Modified fractional order social media addiction modeling and sliding mode control considering a professionally operating population

1.   Introduction

With the development of technology, the forms and types of social media (SM) are colorful and complex, such as Facebook, Twitter, Instagram, etc. Social media are bringing great convenience to people's lives, and these advantages include information sharing and communication, teamwork and working from home, data sharing, sharing equipment and peripherals, services, and education ^[1,2,3]. However, it raises concerns. Numerous studies have shown that social media addiction can lead to many negative effects, including depression, worry, anxiety, mental discomfort, and academic decline ^[4,5,6]. In particular, during the COVID-19 pandemic, people were isolated in their homes and relied heavily on social media, which not only led to addiction, but also to anxiety ^[7]. This indicates an urgent need to investigate the spreading mechanisms of social media addiction and to develop effective control measures.

Mathematical modeling has achieved outstanding results in many areas, such as addiction modeling ^{[8,9,10,11,12]}, infectious disease modeling ^{[13,14,15,16,17,18,19]}, and ecosystem modeling ^[20,21,22]. Therefore, it is essential that we use mathematical modeling to study the transmission mechanisms of social media addiction (SMA). We can apply the SIR model from infectious disease dynamics to the study of SMA. The reason is that the mechanism of transmission of infectious diseases is from infected individuals to healthy individuals through contact, which is the same as the mechanism of transmission from addicted individuals to healthy individuals through contact. Recently, results have been achieved in mathematical modeling of SMA mechanisms by applying the SIR model ^[23,24]. In ^[24], Alemneh et al. developed a 5-dimensional integer-order SMA model and designed optimal controls to decay the SMA propagation.

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]. 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. Detailed answers are given on fractional order effects and numerical simulation techniques ^[27]. Momani et al. considered fractional order calculus with singular kernels and fractal-fractional operators to study SMA models ^[28]. Malik et al. explored the fractional-order SMA mathematical model, which was interpreted by applying the q-homology analysis Sumudu transformation method ^[29]. We utilize Caputo type fractional order derivatives to model the propagation of SMA. This is because the Caputo type fractional order model has the same initial conditions as the integer order model and its constant derivative is 0. In addition, the fractional order model will make the dynamical behavior of the model more complex ^[31,32].

As the market expands, social media is becoming more commercialized and standardized. This has directly led to the emergence of professionals specializing in social media operations, such as social media experts and social media marketers. During the process of social media addiction, groups professionally engaged in social media dissemination influence users' social media usage habits and frequency by spreading content through social media, which may indirectly cause healthy individuals to become dependent on social media. Their activities include creating appealing content, encouraging user interaction, and enhancing user engagement with social media through various means ^[33,34]. This is consistent with the introduced professional population in ^[12]. To study the spread of social media addiction in the real world, unlike previous literature, we for the first time introduce a type of population, namely, the professionally operated population (marketing population), into the model of social media addiction. This not only greatly enriches the realism of the model, but also makes the dynamical behavior of the model more complex.

Similar to the multiple forms of incidence in infectious diseases ^{[35,36,37,38,39,40]}, we also introduce nonlinear transmission rates in our model of social media addiction. Due to a large increase in the population with addiction or conscious protective measures by the healthy population, the valid contact rate between the addicted population and the healthy population may reach saturation at high levels of transmission. If the addiction population increases, the transmission rate decreases, which can be used to explain the "psychological" effect ^[41,42], that when there is a large number of addicted people, the transmission power may decrease. This is because in the presence of a high number of addicts, people may be inclined to reduce their contact with them, to actively and consciously restrain themselves from social media, or to adopt a strict upbringing in the family.

Designing appropriate controllers to control the spread of addiction is a crucial part of the process. Here, we design a fractional order sliding mode control that can eliminate addiction in a finite time. This approach is chosen because classical integer order sliding mode control exhibits high frequency chattering and the conventional controllers do not ensure fast convergence of the system ^[43,44]. Additionally, sliding mode control based on fractional order calculus can offer better properties for the control system, such as robustness and convergence. However, fractional order sliding mode controls have primarily been used for chaos control in fractional order systems ^{[43,45,46,47]}. Dadras et al. design sliding mode controllers for fractional order chaotic financial systems, ensuring robustness to system uncertainty and asymptotic stability under external disturbances ^[47]. Recently, researchers have begun applying fractional-order sliding mode controllers to a wider range of scenarios ^{[48,49,50,51]}. For example, Boonyaprapasorn et al. introduced fractional-order sliding mode controllers for fractional-order HBV epidemic models to control susceptible and infected populations at desired levels, reducing the occurrence of chattering ^[48]. Pourhashemi et al. proposed two new fractional sliding mode surfaces, achieving stabilization with/without uncertainty and external disturbances by designing fractional sliding mode controls. These controls are applied to fractional order nonlinear biological systems for further study ^[50]. Borah et al. studied six fractional order models of chaotic diseases, designing and analyzing suitable fractional order sliding mode controllers to control chaos in biomathematical models that require stability and anti-controllers to generate chaos where there was none ^[51].

Inspired by the above viewpoints, we innovatively proposed a 6-dimensional fractional-order designed media addiction model (FOSMA). The following are the major contributions.

1) For the first time, we incorporated groups that professionally operate social media into the SMA model, correcting the fractional-order dimension mismatch problem and examining the influence of these groups on the model's dynamic behavior. 2) To investigate bifurcation behavior, we generalized the Sotomayor theorem from integer-order models to fractional-order systems. 3) For the first time, we applied fractional-order sliding mode control to fractional-order social media addiction. We introduced a novel frequency distribution model, converting the fractional-order model into an equivalent partial differential model. Furthermore, we designed a new fractional-order sliding mode surface. It has been demonstrated that fractional-order sliding mode control converges in finite time, significantly reducing chattering and aiding in addiction elimination.

The rest of the article is summarized as follows. In Section 2, we provide the fundamentals. In Section 3, we construct the FOSMA model and discuss its positivity and boundedness of the model. In Section 4, we compute the basic regeneration number and the equilibrium point, study the stability of the equilibrium point, and analyze the generation of bifurcations. In Section 5, we transform the fractional order system into a partial differential equation model. Moreover, we design the fractional order sliding mode surface and give the fractional order sliding mode control law. The numerical simulation results are given in Section 6 to verify the correctness of our conclusions. Finally, the results of this paper are summarized.

2.   Basic knowledge

First, we introduce the basics of the article.

Definition 2.1. ^[52,53] The Caputo fractional order derivative for order $\alpha > 0$ is

where $g \in {C^n}([a, \infty), R)$, $a$ is the initial moment, $t \ge a, \; \Gamma (.)$ is the gamma function and there exists $n \in {Z^ + }$ such that $\alpha \in (n - 1, n).$

Remark 2.1. $\Gamma (\cdot)$ is the Euler gamma function, which is defined as

and the following equation holds

Definition 2.2. ^[52,53] We give integrals of the Riemann-Liouville form:

where ${{\cal L}^1}$ is the set of the Lebesgue integrable function, $\alpha > 0, \; a$ is the starting point.

Theorem 2.1. ^[52,53] If $g(t)$ is continuously derivable of order $n$ and ${}_a{\cal D}_t^\alpha g(t)$ is continuous piecewise on $[a, \infty)$ while there exists $a > 0, \; n - 1 < a < n, \; n \in N,$ then its Laplace transform is

where $G(s) = L({}_a{\cal D}_t^\alpha g(t), s).$

Theorem 2.2. ^[52,53] The two-parameter form of the Mittag-Leffler function is ${E_{\alpha, \beta }}(z) = \sum\limits_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (\alpha k + \beta)}}}; (\alpha, \beta > 0);$ the one-parameter form is ${E_{\alpha, 1}}(z) = \sum\limits_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (\alpha k + 1)}}} = {E_1}(z).$ For any $\lambda \in C$ and $m, n > 0,$ it will have the following formula

for ${\mathop{\rm Re}\nolimits} (x) > {\left\| \lambda \right\|^{\frac{1}{m}}},$ where ${\mathop{\rm Re}\nolimits} (x)$ represents the real part of the imaginary number $x.$

Theorem 2.3. ^[52,53] If $m, n > 0$ and $x \in C,$ then

Theorem 2.4. ^[52,53] The equilibrium point ${E^*}$ is locally asymptotically stable if all eigenvalues ${\lambda _i}(i = 1, 2, ...\; n)$ of the Jacobi matrix ${\cal J} = \frac{{\partial g}}{{\partial x}},$ where $g = {[{g_1}, {g_2}...{g_n}]^T},$ computed at ${E^*},$ satisfy $\left| {\arg ({\lambda _i})} \right| > \alpha \frac{\pi }{2}, \; i = 1, 2, ...\; n.$

Remark 2.2 It is mentioned in ^[52] that exponential stability cannot be applied to prove the asymptotic stability of fractional order model. Therefore, it is proposed that the state vector decays slowly with ${t^{ - \alpha }}$ to the equilibrium point, and it is this phenomenon that results in fractional systems with long memory effects, where this power law stability ${t^{ - \alpha }}$ is a particular case of Mittag-Leffler stability.

3.   FOSMA model

First, we give a modified fractional order social media addiction model. We assume that the population is homogeneously mixed, meaning that everyone has an equal chance of interacting with others. This is easy to analyze and understand, providing quick estimates of addiction trends.

If the fractional order differential operator is directly added to the left end of the integer order model as in ^[27], we find that the dimensions of the equations are inconsistent at both ends. Since the dimension of the left side of the equation is $-\alpha$ and the dimension of the right side is $-1,$ we adjust the non-negative parameter of the right side of the equation to be a $\alpha$-exponent to balance out the inconsistency in dimension. We propose the following equation:

The initial conditions of model (3.1) are

The parameters in the model (3.1) are non-negative and their actual practical significance is shown in Table 1. The flow chart of the model (3.1) is shown in Figure 1.

In the FOSMA model (3.1), $S(t)$ denotes the susceptible population, $E(t)$ represents the population that can develop addiction, $A(t)$ denotes the addicted population, $P(t)$ represents the population that operates social media professionally, $R(t)$ represents the recovered population, and $Q(t)$ stands for the population that permanently stops using social media.

Susceptible individuals $S(t)$ enter the system (3.1) at a recruitment rate of $\Pi.$ They are infected by $A(t)$ and $P(t)$ at a non-linear transmission rate of $\frac{{{\beta _1}{\sigma _1}A(t)}}{{1 + aA(t)}} + \frac{{{\beta _2}{\sigma _2}P(t)}}{{1 + bP(t)}}$ into $E(t),$ and also enter $(Q(t))$ at a rate of $\delta.$ $E(t)$ will enter $A(t),$ $(P(t)),$ and $R(t)$ at a rate of $\alpha_1,$ $\alpha_2,$ and $\alpha_3,$ respectively. $A(t)$ will enter $R(t)$ at a rate of $\lambda$ and enter $P(t)$ at a rate of $\omega.$ $P(t)$ will enter $R(t)$ at a rate of $\theta.$ $R(t)$ will transition to $Q(t)$ at a rate of $\gamma_1$ and revert to $S(t)$ at a rate of $\gamma_2.$ The natural mortality rate for each population is $\mu,$ and the addicted population will have a mortality rate of $\nu$ due to the overall decline in physical function that will result from overuse of SM.

The fundamental properties of the FOSMA model (3.1) solution are given next, which is to ensure that model (3.1) is practically meaningful. We give the existence uniqueness theorem for the solutions of the model (3.1).

Theorem 3.1. For any non-negative initial condition (3.2), there is a unique solution of the FOSMA model (3.1).

The method for proving this Theorem 3.1 is given in detail in ^[54] and is therefore omitted here. Immediately after, the positivity and boundedness of the FOSMA model (3.1) are given.

3.1.   Positivity of the solution

Theorem 3.2. The solution of FOSMA (3.1) is non-negative when the initial condition (3.2) is satisfied.

Proof. By $\Pi^\alpha$ is a non-negative constant, so the following equation holds

The Laplace transform of Eq (3.3) is obtained

By the conclusion in ^[55], the rewriting of Eq (3.4) yields

We prove that the following equation also holds by the above way

Therefore, FOSMA (3.1) has a non-negative solution when condition (3.2) is satisfied. □

3.2.   Boundedness of the solution

Theorem 3.3. When the initial condition (3.2) is in $\Omega = \{ (S, E, A, P, R, Q) \in {\cal R}_ + ^6, \; N(t) \le N\},$ where $N = \max \{ \frac{\Pi^\alpha }{\mu^\alpha }, \; N(0)\},$ then the solution of FOSMA (3.1) is also always in the set $\Omega.$

Proof. We let $(S, E, A, P, R, Q)$ be the solution of FOSAM (3.1), then there will be $N = S + E + A + P + R + Q.$ Therefore, we have

The Laplace transformation of Eq (3.6) yields

After simplification, we get

Next, using the Laplace inverse transform and the conclusion in ^[54] yields

Therefore $N(t)$ is bounded, then $S, E, A, P, R, Q$ are also bounded. □

At this point, we proved the feasibility of the model in a realistic sense. It lays the foundation for the next step of the analysis.

4.   Stability analysis of equilibrium points

First, all possible equilibria of the FOSMA model (3.1) are derived theoretically and the basic regeneration number $(R_0)$ is computed using the next generation matrix. Furthermore, the local asymptotic stability of all equilibrium points of the model (3.1) is proved. Finally, the bifurcation behavior at $R_0 = 1$ is proved by extending the Sotomayor's theorem for integer order systems to fractional order systems.

4.1.   No addiction equilibrium (NAE)

For the FOSMA model (3.1), when there is no addicted population, i.e., ($E = A = P = R = 0$). At this point, the NAE is expressed by the following equation

Similar to the infectious disease model, the $R_0$ of the FOSMA model (3.1) represents the number of individuals infected by an addicted individual in a susceptible population. This calculation uses the same next generation matrix, as detailed in ^[54,55]. ${R_0} = \rho (F{V^{ - 1}}),$ where, the infection matrice $F$ and transfer matrice $V$ are

where ${e_1} = {\alpha _1^\alpha} + {\alpha _2^\alpha} + {\alpha _3^\alpha} + \mu^\alpha, \; {e_2} = \mu^\alpha + \nu ^\alpha + \lambda^\alpha + \omega^\alpha, \; {e_3} = \mu^\alpha + \theta^\alpha, \; {e_4} = \mu^\alpha + {\gamma _1}^\alpha + {\gamma _2}^\alpha.$

After simple calculations, we obtain

Remark 4.1. We can find that $R_0$ is composed of two parts: ${R_0} = {R_{01}} + {R_{02}}, \; {R_{01}} = \frac{{{\alpha _1^\alpha}}}{{{e_1}{e_2}}}{\beta _1^\alpha}{\sigma _1^\alpha}\frac{\Pi^\alpha }{{\mu^\alpha + \delta^\alpha }}, \; {R_{02}} = \frac{{{\alpha _1^\alpha}}}{{{e_1}{e_2}}}{\beta _2^\alpha}{\sigma _2^\alpha}\frac{\Pi^\alpha }{{\mu^\alpha + \delta^\alpha }}(\frac{{{\alpha _1^\alpha}\omega^\alpha + {\alpha _2^\alpha}{e_2}}}{{{\alpha _1^\alpha}{e_3}}}),$ 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.

4.2.   Addiction equilibrium $(AE)$

Let the right end of the FOSMA model (3.1) be all zeros. After a simple calculation, we give the following theorem:

Theorem 4.1.

(i) For the FOSMA model (3.1), there always exists a no-addiction equilibrium point (NAE);

(ii) There exists a unique addiction equilibrium point (AE) for the FOSMA model (3.1) when $R_0 > 1;$

where ${E^*} = {c_1}{A^*}, \; {P^*} = {c_2}{A^*}, \; {R^*} = {c_3}{A^*}$, ${S^*} = \frac{\Pi^\alpha }{{\mu ^\alpha + \delta^\alpha }} - {c_4}{A^*}; {Q^*} = \frac{{\delta^\alpha \Pi^\alpha }}{{\mu^\alpha (\mu ^\alpha + \delta^\alpha)}} - {c_5}{A^*}$ and ${c_1} = \frac{{{e_2}}}{{{\alpha _1^\alpha}}}$, ${c_2} = \frac{{{\alpha _1^\alpha}\omega^\alpha + {\alpha _2^\alpha}{e_2}}}{{{\alpha _1^\alpha}{e_3}}}, \; {c_3} = \frac{{{\alpha _3^\alpha}{e_2}{e_3} + \lambda^\alpha {\alpha _1^\alpha}{e_3} + \theta^\alpha ({\alpha _1^\alpha}\omega^\alpha + {\alpha _2^\alpha}{e_2})}}{{{\alpha _1^\alpha}{e_3}{e_4}}}$, ${c_4} = \frac{{{e_1}{c_1} + {\gamma _2^\alpha}{c_3}}}{{\mu^\alpha + \delta^\alpha }}, \; {c_5} = \frac{{{\gamma _1^\alpha}{c_3} - \delta^\alpha {c_4}}}{\mu^\alpha }.$

Proof. Bringing addiction equilibrium point (AE) (4.2) into the first equation of the FOSMA model (3.1), a simplification leads to:

where

We easily find that $d_1 > 0,$ and when $R_0 > 1$, the relationship between the roots and coefficients of the quadratic equation shows that there exists a unique AE for the FOSMA model (3.1). □

The stability of the fractional-order nonlinear model (3.1) is difficult to analyze, so we study the nonlinear model (3.1) using the linearization technique following the analytical method of the integer-order model. This is because the linear system can be accurately approximated as a substitute for the nonlinear system, and the difference between the two is negligible. We first analyze the local stability of NAE.

4.3.   Stability analysis of NAE

The Jacobi matrix characterization matrix for model (3.1) at NAE is shown below:

It can be easily found that the characteristic equation has three negative real roots, which are $- (\mu^\alpha + \delta^\alpha), \; - \mu^\alpha, \; - {e_4}.$

Let $\chi = {s^\alpha }.$ The characteristic equation of (4.4) reduces to

where ${f_1} = {e_1} + {e_2} + {e_3}, \; {f_2} = {e_1}{e_2} + {e_1}{e_3} + {e_2}{e_3} - {\alpha _2^\alpha}{\beta _2^\alpha}{\sigma _2^\alpha}{S^0} - {\alpha _1^\alpha}{\beta _1^\alpha}{\sigma _1^\alpha}{S^0}$, ${f_3} = {e_1}{e_2}{e_3} - {S^0}({\alpha _1^\alpha}{\beta _1^\alpha}{\sigma _1^\alpha}{e_3} + {\beta _2^\alpha}{\sigma _2^\alpha}({\alpha _1^\alpha}\omega^\alpha + {\alpha _2}{e_2})) = {e_1}{e_2}{e_3}(1 - {R_0}).$

After that, we will have the following conclusion:

Theorem 4.2. When the condition ${R_0} < 1$ and ${f_1}{f_2} > {f_3}$ are satisfied, the FOSMA model (3.1) is locally asymptotically stable at the NAE.

Proof. We can find that ${f_1} > 0.$ When ${R_0} < 1,$ we will have ${f_3} > 0.$ Condition ${f_1}{f_2} > {f_3}$ is also satisfied, then by the Routh–Hurwitz criterion ^[52,53], the condition $\left| {\arg ({\chi _i})} \right| > \frac{{\alpha \pi }}{2}$ is satisfied, the FOSMA model (3.1) is locally asymptotically stable at the NAE. □

Next, similar to the approach used to analyze the stability around the NAE, we study the stability of the AE.

4.4.   Stability analysis of AE

The characterization matrix of model (3.1) Jacobi matrix at AE is shown below:

where ${g_1} = \frac{{{\beta _1^\alpha}{\sigma _1^\alpha}{A^*}}}{{1 + a{A^*}}}, \; {g_2} = \frac{{{\beta _2^\alpha}{\sigma _2^\alpha}{P^*}}}{{1 + b{P^*}}}, {g_3} = \mu^\alpha + \delta^\alpha + {g_1} + {g_2}, \; {g_4} = \frac{{{\beta _1^\alpha}{\sigma _1^\alpha}{S^*}}}{{{{(1 + a{A^*})}^2}}}, \; {g_5} = \frac{{{\beta _2^\alpha}{\sigma _2^\alpha}{S^*}}}{{{{(1 + b{P^*})}^2}}}.$

It can be easily found that the characteristic equation has one negative real root, which is $- \mu^\alpha.$ Let $\chi = {s^\alpha }.$ The characteristic equation of (4.6) reduces to

where ${k_1} = {f_1} + {g_3} + {e_4}, \; {k_2} = {h_1} + {f_1}({g_3} + {e_4}) + {g_3}{e_4}$, ${k_3} = {h_2} + {h_1}({g_3} + {e_4}) + {g_3}{e_4}{f_1} - {\alpha _3^\alpha}{\gamma _2^\alpha}({g_1} + {g_2}) + ({g_1} + {g_2})({\alpha _2^\alpha}{g_5} + {\alpha _1^\alpha}{g_4})$, ${k_4} = {h_2}({g_3} + {e_4}) + {g_3}{e_4}{h_1} - {\gamma _2^\alpha}({g_1} + {g_2})({\alpha _3^\alpha}({e_2} + {e_3}) + {\alpha _2^\alpha}\theta^\alpha$ $+ {\alpha _1^\alpha}\lambda^\alpha)+ {\alpha _1^\alpha}\omega^\alpha {g_5} + {\alpha _1^\alpha}{e_3}{g_4} + {\alpha _2^\alpha}{e_2}{g_5} + {e_4}({\alpha _2^\alpha}{g_5} + {\alpha _1^\alpha}{g_4})$, ${k_5} = {g_3}{e_4}{h_2} - ({\alpha _1^\alpha}\omega^\alpha \theta^\alpha + {\alpha _2^\alpha}\theta^\alpha {e_2} + {\alpha _2^\alpha}\lambda^\alpha {e_3})$ $+ {e_4}({\alpha _1^\alpha}\omega^\alpha {g_5} + {\alpha _1^\alpha}{e_3}{g_4} + {\alpha _2^\alpha}{e_2}{g_5}).$ and ${h_1} = {e_1}{e_2} + {e_1}{e_3} + {e_2}{e_3} - {\alpha _2^\alpha}{g_5} - {\alpha _1^\alpha}{g_4}, \; {h_2} = {e_1}{e_2}{e_3} - {\alpha _1^\alpha}\omega^\alpha {g_5} - {\alpha _1^\alpha}{e_3}{g_4} - {\alpha _2^\alpha}{e_2}{g_5}.$

We assume the following conditions

Then, we will have the following conclusion:

Theorem 4.3. When the condition ${R_0} > 1,$ $\mathbf{H1}$ and $\mathbf{H2}$ are satisfied, the FOSMA model (3.1) is locally asymptotically stable at the AE.

Proof. When the condition ${R_0} > 1,$ $\mathbf{H1}$ and $\mathbf{H2}$ are satisfied. By the Routh–Hurwitz criterion ^{[52,53,54,55]}, we can obtain $\left| {\arg ({\chi _i})} \right| > \frac{{\alpha \pi }}{2},$ the FOSMA model (3.1) is locally asymptotically stable at the AE. □

4.5.   Bifurcation analysis

From the dynamical point of view: as parameters are transformed, the state of the system may undergo abrupt changes or transitions. When a parameter reaches a certain threshold, the equilibrium point of the system changes from one to multiple (forward bifurcation). Alternatively, multiple equilibria merge into one or more new equilibria (backward bifurcation). Starting from real addiction scenarios: Addiction is also complicated by bifurcation behavior.

Classical integer-order systems have well-established theories about forward bifurcation or backward bifurcation, such as the central manifold theorem ^[56]. However, the theory of systems at fractional order is not yet perfect. Therefore, we draw on ^[27,57] to study bifurcation in fractional order systems. We have the following conclusions for the FOSMA (3.1) model.

As expressed in the paper ^[27], the addiction-free equilibrium will lose its stability when ${R_0} = 1.$ When satisfying ${f_2} \ge 0, \; f_1^2 \ge 4{f_2},$ the characteristic equation (4.5) will have two roots with negative real part and one zero root. Therefore, bifurcation behavior might occur at this point. The addicted population plays a decisive role in the propagation process, so we choose ${\beta _1}^{\alpha} = {\beta _1}{^{\alpha*}}$ as the bifurcation parameter. At this point, when ${R_0} = \frac{{{\alpha _1}^\alpha }}{{{e_1}{e_2}}}\frac{{{\Pi ^\alpha }}}{{{\mu ^\alpha } + {\delta ^\alpha }}}({\beta _1}^\alpha {\sigma _1}^\alpha + {\beta _2}^\alpha {\sigma _2}^\alpha (\frac{{{\alpha _1}^\alpha {\omega ^\alpha } + {\alpha _2}^\alpha {e_2}}}{{{\alpha _1}^\alpha {e_3}}})) = 1,$ we would have

When the eigenvalue of the eigenmatrix ${{\cal J}^*}(NAE)\left| {_{{\beta _1}^\alpha = {\beta _1}{{^\alpha }^*}}} \right.$ (4.4) is 0, we have

where ${{\cal A}_1} = {\beta _1}^\alpha {\sigma _1}^\alpha \frac{{{\Pi ^\alpha }}}{{{\mu ^\alpha } + {\delta ^\alpha }}}, \; {{\cal A}_2} = {\beta _2}^\alpha {\sigma _2}^\alpha \frac{{{\Pi ^\alpha }}}{{{\mu ^\alpha } + {\delta ^\alpha }}}, \; {{\cal A}_3} = {\mu ^\alpha } + {\delta ^\alpha }.$

Next, the left and right eigenvectors of the matrix (4.11) are computed. Let the right eigenvector be $\tau = {({\tau _1}, {\tau _2}, {\tau _3}, {\tau _4}, {\tau _5}, {\tau _6})^T}$ and the left eigenvector be $\rho = {({\rho _1}, {\rho _2}, {\rho _3}, {\rho _4}, {\rho _5}, {\rho _6})}.$ We have ${{\cal J}^*}(NAE)\tau = 0, {{\cal J}^*}^T(NAE)\rho = 0.$ where

After a simple calculation, we can obtain

where

Next, according to Sotomayor's theorem ^[58], we apply to the fractional order system to obtain:

where ${{\cal B}_1} = \alpha {\beta _1}^{\alpha - 1}{\sigma _1}^\alpha \frac{{{\Pi ^\alpha }}}{{{\mu ^\alpha } + {\delta ^\alpha }}}.$ The above calculations lead us to the following theorem.

Theorem 4.4. When ${\beta _1}^{\alpha} = {\beta _1}{^{\alpha*}},$ and ${\tau _1} \ne a{S^0}{\tau _3}, b{S^0}{\tau _4}$ hold, the model (3.1) undergoes forward (transcritical) bifurcation at ${R_0} = 1.$

Remark 4.2. We find that the introduced nonlinear propagation rate and the addicted population have an important role in changing the dynamical behavior of the FOSMA model (3.1). Also, the occurrence of forward bifurcation at ${R_0} = 1$ is consistent with the findings in ^[27]. In this paper, we give a detailed proof.

Remark 4.3. The central manifold theorem for integer order models was utilized in ^[56], and Xu et al. extended the reference to the central manifold theorem to fractional order systems in ^[57]. We generalize the integer order Sotomayor theorem in ^[58,59] to fractional order systems in our proof. This has important implications for the study of bifurcation phenomena in fractional order systems.

5.   Fractional-order sliding mode control

First, we first give partial differential models that are equivalent to fractional order systems. Then, we design the fractional order sliding mode surface of the control system. Finally, the fractional order sliding mode control law of the control system is given. To simplify the complexity of the control model, internal and external disturbance factors are omitted.

5.1.   Frequency distributed model transformation

To establish the effectiveness of the fractional order sliding mode control, the following theorem is presented first.

The fractional-order system (3.1) could be reformulated in the following form:

where $Y(t) = (S(t), E(t), A(t), P(t), Q(t), R(t)), F(t, Y(t)) = ({F_1}(t),$ ${F_2}(t), {F_3}(t), {F_4}(t), {F_5}(t), {F_6}(t)).$

We employ the fractional-order numerical approximation technique as presented in ^[60]. We make the following assumption.

where $\varphi :(0, \infty) \times [0, T] \to {R^n}.$

Theorem 5.1. Under the assumption (5.2), the system (5.1) can be rewritten as:

where $\varpi (w) = \frac{{2\sin (\pi \alpha)}}{\pi }{w^{1 - 2\alpha }}, \alpha \in (0, 1).$

Proof. The proof of the theorem is divided into two steps.

Step 1: To prove Theorem 5.1, we first rewrite Eq (5.2) as follows:

Differentiating both sides of the Eq (5.4), we obtain

Step 2: Utilizing the Gamma function and its properties, we obtain

We define the variables as follows

then $dz = 2w(t - \tau)dw.$

Substituting Eq (5.7) into Eq (5.6), we can derive

Furthermore, we can deduce

where $\varpi (w) = \frac{2}{{\Gamma (\alpha)\Gamma (1 - \alpha)}}{w^{1 - 2\alpha }} = \frac{{2\sin (\alpha \pi)}}{\pi }{w^{1 - 2\alpha }}.$

Equation (5.9) can be transformed into

For system (5.1), the Caputo type fractional order integral expression for the function $Y(t)$ is identical to the R-L type (2.3), see (^[52,53]) for details. We can obtain

At this point, the proof of Theorem 5.1 is concluded.

□

5.2.   Sliding mode control

The sliding mode controller is designed to converge the system towards $NAE.$ The error is defined as follows:

Substituting the error (5.12) into the model (3.1), we obtain:

where, $u(t) = {({u_1}, {u_2}, {u_3}, {u_4}, {u_5}, {u_6})^T} \in {R^6}$ represent the sliding mode control law, Control $u(t)$ means family, school and social education, reducing exposure to social media, and promoting the dangers of social media. $e(t) = ({e_1(t)}, {e_2}(t), {e_3}(t), {e_4}(t), {e_5}(t), {e_6}(t)) \in {R^6}$ represents the state vector of system (5.13).

Generally, the design process of sliding mode control can be divided into two steps. First, select a suitable sliding surface that can represent the required dynamic characteristics of the system. In this paper, we define a new fractional-order sliding surface based on the sliding surface proposed in ^[43,50].

where $s(t) = {({s_1}, {s_2}, {s_3}, {s_4}, {s_5}, {s_6})^T} \in {R^6}$ are the sliding surfaces, and ${k_1}, {k_2}, r$ are parameters within the sliding surface, with ${k_1} > 0, {k_2} > 0, 0 < r < 1.$

In (5.14), we choose the $tanh$ function due to the fact that it is continuous and smooth without any discontinuity points compared to the sign functions. In addition, the $tanh$ function is a nonlinear activation function that is more expressive and capable of capturing nonlinear relationships in the input data ^[61,62]. Most importantly, the tanh function plays a crucial role in reducing oscillations. Figure 2 provides a comparison of various functions. Following the reference^[62], we present the following lemma.

Lemma 5.1. There is a constant $j > 0$ for all $s_i \ne 0$ such that ${s_i}\tanh ({s_i}) > j{s_i}sign({s_i}).$

The proof of Lemma 5.1 is detailed in ^[62], which we omit here. We find that using smooth continuous functions to design the controller can significantly reduce chattering.

When the system operates on the sliding surface, according to ^[43], it satisfies the following equation

Combining (5.14) and (5.15), we obtain

Then,

According to the properties of Caputo fractional order described in ^[52,53], we have

Theorem 5.2. If the terminal sliding surface is chosen for the Eq (5.14), then the sliding mode system (5.18) is stable, and its state trajectories will converge to 0.

Proof. Using the Theorem 5.1, we can transform the system (5.18) into the following equation

We construct a positive definite Lyapunov function

Differentiating both sides of Eq (5.20), we get

According to the properties of the tanh function, when $t > 0,$ $tanh(t) > 0.$ Therefore,

By applying the Lyapunov stability theory, we can conclude that the error signal in the sliding mode dynamics system (5.18) will locally asymptotically converge to $0,$ thus completing the proof. □

After finalizing the sliding surface design, the subsequent task is to design the control input. This is to assure that the error signal reaches the sliding surface and remains on it continuously. The sliding mode control law is given as follows:

where $f(e) = ({f_1}(e), {f_2}(e), {f_3}(e), {f_4}(e), {f_5}(e), {f_6}(e)),$ ${\eta _1}, {k_3}, L, \vartheta$ are positive parameters with ${\eta _1} > 0,$ $L > 0,$ (Adaptive law coefficients) $0 < \vartheta < 1,$ ${k_3} > 0.$ Then,

Theorem 5.3. For a given error dynamics system (5.13) and sliding surface (5.16), when the system (5.13) is subjected to the control law (5.23), the error signal of the system (5.13) will converge to the sliding surface $s(t) = 0$ within a finite duration.

Proof. Constructing a Lyapunov function ${{\cal V}_2}(t) = \left| s \right|$ and differentiating it with respect to $t,$ we obtain

Differentiating Eq (5.24), we obtain

Substituting Eq (5.25) into Eq (5.24), Eq (5.24) can be rewritten as

Introducing the control law (5.23), we obtain

Based on $s \times sign(s) = \left| s \right|$ and $sign(s) \times sign(s) = 1,$ after simple calculation, we have

According to $\tanh (s) \times sign(s) \ge 0,$ we obtain

Based on the Lyapunov stability theory, the error signal in the error system (5.13) locally asymptotically converges to $s(t) = 0.$

From Eq (5.28) we have:

To substitute ${{\cal V}_2}(t) = \left| s \right|$ into Eq (5.30)

Simplify Eq (5.31)

Integrate both sides of Eq (5.32)

Calculate the result for Eq (5.33)

At this point, we have demonstrated that the sliding mode surface converges to $s(t) = 0$ within a finite duration.

□

Therefore, we can maintain the equilibrium point of the FOSMA model (3.1) around the NAE at all times by applying fractional-order sliding mode control, which serves the purpose of eliminating addiction.

5.3.   Convergence-time analysis

The convergence time of stable sliding mode control can be finite or infinite. The factors that usually affect the convergence time are: the dynamic behavior of the model, the control law, or the initial conditions. Here we briefly discuss the convergence time. Assuming the system has reached a steady state, $S = 0.$ To calculate the convergence time of the sliding surface, the sliding surface (5.16) is linearized near the equilibrium point $(e = 0).$

To determine the eigenvalues of system (5.35), we apply the Laplace transform to both sides of Eq (5.35), yielding:

where $E(s)$ is the Laplace transform of $e(t).$ Equation (5.36) simplifies to:

Taking the inverse Laplace transform of Eq (5.37) and simplifying, we obtain:

Hence, the characteristic equation is:

Based on reference ^[62], it is observed that the convergence time is dependent on the fractional order $\alpha$ and the constant $k_1.$

Remark 5.1. Generally, the larger the fractional order $\alpha,$ the slower the convergence, and the larger the $k_1$ value, the faster the convergence. In reference ^[62], the convergence time of the sliding surface dynamic attractor is calculated using a simple first-order differential equation; see reference ^[62] for a detailed explanation.

6.   Numerical simulation

First, we analyze the influence of key parameters in the FOSMA model (3.1) on the basic reproduction number. Then, the power series expansion method ^[54] will be used to numerically simulate the FOSMA model (3.1) to verify the correctness of the conclusions in Sections 4 and 5.

6.1.   Effect of key parameters

First, the parameters in model (3.1) are detailed in Table 2. To study the influence of the fractional order $\alpha$, the propagation rates $\beta_1$ and $\beta_2$, and the parameters $\omega$ and $\theta$ on the basic reproduction number $R_0$, we will use pseudo-color plots. These plots will help visualize how changes in these parameters affect the value of $R_0$.

The warm color scheme indicates the larger value of the $R_0.$ Through Figure 3$(a)$ we can find that: as $\omega$ increases and $\theta$ decreases, the $R_0$ increases. We can reduce transmission by decreasing the ratio of addicted to professionally operated people and increasing the conversion rate from professionally operated people to recovered people. Figure 3$(b)$, $(c)$ shows that the basic regeneration number $R_0$ increases as the value of the fractional order $\alpha$ increases. This indicates that the basic regeneration number of our modified model (3.1) is related to the fractional order. In the real addiction propagation process, the memory effect of fractional order $\alpha$ significantly impacts addiction formation. After the same analysis, Figures 4 and 5 show that we can reduce the spread rate by decreasing the contact with addicted and professionally operated people to reduce large-scale addiction. Also, the value of fractional order $\alpha$ is important for social media addiction.

6.2.   Fractional order systems

First, we keep the parameter values in model (3.1) consistent with Subsection 6.1. We choose the step size $h$ to be $0.001.$ Figure 6 represents that FOSMA model (3.1) converges to NAE at different fractional order values when $R_0 < 1.$

The Figure 7 indicates that the FOSMA model (3.1) converges to AE at different fractional order values when $R_0 > 1.$ Table 3 presents the data we used and shows the different results.

In the Figures 6 and 7, it can be seen from $(a)$ that the convergence rate of the susceptible population increases as the fractional order value decreases. Panels $(b)$ to $(f)$ show that, with a decrease in the fractional order value, the peaks of initial exposure population, addiction population, professional operation population, recovery population and quit population all decrease, and the final convergence speed gradually increases. This shows that fractional order values significantly impact the convergence rate and peak value of the model (3.1). This also reflects the memory effect of fractional order. This is consistent with the conclusions in ^[63,64].

We choose FOSMA model (3.1) parameters and initial conditions respectively $\Pi = 10, \; {\beta _1} = 0.7, \; {\sigma _1} = 0.8, \; {\beta _2} = 0.9;$ $S(0) = 1000;\; E(0) = 10;\; A(0) = 50;\; P(0) = 20;\; R(0) = 0;\; Q(0) = 100$. The other parameter values are consistent with the subsection 6.1. Moreover, we keep the numerical value of the fractional order fixed at $0.96.$ From Figures 8 and 9, we can find that as the saturation parameter $(a, b)$ increases, the rate of convergence of the susceptible population slows down, and the peaks of the exposed, addicted, professionally operated, recovered, and quit populations decrease. This suggests that the nonlinear incidence in the FOSMA model (3.1) has an inhibitory or psychological effect, which would inhibit the widespread spread of addiction.

6.3.   The application of sliding mode control

We choose the parameter values of the model (3.1) to be consistent with the case $R_0 > 1$ in the Subsection (6.1). The initial conditions for the control system (5.13) are $e(0) = (25, 12, 15, 14, 10, 16).$ The initial conditions for the sliding mode surface system (5.14) is $s(0) = (10, 8, 12, 8, 9, 10).$ The initial condition for the control input (5.23) is $u(0) = (11, 14, 12, 13, 9, 14).$ We respectively choose the values of the fractional order to be $0.96$ and $0.90.$ According to (5.14) and (5.23), we choose the gain parameters as ${k_1} = 0.8, {k_2} = 0.5, {k_3} = 1, r = 0.5, L = 1.5, \vartheta = 0.6, {\eta _1} = 1$ and a step size of $h = 0.01,$ then the sliding mode surface and the control law are shown as follows:

Figure 10 shows the response images of the fractional order sliding mode surface system at various fractional orders ($\alpha = 0.96, \; 0.90$), indicating that all sliding mode surfaces converge asymptotically to 0 within a finite duration. Moreover, as the fractional order decreases, the oscillation amplitude not only advances but also decreases.

Figure 11 demonstrates the response image of the error system using fractional order sliding mode control, showing that all the error signals converge asymptotically to 0 in finite time. As the fractional order value decreases, the amplitude of the error signals decreases and the time to converge to zero decreases. We find that the model (3.1) converges to NAE with the use of the controller, then the addiction phenomenon will disappear.

As depicted in Figure 12, it is evident that all control inputs converge to zero within a finite duration. Moreover, the rate of convergence of the control inputs to zero increases as the fractional order value decreases.

Table 4 shows the coordinates of the sliding surface, error signal, and sliding mode control under different fractional orders. It can be observed that the convergence speed increases as the fractional order decreases, which is consistent with the conclusion obtained in Remark 5.1.

The above numerical simulation results illustrate that the classical integer-order sliding mode control has strong chattering, while the fractional-order sliding mode control just compensates for the drawback.

7.   Conclusions

For the first time, we include a nonlinear incidence and a professionally operating population in a modified fractional-order social media addiction model. The basic regeneration number and equilibrium points of the FOSMA model (3.1) are computed. Sufficient conditions for NAE and AE local asymptotic stability are derived. We generalize the Sotomayor theorem to fractional order systems and demonstrate the existence of forward bifurcation at $R_0 = 1.$ Subsequently, the fractional order system is transformed into a partial differential model using the frequency distribution method. A fractional order sliding mode surface system is formulated, and the system's stability is demonstrated. A fractional order sliding mode control law is designed for the error signal system, demonstrating that the error signal system converges to the sliding mode surface under this control law. Finally, we find that fractional order value, propagation rate and specialized operating population have a significant effect on the basic regeneration number. Numerical simulations of the FOSMA model (3.1) reveal the importance of the fractional order value and the inhibitory factor in the nonlinear infection rate for eliminating social media addiction. With the fractional-order sliding mode control law, the error signal system will reach the sliding mode surface and further converge to 0. This also ensures that the model (3.1) will converge to the NAE, thus eliminating addiction.

In the future, we will extend our approach to modeling social media addiction to more scenarios, such as: Infectious disease modeling, ecosystem modeling, and crime modeling. Furthermore, we plan to develop a suitable fractional-order mathematical model and design fractional-order sliding mode control to address uncertainty and internal disturbances.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.