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

Authors: Changjin Xu; Maoxin Liao; Peiluan Li
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In recent years, with the rapid development of the Internet and the Internet of Things, 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 few decades, numerous researchers put up various kinds of malware propagation models to analyze the dynamic interaction. However, many works are only concerned with the integer-order malware 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 propagation model. 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.


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## Body

## 1. 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 [7–11].The fractional calculus is a generalization of ordinary differentiation and integration to random order (noninteger)[12–19]. 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 [22–28]. 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, 32–35].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:(1)du1tdt=a1-a2u1tu3t-ρ+u2t-a3u1t+a4u4t,du2tdt=a2u1tu3t-ρ+u2t-a5u2t,du3tdt=a2u1tu3t-ρ+u2t-a6u3t-a3u3t,du4tdt=a5u2t+a6u3t+a3u1t-a4u4t,where u1(t) denotes the susceptible population, u2(t) denotes the carrier population, u3(t) denotes the infectious population, u4(t) denotes recovered population, a1 denotes the number of susceptible nodes, a2 denotes the constant contact rate for u1(t),u3(t-ρ), and u2(t), a3 denotes the vaccination rate of nodes, a4 denotes the rate constant for nodes becoming susceptible again, a5 denotes the number of new recovered devices from carrier, and a6 denotes the rate constant for nodes leaving u3(t) for u4(t). When u1(t)+u2(t)+u3(t)+u4(t)=M, where M>0 is a constant, then system (1) becomes(2)du1tdt=a1-a2u1tu3t-ρ+u2t-a3u1t+a4M-u1t-u2t-u3t,du2tdt=a2u1tu3t-ρ+u2t-a5u2t,du3tdt=a2u1tu3t-ρ+u2t-a6u3t-a3u3t.Stimulated by the analysis above, we modify (1) as the following fractional-order delayed malware propagation model:(3)Dpu1t=a1-a2u1tu3t-ρ+u2t-a3u1t+a4M-u1t-u2t-u3t,Dpu2t=a2u1tu3t-ρ+u2t-a5u2t,Dpu3t=a2u1tu3t-ρ+u2t-a6u3t-a3u3t.where p∈(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 fractional-order 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 Section2, 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.


## 2. Preliminary Results

In this section, we introduce three definitions and two lemmas.Definition 1 (see [37]).
The fractional integral of orderp for a function l(ξ) is defined as follows:(4)Iplξ=1Γp∫ξ0ξξ-sp-1lsds,where ξ≥ξ0,p>0,Γ(.) denotes the Gamma function and Γ(s)=∫0∞ξs-1e-ξdξ.Definition 2 (see [37]).
The Caputo fractional-order derivative of orderp for a function l(ξ)∈([ξ0,∞),R) is defined as follows:(5)Dpfξ=1Γn-p∫ξ0ξlnsξ-sp-n+1ds, where ξ≥ξ0 and n is a positive integer such that n-1≤p<n. In particular, when 0<p<1,(6)Dpgv=1Γ1-p∫ξ0ξl′sξ-spds.Definition 3 (see[32]).
For the given fractional-order system(7)Dpuit=giuit,i=1,2,…,n,where p∈(0,1],ui(t)=(u1(t),u2(t),…,un(t)),gi(t)=(g1(t),g2(t),…,gn(t)).  (u1∗,u2∗,…,un∗) is said to be the equilibrium point if gi(ui∗)=0.Lemma 4 (see [38]).
Consider the following autonomous systemDpz=Az,z(0)=z0 where 0<p<1,z∈Rn,A∈Rn×n. Let λi(i=1,2,…,n) be the root of the characteristic equation of Dpz=Az. Then system Dpz=Az is asymptotically stable if and only if argλi>pπ/2(i=1,2,…,n). In this case, each component of the states decays towards 0 like t-p. Also, this system is stable if and only if argλi>pπ/2(i=1,2,…,n) and those critical eigenvalues that satisfy argλi=pπ/2(i=1,2,…,n) have geometric multiplicity one.Lemma 5 (see [39]).
For the given fractional-order delayed differential equation with Caputo derivative,Dpu(t)=A1u(t)+A2u(t-ϱ), where u(t)=ϕ(t),t∈[-ϱ,0],p∈(0,1],u∈Rn,A1,A2∈Rn×n,ϱ∈R+(n×n). Then the characteristic equation of the system is detspI-A1-A2e-sϱ=0. 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.


## 3. 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).Obviously, system (3) has an equilibrium point E0=(a1+a4M/a3+a4,0,0). If the following condition(8)Q1a2a1+a4Ma3+a5+a6>a3+a6holds, then system (3) has a positive equilibrium point E∗=(u1∗,u2∗,u3∗) where(9)u1∗=a5a3+a6a2a3+a5+a6,u2∗=a5a1+a4Ma3+a5+a6-a5a2-1a3+a6a3+a5+a6(a5a6+a3a5+a4a6+a3a4+a4a5,u3∗=a1+a4Ma3+a6a3+a5+a6-a2-1a3+a62a3+a5+a6(a5a6+a3a5+a4a6+a3a4+a4a5.In the sequel, considering the biological meaning, we only consider the positive equilibrium point E∗=(u1∗,u2∗,u3∗). Let u¯i=ui(t)-ui∗(i=1,2,3) and still denote u¯i by ui(t); then system (3) takes the following form:(10)Dpu1t=a1-a2u1t+u1∗u3t-ρ+u3∗+u2t+u2∗-a3u1t+u1∗+a4M-u1t+u1∗-u2t+u2∗-u3t+u3∗,Dpu2t=a2u1t+u1∗u3t-ρ+u3∗+u2t+u2∗-a5u2t+u2∗,Dpu3t=a2u1t+u1∗u3t-ρ+u3∗+u2t+u2∗-a6u3t+u3∗-a3u3t+u3∗.The linearization of (10) near E∗(u1∗,u2∗,u3∗) is(11)Dpu1t=-a2u2∗+a3+a4u1t-a2u1∗+a4u2t-a4u3t-a2u1∗u3t-ρ,Dpu2t=a2u2∗u1t+a2u1∗-a5u2t+a2u1∗u3t-ρ,Dpu3t=a2u2∗u1t+a2u1∗u2t-a3+a6u3t+a2u1∗u3t-ρ.The corresponding characteristic equation of (11) is(12)detsp+a2u2∗+a3+a4a2u1∗+a4a4+a2u1∗e-sρ-a2u2∗sp-a2u1∗-a5-a2u1∗e-sρ-a2u2∗-a2u1∗sp+a3+a6-a2u1∗e-sρ which leads to(13)K1s+K2se-sρ=0, where(14)K1s=s3p+k1s2p+k2sp+k3,K2s=k4s2p+k5sp+k6, where(15)k1=2a3+a6+a2u2∗+a4-a2u1∗+a5,k2=a2u2∗+a3+a4-a2u1∗+a5a3+a6-a2u2∗+a3+a4a2u1∗-a5+a2a4u2∗+a2u2∗a2u1∗+a4,k3=-a2u2∗+a3+a4a2u1∗-a5a3+a6+a22a4u1∗u2∗-a2a4u2∗a2u1∗-a5-a22u1∗u2∗a2u1∗-a5+a2u2∗a2u1∗+a4a3+a6,k4=-a2u1∗,k5=-a2u1∗a2u2∗+a3+a4-a2u1∗+a5+2a2u1∗2,k6=a2u2∗+a3+a4a2u1∗-a5a2u1∗+a2u1∗+a4a22u1∗u2∗+a32u1∗2u2∗-a22u2∗a2u1∗+a4u1∗-a1u1∗2a2u2∗+a3+a4. If ρ=0, then (13) takes the following form:(16)λ3+k1+k4λ2+k2+k5λ+k3+k6=0. If the following condition (17)Q2k1+k4k2+k5>k3+k6,k3+k6>0holds, then all the roots λi(i=1,2,3) of (16) satisfy argλi>pπ/2(i=1,2,3). By Lemma 4, we can conclude that the equilibrium point (u1∗,u2∗,u3∗) of (3) with ρ=0 is locally asymptotically stable.Lets=iϕ=ϕcosπ/2+isinπ/2 be a root of (13). Then(18)Icosϕρ+Jsinϕρ=L1,Jcosϕρ-Isinϕρ=L2,where(19)I=k4ϕ2pcospπ+k5ϕpcospπ2+k6,J=k4ϕ2psinpπ+k5ϕpsinpπ2,L1=-ϕ3pcos3pπ2+k1ϕ2pcospπ+k2ϕpcospπ2+k3,L2=-ϕ3psin3pπ2+k1ϕ2psinpπ+k2ϕpsinpπ2. By (18), we have(20)I2+J2=L12+L22, which leads to(21)ϕ6p+b1ϕ5p+b2ϕ4p+b3ϕ3p+b4ϕ2p+b5ϕp+b6=0, where(22)b1=2b1cos⁡3pπ2cospπ+sin⁡3pπ2sinpπ,b2=b12-b42+2b2cos⁡3pπ2cospπ+sin⁡3pπ2sinpπ,b3=2b3cos3pπ2+2b1b2cos⁡pπ2cospπ+sin⁡pπ2sinpπ-2b4b5cos⁡pπ2cospπ+sin⁡pπ2sinpπ,b4=b22-b52+2b1b3-b4b6cos⁡pπ,b5=2b2b3-b5b6cos⁡pπ2,b6=b32-b62. Let(23)ϱϕ=ϕ6p+b1ϕ5p+b2ϕ4p+b3ϕ3p+b4ϕ2p+b5ϕp+b6. If the following condition (24)Q3b6<0holds, then dϱ(ϕ)/dϕ>0∀ϕ>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.Lemma 6.
For (13), the following results are true:
(1) If (Q2) holds andbi>0(i=1,2,3,4,5,6), then (13) has no root with zero real parts for all ρ≥0.
(2) If (Q3) holds, then (13) has a pair of purely imaginary roots ±iϕ0 when ρ=ρl(l=1,2,⋯), where(25)ρl=1ϕ0arccos⁡L1I+L2JI2+J2+2lπ, where l=0,1,2,…, and ϕ0 is the unique positive zero of the function ϱ(ϕ).The proof of Lemma6 is similar to the proof of Lemma 3.1 in Sun et al. [40]. Here we omit it.In order to obtain the transversality condition of Hopf bifurcation, the following hypothesis is given:(26)Q4U1V1+U2V2>0,where (27)U1=3pϕ03p-1cos3p-1π2+2pk1ϕ02p-1cos2p-1π2+pk2ϕ0p-1cosp-1π2+2pk4ϕ02p-1cos2p-1π2+pk5ϕ0p-1cosp-1π2cosϕ0ρ0+2pk4ϕ02p-1sin2p-1π2+pk5ϕ0p-1sinp-1π2sinϕ0ρ0,U2=3pϕ03p-1sin3p-1π2+2pk1ϕ02p-1sin2p-1π2+pk2ϕ0p-1sinp-1π2,-2pk4ϕ02p-1cos2p-1π2+pk5ϕ0p-1cosp-1π2sinϕ0ρ0+2pk4ϕ02p-1sin2p-1π2+pk5ϕ0p-1sinp-1π2cosϕ0ρ0,V1=k4ϕ02pcospπ+k5ϕ0pcospπ2+k6ϕ0sinϕ0ρ0-k4ϕ02psinpπ+k5ϕ0psinpπ2ϕ0cosϕ0ρ0,V2=k4ϕ02pcospπ+k5ϕ0pcospπ2+k6ϕ0cosϕ0ρ0+k4ϕ02psinpπ+k5ϕ0psinpπ2ϕ0sinϕ0ρ0.Lemma 7.
Lets(ρ)=μ(ρ)+iϕ(ρ) be the root of (13) at ρ=ρ0 satisfying μ(ρ0)=0,ϕ(ρ0)=ϕ0; then Reds/dρρ=ρ0,ϕ=ϕ0>0.Proof.
Differentiating (13) with respect to ρ, one has(28)dsdρ-1=3ps3p-1+2pk1s2p-1+pk2sp-1+2pk4s2p-1+pk5sp-1e-sρk4s2p+k5sp+k6se-sρ-ρs. Hence(29)Redsdρ-1=Re3ps3p-1+2pk1s2p-1+pk2sp-1+2pk4s2p-1+pk5sp-1e-sρk4s2p+k5sp+k6se-sρ. Then (30)Redsdρ-1ρ=ρ0,ϕ=ϕ0=Re3ps3p-1+2pk1s2p-1+pk2sp-1+2pk4s2p-1+pk5sp-1e-sρk4s2p+k5sp+k6se-sρρ=ρ0,ϕ=ϕ0=U1V1+U2V2V12+V22. In view of (Q4), one has(31)Redsdσ-1ρ=ρ0,ϕ=ϕ0>0. The proof of Lemma 7 is completed.According to the analysis above and Lemmas4 and 5, we have the following theorem.Theorem 8.
For system (3), if (Q1)-(Q4) hold, then the equilibrium point (u1∗,u2∗,u3∗) is locally asymptotically stable for ρ∈[0,ρ0) and system (3) undergoes a Hopf bifurcation near the equilibrium point (u1∗,u2∗,u3∗) when ρ=ρ0.Remark 9.
Xu et al. [41–47] studied the Hopf bifurcation of integer-order delayed models. In this article, we investigate the Hopf bifurcation of delayed malware propagation model. All the derived results and analysis ideas [41–47] cannot be applied to (3) to obtain the stability and the existence of Hopf bifurcation for (3). Although the authors of [36, 48] analyzed the Hopf bifurcation, and the stability of integer-order malware propagation model, they did not consider the fractional-order case. From the point, the fruit of this article on the stability and the existence of Hopf bifurcation for (3) is completely new and completes some earlier publications.


## 4. Numerical Simulations

Consider the following fractional-order system:(32)Dpu1t=a1-a2u1tu3t-ρ+u2t-a3u1t+a4M-u1t-u2t-u3t,Dpu2t=a2u1tu3t-ρ+u2t-a5u2t,Dpu3t=a2u1tu3t-ρ+u2t-a6u3t-a3u3t,where a1=1.2,a2=0.5,a3=0.2,a4=0.005,a5=2,a6=1.2,M=1000. It is not difficult to obtain that system (32) has the positive equilibrium point (1.6471,2.9135,4.1621). Let p=0.71. Then the critical frequency ϕ0=0.3368 and the bifurcation point ρ0=0.5739. Then all the conditions (Q1)-(Q4) of Theorem 8 are satisfied. Figure 1 indicates that the positive equilibrium point (1.6471,2.9135,4.1621) of system (32) is locally asymptotically stable for ρ∈[0,ρ0). Figure 2 shows that system (32) loses its stability; Hopf bifurcation occurs when ρ∈[ρ0,+∞). The relation of fractional order p on the critical frequency ϕ0 and bifurcation point ρ0 of (32) is shown in Table 1.Table 1
The impact of fractional-orderp  on the critical frequency ϕ0 and bifurcation point ρ0 of (32).
p ϕ 0 ρ 0 0.18 0.9937 0.1796 0.27 0.8742 0.2581 0.34 0.7238 0.3152 0.48 0.6512 0.4206 0.57 0.5218 0.4832 0.64 0.4425 0.5295 0.71 0.3368 0.5739 0.87 0.2551 0.6694 0.91 0.1724 0.6921Figure 1
ρ = 0.48 < ρ 0 = 0.5739 . The positive equilibrium point (1.6471,2.9135,4.1621) of system (32) is locally asymptotically stable.Figure 2
ρ = 0.62 > ρ 0 = 0.5739 . Hopf bifurcation of system (32) occurs from the positive equilibrium point (1.6471,2.9135,4.1621).


## 5. 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 valueρ0, while, when the time delay exceeds the critical value ρ0, 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.



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