# Nonlinear dynamics of the media addiction model using the fractal-fractional derivative technique

Authors: Saima Rashid; Rehana Ashraf; Ebenezer Bonyah
Journal: Complexity
 (2022)

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## Abstract

Excessive use of social media is a developing concern in the twenty-first century. This issue needs to be addressed before it has any more significant consequences than what we are currently experiencing. As a preventive technique, advertisements and awareness-raising campaigns about the detrimental impact of digital technologies are used. The application of novel mathematical techniques and terminologies in this field of study will have significant potential to enhance healthy living by preventing certain ailments. This is the most compelling justification for conducting a new study with the most up-to-date techniques at our disposal. This study investigates clear and concise transmission in order to generate a deterministic mathematical model of social media addictionSMA using the fractal-fractional (FF) derivative operator. Also, the analysis of the SMA model in terms of the invariant domain, the existence of a positive invariant solution, and equilibria assumptions are stated in a detailed manner. Besides that, the basic reproduction number ℝ0<1 is computed, demonstrating that the proposed methodology is more efficacious. The Atangana-Baleanu FF differential operators are recently defined in FF differential operators that are applied to characterize the SMA model’s mathematical algorithm. We investigated the numerical behaviour of the SMA in three ways: (i) changing the fractional-order α as well as the fractal-dimension ℘; (ii) changing α while keeping ℘ constant; and (iii) changing ℘ while keeping α constant. Our examined visualizations and simulation studies using MATLAB for the numerical modelling of the aforementioned system showed that the novel developed Atangana-Baleanu FF differential operators produce remarkable results when compared to the classical frame.


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## Body

## 1. Introduction

With the popularity of networking communication comes an increase in the psychoactive properties of this technological innovation. Numerous investigations have linked compulsive media platform use to undesirable outcomes, including decreased efficiency, adverse social interactions, and decreased life contentment. The misuse of social networking sites has increased at an alarming rate. For example, Facebook has 68 percent and 73 percent of the adolescent community in the United States, respectively [1]. Over prescription of social media addiction is connected with impaired effectiveness appraisal, problematic interpersonal interactions, insomnia, reduced levels of happiness, and sentiments of hostility, stress, and melancholy, see [2–7].To be consistent with the preponderance of the research, we included the terms “social media addiction” SMA or “addicted social networking usage” (in a non-clinical meaning) in the entirety of that kind of analysis, while acknowledging the concerns surrounding the terminology [8]. Whenever it becomes more appropriate to employ certain concepts, exclusions are created. For the purposes of this evaluation, we characterize excessive social media adoption as becoming extremely apprehensive regarding social media, intrinsically incentivized, and dedicating a significant portion of time and vitality to incorporating social media to the juncture where an individual’s group opportunities, intimate communication, surveys, and/or general well-being and well-being are jeopardized [9].There is now a discrepancy in theorizing SMA, particularly how it progresses. There have been multiple evaluations of conceptualizations to elucidate SMA, but limited research has addressed the mathematical frameworks implemented in actual research on SMA. One evaluation, in particular, barely discussed three extensively acknowledged internet dependency scenarios. One questionnaire range of interventions includes broad scientific viewpoints (for example, neurological aspects and behavioural insights) without addressing relevant paradigms. Although some alternative analyses reviewed various relevant conceptual models, analyzed their flaws, and proposed hypothetical modifications, they did not describe how scientific findings about SMA implement these conceptualizations and the various attributes in these structures. Furthermore, new scientific findings on SMA are appearing at an astonishing rate; there is a need for a comprehensive overview of essential mathematical approaches. Alemneh and Alemu [10] constructed and examined a mathematical formulation for the dissemination and prevention of SMA in global society in 2021, as continues to follow:(1)S.ζ=ς+γRζ−λθAζSζ−κ+νSζ,E.ζ=λθAζSζ−ϕ+νEζ,A.ζ=βϕEζ−ν+ε+ψAζ,R.ζ=1−βϕEζ+εA−ν+Rζ,Q.ζ=κSζ+1−γRζ−νQζ,The current community is categorized into five compartments depending on addiction level in the scheme (1). Compartment 1: individuals that are likely to be exposed are those that are not obsessed but are accessible to SMA is represented by Sζ. Compartment 2: individuals who utilize SM relatively regularly yet do not become obsessed are referred to as exposed individuals Eζ. Compartment 3: compulsive populace are persons who are obsessed to SM and spend a significant amount of time doing it Aζ. Compartment 4: retrieved individuals are persons who have progressed from SMA, Rζ is the number of individuals who have benefited from SMA. Compartment 5: Qζ represents anyone that doesn’t use or stop using SM perpetually. The number of individuals in the community is: N=S+E+A+R+Q. The model’s suppositions are as follows: the dissemination of the SMA issue occurs in a confined area and is not reliant on intimate relations, racial group, or sentient socialist system. Representatives combining uniformly, and social networking dependent individuals will convey to non-addictive people when they are in contact with the compression of habit forming. Furthermore, the mathematical expressions of this scheme are amalgamated by employing the social networking addiction process, which begins with the introduction of vulnerable people into the community at a rate of ς. They are prompted to transfer to the revealed condition by addicted humans having a stress interaction incidence of and a probabilistic transference ratio of θ. At a proportion of κ, certain vulnerable persons migrate to a set of individuals who never access digital platforms. The challenged people are divided into two classes: one that develops obsession and advances to the addictive class at a pace βϕ, and another that recovers using therapy at a pace 1−βϕ. Many dependent individuals transfer to the recovery class at a pace of or die as a result of the overuse of dependence on social networks at a ratio of ψ. People who have overcome become vulnerable again at a ratio of or eventually discontinue access to digital platforms at a ratio of 1−γϱ. Eventually, all of the inhabitants in each group have a spontaneous mortality rate of ν. Alemneh and Alemu [10] further examined the robustness of the equilibria and applied Pontryagin’s maximal principle to design the best monitoring mechanism.In past times, fractional calculus has been consistently proven to be an outstanding approach for illustrating the heredity properties of diverse formations. Furthermore, fractional differential operators have been used in a plethora of distinctive manifestations, such as hydroelectricity, fluid dynamics, chromatography, commerce, and viruses, utilising numerous fractional strategies such as Caputo, Hadamard, Riemann-Liouville, Katugumpola, Caputo–Fabrizio, Atangana-Baleanu and fractal–fractional, see [11–13]. This combo has suddenly accumulated a wealth of traction, partly because fractional differential equations have been shown to be superb tools for displaying a few incredibly challenging phenomena in a wide range of various and endless academic disciplines; assessors are encouraged to [14–20]. In 2019, the African mathematician Abdon-Atangana [21] proposed novel differential operators with fractal derivatives based on the combination of index law, modified Mittag-Leffler rule, and exponential decay law. Novel formulations, according to [20], offer additional non-local physical challenges with fractal characteristics simultaneously. Chen et al. [21] investigated the classification of anomalous propagation by deriving the underlying component of the numerical scheme using fractal derivative. For the purpose of evaluating processing performance and dispersion velocity, they compared fractional and fractal derivatives. Wei et al. [22] employed a scalability transformation technique to offer fractal modelling of elastic deformation. Sania et al. [24] introduced differential operators with fractional order and fractal dimension to describe additional chaotic complexity, wherein various kinds of orientations were mentioned: King Cobra, Shilnikov, Thomas cyclically symmetric, Langford, and Rössler. Rashid et al. [25] discussed the complex oscillatory behaviour of a human liver model via FF derivative operator technique. According to current evidence analysis of the data collected for this study, none of the works have addressed the mathematical framework of SMA involving fractal-fractional derivatives. The novelty of this article is the addition of the FF fractional derivative to the SMA model. The improved SMA transmitting system incorporating the fractal-fractional derivative in the Atangana-Baleanu sense is proposed in the following framework:(2)DFFζα,℘Sζ=ς+γϱRζ−λθAζSζ−κ+νSζ,DFFζα,℘Eζ=λθAζSζ−ϕ+νEζ,DFFζα,℘Aζ=βϕEζ−ν+ϵ+ψAζ,DFFζα,℘Rζ=1−βϕEζ+ϵA−ν+ϱRζ,DFFζα,℘Qζ=κSζ+1−γϱRζ−νQζ.having initial conditions S,E,A,R,Q=S0,E0,A0,R0,Q0, where DFFζα,℘. denotes the FF operator in terms of Atangana-Baleanu derivative operator, whilst α presents the fractional-order and ℘ denotes the fractal-dimension, respectively. Furthermore, Table 1 contains explanations of all attributes. Recently, Alemneh and Alemu [10] demonstrated the mathematical modeling with optimal control analysis of SMA. Kongson et al. [27] expounded the estimates for SMA model pertaining to the Atnagana-Baleanu fractional derivative operator in the Caputo context. Leveraging the aforementioned proclivity, we employ a newly formed arbitrary order derivative in the SMA model. The main purpose of this research is to investigate the SMA model via a pioneering FF derivative operator in the Atangana-Baleanu sense and to characterize the complexities of the uniqueness and existence of the aforementioned framework response by utilizing the Picard-Lindlöf and contraction mapping techniques. Furthermore, to the best of our knowledge, there is no previous paper related to SMA qualitative aspects of fractal–fractional derivative based on the Atangana-Baleanu context. As a result, the main objective of this paper is to bridge that gap. Graphical results demonstrate that while fixing and varying fractional-order and fractal-dimension how they affect the model’s characterizations.This article’s entire tasks are organized into five portions, which are provided as they continue to be implemented: Section 2 summarizes and presents the core notions and relevant formulations of FF derivatives in the Atangana-Baleanu interpretations. Section 3 describes the analysis of the model that corresponds to the positivity, bounded domain, invariant region, and equilibria points via the FF derivative operator technique. Section 4 illustrates and performs a convergence and uniqueness investigation for the above-mentioned SMA with the aid of the FF derivative operator and the numerical formulation of the Newton polynomial approach. Section 5 deals with the numerical outcomes and description of the suggested SMA model’s solution. The prime focus of the planned study is on which we will exhibit simulation consequences. Ultimately, Section 6 summarizes the remarks and discusses the promising possibilities of the fractal-fractional SMA model.Table 1
Table of specified variables and their descriptions
ParametersExplanationData estimatedReferencesςAcquisition of susceptible people0.5Supposedνnatural death rate0.05[39]λAddiction transfer percentage to vulnerable persons0.1-0.8[39]θInteraction proportion of vulnerable people involving addicted people0.2[39]βPercentage of uncovered persons who get obsessed0.7[39]ψinduce mortality rate0.01Supposed.ϕPeople who quit the exposed group0.25[39].ϵAddicts who enter the rehabilitated group as a result of the therapy0.7[40]κSusceptible persons who do not take and/or stop utilizing SM0.01Supposed.γThe percentage of rehabilitated people susceptible to SMA0.35[41]ϱPeople who depart the rehabilitated category0.4[40]


## 2. Preliminaries

In what follows, it is vital to investigate some fundamental FF operator notions prior to continuing on to the formal model. Consider there be a functionyζ, which is continuous and fractal differentiable on c,d and has fractal-dimension ℘ and fractional-order α, in addition to the specifications in [21, 24].Definition 2.1.
(see [21, 24]). Suppose there be a FF operator of yζ having power law kernel in terms of Riemann–Liouville (RL) can be expressed in the form:(3)DFFP0,ζα,pyζ=1Γr−αddζ℘∫0ζζ−ϖr−α−1yϖdϖ,where dyϖ/dϖ℘=limζ↦ϖyζ−yϖ/ζ℘−ϖ℘ and r−1<α,℘≤r∈ℕ.Definition 2.2.
(see [21, 24]). Suppose there be a FF operator of yζ having exponential kernel in terms of RL can be expressed in the form:(4)DFFE0,ζα,℘yζ=Mα1−αddζ℘∫0ζexp−α1−αζ−ϖyϖdϖ,such that M0=M1=1 with α>0,℘≤r∈ℕ.Definition 2.3.
(see [21, 24]). Suppose there be a FF operator of yζ with Mittag-Leffler kernel in terms of RL can be expressed in the form:(5)DFFE0,ζα,℘yζ=ABCα1−αddζ℘∫0ζEα−α1−αζ−ϖyϖdϖ,such that ABCα=1−α+α/Γα with α>0,℘≤1∈ℕ.Definition 2.4.
(see [21, 24]). The corresponding FF integral form of (3) is described as:(6)JFFP0,ζαyζ=℘Γα∫0ζζ−ϖα−1ϖ℘−1yϖdϖ.Definition 2.5.
(see [21, 24]). The corresponding FF integral version of (4) is described as:(7)JFFP0,ζαyζ=α℘Mα∫0ζϖ℘−1yϖdϖ+℘1−αζ℘−1yζMα.Definition 2.6.
(see [21, 24]). The corresponding FF integral form of (5) is described as:(8)JFFP0,ζαyζ=α℘ABCα∫0ζϖ℘−1ζ−ϖα−1yϖdϖ+℘1−αζ℘−1yζABCα.Definition 2.7.
(see [14]). Let y∈H1δ,γ,δ<γ and the Atangana-Baleanu-Caputo derivative operator is defined as:(9)DcABCζαyζ=ABCα1−α∫cζy′ϖEα−αζ−ϖα1−αdϖ,α∈0,1,where ABCα represents the normalization function.


## 3. Qualitative analysis ofSMA

In this section, we demonstrate thatSMA presented in (1) is epidemiologically viable by ensuring that the system’s corresponding model parameters are non-negative for every time-step ζ. This is based on the more straightforward argument that the SMA model with non-negative ICs becomes non-negative for all ζ>0. The preceding is a lemma.Lemma 3.1.
Suppose there be the initial dataG≥0, whereGζ=Sζ,Eζ,Aζ,Rζ,Qζ. Thus theSMAsystem (1) are positive for allζ>0. Also,limζ↦∞Nζ≤ς/νhavingNζ=Sζ+Eζ+Aζ+Rζ+Qζ.Proof.
Assume thatΞ=supζ>0:Gζ>0∈0,ζ. Therefore, Ξ>0, the first equation of the framework (1) consists of following(10)Sζdζ=ς+γϱRζ−λθAζSζ−κ+νSζ,
withη=λθAζ, then (10) diminishes to(11)Sζdζ=ς−ηSζ−κ+νSζ.
It follows that(12)ddζSζexpκ+μζ+∫0Ξηϕ1dϕ1=ςκ+μζ+∫0Ξηϕ1dϕ1.
Consequently, we have(13)SΞexpκ+μΞ+∫0Ξηϕ1dϕ1−S0=ςexpκ+μy1+∫0Ξηϕ2dϕ2dy1.
Note that(14)SΞ=S0exp−κ+μΞ−∫0Ξηϕ1dϕ1+exp−κ+μΞ−∫0Ξηϕ1dϕ1×∫0Ξςexpκ+μy1+∫0Ξηϕ2dϕ2dy1>0.
Thus, we can obtainGζ>0 for any ζ>0 by repeating the previous methods for the leftover equations of model (1).
Now adding theSMA cohorts lead to the subsequent(15)dNdζ≤ς−νN−ψA.
If there is no death to theSMA, then(16)dNdζ≤ς−νN.
Therefore, we have(17)limζ↦∞Nζ≤ςν,
which is the desired proof.
Further, in order to show the invariant region for the proposedSMA system (1), suppose(18)∇=S,E,A,R,Q∈ℝ+5:0<Nζ≤ςν.Lemma 3.2.
The domain represented by∇is positively invariant for theSMAsystem (1) along with non-negative ICsS,E,A,R,Q>0for allζ≥0.Proof.
In view of (12), then we have(19)dNdζ=ς−νN.
Therefore, we havedN/dζ≤0, if N0≥ς/ν. Thus, we have(20)Nζ≤N0exp−νζ+ςν1−exp−νζ,
which shows that, the domain presented by∇ is positively invariant. Moreover, if N0>ς/ν or Nζ approaches to ς/ν asymptotically. Hence, the domains presented by ∇ capture all of the possibilities in ℝ+5.


### 3.1. Existence and nonnegativity of the solution

Further, we investigate the existence and nonnegativity of theSMA system (2).Theorem 3.3.
If there be a unique solution of the system (2) and capture the solution inℝ+5.Proof.
In order to prove the solution of the system (2) is positive, we have(21)DFF0,ζα,℘Sζ|S=0=ς≥0,DFF0,ζα,℘Eζ|E=0=0,DFF0,ζα,℘Aζ|A=0=βϕEζ≥0,DFF0,ζα,℘Rζ|R=0=1−βϕE+ϵAζ≥0,DFF0,ζα,℘Qζ|Q=0=κSζ+1−γϱRζ≥0,
which indicates that the system (2) solution exist in ℝ+5∀ζ>0. Summing up all cohorts in (2), we have(22)DFF0,ζα,℘Nζ=ς−νN.
Moreover, we have(23)limn↦∞supNζ≤ςν
and hence, the biologically viable domain for the system (2) can be represented by(24)∇1=S,E,A,R,Q∈ℝ+5:0<Nζ≤ςν.The framework forSMA mentioned-above (2) in the FF operator in the Atangana-Baleanu sense is implemented to provide the outcomes in the next subsection.


### 3.2. Stability result for disease free case

The stability outcomes for theSMA framework introduced by at disease free equilibrium (DFE) E0 are explored in this section. We can get the respective formulas by changing the right side terms of the SMA (2) to zero, as(25)DFF0,ζα,℘Sζ=0,DFF0,ζα,℘Eζ=0,DFF0,ζα,℘Aζ=0,DFF0,ζα,℘Rζ=0,DFF0,ζα,℘Qζ=0,we have the followingDFE as follows(26)E0=ςκ+ν,0,0,0,κςνν+κ.Furthermore, the fundamental reproduction numberℝ0, which may be determined by applying the next generation methodology for the scheme, can be used to examine the robustness of DFE at E0. The infectious cohorts in the SMA system (2) are E,A, and the matrices F and V are obtained as follows:(27)F=λθAS00andV=ν+ϕE−ϕβE+ν+ϵ+ψA−1−βϕE+ν+ϱR.Hence, the fundamental reproductive number can be calculated as(28)ℝ0=ρ˜ℱV−1=λςβϕθκ+νϕ+νν+ϵ+ψ.


### 3.3. Strength number

Following the work [28], we will present the strength number.In recent years, the concept of reproduction in a specific infectious problem has been extensively used in epidemiology modelling. As predicted by the concept, two components, F and V, will be identified in (27)(29)FV−1−μI=0will be employed to generate the reproductive number [29]. The nonlinear portion of the classes that are infected is how the component F, which is quite intriguing, gets derived(30)∂∂AAN=N−AN2.Again, we have(31)∂2∂A2N−1N3=−2N−AN3=−2ς−νSζ+Eζ+Rζ+Qζ−βϕEζ+ϵAζSζ+Eζ+Aζ+Rζ+Qζ3.At DFEE0=ς/κ+ν,0,0,0,κς/νν+κ, we have(32)∂2∂A2N−1N3=−2ς−νS0+E0+R0+Q0−βϕE0+ϵA0S0+E0+A0+R0+Q03=−2ς−νS0+Q0S0+Q03.In this case, we have the following(33)FA=−2λς−νS0+Q0S0+Q0300.Then detFAV−1−μI=0 leads to(34)A0=−2λν3κ+ν−1ς2<0.Therefore, the dispersion will have a single magnitude and fade out if there is no regeneration mechanism, which is indicated by a value of A0=0. Furthermore, A0>0 denotes a strength that will trigger a renewing mechanism, indicating that the expansion will contain multiple waves. However, biologists will give a precise explanation of the aforementioned number.Theorem 3.4.
The DFE atE0forSMAmodel (2) is locally asymptotically stable whenℝ0<1satisfying the assumption(35)argμj>φπ2.Proof.
To illustrate the provided hypothesis, we must first acquire the Jacobian matrix by evaluatingSMA system (2) at the DFE E0, we have(36)JE0=−κ+ν0−λςθκ+νγϱ00−ϕ+νλςθκ+ν000βϕ−ϵ+ψ+ν0001−βϕϵ−ϱ+ν0κ001−γϱ−ν.
The negative eigenvalues are−ν,−ν+κ,−ν+ϱ,−ν+ϕ and remaining eigenvalues can be achieved from the following expression μ2+ϵ+ψ+ϕ+2νμ+ϕ+νϵ+ψ+ν−λςβϕθ/κ+ν.the coefficients ϵ+ψ+ϕ+2ν and ϕ+νϵ+ψ+ν−λςβϕθ/κ+ν are positive, for the DFE case, the value of ℝ0 should be less than 1. So that the Rough-Hurtwiz condition is satisfied for the assumptions presented if and only if ϵ+ψ+ϕ+2ν>0,ϕ+νϵ+ψ+ν−λςβϕθ/κ+ν>0 and ϵ+ψ+ϕ+2νϕ+νϵ+ψ+ν−λςβϕθ/κ+ν>0. Thus, the Rough-Hurtwiz condition promise the local asymptotic stability of the SMA system presented (2) at DFE E0. The aforesaid results were achieved utilizing the FF framework as described and utilized in [30].


### 3.4. Endemic equilibrium and their stability

The endemic equilibria of theSMA (2) designated by E1=S∗,E∗,A∗,R∗,Q∗ and the outcome indicated in (2) are presented in this subsection as(37)S∗=ν+ϵ+ψν+ϕλβϕθ,E∗=ν+ϵ+ψςβϕλθϱ+ν−κ+νϵϕϱ+ν+ϵϱν−ψ+νκ+νν+ϕν+ϱ−ϵν2ν+κλβϕθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν,A∗=ςβϕλθϱ+ν−κ+νϵϕϱ+ν+ϵϱν−ψ+νκ+νν+ϕν+ϱ−ϵν2ν+κλθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν,R∗=α−1ν+ψ−ϵςβλϕθ−ν+ϕκ+νν+ϵ+ψλβθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν,Q∗=κνν+ϵ+ψν+ϕλβϕθ+ϱ1−γνα−1ν+ψ−ϵςβλϕθ−ν+ϕκ+νν+ϵ+ψλβθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν.Theorem 3.5.
TheSMA presented by (2) has the following assertions:(a)
ifℝ0>1, then system (2) exhibits unique endemic equilibrium.(b)
ifℝ0=1, then system (2) has forward bifurcation.(c)
ifℝ0<1, then system (2) does not contain the endemic equilibrium point or has backward bifurcation.


## 3.1. Existence and nonnegativity of the solution

Further, we investigate the existence and nonnegativity of theSMA system (2).Theorem 3.3.
If there be a unique solution of the system (2) and capture the solution inℝ+5.Proof.
In order to prove the solution of the system (2) is positive, we have(21)DFF0,ζα,℘Sζ|S=0=ς≥0,DFF0,ζα,℘Eζ|E=0=0,DFF0,ζα,℘Aζ|A=0=βϕEζ≥0,DFF0,ζα,℘Rζ|R=0=1−βϕE+ϵAζ≥0,DFF0,ζα,℘Qζ|Q=0=κSζ+1−γϱRζ≥0,
which indicates that the system (2) solution exist in ℝ+5∀ζ>0. Summing up all cohorts in (2), we have(22)DFF0,ζα,℘Nζ=ς−νN.
Moreover, we have(23)limn↦∞supNζ≤ςν
and hence, the biologically viable domain for the system (2) can be represented by(24)∇1=S,E,A,R,Q∈ℝ+5:0<Nζ≤ςν.The framework forSMA mentioned-above (2) in the FF operator in the Atangana-Baleanu sense is implemented to provide the outcomes in the next subsection.


## 3.2. Stability result for disease free case

The stability outcomes for theSMA framework introduced by at disease free equilibrium (DFE) E0 are explored in this section. We can get the respective formulas by changing the right side terms of the SMA (2) to zero, as(25)DFF0,ζα,℘Sζ=0,DFF0,ζα,℘Eζ=0,DFF0,ζα,℘Aζ=0,DFF0,ζα,℘Rζ=0,DFF0,ζα,℘Qζ=0,we have the followingDFE as follows(26)E0=ςκ+ν,0,0,0,κςνν+κ.Furthermore, the fundamental reproduction numberℝ0, which may be determined by applying the next generation methodology for the scheme, can be used to examine the robustness of DFE at E0. The infectious cohorts in the SMA system (2) are E,A, and the matrices F and V are obtained as follows:(27)F=λθAS00andV=ν+ϕE−ϕβE+ν+ϵ+ψA−1−βϕE+ν+ϱR.Hence, the fundamental reproductive number can be calculated as(28)ℝ0=ρ˜ℱV−1=λςβϕθκ+νϕ+νν+ϵ+ψ.


## 3.3. Strength number

Following the work [28], we will present the strength number.In recent years, the concept of reproduction in a specific infectious problem has been extensively used in epidemiology modelling. As predicted by the concept, two components, F and V, will be identified in (27)(29)FV−1−μI=0will be employed to generate the reproductive number [29]. The nonlinear portion of the classes that are infected is how the component F, which is quite intriguing, gets derived(30)∂∂AAN=N−AN2.Again, we have(31)∂2∂A2N−1N3=−2N−AN3=−2ς−νSζ+Eζ+Rζ+Qζ−βϕEζ+ϵAζSζ+Eζ+Aζ+Rζ+Qζ3.At DFEE0=ς/κ+ν,0,0,0,κς/νν+κ, we have(32)∂2∂A2N−1N3=−2ς−νS0+E0+R0+Q0−βϕE0+ϵA0S0+E0+A0+R0+Q03=−2ς−νS0+Q0S0+Q03.In this case, we have the following(33)FA=−2λς−νS0+Q0S0+Q0300.Then detFAV−1−μI=0 leads to(34)A0=−2λν3κ+ν−1ς2<0.Therefore, the dispersion will have a single magnitude and fade out if there is no regeneration mechanism, which is indicated by a value of A0=0. Furthermore, A0>0 denotes a strength that will trigger a renewing mechanism, indicating that the expansion will contain multiple waves. However, biologists will give a precise explanation of the aforementioned number.Theorem 3.4.
The DFE atE0forSMAmodel (2) is locally asymptotically stable whenℝ0<1satisfying the assumption(35)argμj>φπ2.Proof.
To illustrate the provided hypothesis, we must first acquire the Jacobian matrix by evaluatingSMA system (2) at the DFE E0, we have(36)JE0=−κ+ν0−λςθκ+νγϱ00−ϕ+νλςθκ+ν000βϕ−ϵ+ψ+ν0001−βϕϵ−ϱ+ν0κ001−γϱ−ν.
The negative eigenvalues are−ν,−ν+κ,−ν+ϱ,−ν+ϕ and remaining eigenvalues can be achieved from the following expression μ2+ϵ+ψ+ϕ+2νμ+ϕ+νϵ+ψ+ν−λςβϕθ/κ+ν.the coefficients ϵ+ψ+ϕ+2ν and ϕ+νϵ+ψ+ν−λςβϕθ/κ+ν are positive, for the DFE case, the value of ℝ0 should be less than 1. So that the Rough-Hurtwiz condition is satisfied for the assumptions presented if and only if ϵ+ψ+ϕ+2ν>0,ϕ+νϵ+ψ+ν−λςβϕθ/κ+ν>0 and ϵ+ψ+ϕ+2νϕ+νϵ+ψ+ν−λςβϕθ/κ+ν>0. Thus, the Rough-Hurtwiz condition promise the local asymptotic stability of the SMA system presented (2) at DFE E0. The aforesaid results were achieved utilizing the FF framework as described and utilized in [30].


## 3.4. Endemic equilibrium and their stability

The endemic equilibria of theSMA (2) designated by E1=S∗,E∗,A∗,R∗,Q∗ and the outcome indicated in (2) are presented in this subsection as(37)S∗=ν+ϵ+ψν+ϕλβϕθ,E∗=ν+ϵ+ψςβϕλθϱ+ν−κ+νϵϕϱ+ν+ϵϱν−ψ+νκ+νν+ϕν+ϱ−ϵν2ν+κλβϕθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν,A∗=ςβϕλθϱ+ν−κ+νϵϕϱ+ν+ϵϱν−ψ+νκ+νν+ϕν+ϱ−ϵν2ν+κλθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν,R∗=α−1ν+ψ−ϵςβλϕθ−ν+ϕκ+νν+ϵ+ψλβθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν,Q∗=κνν+ϵ+ψν+ϕλβϕθ+ϱ1−γνα−1ν+ψ−ϵςβλϕθ−ν+ϕκ+νν+ϵ+ψλβθβϕγϱψ+ν−ϕγϱϵ+ϱ+ν+ϵ+ψ+νν+ϕϱ+ν.Theorem 3.5.
TheSMA presented by (2) has the following assertions:(a)
ifℝ0>1, then system (2) exhibits unique endemic equilibrium.(b)
ifℝ0=1, then system (2) has forward bifurcation.(c)
ifℝ0<1, then system (2) does not contain the endemic equilibrium point or has backward bifurcation.


## 4. The fractal-fractionalSMA model

Here, we used the novel FF methodology in this section to reassemble the classical integer-orderSMA system with a non-singular and nonlocal kernel (2). The SMA framework that ensues when the FF operator is taken into account is (2).


### 4.1. Existence-uniqueness outcomes of FF-SMA

Now, the existence-uniqueness of theSMA obtained in the FF operator are succinctly discussed in (2). To do so, we shall use a FF derivative to generate the generic Cauchy problem:(38)DFFα,℘0,ζΛζ=Υζ,Λζ,Λ0=Λ0.In view of Definition (12), the right hand side of (38) yields:(39)ABCα1−αddζ∫0ζΥu,ΛuE¯α−α1−αζ−uαdu=℘ζ℘−1Υζ,Λζ.Considering the implementation of the appropriate integral, the following conclusions are drawn as:(40)Λζ=1−αABCα℘ζ℘−1Υζ,Λζ+℘αABCαΓα∫0ζζ−uα−1Υu,Λuu℘−1du+Λ0.Employing the Picard-Lindelöf method, we have(41)∏η1η2=ℑpζp×ℳ0Λ0¯,where ℑpζp¯=ζp−μ1,ζp+μ1,ℳ0Λ0¯=ζ0−ν1,ζ0+ν1.Accordingly, surmise that(42)ℏ=supζ∈∏η1η2ΥFurthermore, the norm is written as follows:(43)ℑ∞=supζ∈∏η1η2ℑ,and consider the operations(44)ΘCℑpζp,ℳbζp⟶Cℑpb,ℳbζp,described as(45)ΘΥζ=Υ0+1−αABCα℘ζ℘−1Υζ,Λζ+α℘ABCαΓα∫0ζζ−uα−1Υu,Λuu℘−1du.The essential objective is to illustrate that the aforementioned operator can convert a completely empty metric space onto itself. We also aim to illustrate that it has the potential to map contractions. First and foremost, we show that(46)ΘΛζ−Λ0≤b,ΘΛζ−Λ0≤1−αABCα℘ζ℘−1Υζ,Λζ∞+α℘ABCαΓα∫0ζζ−uα−1Υu,Λuu℘−1du≤1−αABCα℘ζ℘−1ℑ+α℘ABCαΓαℑ∫0ζζ−uα−1u℘−1du.Insertingu=ζx, then produces the foregoing(47)ΘΛζ−Λ0≤1−αABCα℘ζ℘−1ℑ+α℘ABCαΓαℑζα+℘−1B1℘,α.Therefore,(48)ΘΛζ−Λ0≤b↦ℑ<bB1℘,α1−α/ABCα℘ζ℘−1+α℘/ABCαΓαζα+℘−1.Then, surmisingΛ1,Λ2∈Cℑpζp,ℳbζp. To obtain at the following result, apply the Banach fixed point theorem:(49)ΘΛ1−ΘΛ2≤LΩΛ−Λ2∞,where LΩ<1.(50)ΘΛ1−ΘΛ2≤1−αABCα℘ζ℘−1Υζ,Λ1−Υζ,Λ2+α℘ABCαΓα∫0ζζ−uα−1u℘−1Υζ,u1−Υζ,s2du,Owing to the contraction mappingΥ, we have(51)ΘΛ1−ΘΛ2≤1−αABCα℘ζ℘−1LΛΛ1−Λ2∞+α℘ABCαΓαLΛΛ1−Λ2∞∫0ζζ−uα−1u℘−1du≤1−αABCα℘ζ℘−1LΛΛ1−Λ2∞+α℘ABCαΓαLΛΛ1−Λ2∞ζα+℘−3B1℘,α.Consequently, we have(52)ΘΛ1−ΘΛ2≤1−αABCα℘ζ℘−1LΛ+α℘ABCαΓαLΛζα+℘−3B1℘,αΛ1−Λ2∞<1−αABCα℘a℘−1LΛ+α℘ABCαΓαLΛaα+℘−3B1℘,αΛ1−Λ2∞.If the supposition made is correct, then(53)LΛ<1−αABCα℘a℘−1LΛ+α℘ABCαΓαLΛaα+℘−3B1℘,α,then the contraction criterion is achieved, i.e.,(54)ΘΛ1−ΘΛ2≤Λ1−Λ2∞.In a nutshell, the proof is completed by demonstrating that there is only one solution.In the next, we describes the numerical solutions for the proposedSMA system.


### 4.2. Newton polynomial approach

Here, we configure a comprehensive analysis of the numerical approach, which relies on an efficient Newton polynomial method. This methodology, which was also originally envisioned in [34], is more effective than some of the previous methods available in the analysis. To continue further with the approach, we apply the equation:(55)DFFζα,℘Λζ=Υζ,Λζ.Integrating (55) with respect to ζ, produces(56)Λζ−Λ0=1−αABCα℘ζ℘−1Υζ,Λζ+α℘ABCαΓα∫0ζζ−uα−1u℘−1Υζ,Λζdu.TakingWζ,Λζ=℘ζ℘−1Υζ,Λζ, then (56) reduces to(57)Λζ−Λ0=1−αABCαWζ,Λζ+αABCαΓα∫0ζζ−uα−1Wu,Λudu.Atζp+1=n+1Δζ, we have(58)Λζp+1−Λ0=1−αABCαWζp,Λζp+αABCαΓα∫0ζp+1ζp+1−uα−1Wu,Λudu.Therefore, we have(59)Λζp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2p∫ζιζι+1ζp+1−uα−1Wu,Λudu.To estimate the mapping, employ the Newton polynomialWζ,Λζ, we have(60)Upu=Wζp−2,Λζp−2+Wζp−1,Λζp−1−Wζp−2,Λζp−2Δζu−ζp−2+Wζp,Λζp−2Wζp−1,Λζp−1+Wζp−2,Λζp−22Δζ2u−ζp−2u−ζp−1.Substituting (60) into (57), yields(61)Λp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2p∫ζιζι+1ζp+1−uα−1Wζℓ−2,Λℓ−2+Wζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2Δζu−ζℓ−2+Wζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−22Δζ2u−ζℓ−2u−ζℓ−1du.Simple computations yield(62)Λp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2p∫ζιζι+1ζp+1−uα−1Wζℓ−2,Λℓ−2du+∫ζιζι+1ζp+1−uα−1Wζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2Δζu−ζℓ−2du+∫ζιζι+1ζp+1−uα−1Wζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−22Δζ2u−ζℓ−2u−ζℓ−1du.Note that(63)Λp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2pWζℓ−2,Λℓ−2∫ζιζι+1ζp+1−uα−1du+αABCαΓα∑ι=2pWζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2Δζ∫ζιζι+1ζp+1−uα−1u−ζℓ−2du+αABCαΓα∑ι=2pWζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−22Δζ2∫ζιζι+1ζp+1−uα−1u−ζℓ−2u−ζℓ−1du.Using the fact that(64)∫ζιζι+1ζp+1−uα−1du=Δζαp−ι+1α−p−ιαα,∫ζιζι+1u−ζι−2ζp+1−uα−1du=Δζα+1p−ι+1αp−ι+2α+3−p−ι+1αp−ι+3α+3αα+1,∫ζιζι+1ζp+1−uα−1u−ζℓ−2u−ζℓ−1du=Δζα+2αα+1α+2×p−ι+1α2p−ι2+3α+10p−ι+2α2+9α+12−p−ια2p−ι2+5α+10p−ι+6α2+18α+12.Furthermore, we have(65)Λp+1=Λ0+1−αABCαWζp,Λζp+αΔζαABCαΓα+1∑ι=2pWζℓ−2,Λℓ−2p−ι+1α−p−ια+αΔζαABCαΓα+2∑ι=2pWζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2×p−ι+1αp−ι+2α+3−p−ι+1αp−ι+3α+3+αΔζα2ABCαΓα+2∑ι=2pWζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−2×p−ι+1α2p−ι2+3α+10p−ι+2α2+9α+12−p−ια2p−ι2+5α+10p−ι+6α2+18α+12.Therefore, a general approximate solution of theSMA is as follows:(66)Λp+1=Λ0+1−αABCα℘ζp℘−1Wζp,Λζp+αΔζαABCαΓα+1∑ι=2p℘ζι−2℘−1Wζℓ−2,Λℓ−2p−ι+1α−p−ια+℘αΔζαABCαΓα+2∑ι=2pζι−1℘−1Wζℓ−1,Λℓ−1−ζι−2℘−1Wζℓ−2,Λℓ−2×p−ι+1αp−ι+2α+3−p−ι+1αp−ι+3α+3+℘αΔζα2ABCαΓα+2∑ι=2pζι℘−1Wζℓ,Λℓ−2tι−1℘−1Wζℓ−1,Λℓ−1+ζι−2℘−1Wζℓ−2,Λℓ−2×p−ι+1α2p−ι2+3α+10p−ι+2α2+9α+12−p−ια2p−ι2+5α+10p−ι+6α2+18α+12.


### 4.3. New numerical technique forSMA FF-AB derivative model

The objective of this task is to provide a structured approach technique for interacting with the (1) social media framework, using the FF operator in the Atangana-Baleanu context. converting the (2) system to the FF-Atangana-Baleanu derivative configuration as follows:(67)DABR0,ζα,℘Sζ=℘ζ℘−1ℱ1S,E,A,R,Q,ζ,DABR0,ζα,℘Eζ=℘ζ℘−1ℱ2S,E,A,R,Q,ζ,DABR0,ζα,℘Aζ=℘ζ℘−1ℱ3S,E,A,R,Q,ζ,DABR0,ζα,℘Rζ=℘ζ℘−1ℱ4S,E,A,R,Q,ζ,DABR0,ζα,℘Qζ=℘ζ℘−1ℱ5S,E,A,R,Q,ζ.Employing the AB fractional integral operator, the preceding conclusions were made as(68)Sζ=S0+℘ζ℘−11−αABCαℱ1S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ1S,E,A,R,Q,ζdξ,Eζ=E0+℘ζ℘−11−αABCαℱ2S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ2S,E,A,R,Q,ζdξ,Aζ=A0+℘ζ℘−11−αABCαℱ3S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ3S,E,A,R,Q,ζdξ,Rζ=R0+℘ζ℘−11−αABCαℱ4S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ4S,E,A,R,Q,ζdξ,Qζ=Q0+℘ζ℘−11−αABCαℱ5S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ5S,E,A,R,Q,ζdξ.Atζn+1, We obtain the below(69)Sn+1ζ=S0+℘ζ℘−11−αABCαℱ1Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ1S,E,A,R,Q,ξdξ,En+1ζ=E0+℘ζ℘−11−αABCαℱ2Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ2S,E,A,R,Q,ξdξ,An+1ζ=A0+℘ζ℘−11−αABCαℱ3Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ3S,E,A,R,Q,ξdξ,Rn+1ζ=R0+℘ζ℘−11−αABCαℱ4Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ4S,E,A,R,Q,ξdξ,Qn+1ζ=Q0+℘ζ℘−11−αABCαℱ5Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ5S,E,A,R,Q,ξdξ.(69) has been modified significantly, yielding the following results:(70)Sn+1ζ=S0+℘ζ℘−11−αABCαℱ1Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ1S,E,A,R,Q,ξdξ,En+1ζ=E0+℘ζ℘−11−αABCαℱ2Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ2S,E,A,R,Q,ξdξ,An+1ζ=A0+℘ζ℘−11−αABCαℱ3Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ3S,E,A,R,Q,ξdξ,Rn+1ζ=R0+℘ζ℘−11−αABCαℱ4Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ4S,E,A,R,Q,ξdξ,Qn+1ζ=Q0+℘ζ℘−11−αABCαℱ5Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ5S,E,A,R,Q,ξdξ.Moreover, implementingξα−1ℱιS,E,A,R,Q,ξforι=1,2,…,5 to describe the expressions in (70), in the defined interval ζℓ,ζℓ+1, the relevant numerical strategy is constructed as(71)Sn+1ζ=S0+℘ζ℘−11−αABCαℱ1Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ1Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ1Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n+1−ℓα+1−n−ℓαn−ℓ+1+α,En+1ζ=E0+℘ζ℘−11−αABCαℱ2Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ2Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ2Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α,An+1ζ=A0+℘ζ℘−11−αABCαℱ3Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ3Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ3Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α,Rn+1ζ=R0+℘ζ℘−11−αABCαℱ4Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ4Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ4Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α,(72)Qn+1ζ=Q0+℘ζ℘−11−αABCαℱ5Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ5Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ5Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α.


## 4.1. Existence-uniqueness outcomes of FF-SMA

Now, the existence-uniqueness of theSMA obtained in the FF operator are succinctly discussed in (2). To do so, we shall use a FF derivative to generate the generic Cauchy problem:(38)DFFα,℘0,ζΛζ=Υζ,Λζ,Λ0=Λ0.In view of Definition (12), the right hand side of (38) yields:(39)ABCα1−αddζ∫0ζΥu,ΛuE¯α−α1−αζ−uαdu=℘ζ℘−1Υζ,Λζ.Considering the implementation of the appropriate integral, the following conclusions are drawn as:(40)Λζ=1−αABCα℘ζ℘−1Υζ,Λζ+℘αABCαΓα∫0ζζ−uα−1Υu,Λuu℘−1du+Λ0.Employing the Picard-Lindelöf method, we have(41)∏η1η2=ℑpζp×ℳ0Λ0¯,where ℑpζp¯=ζp−μ1,ζp+μ1,ℳ0Λ0¯=ζ0−ν1,ζ0+ν1.Accordingly, surmise that(42)ℏ=supζ∈∏η1η2ΥFurthermore, the norm is written as follows:(43)ℑ∞=supζ∈∏η1η2ℑ,and consider the operations(44)ΘCℑpζp,ℳbζp⟶Cℑpb,ℳbζp,described as(45)ΘΥζ=Υ0+1−αABCα℘ζ℘−1Υζ,Λζ+α℘ABCαΓα∫0ζζ−uα−1Υu,Λuu℘−1du.The essential objective is to illustrate that the aforementioned operator can convert a completely empty metric space onto itself. We also aim to illustrate that it has the potential to map contractions. First and foremost, we show that(46)ΘΛζ−Λ0≤b,ΘΛζ−Λ0≤1−αABCα℘ζ℘−1Υζ,Λζ∞+α℘ABCαΓα∫0ζζ−uα−1Υu,Λuu℘−1du≤1−αABCα℘ζ℘−1ℑ+α℘ABCαΓαℑ∫0ζζ−uα−1u℘−1du.Insertingu=ζx, then produces the foregoing(47)ΘΛζ−Λ0≤1−αABCα℘ζ℘−1ℑ+α℘ABCαΓαℑζα+℘−1B1℘,α.Therefore,(48)ΘΛζ−Λ0≤b↦ℑ<bB1℘,α1−α/ABCα℘ζ℘−1+α℘/ABCαΓαζα+℘−1.Then, surmisingΛ1,Λ2∈Cℑpζp,ℳbζp. To obtain at the following result, apply the Banach fixed point theorem:(49)ΘΛ1−ΘΛ2≤LΩΛ−Λ2∞,where LΩ<1.(50)ΘΛ1−ΘΛ2≤1−αABCα℘ζ℘−1Υζ,Λ1−Υζ,Λ2+α℘ABCαΓα∫0ζζ−uα−1u℘−1Υζ,u1−Υζ,s2du,Owing to the contraction mappingΥ, we have(51)ΘΛ1−ΘΛ2≤1−αABCα℘ζ℘−1LΛΛ1−Λ2∞+α℘ABCαΓαLΛΛ1−Λ2∞∫0ζζ−uα−1u℘−1du≤1−αABCα℘ζ℘−1LΛΛ1−Λ2∞+α℘ABCαΓαLΛΛ1−Λ2∞ζα+℘−3B1℘,α.Consequently, we have(52)ΘΛ1−ΘΛ2≤1−αABCα℘ζ℘−1LΛ+α℘ABCαΓαLΛζα+℘−3B1℘,αΛ1−Λ2∞<1−αABCα℘a℘−1LΛ+α℘ABCαΓαLΛaα+℘−3B1℘,αΛ1−Λ2∞.If the supposition made is correct, then(53)LΛ<1−αABCα℘a℘−1LΛ+α℘ABCαΓαLΛaα+℘−3B1℘,α,then the contraction criterion is achieved, i.e.,(54)ΘΛ1−ΘΛ2≤Λ1−Λ2∞.In a nutshell, the proof is completed by demonstrating that there is only one solution.In the next, we describes the numerical solutions for the proposedSMA system.


## 4.2. Newton polynomial approach

Here, we configure a comprehensive analysis of the numerical approach, which relies on an efficient Newton polynomial method. This methodology, which was also originally envisioned in [34], is more effective than some of the previous methods available in the analysis. To continue further with the approach, we apply the equation:(55)DFFζα,℘Λζ=Υζ,Λζ.Integrating (55) with respect to ζ, produces(56)Λζ−Λ0=1−αABCα℘ζ℘−1Υζ,Λζ+α℘ABCαΓα∫0ζζ−uα−1u℘−1Υζ,Λζdu.TakingWζ,Λζ=℘ζ℘−1Υζ,Λζ, then (56) reduces to(57)Λζ−Λ0=1−αABCαWζ,Λζ+αABCαΓα∫0ζζ−uα−1Wu,Λudu.Atζp+1=n+1Δζ, we have(58)Λζp+1−Λ0=1−αABCαWζp,Λζp+αABCαΓα∫0ζp+1ζp+1−uα−1Wu,Λudu.Therefore, we have(59)Λζp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2p∫ζιζι+1ζp+1−uα−1Wu,Λudu.To estimate the mapping, employ the Newton polynomialWζ,Λζ, we have(60)Upu=Wζp−2,Λζp−2+Wζp−1,Λζp−1−Wζp−2,Λζp−2Δζu−ζp−2+Wζp,Λζp−2Wζp−1,Λζp−1+Wζp−2,Λζp−22Δζ2u−ζp−2u−ζp−1.Substituting (60) into (57), yields(61)Λp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2p∫ζιζι+1ζp+1−uα−1Wζℓ−2,Λℓ−2+Wζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2Δζu−ζℓ−2+Wζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−22Δζ2u−ζℓ−2u−ζℓ−1du.Simple computations yield(62)Λp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2p∫ζιζι+1ζp+1−uα−1Wζℓ−2,Λℓ−2du+∫ζιζι+1ζp+1−uα−1Wζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2Δζu−ζℓ−2du+∫ζιζι+1ζp+1−uα−1Wζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−22Δζ2u−ζℓ−2u−ζℓ−1du.Note that(63)Λp+1=Λ0+1−αABCαWζp,Λζp+αABCαΓα∑ι=2pWζℓ−2,Λℓ−2∫ζιζι+1ζp+1−uα−1du+αABCαΓα∑ι=2pWζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2Δζ∫ζιζι+1ζp+1−uα−1u−ζℓ−2du+αABCαΓα∑ι=2pWζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−22Δζ2∫ζιζι+1ζp+1−uα−1u−ζℓ−2u−ζℓ−1du.Using the fact that(64)∫ζιζι+1ζp+1−uα−1du=Δζαp−ι+1α−p−ιαα,∫ζιζι+1u−ζι−2ζp+1−uα−1du=Δζα+1p−ι+1αp−ι+2α+3−p−ι+1αp−ι+3α+3αα+1,∫ζιζι+1ζp+1−uα−1u−ζℓ−2u−ζℓ−1du=Δζα+2αα+1α+2×p−ι+1α2p−ι2+3α+10p−ι+2α2+9α+12−p−ια2p−ι2+5α+10p−ι+6α2+18α+12.Furthermore, we have(65)Λp+1=Λ0+1−αABCαWζp,Λζp+αΔζαABCαΓα+1∑ι=2pWζℓ−2,Λℓ−2p−ι+1α−p−ια+αΔζαABCαΓα+2∑ι=2pWζℓ−1,Λℓ−1−Wζℓ−2,Λℓ−2×p−ι+1αp−ι+2α+3−p−ι+1αp−ι+3α+3+αΔζα2ABCαΓα+2∑ι=2pWζℓ,Λℓ−2Wζℓ−1,Λℓ−1+Wζℓ−2,Λℓ−2×p−ι+1α2p−ι2+3α+10p−ι+2α2+9α+12−p−ια2p−ι2+5α+10p−ι+6α2+18α+12.Therefore, a general approximate solution of theSMA is as follows:(66)Λp+1=Λ0+1−αABCα℘ζp℘−1Wζp,Λζp+αΔζαABCαΓα+1∑ι=2p℘ζι−2℘−1Wζℓ−2,Λℓ−2p−ι+1α−p−ια+℘αΔζαABCαΓα+2∑ι=2pζι−1℘−1Wζℓ−1,Λℓ−1−ζι−2℘−1Wζℓ−2,Λℓ−2×p−ι+1αp−ι+2α+3−p−ι+1αp−ι+3α+3+℘αΔζα2ABCαΓα+2∑ι=2pζι℘−1Wζℓ,Λℓ−2tι−1℘−1Wζℓ−1,Λℓ−1+ζι−2℘−1Wζℓ−2,Λℓ−2×p−ι+1α2p−ι2+3α+10p−ι+2α2+9α+12−p−ια2p−ι2+5α+10p−ι+6α2+18α+12.


## 4.3. New numerical technique forSMA FF-AB derivative model

The objective of this task is to provide a structured approach technique for interacting with the (1) social media framework, using the FF operator in the Atangana-Baleanu context. converting the (2) system to the FF-Atangana-Baleanu derivative configuration as follows:(67)DABR0,ζα,℘Sζ=℘ζ℘−1ℱ1S,E,A,R,Q,ζ,DABR0,ζα,℘Eζ=℘ζ℘−1ℱ2S,E,A,R,Q,ζ,DABR0,ζα,℘Aζ=℘ζ℘−1ℱ3S,E,A,R,Q,ζ,DABR0,ζα,℘Rζ=℘ζ℘−1ℱ4S,E,A,R,Q,ζ,DABR0,ζα,℘Qζ=℘ζ℘−1ℱ5S,E,A,R,Q,ζ.Employing the AB fractional integral operator, the preceding conclusions were made as(68)Sζ=S0+℘ζ℘−11−αABCαℱ1S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ1S,E,A,R,Q,ζdξ,Eζ=E0+℘ζ℘−11−αABCαℱ2S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ2S,E,A,R,Q,ζdξ,Aζ=A0+℘ζ℘−11−αABCαℱ3S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ3S,E,A,R,Q,ζdξ,Rζ=R0+℘ζ℘−11−αABCαℱ4S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ4S,E,A,R,Q,ζdξ,Qζ=Q0+℘ζ℘−11−αABCαℱ5S,E,A,R,Q,ζ+α℘ABCαΓα∫0ζξ℘−1ζ−ξα−1ℱ5S,E,A,R,Q,ζdξ.Atζn+1, We obtain the below(69)Sn+1ζ=S0+℘ζ℘−11−αABCαℱ1Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ1S,E,A,R,Q,ξdξ,En+1ζ=E0+℘ζ℘−11−αABCαℱ2Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ2S,E,A,R,Q,ξdξ,An+1ζ=A0+℘ζ℘−11−αABCαℱ3Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ3S,E,A,R,Q,ξdξ,Rn+1ζ=R0+℘ζ℘−11−αABCαℱ4Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ4S,E,A,R,Q,ξdξ,Qn+1ζ=Q0+℘ζ℘−11−αABCαℱ5Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∫0ζn+1ξ℘−1ζn+1−ξα−1ℱ5S,E,A,R,Q,ξdξ.(69) has been modified significantly, yielding the following results:(70)Sn+1ζ=S0+℘ζ℘−11−αABCαℱ1Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ1S,E,A,R,Q,ξdξ,En+1ζ=E0+℘ζ℘−11−αABCαℱ2Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ2S,E,A,R,Q,ξdξ,An+1ζ=A0+℘ζ℘−11−αABCαℱ3Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ3S,E,A,R,Q,ξdξ,Rn+1ζ=R0+℘ζ℘−11−αABCαℱ4Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ4S,E,A,R,Q,ξdξ,Qn+1ζ=Q0+℘ζ℘−11−αABCαℱ5Sn,En,An,Rn,Qn,ζn+α℘ABCαΓα∑ℓ=0n∈ζζℓζℓ+1ξ℘−1ζn+1−ξα−1ℱ5S,E,A,R,Q,ξdξ.Moreover, implementingξα−1ℱιS,E,A,R,Q,ξforι=1,2,…,5 to describe the expressions in (70), in the defined interval ζℓ,ζℓ+1, the relevant numerical strategy is constructed as(71)Sn+1ζ=S0+℘ζ℘−11−αABCαℱ1Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ1Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ1Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n+1−ℓα+1−n−ℓαn−ℓ+1+α,En+1ζ=E0+℘ζ℘−11−αABCαℱ2Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ2Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ2Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α,An+1ζ=A0+℘ζ℘−11−αABCαℱ3Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ3Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ3Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α,Rn+1ζ=R0+℘ζ℘−11−αABCαℱ4Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ4Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ4Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α,(72)Qn+1ζ=Q0+℘ζ℘−11−αABCαℱ5Sn,En,An,Rn,Qn,ζn+Δζα℘ABCαΓα+2∑ℓ=0nζℓα−1ℱ5Sℓ,Eℓ,Aℓ,Rℓ,Qℓ,ζℓ×n+1−ℓαn−ℓ+2α−n−ℓαn−ℓ+2+2α−ζℓ−1℘−1ℱ5Sℓ−1,Eℓ−1,Aℓ−1,Rℓ−1,Qℓ−1,ζℓ−1×n−ℓ+1α+1−n−ℓαn−ℓ+1+α.


## 5. Numerical results and description

In this part, we exhibit simulation results for the FF derivative operator for theSMA model (2) assuming the numerical methods proposed by [34], as mentioned previously. Such numeric findings are obtained using non-negative factors, as indicated in Table 1. We examine a FF SMA model, including the Atangana-Baleanu approach in the Caputo context, to demonstrate the reliability and usefulness of the new efficient and Newton polynomial approach. We can effortlessly acquire approximate methods for it using new efficient and Newton polynomial approach. The primary goal of this review is to identify individuals who are vulnerable and exposed to various fractional Brownian patterns as well as classical motion.The simulation results of the scheme (2) for various Brownian motions are α=1.0,0.9,0.8,0.7,0.6,0.5 are presented in Figure11(a)-1(b), Figure 2(a)-2(b) and Figure 3(a)-3(b). As seen in Figure 1(a), the susceptible group diminishes as fractional orders increase and reach 1 while increasing the fractal-dimension ℘, and finally remains consistent including all Brownian motion at S=8.1212. Furthermore, when E=0.0450 and A=0.0104, Figure 1(b) and Figure 2(a) illustrate that the unprotected and addictive groups rise dramatically and decline, respectively, exhibiting behaviour of different fractional orders eventually reaching 1. When R=0.0236 and Q=1.7517, Figure 2(b) and Figure 3 indicate that the community of individuals who are rehabilitated and consistently do not utilize and discontinue using SM significantly improves and drops, respectively, using multiple fractional-orders changes, reaching 1.Figure 1
(a) Displays of susceptible individualsSζ (b) Displays of exposed individuals Eζ using a Newton polynomial approach for decreasing fractional-order α and increasing fractal-dimension ℘.
(a)(b)Figure 2
(a) Displays of addicted individualsAζ (b) Displays of recovered individuals Rζ using a Newton polynomial approach for decreasing fractional-order α and increasing fractal-dimension ℘.
(a)(b)Figure 3
Displays of individuals who quit or not usingQζ social media using a Newton polynomial approach for decreasing fractional-order α and increasing fractal-dimension ℘.In a similar way, we can discuss the behaviour of Figure4(a)-4(b), Figure 5(a)-5(b) and Figure 6(a)-6(b), by incorporating the values of λ=0.5, the induced mortality rate of 0.05 and κ=0.06 via the FF operator in the Atangana-Baleanu sense. In this case, the graphical representation has a lower fractional-order α while keeping the fractal-dimension ℘=0.7 constant. The fundamental target of this article is that subtle improvements in the fractional derivative order have no impact on the general behaviour of the consequent structures; simply simulation studies are altered.Figure 4
(a) Displays of susceptible individualsSζ (b) Displays of exposed individuals Eζ using a Newton polynomial approach for decreasing fractional-order α and fractal-dimension ℘=0.7.
(a)(b)Figure 5
(a) Displays of addicted individualsAζ (b) Displays of recovered individuals Rζ using a Newton polynomial approach for decreasing fractional-order α and fractal-dimension ℘=0.7.
(a)(b)Figure 6
Displays of individuals who quit or not usingQζ using a Newton polynomial approach for decreasing fractional-order α and fractal-dimension ℘=0.7.Figure7(a)-7(b), Figure 8(a)-8(b) and Figure 9(a)-9(b) demonstrates that the percentage of unprotected people appears to have decreased in the first two years, but affected people revert to utilizing social platforms owing to the scheme’s ineffectiveness via FF derivative operator in the Atangana-Baleanu sense. In this case, the graphical representation has a lower fractal-dimension ℘ while retaining a constant fractional-order α=0.7. As a result, fighting SMA in the community is ineffective.Figure 7
(a) Displays of susceptible individualsSζ (b) Displays of exposed individuals Eζ using a Newton polynomial approach for decreasing fractal-dimension ℘ and fixed α=0.7.
(a)(b)Figure 8
(a) Displays of addicted individualsSζ (b) Displays of recovered individuals Eζ using a Newton polynomial approach for decreasing fractal-dimension ℘ and fixed α=0.7.
(a)(b)Figure 9
Displays of individuals who quit or not usingQζ employing the Newton polynomial approach for decreasing fractal-dimension ℘ and fixed α=0.7.As shown in Figure10(a)-10(b), Figure 11(a)-11(b) and Figure 12(a)-12(b), the proportion of affected and intoxicated people was diminished when the method was used against no approach within the FF derivative operator in the Atangana-Baleanu sense. The graphical illustration in this approach has a lower fractional-order α while maintaining a constant fractal-dimension ℘=1. The proposed technique appears to be efficacious in diminishing dependency load during the implementation, and hence can be considered an ideal contender for managing the stress of SMA.Figure 10
(a) Displays of susceptible individualsSζ (b) Displays of exposed individuals Eζ using an another numerical approach for decreasing fractional-order α and fixed fractal-dimension ℘=1.
(a)(b)Figure 11
(a) Displays of addicted individualsSζ (b) Displays of recovered individuals Eζ using a another numerical approach for decreasing fractional-order α and fixed fractal-dimension ℘=1.
(a)(b)Figure 12
Displays of individuals who quit or not usingQζ applying an another numerical approach for decreasing fractional-order α and fixed fractal-dimension ℘=1.


## 6. Conclusion

In this investigation, we presented a mathematical framework for the prevalence and distribution of theSMA model, including the fractal-fractional derivative operator in the Atangana-Baleanu sense. According to the analysis, the system’s disease-free equilibrium is locally asymptotically stable when the ℝ0<1, but generally unstable. The stability of equilibria points was explored utilizing ℝ0. Bifurcation investigation shows that the model demonstrates forward bifurcation at ℝ0=1. Furthermore, this study proposes two effective mathematical approaches for numerically solving a fractional SMA model in the fractal-fractional derivatives perspective. One initial approach relies on product integration formulation, while the other is focused on the numeric Newton polynomial approach. When the obtained simulations from the two methodologies are compared, it is clear that their respective behaviour in fixing the challenges is remarkably analogous. The findings demonstrate that our generated results are in close harmony with the precise outcomes. In a simulation study, the responses from the two different techniques exhibit the same behaviour for fractional order and fractal-dimension. Such mathematical approaches can also be leveraged to generate analytical results for other complex scientific systems of any complexity. The numerical behaviours produced by the aforesaid approaches are perfectly compatible with the model’s projected rational behaviour. [30–34].



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