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 [1–3]. Social media, when utilized properly, can help people develop essential abilities such as information access [4], self-directed learning [5], problem-solving [6], and business, but those who misuse it have a detrimental impact on them [7, 8]. The most important negative impact is social media addiction [8]. In the beginning, social media addiction appears to be safe, similar to addiction to gambling, alcohol, and narcotics, but it is a serious trend that has to be addressed [9]. The individuals who are spending much time on social media such as YouTube, Facebook, Twitter, Instagram, and other such platforms are called social media addicted individuals [6]. Despite the myriad benefits social media offers, it is crucial to exercise responsible usage and remain aware of potential downsides, including privacy issues, information overload, cyberbullying, and the risk of addiction. Signs and symptoms of social media addiction may include compulsive checking, neglecting responsibilities, tolerance, neglecting relationship, disturbance in sleeping, and impact on mental health.

Numerous methods have been presented in academic literature to comprehend various real-world situations [10–12]. Social media addiction has emerged as a prominent and pressing problem, garnering significant attention and concern within the academic and scientific community [13]. As a result of this, substantial research on this topic has been done in multiple nations [8, 13] and attracted many researchers. Affinity-based information diffusion in social networks can be analyzed using tools from Lie algebra, particularly through the concept of graph representations. In [14], the authors proposed a dynamical information diffusion model that incorporates the concept of affinity among individuals toward the information being disseminated in social networks. The activities associated with addiction can cause serious issues at home, work, and school which can have a negative impact on society as a whole [3, 4]. Before it has any more negative impacts than what we can now see, this issue needs to be addressed [9, 13]. The easiest approach to do this is to think about how social media addiction will affect your life. Advertising and informational campaigns regarding the harmful consequences of social media are seen as a form of control. The alternative strategy involves employing therapeutic techniques such as disabling notifications [6], setting time limits for social media use [7], deleting apps [15], disconnecting from technology [16], never using a smartphone in bed, and others. It has been acknowledged that there is a high negative impact of social media addiction; moreover, protective policies are required to reduce and manage its influence. Hence, we propose a mathematical model to comprehend the dynamics of social media dissemination, aiming to conceptualize the comprehensive phenomena and forecast potential control strategies that hold the capacity to ameliorate the adverse repercussions arising from social media addiction and foster more conducive usage patterns.

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 [17–19]. 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.

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 Y(t)∈H1(c1,c2) 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∉H1(c1,c2), then we can write

Remark 2.1. If φ=1-ϖϖ∈[0,∞) and ϖ=11+φ∈[0,1], then Equation (2) states that

Moreover, we can write

Definition 2.2. [22]. The integral in fractional framework for 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(ϖ)=22-ϖ, 0 < ϖ < 1. By utilizing (6) through the study [22], we obtained the following result:

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 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

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

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

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

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

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 Table 1 in detail. The following result provides the SMA fractional system positive invariant region stated as follows.

Theorem 3.1. The set Γ will be a positive invariant for system (9) of SMA if Γ={(𝔖,𝔈,𝔄,ℜ,𝔔)∈R+5:0<𝔑(t)≤αρ}.

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

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(t)≤𝔑(0)exp-ρt+αρ(1-exp-ρt) is the solution of Equation (11), as t → ∞, then 𝔑(t)→αρ. Therefore, the set Γ={(𝔖,𝔈,𝔄,ℜ,𝔔)∈R+5:0<𝔑(t)≤αρ} is a positive invariant of system (9).

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:

For a reproduction parameter, we proceed as follows:

Evaluating Jacobian of matrix F and V at 𝔈₀, we get the following:

Hence, we get that Ξ(ϜV-1)=ϕαυτθ(μ+ρ)(τ+ρ)(ρ+ζ+ψ) , which gives the reproduction number as

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 J𝔈0 are

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

where P1=ζ+ψ+τ+2ρ and P2=(τ+ρ)(ζ+ψ+ρ)-ϕαυτθμ+ρ.

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 P1>0,P2>0,  and  P1P2>0. Obviously, P1>0, we may write P2 in the following form:

which shows that P2>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=(ρ+τ)(ρ+ζ+ψ)υϕτθ,

𝔈=(ζ+ψ+ρ)ℵ1ϕτυθ,

𝔄=ℵ1ϕθℵ2,

ℜ=(υρ+υψ-ζ-ρ-ψ)[αυϕτθ-(ρ+τ)(μ+ρ)(ρ+ζ+ψ)]υϕθℵ2,

𝔔=μ𝔖+(1-ω)ξℜρ.

where

ℵ1=αυτϕθ(ξ+ρ)-(μ+ρ)(τζξ+τζρ+ζξρ)-(ψ+ρ)(μ+ρ)(ρ+τ)(ρ+ξ)-ζρ2(ρ+μ),

ℵ₂ = 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.

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≤ϕθA+μ+ρ<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 Ϝ1=(ϕθA+μ+ρ), where ||𝔄||≤A due to boundedness, we achieve the following result

Hence, the Lipschitz condition is obtained for 𝔏₁, from the condition 0≤(ϕθA+μ+ρ)<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

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 t₀ such that it satisfies the following condition

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: