Dynamic analysis and optimal control of a novel fractional-order 2I2SR rumor spreading model

. In this paper, a novel fractional-order 2I2SR rumor spreading model is investigated. Firstly, the boundedness and uniqueness of solutions are proved. Then the next-generation matrix method is used to calculate the threshold. Furthermore, the stability of rumor-free/spreading equilibrium is discussed based on fractional-order Routh–Hurwitz stability criterion, Lyapunov function method, and invariance principle. Next, the necessary conditions for fractional optimal control are obtained. Finally, some numerical simulations are given to verify the results


Introduction
Rumor refers to the remarks that have no corresponding factual basis but are fabricated and promoted to spread by certain ways.With the development of science and technology, the rapid spread of rumors causes huge economic losses, disturb the normal order, and undermine social stability [6,8].Therefore, it is of great practical significance to study the dynamics of rumor propagation in social networks.
The mechanism of rumor transmission is similar to the process of epidemic transmission.In the 1960s, the classic DK model was proposed by Daley and Kendall, which divided the total population into three categories: ignorant, spreader, and removed, then numerical method is used to study the spread process of rumors, which is similar to those in infectious diseases [2,3].Based on the above research, the DK model was improved, and then MK model was obtained in 1973 [14].With the efforts of many scholars, more and more modified rumor propagation models have been put forward in recent years [13,22,28,31].In 2019, Wang et al. considered the cross-propagation mechanism and established SIR rumor propagation model in a multilanguage environment.Then its stability also was deeply investigated [22].In 2020, considering the network topology, Li et al. analyzed the dynamic behaviors of rumor propagation model with educational mechanism and carried out optimal control in the multilanguage environment [13].In 2020, a time-delay SIR rumor propagation model considering the network topology and forcing silence function was proposed and the stability of the rumor propagation model was analyzed in [31].Yu et al. established 2S2IR model based on multilanguage environment and studied Hopf bifurcation with time delay and the important parameter of the model, respectively [28].
Fractional calculus, called generalized calculus or arbitrary calculus, is a generalization of integral calculus and has short-term memory effect and genetic effect.With the continuous development and improvement of the fractional calculus theory, the fractional differential equation has been widely used in many fields [25,29,30].Huang et al. considered fractional neural networks with double delays.Furthermore, the stability and bifurcation of the system were studied [9].The dynamics of fractional SIR epidemic model with time delay and saturation function were studied by Wang et al. [23].In 2019, Wang et al. proposed a fractional ecoepidemiological model with time delay.Then the Hopf bifurcation of the system was studied, and the control strategies were given [24].In recent years, the problem of fractional optimal control has been studied extensively, and the conditions of fractional optimal control have been obtained [11,15,21].Memory effect on information transmission process is studied in [17,20,26], which shows that multiple redundant contacts of the same rumor will change people's initial thoughts of it, and the cumulative feature will affect the behavior of individuals in social networks.Due to the memory effect of fractional calculus, rumor propagation process can be analyzed accurately by studying the rumor propagation process with fractional calculus.In 2019, Singh considered a SIR rumor propagation model with Atangana-Baleanu derivative, and the effect of fractional order on the population of each warehouse was studied [18].A fractional-order SIR model, which is similar to the epidemic model, was established to examine the adoption and abandonment of online social networks by social network users.Then the properties of the solutions of the system were studied in [7].Ren et al. established a fractional stochastic rumor propagation model in mobile social networks, and the stability conditions of the system were obtained [16].Inspired by [7,28], we consider a fractional-order 2I2SR rumor spreading model and study the properties of the solutions.The optimal control conditions of the system are given at last.The main contributions of this study are as follows: • Compared with [28], a fractional 2I2SR rumor propagation model is generalized, which considers the unit dimension of the equation and memory effect in rumor propagation process.• The properties and dynamics of the solutions to the given rumor propagation model are studied by means of fractional differential equation theory.• The optimal control of the given fractional rumor propagation model is obtained by using the fractional optimal control theory. https://www.journals.vu.lt/nonlinear-analysis The rest of this article is arranged as follows.In Section 2, some preparations related to fractional equations are introduced.In Section 3, a fraction-order 2S2IR rumor spreading model is proposed.In Section 4, the properties of the solutions are disscussed.Furthermore, the fractional-order optimal control strategies are presented.In Section 5, some numerical simulations are illustrated to verify the theoretical results.In Section 6, we have a brief summary for the whole paper.

Preliminaries
In this section, some preparations related to fractional differential equations are given, which will be used in the following discussion.Definition 1. (See [5].)Let f be a function defined on [a, b], and let κ > 0. The Riemann-Liouville fractional integral of order κ for the function f is defined by where Γ(•) is the gamma function.
Definition 2. (See [5].)The Caputo fractional derivative of order κ of a function f (t) is defined as where n is the positive integer, and Nonlinear Anal.Model.Control, 28(5):859-882, 2023 Lemma 2. (See [27].)Let 0 < α < 1 and t 0, then function E α (µ(t − t 0 ) α ) is nonnegative.Furthermore, 0 E α (µ(t − t 0 ) α ) 1 for t t 0 when µ 0. Lemma 3. (See [12].)Assume that w(t) is continuous on [t 0 , +∞) and satisfies where 0 < κ < 1, (λ, µ) ∈ R 2 , and λ = 0. Then Lemma 4. (See [12].)Consider the system with initial condition x t0 , where Then there exists a unique solution of system (1) on [t 0 , ∞) × Ω if f (t, x) satisfies the locally Lipschitz condition with respect to x. Lemma 5. (See [4].)Considering the following n-dimensional linear fractional differential system with multiple time delays: where 0 < κ i < 1, and κ i is real.The initial values x i = φ i (t) are given for − max i,j τ ij = −τ max t 0 and i = 1, 2, . . ., n.In this system, state variables Then the characteristic matrix of system (2) can be labeled as , and det ∆(λ) = 0 is the characteristic equation of the following equation: Then we have the equivalent conditions: Lemma 7. (See [10].)Assume that u(t) ∈ R + is continuous and derivable.Then for any time instant t t 0 and for all κ ∈ (0, 1), Lemma 8. (See [10].)Consider the following autonomous system: Suppose B is a bounded closed set, every solution of system (3) starts from a point in B and remains in B for all time.There exists V (y) : B → R with continuous first partial derivatives satisfying the following condition: , and let M be the largest invariant set of F .Then every solution y(t) originating in B tends to M as t → +∞.Particularly, if M = 0, then y → 0, t → +∞.

Model formulation
Many rumor models have been improved to better understand the rumor spreading process.A 2I2SR rumor propagation model in multilingual environment is introduced in [5].The information may be in Chinese, English, or even other languages since these users come from different countries or regions.Assume that one of them is the official language of this social network, and others are unofficial languages, and all users understand official language.In this model, we consider five types of users: Ignorants-1 (I 1 (t)), Ignorants-2 (I 2 (t)), Spreaders-1 (S 1 (t)), Spreaders-2 (S 2 (t)) and Removers (R(t)).I 1 (t) represents users who can speak both official and other languages, but they prefer to publish information in unofficial languages, and they do not know the rumor information.I 2 (t) stands for users who only can use official language to exchange information, and they also do not know the rumor information.S 1 (t) refers to individuals who have received the rumor and can speak official language and other languages, but they prefer to spread rumor in unofficial languages.S 2 (t) describes the ones who know the rumor and propagate it in Nonlinear Anal.Model.Control, 28(5):859-882, 2023 Table 1.Descriptions of parameters for the model (4).

Symbols Description Units Π i
The immigration rates of The probability of turning The probability of turning The removal rate for each compartment [Unit of time] −1 official language.R(t) represents the rumor recovery individuals who know the rumor and no longer spread it.The population movement between the five warehouses is modeled as follows: with the initial conditions People's acceptance of information and whether they choose to spread information are affected by individual's subjective will.For model (4), it was established with integerorder differential equations.A detailed description of the parameters can be seen in Table 1.However, the state of each moment does not depend on the historical status of the system.The memory effect of rumor transmission was not considered.It can be seen from [7,9,11,15,16,18,23,24] that fractional calculus can better describe the dynamic processes with memory effect than integer calculus.
Fractional calculus is introduced to describe the memory effect.Through the application of fractional differential equations in dynamical systems in recent years [9,11,15,23,24], we can generalize system (4) into the following form in the sense of Caputo derivative: https://www.journals.vu.lt/nonlinear-analysis It is reasonable to generalize system (4) to system (5) because the memory effect of rumor propagation is considered.However, this approach does not take the time dimension into better account.The units on the left-hand side of system (5) are [Number] × [Unit of time] −κ , while the units on the right-hand side of system ( 5) are [Number] × [Unit of time] −1 .In recent years, some scholars have considered the unity of fractional differential equations, which can be observed in [1,7].
Inspired by the unit problem of considering parameters in [7], we generalized the 2I2SR rumor propagation model, which was studied in [29], into fractional (0 < κ < 1) differential equations.Firstly, system (4) is equivalent to the following integral equations: with the initial conditions In order to consider the effect of memory effect on rumor spreading process, we rewrite system (6) into the following form with memory effect: where k(t, s) is the kernel function, and it has the following form: Remark 2. Compared with integer calculus, the memory of fractional calculus is mainly reflected in the power law property of kernel function.
Considering the unity of the units on both sides of this equations and applying Definition 1, we obtain Applying Lemma 1 and the Caputo derivative of order κ to both sides of Eq. ( 7), the following fractional-order 2S2IR rumor propagation model can be obtained: with the initial conditions After a simple analysis of model ( 8), we can easily find that both sides of the equations have the same units.The specific parameters of this model are shown in Table 2.The following discussion and analysis are based on model (8).
Remark 3. When κ → 1 − , system (8) is transformed into system (4) without memory effect.Furthermore, system (4) can be viewed as a special case of system (8), and it can be seen that the order κ is an intuitive embodiment of the memory effect of fractional-order system (8).

Π i
The immigration rate of The probability of turning The probability of turning The probability of turning The removal rate for each compartment [Unit of time] −1

Main results
In this section, we mainly prove the boundedness and uniqueness of the solutions of system (8).Next, the sufficient conditions for the stability of equilibriums are obtained.Furthermore, the necessary conditions for fractional optimal control are obtained.

Properties of the solutions
For convenience, let Theorem 1.All the solutions of system (8), which start in D + , are bounded.
Proof.Define the function as t → +∞.Therefore, all the solutions of system (8), which start in D + , are confined to the region This completes the proof of theorem. where For any Y, Y ∈ Z, Furthermore, we can obtain This completes the proof by applying Lemma 4.

Stability analysis
In this section, we will give the threshold R α 0 and discuss the stability of the equilibrium of system (8).The main results and proofs are as follows.
The threshold of system ( 8) is given by Remark 4. R κ 0 is a threshold quantity.When κ = 1, R κ 0 is the basic reproduction number of model (8).It refers to the number of people that a ignorant can turn into a spreader during the average period of transmission when everyone is ignorant at the initial stage of rumor transmission.
Remark 5.In a multilingual environment, the system threshold R κ 0 is equal to the sum of the thresholds R κ 0i of each group i.In this paper, R κ 01 and R κ 02 are thresholds in two language environments, respectively.A detailed description can be found in [19].
Obviously, R(t) is independent of the first four equations, so we can just consider the first four equations in the following study.
Next, we discuss the stability of rumor-spreading equilibrium, system (8) satisfies the following equations at When where where Combining the expressions of I * 1 , S * 2 and the third equation of system (11), it can be seen that where The following results can be proved by simple calculation: Nonlinear Anal.Model.Control, 28(5):859-882, 2023 M. Ye et al.
So we discuss the roots of Eq. ( 12) in two different cases in the following study.
We make the following hypotheses to obtain our result.
Remark 6.In [13,22,28], rumor propagation models in multilingual environment are established, and the stability of these systems is studied.Different from their work, a fractional-order 2I2SR rumor model is proposed in multilingual environment in which the memory effect in rumor propagation is considered by Caputo fractional derivative.More importantly, the necessary conditions for fractional optimal control of rumor propagation are obtained.
Remark 7. Compared with [11,15,23,24], not only a fractional 2I2SR rumor propagation model is established in a multilingual environment and the stability conditions of the fractional-order system are obtained, but also the unity of units on the left and right sides of the generalized equation is taken into account, which is worth being considered in future research. https://www.journals.vu.lt/nonlinear-analysis

Numerical examples
To solve fractional differential equations, we mainly use predictor-corrector method, which is described in [15].

5.3
The effect of control u i (t) (i = 1, 2) on system (8) To test our theoretical results, we discuss the influence of different orders on system (14), and the influence of control u i (t) (i = 1, 2) on the controlled system is simulated.Choose 6.The trajectories of optimal control u i (t) (i = 1, 2) and consumption J(t) with κ = 0.96.
have a great influence on rumor spreader S i (t) (i = 1, 2), which can effectively control the rumor propagation.Next, the optimal control curve and the cost of official control curve are given in Fig. 6.

Conclusions
A fractional-order 2I2SR rumor spreading model is investigated in this paper.Firstly, the boundedness and uniqueness of the solutions of the fractional-order system are proved.
Then the next-generation matrix method is used to calculate the threshold.Based on generalized fractional-order Routh-Hurwitz judgment, the local asymptotic stability of the rumor-free equilibrium E 0 and the rumor-spreading equilibrium

Definition 3 .
(See [15].)Let f ∈ C[a, b], where C[a, b] represents the space of absolutely continuous functions on [a, b], the left and right Caputo fractional derivatives (CFDs) are as follows:

Π 1 = 25 ,Figure 5 .
Figure 5.Comparison of the number of individuals in controlled and uncontrolled systems.
Nonlinear Anal.Model.Control, 28(5):859-882, 2023 * 1 is studied.The global asymptotic stability of rumor-spreading equilibrium E * 2 is discussed by means of Lyapunov function method and invariance principle.It can be obtained by detailed proof that if R κ 01 < 1 and R κ 02 < 1, E 0 is locally asymptotically stable, if R κ 01 > 1 and R κ 02 < 1 are satisfied, E * 1 is locally asymptotically stable, and if R κ 02 > 1, E * 2 is globally asymptotically stable.Finally, the necessary conditions for fractional optimal control of the rumor spreading model are obtained.