \left(t\right)){+}_{PHC}{\mathfrak{J}}^{k,\sigma }V\left(t\right).\hfill \end{array}$$

(38)

By using both Equations (36) and (38), we have

$$\begin{array}{c}\hfill ∥\zeta \left(t\right)-{\zeta }_0{-}_{PHC}{\mathfrak{J}}^{k,\sigma }\mathsf{\Lambda }(t,\zeta \left(t\right))∥{\le }_{PHC}{\mathfrak{J}}^{k,\sigma }\left|\right|V\left(t\right)\left|\right|\le \lambda {\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right),\end{array}$$

(39)

where ${\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}=\frac{{(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}.$ Thus, inequality (41) is established. □

Lemma 5.

Let $\sigma =1$ in Equations (4) and (5), and consider the following condition:

$\left({\mathcal{G}}_4^{\prime }\right)$:

∃ a non-decreasing operator ${\mathsf{\Psi }}_{\mathsf{\Lambda }}\in \mathbb{A}$ and ∃${\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}>0$, satisfies:

$${}_{HC}{\mathfrak{J}}^k{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right)\le {\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right)t\in [1,T].$$

(40)

If $k>0$ and $\sigma \in (0,1]$, and the subsequent inequality

$$∥\zeta \left(t\right)-{\zeta }_0{-}_{HC}{\mathfrak{J}}^k\mathsf{\Lambda }(t,\zeta \left(t\right))∥\le \lambda {\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right).$$

(41)

is satisfied for each $\zeta \in \mathbb{A}$ which is a solution of inequality (22) with respect to Hadamard-Caputo operators.

Proof. 

The proof can be concluded by adding $\sigma =1$ to the proof of lemma 4. □

Lastly, UHRS and extended UHRS will then be shown for the HIV model (9).

Theorem 4.

Consider $\mathsf{\Lambda }(t,\zeta \left(t\right))\in \mathcal{C}([1,T],{\mathbf{R}}^{+})$ for each $\zeta \in \mathbb{A}$. The HIV model (9) or problem (13) is stable under UHR and extended UHR conditions if $\left({\mathcal{G}}_3\right),\left({\mathcal{G}}_4\right)$ and Equation (16) are all met.

Proof. 

Choose $x\in \mathbb{A}$ is a unique solution of the problem (13). Also, consider $\lambda >0$ and $\zeta \in \mathbb{A}$ satisfies the inequality (21). Utilizing Lemma 4 and Equation (13), we obtain

$$\begin{array}{ccc}\hfill \parallel \zeta \left(t\right)-x\left(t\right)\parallel & \le & ∥\zeta \left(t\right)-x_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,x(t\left)\right)}{v}dv∥\hfill \\ & \le & ∥\zeta \left(t\right)-{\zeta }_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv∥\hfill \\ & +& \frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\left|\right|\mathsf{\Lambda }(t,\zeta (t\left)\right)-\mathsf{\Lambda }(t,x(t\left)\right)\left|\right|}{v}dv\hfill \\ & \le & ∥\zeta \left(t\right)-{\zeta }_0-\frac{1}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{e}^{\frac{\sigma -1}{\sigma }ln\left(\frac{t}{v}\right)}{(lnt-lnv)}^{k-1}\frac{\mathsf{\Lambda }(t,\zeta (t\left)\right)}{v}dv∥\hfill \\ & +& \frac{{\delta }_{\mathsf{\Lambda }}}{{\sigma }^k\mathsf{\Gamma }\left(k\right)}{\int }_1^t{(lnt-lnv)}^{k-1}\frac{\left|\right|\zeta \left(t\right)-x\left(t\right)\left|\right|}{v}dv\hfill \\ & \le & \lambda {\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right)+\frac{{\delta }_{\mathsf{\Lambda }}{(lnT)}^k}{{\sigma }^k\mathsf{\Gamma }(k+1)}\parallel \zeta \left(t\right)-x\left(t\right)\parallel .\hfill \end{array}$$

(42)

Hence, $\parallel \zeta \left(t\right)-x\left(t\right)\parallel \le \lambda {\mathcal{M}}_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}{\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(t\right)$, where

$${\mathcal{M}}_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}=\frac{{\sigma }^k\mathsf{\Gamma }(k+1){\eta }_{{\mathsf{\Psi }}_{\mathsf{\Lambda }}}}{{\sigma }^k\mathsf{\Gamma }(k+1)-{\delta }_{\mathsf{\Lambda }}{(lnT)}^k}.$$

(43)

Therefore, model (9) is UHR stable. Additionally, it is implied that model (9) is extended UHR stable by setting $\lambda =1$ such that ${\mathsf{\Psi }}_{\mathsf{\Lambda }}\left(1\right)=0.$ The proof is now complete. □

The Hadamard-Caputo HIV model’s UHRS and extended UHRS are examined in the following result using Corollary 1 and Theorem 4.

Corollary 4.

Consider $\mathsf{\Lambda }(t,\zeta \left(t\right))\in \mathcal{C}([1,T],{\mathbf{R}}^{+})$ for each $\zeta \in \mathbb{A}$. The Hadamard-Caputo HIV model is stable under UHR and extended UHR conditions if $\left({\mathcal{G}}_3^{\prime }\right),\left({\mathcal{G}}_4^{\prime }\right)$ and Equation (16) with respect to Hadamard-Caputo operators are all met.

5. Conclusions

Evidently, fractional models have many benefits that classical mathematical models do not. Modeling real phenomena via fractional operators involves memory influences in contrast to modeling by integer-order derivative. This makes fractional mathematical models more beneficial because, in most events, the models rely not only on the current status but also on the previous attitude of the model. This paper has justified the validity of employing the generalized proportional Hadamard-Caputo fractional operators to introduce a new fractional HIV model. The solution of this model has been proven to exist and be unique via Banach’s and (L-SNLA) fixed-point theorems. Further, for the suggested fractional HIV model, diverse kinds of Ulam’s stability results have been provided and analyzed. The technique proposed in this study, combined with the generalized proportional Hadamard-Caputo fractional operators, can be utilized to model and analyze several infectious diseases, like measles, strep throat, COVID-19, and salmonella. Moreover, the obtained existence, uniqueness, and stability results are valid to analyze the solution of the fractional HIV under Hadamard-Caputo fractional operators [24], when $\sigma =1$, and, in the traditional HIV model, when $\sigma =k=1$ [25]. Recently, Hyder et al. [26] derived a fractional HIV model via the improved fractional operators. They investigated the existence, uniqueness, and stability of the solution on the time interval [0,T]. Obviously, the generalized proportional Hadamard-Caputo operators and the improved fractional operators are alternative fractional operators for generalizing the Hadamard-Caputo fractional operators. Hence, in comparing the results of the current study with the results of Hyder et al. [26], one can conclude that both results are different and give alternative qualitative fractional characteristics for the solution of the fractional HIV model.