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A mathematical model for the dynamic systems of $\mathbb{SMA}$ SMA involving the $\mathbb{ABC}$ ABC -fractional derivative is considered in this manuscript. We examine the basic reproduction number and analyze the stability of the equilibrium points. We prove the theoretical results of the existence and Ulam’s stability of the solutions for the pro...
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... main point of this manuscript is that tiny changes in the fractional derivative order do not affect the overall behavior of the resultant functions; only the numerical simulations are affected. In addition, the absolute errors of the numerical results of the population in five groups for all fractional orders comparing with φ = 1 in the case of β = 0.30 are shown in Table 2-Table 6 and in the case of β = 0.80 are shown in Table 7-Table 11. ...
Context 2
... main point of this manuscript is that tiny changes in the fractional derivative order do not affect the overall behavior of the resultant functions; only the numerical simulations are affected. In addition, the absolute errors of the numerical results of the population in five groups for all fractional orders comparing with φ = 1 in the case of β = 0.30 are shown in Table 2-Table 6 and in the case of β = 0.80 are shown in Table 7-Table 11. ...
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In this paper, the ABC fractional derivative is used to provide a mathematical model for the dynamic systems of substance addiction. The basic reproduction number is investigated, as well as the equilibrium points' stability. Using fixed point theory and nonlinear analytic techniques, we verify the theoretical results of solution existence and uniq...
Citations
... Te application of Liouville-Caputo fractional-order derivative (LC), Caputo-Fabrizio fractional-order derivative (CF), and others has recently been applied to mathematical models in recent times. For the fractionalorder models, we recommend the reader explore the following references and the references therein [33][34][35][36][37][38][39][40][41][42]. Additionally, a few mathematicians have used the fractionalorder derivative to address social-related problems. ...
... Fractional derivatives of SM addiction have been formulated and studied [33,66,67]. However, none of these studies have included the impacts of SM despite its tremendous infuence on individuals. ...
... In this section, we present the basic results necessary for the analysis of the problem in the work. We rely on defnitions and lemma from [33,48,70]. ...
The advent of social media (SM) platforms has transformed communications, information dissemination, and interpersonal relationships on a global scale. As SM continues to evolve and proliferate, its impact on various aspects of society has become increasingly complex and multifaceted. For this reason and over the past decades, several controversies have been held about whether SM is good or bad. However, the mathematical modeling technique inculcating SM impacts (positive and negative) has not been studied in the existing works. This article considers a mathematical model approach using the ABC-fractional derivative technique to study the dynamics of SM impacts. We provide the various definitions and the properties needed to study the model. Also, we use the fixed point theorem and a nonlinear analytic approach to demonstrate the theoretical solutions of the existence of solutions for the proposed model. Furthermore, the fundamental reproduction number is computed, and the stability of the model is achieved using the Ulam–Hyers (HU) criteria. We again perform a sensitivity study for the SM impact model and the effects of the sensitive parameters are presented in 3D and contour plots. In addition, a numerical algorithm of the predictor–corrector type of the Adams–Bashforth method for determining the approximate solution of the model is developed and the results are discussed. The effects of the most sensitive parameters on affected individuals in the model with a constant fractional order are shown and discussed. The simulation results indicate that as individuals become aware of the negative impacts of SM, the number of positively impacted individuals rises.
... Modeling of tobacco smoking using Caputo derivatve has been developed and studied by [23]. ABC fractional derivatives of social media addiction have been formulated and studied in [24,25]. The authors in [26] have used fractional order to study computer viruses pertaining. ...
... Assuming 1 > 0, we can conclude that * * ∈ is true for Eq. (25). Furthermore, we assume that ( ) ∈ is a unique solution of the suggested model (13) Therefore, taking into account the triangular inequality features and utilizing Lemma 2, we estimate, ...
... c À a ðð1 þ ðF À 1ÞzÞr À u ðz À 1ÞÞÞ B ÞwÞ: [45] λ Individuals that leave recovered class 0.4 [42] τ Natural death rate 0.05-0.25 [42,45] α The rate at which depression is brought on by media impact 0.3-0.5 Approximate β Susceptibles who avoid and/or stop using social media 0.01 Assume χ Contact rate of susceptibles with addicted population 0.25 [42] υ Depressed individuals that move to recovered individuals through the treatment 0.7 [40] B Individuals that leave exposed class 0.25 [40] ω Probability that treatment is successful 0.8 [43] ψ ...
... c À a ðð1 þ ðF À 1ÞzÞr À u ðz À 1ÞÞÞ B ÞwÞ: [45] λ Individuals that leave recovered class 0.4 [42] τ Natural death rate 0.05-0.25 [42,45] α The rate at which depression is brought on by media impact 0.3-0.5 Approximate β Susceptibles who avoid and/or stop using social media 0.01 Assume χ Contact rate of susceptibles with addicted population 0.25 [42] υ Depressed individuals that move to recovered individuals through the treatment 0.7 [40] B Individuals that leave exposed class 0.25 [40] ω Probability that treatment is successful 0.8 [43] ψ ...
We formulate a mathematical model of social media addiction and depression (SMAD) in this study. Key aspects, such as social media addiction and depression disease-free equilibrium point (SMADDFEP), social media addiction and depression endemic equilibrium point (SMADEEP), and basic reproduction number (R0), have been analyzed qualitatively. The results indicate that if R0 < 1, the SMADDFEP is locally asymptotically stable. The global asymptotic stability of the SMADDFEP has been established using the Castillo-Chavez theorem. On the other hand, if R0 > 1, the unique endemic equilibrium point (SMADEEP) is locally asymptotically stable by Lyapunov theorem, and the model exhibits a forward bifurcation at R0 = 1 according to the Center Manifold theorem. To examine the model’s sensitivity, we calculated the normalized forward sensitivity index and conducted a Partial Rank Correlation Coefficient (PRCC) analysis to describe the influence of parameters on the SMAD. The numerical results obtained using the Fourth-order Runge-Kutta (RK-4) scheme show that increasing the number of addicted individuals leads to an increase in the number of depressed individuals.
... Modeling of tobacco smoking using Caputo derivatve has been developed and studied by [23]. ABC fractional derivatives of social media addiction have been formulated and studied in [24,25]. The authors in [26] have used fractional order to study computer viruses pertaining. ...
... Assuming 1 > 0, we can conclude that * * ∈ is true for Eq. (25). Furthermore, we assume that ( ) ∈ is a unique solution of the suggested model (13) Therefore, taking into account the triangular inequality features and utilizing Lemma 2, we estimate, ...
... With the continuous improvement and development of fractional calculus, detailed results on fractional-order social media addiction modeling can be found in [25][26][27][28][29][30]. For example, Maayah et al. used the Caputo differential operator to study the Hilbert approximate solutions of the SMA model and the geometric behavior of fractions [25]. ...
... For example, Maayah et al. used the Caputo differential operator to study the Hilbert approximate solutions of the SMA model and the geometric behavior of fractions [25]. Kongson et al. modeled SMA using the Atangana-Baleanu-Caputo type derivative and proposed a numerical simulation technique to solve the model [26]. Rashid et al. modeled SMA using fractal-fractional order derivatives. ...
... , where R 01 represents the transmission from the addicted population to the susceptible population; R 02 represents the transmission from the professional population to the susceptible population. Therefore, our introduction of P(t) will make R 0 further increase compared to [26][27][28][29], which is also more realistic. ...
With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction.
... Furthermore, fractional-order derivatives outperform integer-order derivatives in many applications of sciences where memory plays an important role [20][21][22][23][24][25]. As a result, in 2021, Kongson et al. proposed a nonlinear fractional social media addiction model utilizing Atangana Baleanu Caputo derivative [26]. Most of the earlier systems did not include the impact of social media addiction on academics as well as memory/past history. ...
Due to the COVID pandemic and lockdown, usage of social platforms increased for academic and non-academic purposes. As a result, students are at significant risk of developing social media addiction, so techniques to control social media addiction throughout society are required. There are several positive and negative ways in which social media affects the academic performance of a student. Most of the mathematical models exclude the past of an individual, which is critical for controlling social media consumption. Hence, this study offers a fractional-order mathematical model to analyze the impact of social media on academics. There are two equilibrium points for the proposed model: social web-free and endemic equilibrium. Based on an evaluation of the threshold value, the social web-free equilibrium point is globally asymptotically stable whenever the threshold value is less than one. Endemic equilibrium points exist when the threshold value is greater than 1. Additionally, numerical simulations have been performed to examine changes in population dynamics and validate analytical outcomes. In summary, the findings of this research reveal that social media addiction decreases as the order of the derivative decreases, demonstrating the high efficiency of a fractional-order model over an integer-order model.
... Seminars are essential to educate graduate practitioners at institutes on IA and its effects on sleep (Mahmoud et al. 2022). Kongson et al. (2021) have presented research that takes into account a statistical method of the ABC-fractional differentiation for the dynamical system of SMA. The fundamental ratio is looked at in the suggested analysis, along with the sustainability of the equilibrium position. ...
... Addictive use, productivity, creativity, unhealthy relationships, life satisfaction, prevention, self-regulation, and psychological dependency (Sun and Zhang 2021) Internet addiction, psychological function, quality index, demographic information, academic performance, and dysfunctional pattern (Mahmoud et al. 2022) Mathematical model, dynamic, stability, analytic technique, transmission rate, uniqueness, and creativity (Kongson et al. 2021) Social media usage, Internet addiction, academic learning, addictive behavior, maintaining relationships, product inquiry, entertainment, and compulsive (Moghavvemi et al. 2017) Standardized, general information, human factors, social networking, time wastage, public awareness, and file sharing (Host'ovecký and Ivanová 2016) Topical area, rigorous analysis, innovation opportunities, atmospheric measurement, and literature analysis (Sheibani 2015) Performance impact, feasibility study, online training, data processing, and the mediating role (Saidin et al. 2015) Usefulness, students' perception, interactive learning, academic performance, validation, knowledge sharing, creative, and researchoriented (Ansari and Khan 2020) ...
People believe that social networks and the Internet are the two most widely used media platforms worldwide. The world's use of the Internet has substantially expanded, and it is anticipated that this trend will continue as it becomes an indispensable part of daily life. The Internet and social media are frequently used for interacting, playing video games, reading and creating posts, exchanging files, and posting images. Due to their rapid growth, teenagers are drawn to the Internet and social media. As a result, they use the Internet and social media excessively and develop negative views about them. This is called "Internet addiction" (IA) and "social media addiction" (SMA). People with IA and SMA illnesses may use social media excessively, which might interfere with daily chores and make functioning difficult. IA and SMA are having an increasing impact on young people worldwide. The Internet offers many educational advantages, but spending too much time on social media can result in issues such as isolation, loneliness, and poor academic performance. The proposed study goals are to figure out how IA, SMA, and other factors affect people all over the world. The current study has used the Fuzzy Analytical Hierarchy Process (FAHP) and Evaluation Based on Distance from Average Solution (EDAS), which are both decision-making tools, to figure out how IA and SMA affect things. The results of the study show that the SMA with the highest score gets the first spot and is called the "best option," while the SMA with the lowest score gets the last spot and is called the "worst option." The proposed systems might help us accurately weigh the different criteria and make a good choice when there are many options. Also, the solutions that were put in place are easy to use and might help us solve problems that are not clear or certain. According to the research, IA and SMA have a big effect on people's daily lives, and the main goal of the suggested evaluation, which was to find the best solution, has been met. In tough situations where making a choice is hard, the offered assessments may be a big help.
... Many important processes and phenomena in real-world situations can be mathematically modeled by autonomous dynamical systems described by differential equations associated with the classical and fractional derivative operators [1][2][3][4][5][6][7][8]. While differential equation models with the classical derivatives have been formed and studied for a long time [1,3,5,6,8], mathematical models based on fractional differential equations have been strongly developed in recent years (see, for example, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]). The stability analysis of differential equation models has been a central and prominent problem with many useful applications. ...
In this work, we introduce a simple method for investigating the asymptotic stability of discrete dynamical systems, which can be considered as an extension of the classical Lyapunov's indirect method. This method is constructed based on the classical Lyapunov's indirect method and the idea proposed by Ghaffari and Lasemi in a recent work. The new method can be applicable even when equilibia of dynamical systems are non-hyperbolic. Hence, in many cases, the classical Lyapunov's indirect method fails but the new one can be used simply. In addition, by combining the new stability method with the Mickens' methodology, we formulate some nonstandard finite difference (NSFD) methods which are able to preserve the asymptotic stability of some classes of differential equation models even when they have non-hyperbolic equilibrium points. As an important consequence, some well-known results on stability-preserving NSFD schemes for autonomous dynamical systems are improved and extended. Finally, a set of numerical examples are performed to illustrate and support the theoretical findings.
... Fractional calculus (FC) is discussed as the fractional integral operator (FIO) and fractional derivative operator (FDO), which have a long and illustrious history. FC is popularly used to analyze phenomena in the branch of mathematical analysis, which is noticed to be of outstanding assistance in modifying complex real-world problems in many fields, such as physical sciences [1], financial economics [2], dynamics of particles, fields and media [3], bio-engineering [4], Zika [5], HIV [6], COVID-19 [7], ecology [8], continuum mechanics [9], Navier-Stokes problem [10], social media addiction [11], and references cited therein. For more theoretical details on this topic, see: [12][13][14][15][16]. ...
... The remaining sections of this work are structured as follows: in Section 2, some concepts of the (ρ, φ)-Hilfer fractional operators related to our discussion are defined along with some essential lemmas are proved. Additionally, the solution of the linear variant of the (ρ, φ)-Hilfer fractional Cauchy problem (11) is derived in the form of the generalized Mittag-Leffler kernel. After that, an equivalent integral equation to the impulsive (ρ k , φ k )-Hilfer FIDE-NMP-FIBCs (4). ...
... . . , n, and λ < 0. By applying the Picard's successive approximation technique, we derive to construct an explicit solution to the problem (11) in form of the Mittag-Leffler kernel. ...
In this paper, we establish the existence and stability results for the (ρk,φk)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρk , φk )-Hilfer fractional differential equation with constant coefficients in terms of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results.
... Many important processes and phenomena in real-world situations can be mathematically modeled by autonomous dynamical systems described by differential equations associated with the classical and fractional derivative operators [4,18,33,35,38,54,55,62]. While differential equation models with the classical derivatives have been formed and studied for a long time [4,33,38,54,62], mathematical models based on fractional differential equations have been strongly developed in recent years (see, for example, [5,6,7,8,17,26,31,32,37,39,41,47,49,50,51,56,57,58,59,65]). The stability analysis of differential equation models has been a central and prominent problem with many useful applications. ...
In this work, we introduce a simple method to investigate the asymptotic stability of discrete dynamical systems, which can be considered as an extension of the classical Lyapunov's indirect method. This method is constructed based on the classical Lyapunov's indirect method and the idea proposed by Ghaffari and Lasemi in a recent work. The new method can be applicable even when equilibia of dynamical systems are non-hyperbolic. Hence, in many cases, the classical Lyapunov's indirect method fails but the new one can be used simply. In addition, by combining the new stability method with the Mickens' methodology, we formulate some nonstandard finite difference (NSFD) methods which are able to preserve the asymptotic stability of some classes of differential equation models even when they have non-hyperbolic equilibrium points. As an important consequence, some well-known results on stability-preserving NSFD schemes for autonomous dynamical systems are improved and extended. Finally, a set of numerical examples are performed to illustrate and support the theoretical findings.
