ORIGINAL RESEARCH article

Front. Appl. Math. Stat., 09 August 2023
Sec. Dynamical Systems
Volume 9 - 2023 | https://doi.org/10.3389/fams.2023.1210404

Modeling and analysis of the addiction of social media through fractional calculus

Meshal Shutaywi1 Ziad Ur Rehman2 Zahir Shah3 Narcisa Vrinceanu4* Rashid Jan5 Wejdan Deebani1 Oana Dumitrascu4
• 1Department of Mathematics, College of Science & Arts, King Abdulaziz University, Rabigh, Saudi Arabia
• 2Department of Mathematics, University of Swabi, Swabi, Pakistan
• 3Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat, Pakistan
• 4Department of Industrial Machines and Equipments, Faculty of Engineering, “Lucian Blaga” University of Sibiu, Sibiu, Romania
• 5Department of Civil Engineering, College of Engineering, Institute of Energy Infrastructure, Universiti Tenaga Nasional, Kajang, Selangor, Malaysia

Social media addiction (SMA) is the excessive use of social media platforms, resulting in negative consequences for individuals. It is characterized by an uncontrollable urge to use social media, leading to negative effects in human's life. This study aims to construct a mathematical model to conceptualize the transmission dynamics of SMA and explore the underlying mechanisms of this harmful addiction in the framework of fractional derivative. The fundamentals of fractional calculus are listed for examining the model. Equilibrium points are identified, and the reproduction parameter R0 is computed to understand the dynamics of SMA spread. Stability analysis of the equilibria is performed, and the impact of various input parameters is numerically investigated. The existence and uniqueness of the proposed SMA model are demonstrated through simulations, which also study the intricate dynamics with respect to different input factors. To develop effective control strategies, the system's dynamical behavior is examined, and the influence of fractional derivative order on fluctuations is explored. This research offers a range of suggestions aimed at reducing the occurrence of social media addiction.

1. Introduction

Fractional calculus offers a valuable mathematical framework for modeling complex phenomena, analyzing intricate systems, and providing more accurate descriptions of various natural and engineered systems [1719]. It enables a deeper understanding and improved control of processes with fractional-order dynamics that go beyond the capabilities of traditional integer-order calculus. In literature [20, 21], novel fractional operators were developed that were successfully applied to the simulation of numerous real-world scenarios. Ordinary differential systems are widely recognized as suitable for local systems that are not influenced by external factors. However, conventional differential operators are unable to handle the complexity and crossover behavior of natural phenomena. Fractional calculus have numerous applications in control theory, finance, image processing, electrochemistry, and many other branches of science and engineering. Its ability to describe and analyze systems with non-local, non-linear, and memory-dependent behavior makes it a valuable tool in many scientific, engineering, and mathematical fields. In this study, we will utilize the concept of full memory to interrogate and understand the intricate phenomena of social media addiction. We opt to find out the most sensitive factors of the system and suggest to the concerned officials.

We have decided to propose a mathematical model for SMA using Caputo-Fabrizio of order ϖ∈(0, 1] as a consequence of the study mentioned above. The remaining article is structured as follows: Section 2 includes the fundamental results and claims of the fractional-calculus. In Section 3, we construct a fractional Caputo-Fabrizio (CF) derivative-based model of social media addiction (SMA). We examined the equilibriums and ascertained the reproduction number for the recommended system in Section 4. Existence and uniqueness of the solution of our system are examined with the use of fixed-point theory in Section 5. We also established a numerical framework in Section 5 to examine how various input factors affect the system's dynamics. The study's conclusion is provided in Section 6.

2. Fractional concepts

In this study, the rudimentary results of Caputo-Fabrizio (CF) fractional derivative will be presented for the analysis of the recommended model. The basic concepts are given below.

Definition 2.1. Let a function $\mathcal{Y}\left(t\right)\in {\mathcal{H}}^{1}\left({c}_{1},{c}_{2}\right)$ then its CF fractional derivative [21] is given as

where c2 > c1 and 𝔘(τ) represents the normality [21] such that 0 ≤ ϖ ≥ 1. If $j\notin {\mathcal{H}}^{1}\left({c}_{1},{c}_{2}\right)$, then we can write

Remark 2.1. If $\phi =\frac{1-\varpi }{\varpi }\in \left[0,\infty \right)$ and $\varpi =\frac{1}{1+\phi }\in \left[0,1\right]$, then Equation (2) states that

Moreover, we can write

Definition 2.2. [22]. The integral in fractional framework for $\mathcal{Y}$ is stated below as

in which the order of fractional derivative is represented by ϖ with the restriction that 0 < ϖ < 1.

Remark 2.2. On the basis of Definition 2.2, we can write

which states that $𝔘\left(\varpi \right)=\frac{2}{2-\varpi }$, 0 < ϖ < 1. By utilizing (6) through the study [22], we obtained the following result:

where fractional order is ϖ with 0 < ϖ < 1.

3. Formulation of the non-integer model

In this formulation, we classified the humans population into five subclasses symbolized in the following manner; 𝔖 represents the susceptible individuals, exposed individuals are represented by 𝔈, addicted individuals are indicated by 𝔄, recovered class of individuals are denoted by ℜ while the individuals not permanently addicted with SMA are represented by 𝔔.

The susceptible class grows due to the recruitment of individuals at the rate of α. Moreover, the term ωξℜ increases the population, while the terms ϕθ𝔄𝔖 and (μ+ρ)𝔖 lowers the population where ω is the proportion of recovered individuals that were susceptible to SMA, ξ is the transmission rate from the recovered class, ϕ is the rate at which addiction is transmitted to individuals, θ is the rate at which susceptible individuals came in contact with addicted individuals, μ represents the susceptible individuals that were not addicted to social media, and ρ is the natural fatality rate. Therefore, the susceptible individuals class can be stated as

$d𝔖dt=α+ωξℜ-ϕθ𝔄𝔖-(μ+ρ)𝔖.$

The exposed class increases due to ϕθ𝔄𝔖, while the term (τ + ρ)𝔈 decreases the population, where τ is the treatment rate of addicted individuals. Thus, the exposed class is mathematically represented as

$d𝔈dt=ϕθ𝔄𝔖-(τ+ρ)𝔈.$

The term υτ𝔈 increases the addicted class population, while (ρ + ζ + ψ)ℜ lowers the population, where ζ is the progression rate from the addicted to the recovered class and ψ is the fatality rate due to SMA. Hence, the addicted class is mathematically given as

$d𝔄dt=υτ𝔈-(ρ+ζ+ψ)𝔄.$

The recovered class grows due to τ𝔈 and ζ𝔄, while it decreases due to υτ𝔈 and (ρ+ξ)ℜ. Thus, the recovered class is represented by

$dℜdt=(1-υ)τ𝔈+ζ𝔄-(ρ+ξ)ℜ.$

The class of not permanently addicted individuals increases by μ𝔖 and ξℜ, while it decreases due to ωξℜ and ρθ. Therefore, the class of individuals permanently not addicted with social media is stated as

$d𝔔dt=μ𝔖+(1-ω)ξℜ-ρ𝔔.$

Thus, the above-stated assumptions lead us to derive an ODE model for SMA, which is stated below as

having state-variables

The total population is stated by

$𝔑(t)=𝔖(t)+𝔈(t)+𝔄(t)+ℜ(t)+𝔔(t).$

In particular, the realistic problem can benefit greatly from the use of fractional calculus. It has been demonstrated that fractional frameworks can more precisely depict the dynamics of real-world problems than traditional integer-order derivatives. The fractional framework equivalent to the aforementioned SMA model is stated below:

where CF fractional operator order is ϖ. The parameters of the model are described in Table 1 in detail. The following result provides the SMA fractional system positive invariant region stated as follows.

TABLE 1

Table 1. Detail descriptions of the parameters of the model used in the study.

Theorem 3.1. The set Γ will be a positive invariant for system (9) of SMA if $\Gamma =\left\{\left(𝔖,𝔈,𝔄,\Re ,𝔔\right)\in {\mathcal{R}}_{+}^{5}:0<𝔑\left(t\right)\le \frac{\alpha }{\rho }\right\}$.

Proof Let us consider any solution of the system (9) for the set (𝔖, 𝔈, 𝔄, ℜ, 𝔔). Since we know that

$𝔑=𝔖+𝔈+𝔄+ℜ+𝔔.$

Then adding the equations of system (9), we get that

If no death occurs due to SMA, then Equation (10) can be rewritten as

Then, $𝔑\left(t\right)\le 𝔑\left(0\right){exp}^{-\rho t}+\frac{\alpha }{\rho }\left(1-{exp}^{-\rho t}\right)$ is the solution of Equation (11), as t → ∞, then $𝔑\left(t\right)\to \frac{\alpha }{\rho }.$ Therefore, the set $\Gamma =\left\{\left(𝔖,𝔈,𝔄,\Re ,𝔔\right)\in {\mathcal{R}}_{+}^{5}:0<𝔑\left(t\right)\le \frac{\alpha }{\rho }\right\}$ is a positive invariant of system (9).

4. Analysis of fractional dynamics

Now, we will examine the steady-state and stability of the recommended fractional model (9) of SMA. First, we investigate the system's addiction-free equilibrium, represented by 𝔈0. Consider the fractional model of SMA described above in steady states without addiction. Thus, we get the SMA addiction-free equilibrium as follows:

$𝔈0=(α(ρ+μ),0,0,0,αμρ(ρ+μ)).$

For a reproduction parameter, we proceed as follows:

Evaluating Jacobian of matrix $\mathcal{F}$ and $\mathcal{V}$ at 𝔈0, we get the following:

Hence, we get that $\Xi \left({ϜV}^{-1}\right)=\frac{\varphi \alpha \upsilon \tau \theta }{\left(\mu +\rho \right)\left(\tau +\rho \right)\left(\rho +\zeta +\psi \right)}$ , which gives the reproduction number as

$R0=ϕαυτθ(μ+ρ)(τ+ρ)(ρ+ζ+ψ).$

Theorem 4.1. If R0 < 1, then the system (9) addiction-free equilibrium is locally asymptotically stable and unstable otherwise.

Proof To get the required results, first we will evaluate the Jacobian matrix of the system (9) at 𝔈0 as follows:

The eigenvalues of the matrix ${\mathcal{J}}_{{𝔈}_{0}}$ are

$-ρ,-(μ+ρ),-(ξ+ρ),$

and the remaining two eigenvalues will be obtained by solving the below stated quadratic equation:

$λ2+P1λ+P2=0,$

where ${\mathcal{P}}_{1}=\zeta +\psi +\tau +2\rho$ and ${\mathcal{P}}_{2}=\left(\tau +\rho \right)\left(\zeta +\psi +\rho \right)-\frac{\varphi \alpha \upsilon \tau \theta }{\mu +\rho }.$

To check that the roots of the quadratic equation are negative, we utilize the Routh-Hurwitz criteria which state that the Equation (14) has negative real roots if . Obviously, ${\mathcal{P}}_{1}>0$, we may write ${\mathcal{P}}_{2}$ in the following form:

$P2=(τ+ρ)(ζ+ψ+ρ)(1-ϕαυτθ(μ+ρ)(τ+ρ)(ζ+ψ+ρ)),P2=(τ+ρ)(ζ+ψ+ρ)(1-𝔈0),$

which shows that ${\mathcal{P}}_{2}>0$ if 𝔈0 < 1. Therefore, we proved that if 𝔈0 < 1, then the SMA addiction-free equilibrium is locally asymptotically stable.

The equilibrium point with addiction can be obtained by supposing Equation (9) equal to zero as follows:

and on further simplification, we get that

$𝔖=\frac{\left(\rho +\tau \right)\left(\rho +\zeta +\psi \right)}{\upsilon \varphi \tau \theta },$

$𝔈=\frac{\left(\zeta +\psi +\rho \right){\aleph }_{1}}{\varphi \tau \upsilon \theta }$,

$𝔄=\frac{{\aleph }_{1}}{\varphi \theta {\aleph }_{2}},$

$\Re =\frac{\left(\upsilon \rho +\upsilon \psi -\zeta -\rho -\psi \right)\left[\alpha \upsilon \varphi \tau \theta -\left(\rho +\tau \right)\left(\mu +\rho \right)\left(\rho +\zeta +\psi \right)\right]}{\upsilon \varphi \theta {\aleph }_{2}},$

$𝔔=\frac{\mu 𝔖+\left(1-\omega \right)\xi \Re }{\rho }.$

where

${\aleph }_{1}=\alpha \upsilon \tau \varphi \theta \left(\xi +\rho \right)-\left(\mu +\rho \right)\left(\tau \zeta \xi +\tau \zeta \rho +\zeta \xi \rho \right)-\left(\psi +\rho \right)\left(\mu +\rho \right)\left(\rho +\tau \right)\left(\rho +\xi \right)-\zeta {\rho }^{2}\left(\rho +\mu \right),$

2 = υτωξ(ψ+ρ)−τωξ(ζ+ξ+ρ) + (ζ+ψ+ρ)(τ+ρ)(ξ+ρ).

Theorem 4.2. If R0 > 1, then the fractional system (9) has locally asymptotically stable endemic steady-state.

To analyze the local asymptotic stability of the endemic equilibrium in a dynamical system, we typically perform linear stability analysis. This involves linearizing the system's equations around the endemic equilibrium point and examining the eigenvalues of the resulting Jacobian matrix. The local asymptotic stability is determined by the signs of the real parts of the eigenvalues. If all eigenvalues have negative real parts, then the endemic equilibrium is locally asymptotically stable. This means that small perturbations around the equilibrium point will decay over time, and the system will ultimately return to the endemic equilibrium after experiencing minor disturbances.

5. Analysis of the solutions

In this study, we concentrate on the examination of the recommended SMA fractional system solutions. The presence of a solution of system (9) will be investigated using the fixed-point theory. We move forward in the following manner:

Utilizing the similar approach listed in [22], we get the following result

Moreover, we can write

Theorem 5.1. If the condition $0\le \varphi \theta \mathcal{A}+\mu +\rho <1$ holds then the kernels 𝔏1, 𝔏2, 𝔏3, 𝔏4, and 𝔏5 satisfies the condition of Lipschitz and contraction.

Proof 5.1 To obtain the required result, we proceed in the following way:

Taking norm on Equation (19) and further evaluating, we get

Let ${Ϝ}_{1}=\left(\varphi \theta \mathcal{A}+\mu +\rho \right)$, where $||𝔄||\le \mathcal{A}$ due to boundedness, we achieve the following result

Hence, the Lipschitz condition is obtained for 𝔏1, from the condition $0\le \left(\varphi \theta \mathcal{A}+\mu +\rho \right)<1$, we can also get the contraction. Similarly, the Lipschitz condition for 𝔏2, 𝔏3, 𝔏4, and 𝔏5 can be derived as

After simplifying Equation (17), we get

moreover, we obtain that

initial values are given as

$𝔖0(t)=𝔖(0),𝔈(t)=𝔈(0),𝔄0(t)=𝔄(0),ℜ0(t)=ℜ(0),𝔔0(t)=𝔔(0).$

We get the difference terms in the following manner:

Observing that

Similarly, we obtain

Equation (27) states that

which gives the following equation

In similar manner, we get the following results

Theorem 5.2. If we can find t0 such that it satisfies the following condition

$2(1-ϖ)(2-ϖ)G(ϖ)Ϝ1+2ϖ(2-ϖ)G(ϖ)Ϝ1t0<1,$

then, the proposed fractional system (9) have an exact coupled-solutions.

Proof 5.2 Since the Lipschitz condition holds and 𝔖(t), 𝔈(t), 𝔄(t), ℜ(t), and 𝔔(t) are bounded. Then, Equations (30) and (31) implies

Hence, the continuity and existence of the solution are proved. In order to prove that the above mentioned is a solution of system (9), we take the following steps:

Now consider

Moreover, we have

At time t0, we have

Proceeding in the same manner and utilizing (36), we obtain

Moving in the same manner as stated above, we get that $\mathcal{Z}{2}_{q}\left(t\right),\mathcal{Z}{3}_{q}\left(t\right),\mathcal{Z}{4}_{q}\left(t\right),\mathcal{Z}{5}_{q}\left(t\right),$ and $\mathcal{Z}{6}_{q}\left(t\right)$ approaches to 0 as q approaches ∞.

Theorem 5.3. If the below stated condition holds

then the solution of system (9) will be unique.

Proof 5.3 To prove the required result, let the above (37) holds true and (𝔖1(t), 𝔈1(t), 𝔄1(t), ℜ1(t), 𝔔1(t)) is any other solution of system (9), then

Taking norm on (38), we obtain

Hence, by using Lipschitz condition, we get the following result:

This shows that

Moreover, we have

$(1−2(1−ϖ)(2−ϖ)G(ϖ)Ϝ1−2ϖ(2−ϖ)G(ϖ)Ϝ1t)>0,$

and the above inequality 41 implies that

$||𝔖(t)-𝔖1(t)||=0,$

which gives

Similarly, we get the following results

Hence, the fractional system (9) of social media addiction has a unique solution.

In the upcoming subsection, we will present a numerical scheme to visualize the solution pathways of the proposed model. We will perform different numerical simulations to show the variation in the system with different values of input factors.

5.1. Numerical scheme for the system

In this study, our objective is to evaluate the significance of input factors on the output of the proposed fractional system through various simulations. The primary goal is to gain insights into the critical parameters driving the recommended dynamics. First, we provide the numerical solution for the Caputo-Fabrizio model (6) using the scheme introduced in reference [23]. The numerical scheme consists of the following steps:

For t = tn+1, n = 0, 1, 2, ..., we have

The expression for the difference between successive terms is

The function ${\mathcal{G}}_{1}\left(t,𝔖\right)$ is approximated in the time interval [tm, t(m+1)] through the interpolation polynomial as follows:

in which h = tmtm−1. Using the above approximation, the integral in (45) can be calculated as

Thus, we obtained the following:

Similarly, we can extend the approach to calculate for other compartments within the system. The numerical scheme described above is employed to obtain the numerical results. Additionally, we make assumptions for the initial values of state-variables and input parameters to facilitate the computations. The outcome of the most sensitive scenario will be proposed to relevant authorities as a preventive measure against social media addiction.

In the first scenario presented in Figures 13, we illustrated the time series of the exposed, addicted, and recovered class with different values of fractional parameter. We assumed the value of the fractional order to be 1.0, 0.9, and 0.8 in the first simulation shown in Figure 1, 0.6, 0.7, and 0.8 in the second simulation represented in Figure 2, and 0.4, 0.5, and 0.6 in the third simulation demonstrated in Figure 3. It has been observed that the fractional order significantly contribute to the proposed dynamics and can lower exposed and addicted individuals in the society by lowering the fractional parameter. In these figures, the arrows indicate that decreasing the fractional parameter leads to a reduction in the number of exposed and addicted individuals within the society. Second, we checked the influence of the contact rate of addicted to the susceptible individuals presented in Figure 4. In this simulation, the contact rate ϕ is taken to be 0.255, 0.355, and 0.455. Finally, we presented the influence of the input parameters ρ and ξ in Figures 5, 6, respectively. The value of ρ is assumed to be 0.221, 0.321, and 0.421, while the value of ξ is considered to be 0.046, 0.056, and 0.066. The role of these parameters has been visualized on the time series of exposed, addicted, and recovered individuals of the system. Our analysis predicts that how the input factors influence the output of the system of social media addiction.

FIGURE 1

Figure 1. Illustration of the solution pathways of (A) exposed individuals, (B) addicted individuals, and (C) recovered individuals of the recommended system with fractional order 1.0, 0.9, 0.8, where the arrow shows a decrease in the addiction level with the decrease in fractional parameter.

FIGURE 2

Figure 2. Illustration of the solution pathways of (A) exposed individuals, (B) addicted individuals, and (C) recovered individuals of the recommended system with fractional order 0.8, 0.7, 0.6 where the arrow shows a decrease in the addiction level with the decrease in fractional parameter.

FIGURE 3

Figure 3. Illustration of the solution pathways (A) exposed individuals, (B) addicted individuals, and (C) recovered individuals of the recommended system with fractional order 0.6, 0.5, and 0.4 where the arrow shows a decrease in the addiction level with the decrease in fractional parameter.

FIGURE 4

Figure 4. Graphical view analysis of the solution pathways of (A) exposed individuals, (B) addicted individuals, and (C) recovered individuals of the recommended system with contact rate 0.255, 0.355, and 0.455.

FIGURE 5

Figure 5. Graphical view analysis of the solution pathways of (A) exposed individuals, (B) addicted individuals, and (C) recovered individuals of the recommended system with input parameter ρ = 0.221, 0.321, 0.421.

FIGURE 6

Figure 6. Dynamical behavior of (A) exposed individuals, (B) addicted individuals, and (C) recovered individuals of the recommended system with input parameter ξ = 0.046, 0.056, 0.066.

It is essential to recognize that while social media can offer many benefits, excessive and uncontrolled use can lead to addiction and negative effects. Individuals should be aware of their social media habits and take necessary steps to maintain a healthy balance between their online and offline lives. If someone suspects they or someone they know is struggling with social media addiction, seeking support from mental health professionals or support groups can be beneficial.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This project was financed by Lucian Blaga University of Sibiu through the research grant LBUS-IRG-2023.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: fractional calculus, social media addiction, mathematical model, Caputo-Fabrizio operator, numerical scheme, dynamical behavior

Citation: Shutaywi M, Rehman ZU, Shah Z, Vrinceanu N, Jan R, Deebani W and Dumitrascu O (2023) Modeling and analysis of the addiction of social media through fractional calculus. Front. Appl. Math. Stat. 9:1210404. doi: 10.3389/fams.2023.1210404

Received: 22 April 2023; Accepted: 24 July 2023;
Published: 09 August 2023.

Edited by:

Christos Volos, Aristotle University of Thessaloniki, Greece

Reviewed by:

Omar Abu Arqub, Al-Balqa Applied University, Jordan
Guoyong Yuan, Hebei Normal University, China
Yilun Shang, Northumbria University, United Kingdom

Copyright © 2023 Shutaywi, Rehman, Shah, Vrinceanu, Jan, Deebani and Dumitrascu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Narcisa Vrinceanu, vrinceanu.narcisai@ulbsibiu.ro