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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

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.

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 R_{0} 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.

Social media and fractional calculus, social media addiction, mathematical model, Caputo-Fabrizio operator, numerical scheme, dynamical behavior internet are specific platforms and websites that facilitate social interactions, content sharing, and networking among users. These platforms typically have features that allow individuals to create profiles, connect with friends, share updates, photos, videos, and engage in conversations and communities [

Numerous methods have been presented in academic literature to comprehend various real-world situations [

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 [

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.

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

where _{2} > _{1} and 𝔘(τ) represents the normality [

Remark 2.1. If

Moreover, we can write

Definition 2.2. [

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

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

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 ω

The exposed class increases due to ϕ

The term υ

The recovered class grows due to τ𝔈 and ζ𝔄, while it decreases due to υ

The class of not permanently addicted individuals increases by μ𝔖 and ξℜ, while it decreases due to ω

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

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

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

α | Susceptible individuals recruitment rate |

ρ | Natural death rate |

ϕ | The rate at which addiction is transmitted to individuals |

θ | The rate at which susceptible individuals came in contact with addicted individuals |

υ | Portion of recovered individuals transferred to addicted class |

ψ | Death occurs due to social media addiction |

τ | Treatment rate of addicted individuals |

μ | Susceptible individuals that were not using social media |

ω | Recovered individuals that were susceptible to SMA |

ξ | Transition rate from recovered class |

ζ | Progression rate from addicted to recovered class |

Theorem 3.1. The set Γ will be a positive invariant for system (9) of SMA if

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,

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 𝔈_{0}. Consider the fractional model of SMA described above in steady states without addiction. Thus, we get the SMA addiction-free equilibrium as follows:

For a reproduction parameter, we proceed as follows:

Evaluating Jacobian of matrix _{0}, we get the following:

Hence, we get that

Theorem 4.1. If _{0} < 1, then the system (9) addiction-free equilibrium is locally asymptotically stable and unstable otherwise.

_{0} as follows:

The eigenvalues of the matrix

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

where

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

which shows that _{0} < 1. Therefore, we proved that if 𝔈_{0} < 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

where

ℵ_{2} = υ

Theorem 4.2. If _{0} > 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.

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 [

Moreover, we can write

Theorem 5.1. If the condition _{1}, 𝔏_{2}, 𝔏_{3}, 𝔏_{4}, _{5} satisfies the condition of Lipschitz and contraction.

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

Let

Hence, the Lipschitz condition is obtained for 𝔏_{1}, from the condition _{2}, 𝔏_{3}, 𝔏_{4}, _{5} can be derived as

After simplifying Equation (17), we get

moreover, we obtain that

initial values are given as

We get the difference terms in the following manner:

Observing that

Similarly, we obtain

Equation (27) states that

which gives the following equation

Additionally,

In similar manner, we get the following results

Theorem 5.2. If we can find _{0} such that it satisfies the following condition

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

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 _{0}, we have

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

Moving in the same manner as stated above, we get that

Theorem 5.3. If the below stated condition holds

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

_{1}(_{1}(_{1}(_{1}(_{1}(

Taking norm on (38), we obtain

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

This shows that

Moreover, we have

and the above inequality 41 implies that

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.

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 [

For _{n+1},

The expression for the difference between successive terms is

The function _{m}, _{(m+1)}] through the interpolation polynomial as follows:

in which _{m}−_{m−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

Illustration of the solution pathways of

Illustration of the solution pathways of

Illustration of the solution pathways

Graphical view analysis of the solution pathways of

Graphical view analysis of the solution pathways of

Dynamical behavior of

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.

The addiction to social media is a significant concern due to its potential adverse effects on various aspects of individuals' lives, such as their social interactions, daily activities, and overall health. Therefore, it is imperative to investigate the influence of different factors on social media addiction. In this study, we constructed a mathematical model for social media addiction using the Caputo-Fabrizio derivative, a fractional calculus approach. To examine the dynamics of the proposed model, we utilized basic results from fractional theory. The model's equilibrium points were analyzed, and the reproduction parameter was determined, denoted as _{0}. We demonstrated that the equilibrium point representing an addiction-free state is locally asymptotically stable when _{0} < 1 and unstable under other conditions. We have shown the existence and uniqueness of the solution of the recommended model of social media addiction. For analyzing the system's time series behavior, we employed a numerical scheme. Through numerical analysis, we illustrated the dynamical behavior of the social media addiction model under various parameter settings. This allowed us to conceptualize how various input parameters can influence the dynamics of social media addiction. In particular, we observed how different factors contribute to the development and progression of social media addiction. The research outcomes provide valuable insights for public health interventions aimed at preventing social media addiction. We predicted crucial scenarios that could lead to severe addiction and recommended key factors that can be targeted to mitigate the risk of addiction. By understanding the dynamics of social media addiction through our model, policymaker professionals can devise effective strategies to promote healthy social media usage and safeguard individuals from the detrimental effects of excessive social media engagement.

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

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

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

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.

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.