Bifurcation of a Fractional-Order Delayed Malware Propagation Model in Social Networks

In recent years, with the rapid development of the Internet and the Internet ofThings, network security is urgently needed.Malware becomes a major threat to network security. Thus, the study on malware propagation model plays an important role in network security. In the past fewdecades, numerous researchersput up various kinds ofmalware propagationmodels to analyze the dynamic interaction. However, many works are only concerned with the integer-ordermalware propagation models, while the investigation on fractional-order ones is very few. In this paper, based on the earlier works, we will put up a new fractional-order delayed malware propagationmodel. Letting the delay be bifurcation parameter and analyzing the corresponding characteristic equations of considered system, we will establish a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model. The study shows that the delay and the fractional order have important effect on the stability and Hopf bifurcation of considered system. To check the correctness of theoretical analyses, we carry out some computer simulations. At last, a simple conclusion is drawn. The derived results of this paper are completely innovative and play an important guiding role in network security.


Introduction
Nowadays, social networks are important platforms for disseminating information and building relationship.Different from the classical approaches of communication, social networks have fast speed of information propagation and diffusion.Furthermore, social networks have important effect on commercial negotiations, social connections, and information-sharing activities.Owing to the potential applications of social networks in many areas, many scholars pay much attention to dynamics of wireless sensor networks.For example, Deng et al. [1] considered the mobility-based clustering protocol of wireless sensor networks, Liu et al. [2] focused on the design and statistical analysis of a new chaotic block cipher for wireless sensor networks, Wang and Tseng [3] discussed the distributed deployment schemes for mobile wireless sensor networks, and Kulkarni and Zambare [4] studied the impact of Houseplants in purification of environment using wireless sensor network.
Malware software (malware) is an important tool to attack cybersecurity.Malware, which widely appears in the Internet, will cause some serious security risks of networks such as network paralysis, instability of society, loss of secret key, and personal information leakage.In order to understand and grasp the damage of malware, it is necessary for us to investigate the cause of malware occurrence, the harmful level to human beings, and the internal mechanism of malware propagation.During the past few decades, some researchers have obtained excellent achievements.For example, Liu et al. [5] analyzed the spread of malware with the influence of heterogeneous immunization.By constructing an appropriate Lyapunov function, the sufficient condition which guarantees the globally asymptotic stability of the equilibrium is obtained.Their research shows that interpretation of malware parameters and prediction of the evolution of future malware outbreaks play an important role in ensuring user safety.Signes Pont et al. [6] modelled the malware propagation in mobile computer devices.Hosseini and Azgomi [7] considered the stability of an SEIRS-QV malware propagation model in heterogeneous networks.In detail, one can see [7][8][9][10][11].
The fractional calculus is a generalization of ordinary differentiation and integration to random order (noninteger) [12][13][14][15][16][17][18][19].Many scholars pointed out that it is more accurate to describe the real object by fractional-order derivatives than integer-order ones.In recent years, the fractional calculus has many potential applications in many areas such as robotics, bioengineering, electroanalytical chemistry, viscoelasticity, heat conduction, and economics [20,21].Thus, the dynamical nature of fractional-order differential systems has attracted much attention and excellent results have been achieved.We refer the readers to [22][23][24][25][26][27][28].In particular, some achievements about Hopf bifurcation of fractional-order differential systems are also available.For example, Rajagopal et al. [29] studied the bifurcation and chaos of delayed fractional-order chaotic memfractor oscillator, Huang [30] dealt with the bifurcation in a delayed van der Pol oscillator, and Xiao et al. [31] discussed the bifurcation control of a fractional-order van der pol oscillator.In detail, one can see [29,[32][33][34][35].
Here we must point out that all the above works on Hopf bifurcation of fractional-order (integer-order) differential models are only concerned with neural networks and predator-prey models.Up to now, there are few articles that focus on the effect of delays on the Hopf bifurcation of fractional-order delayed malware propagation model.
In 2018, Du et al. [36] investigated the following malware propagation model with delay: where Stimulated by the analysis above, we modify (1) as the following fractional-order delayed malware propagation model: where  ∈ (0, 1].All the variables and coefficients have the same implications as those in (1).
The main objective of this article is to handle two problems: (i) the sufficient conditions that guarantee the stability and existence of Hopf bifurcation of system (3) are established; (ii) the effect of the delay and fractional order on Hopf bifurcation of model ( 3) is shown.
The highlights of this paper consist of four points: (i) The integer-order delayed malware propagation model in social networks has been extended to fractional-order delayed malware propagation model in social networks, which can better describe the memory and hereditary properties of the model.
(ii) A sufficient criterion of the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model in social networks are derived.The effect of delay and fractional order on the stability and Hopf bifurcation of (3) is illustrated.
(iii) To the best of our knowledge, there are no articles that focus on the Hopf bifurcation of fractional-order malware propagation model.The obtained results of this article will enrich and develop the Hopf bifurcation theory of fractionalorder delayed differential equations and supplement the previous publications.
(iv) The idea of this manuscript will provide a good reference to investigate many other fractional-order systems with delays.
The rest of this paper is organized as follows.In Section 2, several notations and preliminary results on fractional calculus are listed.In Section 3, a sufficient condition to ensure the stability and the existence of Hopf bifurcation of system (3) is presented.The effect of the delay on the stability and Hopf bifurcation of the considered system (3) is discussed.In Section 4, some numerical simulations are carried out to check the theoretical findings.In Section 5, we give a brief conclusion.

Preliminary Results
In this section, we introduce three definitions and two lemmas.
In particular, when 0 <  < 1, Definition 3 (see [32] ).For the given fractional-order system where  ∈ (0,1],  () = ( Lemma 5 (see [39]).For the given fractional-order delayed differential equation with Caputo derivative, ×) .Then the characteristic equation of the system is If all the roots of the characteristic equation of the system have negative real roots, then the zero solution of the system is asymptotically stable.

Effect of the Delay on Bifurcation for Model (3)
In this section, we will investigate the impact of the delay on Hopf bifurcation for system (3).
Let  =  = ( cos (/2) +  sin (/2)) be a root of (13).Then where By (18), we have which leads to where Let If the following condition holds, then ()/ > 0 ∀  > 0; then Eq. ( 21) has at least one positive real root.Thus, Eq. ( 13) has at least one pair of pure roots.In view of Sun et al. [40], we have the following result.
The proof of Lemma 6 is similar to the proof of Lemma 3.1 in Sun et al. [40].Here we omit it.
Proof.Differentiating (13) with respect to , one has Hence Then In view of (Q4), one has The proof of Lemma 7 is completed.

Numerical Simulations
Consider the following fractional-order system:  32) is locally asymptotically stable for  ∈ [0,  0 ). Figure 2 shows that system (32) loses its stability; Hopf bifurcation occurs when  ∈ [ 0 , +∞).The relation of fractional order  on the critical frequency  0 and bifurcation point  0 of ( 32) is shown in Table 1.

Conclusions
In real world, malware plays an important role in economy and society.Based on the previous works, in this paper, we propose a new fractional-order delayed malware propagation model in social networks.By choosing the time delay as bifurcation parameter, we establish a sufficient condition to ensure the stability and the existence of Hopf bifurcation of fractional-order delayed malware propagation model.The research shows that the positive equilibrium point of the involved model is locally asymptotically stable when the time delay is less than the critical value  0 , while, when the time delay exceeds the critical value  0 , the system will lose its stability and Hopf bifurcation will appear.The investigation also shows the effect of the fractional order on the stability and Hopf bifurcation of involved model.Furthermore, the relationship of fractional-order and bifurcation point is discussed.The derived results have important theoretical significance and practical value in controlling the malware propagation.

Table 1 :
The impact of fractional-order  on the critical frequency  0 and bifurcation point  0 of (32).