DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2015/101874 101874 Research Article Hopf Bifurcation of an SIQR Computer Virus Model with Time Delay http://orcid.org/0000-0002-2879-4434 Zhang Zizhen 1, 2 Yang Huizhong 2 Iqbal Muhammad Naveed 1 Anhui University of Finance and Economics School of Management Science and Engineering, Caoshan Road 962, Bengbu 233030 China aufe.edu.cn 2 Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122 China jiangnan.edu.cn 2015 1512015 2015 13 06 2014 14 11 2014 1512015 2015 Copyright © 2015 Zizhen Zhang and Huizhong Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A delayed SIQR computer virus model is considered. It has been observed that there exists a critical value of delay for the stability of virus prevalence by choosing the delay as a bifurcation parameter. Furthermore, the properties of the Hopf bifurcation such as direction and stability are investigated by using the normal form method and center manifold theory. Finally, some numerical simulations for supporting our theoretical results are also performed.

1. Introduction

Recently, many scholars have been studying the prevalence of computer viruses by establishing reasonable mathematics models . In , Piqueira and Araujo established a modified version of SIR model for the computer viruses in network and they got the stability and bifurcation conditions of the model. In , Gan et al. proposed an epidemic model of computer viruses by incorporating a vaccination probability in the SIRS model with generalized nonlinear incidence rate.

As is known, many computer viruses have different kinds of delays when they spread, such as latent period delay [6, 7], temporary immunity period delay , and other types . In , Yang proposed the following SIQR computer virus model with time delay: (1)dS(t)dt=(1-p)b-βS(t-τ)I(t-τ)-dS(t),dI(t)dt=βS(t-τ)I(t-τ)-(δ+d+α1+γ)I(t),dQ(t)dt=δI(t)-(ɛ+d+α2)Q(t),dR(t)dt=γI(t)+pb+ɛQ(t)-dR(t), where S(t), I(t), Q(t), and R(t) denote the numbers of nodes in states susceptible, infectious, quarantined, and recovered at time t, respectively. b is the new number of nodes. p is the proportion of new nodes who are immunized directly. β is the probability for a susceptible node to be infected. d is the natural death rate of nodes. α1 and α2 are the death rates due to the virus for the nodes in states infectious and quarantined, respectively. γ, δ, and ɛ are the coefficients of state transmission. τ is the latent period of the virus.

Gan et al. investigated global attractivity and sustainability of system (1) in . However, studies on dynamical systems not only involve a discussion of attractivity and sustainability but also involve many dynamical behaviors such as stability, bifurcation, and chaos. In particular, the existence and properties of the Hopf bifurcation for the delayed dynamical systems have been studied by many authors [810, 12]. In , Feng et al. investigated the Hopf bifurcation of a delayed SIRS viral infection model in computer networks by regarding the delay as a bifurcation parameter. In , Zhuang and Zhu investigated the Hopf bifurcation of an improved HIV model with time delay and cure rate. It is well known that the occurrence of Hopf bifurcation means that the state of virus prevalence changes from an equilibrium point to a limit cycle, which is not welcomed in networks. To the best of our knowledge, few papers deal with the research of Hopf bifurcation of system (1). Simulated by this reason and motivated by work above, we consider the Hopf bifurcation of system (1) in this paper.

This paper is organized as follows. In Section 2, we show that the complex Hopf bifurcation phenomenon at the positive equilibrium of the system (1) can occur as the delay crosses a critical value by choosing the delay as a bifurcation parameter. In Section 3, explicit formulae for the direction and stability of the Hopf bifurcation are derived by using the normal form theory and center manifold theorem. In Section 4, some numerical simulations are carried out to verify the theoretical results. A brief discussion is given to conclude this work in Section 5.

2. Stability and Existence of Local Hopf Bifurcation

In this section, we mainly focus on the local stability of positive equilibrium and existence of local Hopf bifurcation. It is not difficult to verify that if the basic reproduction number R0=(1-p)bβ/d(δ+dα1+γ)>1, system (1) has a unique positive equilibrium D*(S*,I*,Q*,R*), where (2)S*=δ+d+α1+γβ,I*=(1-p)bβ-d(δ+d+α1+γ)β(δ+d+α1+γ),Q*=δI*ɛ+d+α2,R*=pb+γI*+ɛQ*d. The linearization of system (1) about the positive equilibrium D* is (3)dS(t)dt=a1S(t)+b1S(t-τ)+b2I(t-τ),dI(t)dt=a2I(t)+b3S(t-τ)+b4I(t-τ),dQ(t)dt=a3I(t)+a4Q(t),dR(t)dt=a5I(t)a6Q(t)+a7R(t), where (4)a1=-d,a2=-ɛ+d+α1+γ,a3=δ,a4=-(ɛ+d+α2),a6=ɛ,a7=-d,b1=-βI*,b2=-βS*,b3=βI*,b4=βS*. The characteristic equation of system (1) is (5)λ4+A3λ3+A2λ2+A1λ+A0+(B3λ3+B2λ2+B1λ+B0)e-λτ=0, where (6)A0=a1a2a4a7,A1=-a1a2(a4+a7)-a4a7(a1+a2),A2=a1a2+a4a7+(a1+a2)(a4+a7),A3=-(a1+a2+a4+a7),B0=a4a7(a1b4+a2b1),B1=-a4a7(b1+b4)-(a4+a7)(a1b4+a2b1),B2=a1b4+a2b1+(a4+a7)(b1+b4),B3=-(b1+b4). For τ=0, (7) reduces to (7)λ4+A3*λ3+A2*λ2+A1*λ+A0*=0, where (8)A0*=A0+B0,A1*=A1+B1,A2*=A2+B2,A3*=A3+B3. By the Routh-Hurwitz criterion, if condition (H1)  (10)–(15) holds, D* is locally asymptotically stable in the absence of delay: (9)Det1=A3*>0,(10)Det2=A3*1A1*A2*>0,(11)Det3=A3*10A1*A2*A3*0A0*A1*>0,(12)Det4=A3*100A1*A2*A3*10A0*A1*A2*000A0*>0. For τ>0, let iω(ω>0) be the root of (7). Then, we can get (13)g1(ω)cosτω-g2(ω)sinτω=g3(ω),g1(ω)sinτω+g2(ω)cosτω=g4(ω), where (14)g1(ω)=B1ω-B3ω3,g2(ω)=B0-B2ω2,g3(ω)=A3ω3-A1ω,g4(ω)=A2ω2-ω4-A0. Then, we can obtain the following equation with respect to ω: (15)ω8+c3ω6+c2ω4+c1ω2+c0, where (16)c0=A02-B02,c1=A12-B12-2A0A2+2B0B2,c2=A22-B22+2A0-2A1A3+2B1B3,c3=A32-B32-2A2. Let ω2=v; then (15) becomes (17)v4+c3v3+c2v2+c1v+c0. If the coefficients of system (1) are given, then one can get the roots of (17) by Matlab software package easily. In order to give the main results in this paper, we make the following assumption.

( H 2 )   Equation (17) has at least one positive root.

If condition (H2) holds, we know that (15) has at least a positive root ω0 such that (7) has a pair of purely imaginary roots ±iω0. The corresponding critical value of the delay is (18)τ0=1ω0arccosm6ω06+m4ω04+m2ω02+m0n6ω06+n4ω04+n2ω02+n0, where (19)m0=-A0B0,m2=A0B2-A1B1+A2B0,m4=A1B3-A2B2+A3B1-B0,m6=B2-A3B3,n0=B02,n2=B12-2B0B2,n4=B22-2B1B3,n6=B32. Substituting λ(τ) into the left side of (7) and taking the derivative with respect to τ, one can obtain (20)dλdτ-1=-4λ3+3A3λ2+2A2λ+A1λ(λ4+A3λ3+A2λ2+A1λ+A0)+3B3λ2+2B2λ+B1λ(B3λ3+B2λ2+B1λ+B0). Thus, (21)Redλdττ=τ0-1=f(v*)g1ω02+g2ω02, where v*=ω02 and f(v)=v4+c3v3+c2v2+c1v+c0.

Thus, if condition H3fv*0, then Re[dλ/dτ]τ=τ0-1. According to the Hopf bifurcation theorem in , we have the following results.

Theorem 1.

If the conditions (H1)(H3) hold, then the positive equilibrium D*(S*,I*,Q*,R*) of system (1) is asymptotically stable for τ[0,τ0); system (1) undergoes a Hopf bifurcation at the positive equilibrium D*(S*,I*,Q*,R*) when τ=τ0 and a family of periodic solutions bifurcating from the positive equilibrium D*(S*,I*,Q*,R*) near τ=τ0.

3. Direction and Stability of the Hopf Bifurcation

In this section, we investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using the normal form theory and the center manifold theorem in .

Let u1(t)=S(t)-S*, u2(t)=I(t)-I*, u3(t)=Q(t)-Q*, u4(t)=R(t)-R*, and τ=τ0+μ, μR and normalize the time delay by t(t/τ). Then system (1) can be transformed into an FDE as (22)u˙(t)=Lμut+F(μ,ut), where (23)ut=u1t,u2t,u3t,u4tTC-1,0,R4,Lμϕ=τ0+μAtrixϕ0+Btrixϕ-1,F(μ,ϕ)=τ0+μ-βϕ1-1ϕ2-1βϕ1-1ϕ2-100, where (24)Atrix=a10000a2000a3a400a5a6a7,Btrix=b1b200b3b40000000000. By the Riesz representation theorem, there is a 4×4 matrix function with bounded variation components η(θ,μ), θ[-1,0] such that (25)Lμϕ=-10dη(θ,μ)ϕ(θ),ϕC-1,0,R4. In fact, we choose (26)η(θ,μ)=(τ0+μ)(Atrixδ(θ)+Btrixδ(θ+1)), where δ is the Dirac delta function.

For ϕC([-1,0],R5), we define (27)A(μ)ϕ=dϕ(θ)dθ,-1θ<0,-10dη(θ,μ)ϕ(θ),θ=0,R(μ)ϕ=0,-1θ<0,F(μ,ϕ),θ=0. Then system (22) is equivalent to the following operator equation: (28)u˙(t)=A(μ)ut+R(μ)ut. Next, we define the adjoint operator A* of A(29)A*(φ)=-dφ(s)ds,0<s1,-10dηT(s,0)φ(-s),s=0, and a bilinear inner product (30)φs,ϕθ=φ¯(0)ϕ(0)-θ=-10ξ=0θφ¯(ξ-θ)dη(θ)ϕ(ξ)dξ, where η(θ)=η(θ,0).

Let ρ(θ)=(1,ρ2,ρ3,ρ4)Teiω0τ0θ be the eigenvector of A(0) corresponding to +iω0τ0 and let ρ*(s)=D(1,ρ2*,ρ3*,ρ4*)eiω0τ0s be the eigenvector of A*(0) corresponding to -iω0τ0. From the definition of A(0) and A*(0) and by a simple computation, we obtain (31)ρ2=iω0-a1-b1e-iω0τ0b2e-iω0τ0,ρ3=a3(iω0-a1-b1e-iω0τ0)b2(iω0-a4)e-iω0τ0,ρ4=a5(iω0-a1-b1e-iω0τ0)b2(iω0-a7)e-iω0τ0+a3a6(iω0-a1-b1e-iω0τ0)b2(iω0-a4)(iω0-a7)e-iω0τ0,ρ2*=-iω0+a1+b1eiω0τ0b3eiω0τ0,ρ3*=-a6(iω0+a2+b4eiω0τ0)ρ2*ia5ω0+a4a5-a3a6,ρ4*=-(iω0+a4)(iω0+a2+b4eiω0τ0)ρ2*ia5ω0+a4a5-a3a6. From the definition of φ(s),ϕ(θ), we can obtain (32)q*s,qθ=D¯1+ρ2ρ¯2*+ρ3ρ¯3*+ρ4ρ¯4*b1+b3ρ¯2*+ρ2b2+b4ρ¯2*τ0e-iω0τ0b1+b3ρ¯2*+ρ2b2+b4ρ¯2*hhhhh+τ0e-iω0τ0b1+b3ρ¯2*+ρ2b2+b4ρ¯2*. Then we choose (33)D¯=1+ρ2ρ¯2*+ρ3ρ¯3*+ρ4ρ¯4*b1+b3ρ¯2*+ρ2b2+b4ρ¯2*τ0e-iω0τ0b1+b3ρ¯2*+ρ2b2+b4ρ¯2*hhh+τ0e-iω0τ0b1+b3ρ¯2*+ρ2b2+b4ρ¯2*-1 such that q*,q=1, q*,q¯=0.

Following the algorithms given in  and using similar computation process in , we can get the coefficients which determine the direction and stability of the Hopf bifurcation: (34)g20=2βτ0D¯ρ(1)(-1)ρ(2)(-1)(ρ¯2*-1),g11=βτ0D¯ρ1-1ρ¯(2)(-1)hhhhh+ρ¯1(-1)ρ2(-1)(ρ¯2*-1),g02=2βτ0D¯ρ¯1-1ρ¯(2)(-1)(ρ¯2*-1),g21=2βτ0D¯(ρ¯2*-1)W11(1)(-1)ρ(2)(-1)+12W202-1ρ¯(1)(-1)hhhhhhhhhhhhhh+12W201-1ρ¯(2)(-1)hhhhhhhhhhhhhh+W11(2)(-1)ρ(1)(-1)hhhhhhhhhhhhhhW11(1)(-1)ρ(2)(-1)+12W202-1ρ¯(1)(-1), with (35)W20θ=ig20q(0)ω0τ0eiω0τ0θ+ig¯02q¯(0)3ω0τ0e-iω0τ0θ+E1e2iω0τ0θ,W11(θ)=-ig11q(0)ω0τ0eiω0τ0θ+ig¯11q¯(0)ω0τ0e-iω0τ0θ+E2, where E1 and E2 can be determined by the following equations, respectively: (36)E1=2a11a1200a21a22000-a3a3300-a5-a6a44-1E1(1)E1(2)00,E2=-b11b200b3b22000a3a400a5a6a7-1E2(1)E2(2)00, with (37)a11=2iω0-a1-b1e-2iω0τ0,a12=-b2e-2iω0τ0,a21=-b3e-2iω0τ0,a22=2iω0-a2-b4e-2iω0τ0,a33=2iω0-a4,a44=2iω0-a7,b11=a1+b1,b22=a2+b4,E1(1)=-βρ(1)(-1)ρ(2)(-1),E1(2)=βρ(1)(-1)ρ(2)(-1),E2(1)=-βρ1-1ρ¯2-1+ρ¯1-1ρ2-1,E2(2)=βρ1-1ρ¯2-1+ρ¯1-1ρ2-1. Then, we can get the following coefficients: (38)C1(0)=i2ω0τ0g11g20-2g112-g0223+g212,μ2=-Re{C1(0)}Re{λ(τ0)},β2=2Re{C1(0)},T2=-ImC10+μ2Imλτ0ω0τ0. In conclusion, we have the following results.

Theorem 2.

For system (1), if μ2>0 (μ2<0), then the Hopf bifurcation is supercritical (subcritical). If β2>0 (β2<0), then the bifurcating periodic solutions are stable (unstable). If T2>0 (T2<0), then the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Simulation and Discussion

In this section, a numerical example is given to support the theoretical results in Sections 2 and 3. Let p=0.2, β=0.1, d=0.01, α1=0.01, γ=0.5, δ=0.1, ɛ=0.1, α2=0.02, and b=10. Then, we get a particular case of system (1): (39)dS(t)dt=8-0.1S(t-τ)I(t-τ)-0.01S(t),dI(t)dt=0.1S(t-τ)I(t-τ)-0.62I(t),dQ(t)dt=0.1I(t)-0.13Q(t),dR(t)dt=0.5I(t)+2+0.1Q(t)-0.01R(t). Then, we can get R0=129.0323>1 and the unique positive equilibrium D*(6.2,12.8032,9.8486,938.646) of system (39). By some complex computation, it can be verified that condition (H1) is satisfied for system (39). Further, we obtain ω0=0.4107, τ0=1.3265. By Theorem 1, we can conclude that when τ[0,1.3265) the positive equilibrium D*(6.2,12.8032,9.8486,938.646) is asymptotically stable. This property can be illustrated by Figures 1, 2, and 3. However, if we choose τ=1.3625>τ0=1.3265, the positive equilibrium D*(6.2,12.8032,9.8486,938.646) becomes unstable and a Hopf bifurcation occurs, which can be illustrated by Figures 4, 5, and 6. Furthermore, we obtain λ(τ0)=1.6465-0.8380i, C1(0)=-11.0407+1.3002i. Then, from (38) we get μ2=6.7482>0, β2=-22.1048, and T2=7.9935>0. Therefore, we can know that the Hopf bifurcation of system (39) is supercritical, the bifurcated periodic solutions are stable, and the period of the periodic solutions increases according to Theorem 2.

The track of the states S, I, Q, and R for τ=1.3025<1.3265=τ0.

The phase plot of the states S, I, and R for τ=1.3025<1.3265=τ0.

The phase plot of the states I, Q, and R for τ=1.3025<1.3265=τ0.

The track of the states S, I, Q, and R for τ=1.3625>1.3265=τ0.

The phase plot of the states S, I, and R for τ=1.3625>1.3265=τ0.

The phase plot of the states I, Q, and R for τ=1.3625>1.3265=τ0.

In addition, according to the numerical simulation, we find that the onset of the Hopf bifurcation can be delayed by decreasing the number of new nodes connected to a network or increasing the immunization rate of the new nodes. Therefore, the managers of a real network should control the number of the new nodes connected to network and strengthen the immunization of the new nodes in order to delay and control the onset of the Hopf bifurcation, so as to make the propagation of computer viruses be predicted and controlled easily.

5. Conclusion

In this paper, the problem of Hopf bifurcation for a delayed SIQR computer virus model has been studied. The stability of the positive equilibrium and the existence of Hopf bifurcation under this model are analyzed. It has been found that when the delay is suitable small (τ<τ0), the computer virus mode is asymptotically stable. In this case, the characteristics of the propagation of computer viruses can be easily predicted and controlled. However, if the delay passes though the critical value τ0, a Hopf bifurcation occurs. Then, the propagation of computer viruses becomes unstable and out of control. Furthermore, the properties of the Hopf bifurcation such as direction and stability have also been investigated in detail. Finally, numerical results have been presented to verify the analytical predictions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous referees and the editor for their valuable comments and suggestions on the paper. The research was supported by the National Nature Science Foundation of China (61273070), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2014A005).

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