MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 319174 10.1155/2013/319174 319174 Research Article Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network Zhang Zizhen 1,2 Yang Huizhong 1 Cannarsa Piermarco 1 Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education) Jiangnan University Wuxi 214122 China jiangnan.edu.cn 2 School of Management Science and Engineering Anhui University of Finance and Economics Bengbu 233030 China aufe.edu.cn 2013 20 11 2013 2013 08 06 2013 04 10 2013 07 10 2013 2013 Copyright © 2013 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 SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

1. Introduction

Since the pioneering work of Kephart and White [1, 2], classical epidemiological models have been used to understand and predict virus propagations in computer network by many authors . In , Thommes and Coates proposed a modified version of the SEI model to predict the virus propagation in a network. In , Yuan and Chen proposed an e-SEIR model and studied the behavior of virus propagation with the presence of antivirus countermeasures. Yuan and Chen  supposed that the recovered computers have a permanent immunization period and can no longer be infected. Considering that there is no permanent recovery from the virus till a node is attached to the computer network, Mishra and Pandey  proposed the following epidemic model for the transmission of worms with vertical transmission: (1)dS(t)dt=b-βI(t)S(t)-pbE(t)-qbI(t)-dS(t)+ζR(t),dE(t)dt=βI(t)S(t)+pbE(t)+qbI(t)-εE(t)-dE(t),dI(t)dt=εE(t)-γI(t)-dI(t)-ηI(t),dR(t)dt=γI(t)-dR(t)-ζR(t), where S(t), E(t), I(t), and R(t) denote the numbers of nodes at time t in susceptible, exposed, infectious and recovered states, respectively. b is the recruitment rate of susceptible nodes to the computer network. d is the crashing rate of nodes due to the reason other than the attack of worms. p and q are the fractions of the new nodes from the exposed and the infectious classes, respectively, that are introduced into the exposed class. β is the transmission rate. ε, γ, and ζ are the state transition rates. η is the crashing rate of nodes due to the attack of worms. Mishra and Pandey  discussed the characteristic of the basic reproduction number and investigated the global stability of system (1) by constructing a new Liyapunov function.

It is well known that time delays can play a complicated role on the dynamics of a system. They can cause the stable equilibrium of a system to become unstable and make a system bifurcate periodic solutions. Dynamical systems with delay have been studied by many scholars . Motivated by the works above and considering that the antivirus software may use a period to clean the worms in one node and that the recovered nodes may have a temporary immunity period due to the antivirus software, we consider the following delayed system in this paper: (2)dS(t)dt=b-βI(t)S(t)-pbE(t)-qbI(t)-dS(t)+ζR(t-τ2),dE(t)dt=βI(t)S(t)+pbE(t)+qbI(t)-εE(t)-dE(t),dI(t)dt=εE(t)-γI(t-τ1)-dI(t)-ηI(t),dR(t)dt=γI(t-τ1)-dR(t)-ζR(t-τ2), where τ1 is the period to clean the worms in one node and τ2 is the temporary immunity period. For the convenience of analysis, throughout this paper, we assume that the period to clean the worms in one node and the temporary immunity period are the same.

Let τ1=τ2=τ, and then system (2) becomes the following form: (3)dS(t)dt=b-βI(t)S(t)-pbE(t)-qbI(t)-dS(t)+ζR(t-τ),dE(t)dt=βI(t)S(t)+pbE(t)+qbI(t)-εE(t)-dE(t),dI(t)dt=εE(t)-γI(t-τ)-dI(t)-ηI(t),dR(t)dt=γI(t-τ)-dR(t)-ζR(t-τ).

The main purpose of this paper is to investigate the effects of the time delay on the dynamics of system (3). We study the stability of the positive equilibrium of system (3) and find the critical value of the time delay where a Hopf bifurcation occurs. We also study the properties of the Hopf bifurcation such as direction and stability.

This paper is organized as follows. In Section 2, we investigate local stability of the positive equilibrium and obtain the sufficient conditions for the existence of local Hopf bifurcation. In Section 3, we determine the direction and the stability of the Hopf bifurcation by using the normal form theory and center manifold theorem. In order to testify the theoretical analysis, a numerical example is presented in Section 4.

2. Stability and Existence of Local Hopf Bifurcation

It is not difficult to verify that if (4)R0=bβε+bdqε+bdp(d+γ+η)d(d+ε)(d+γ+η)>1,(d+ε)(d+γ+η)>pb(d+γ+η)+qbε, system (2) has a unique positive equilibrium D*(S*,E*,I*,R*), where (5)S*=(d+ε)(d+γ+η)-pb(d+γ+η)-qbεβε,I*=(d+ζ)[bβε+d(pb(d+γ+η)222222222+qbε-(d+ε)(d+γ+η))]×(β(d+ε)(d+ζ)(d+γ+η)-βγεζ)-1,E*=d+γ+ηεI*,R*=γd+ζI*.

R 0 is called the basic reproduction number.

The Jacobian matrix of system (3) about the positive equilibrium D* is(6)J(D*)=(λ-a11-a12-a12-b14e-λτ-a21λ-a22-a2300-a32λ-a33-b33e-λτ000-b43e-λτλ-a44-b43e-λτ),where (7)a11=-(d+βI*),a12=-pb,a13=-(qb+βS*),a21=βI*,a22=pb-d-ε,a23=qb+βS*,a32=ε,a33=-(d+η),a44=-d,b14=ζ,b33=-γ,b43=γ,b44=-ζ. Thus, the characteristic equation of system (2) at the positive equilibrium D* is (8)λ4+A3λ3+A2λ2+A1λ+A0+(B3λ3+B2λ2+B1λ+B0)e-λτ+(C2λ2+C1λ+C0)e-2λτ=0, where (9)A0=a11a44(a22a33-a23a32)+a21a44(a13a32-a12a33),A1=(a33+a44)(a12a21-a11a22)+a23a32(a11+a44)-a33a44(a11+a22)+a13a21a32,A2=(a11+a22)(a33+a44)+a11a22+a33a44-a12a21-a23a32,A3=-(a11+a22+a33+a44),B0=(a11a22-a12a21)(a33b44+a44b33)+(a13a21-a11a23)a32b44,B1=(b33+b44)(a12a21-a11a22)+a23a32b44-(a11+a22)(a33b44+a44b33),B2=(a11+a22)(b33+b44)+a33b44+a44b33,B3=-(b33+b44),C2=b33b44,C1=b33b44(a11+a22),C0=(a11a22-a12a21)b33b44-a21a32b14b43.

Multiplying eλτ on both sides of (8), it is easy to get (10)B3λ3+B2λ2+B1λ+B0+(λ4+A3λ3+A2λ2+A1λ+A0)eλτ+(C2λ2+C1λ+C0)e-λτ=0.

When τ=0, (10) reduces to (11)λ4+A13λ3+A12λ2+A11λ+A10=0, where (12)A10=A0+B0+C0,A11=A1+B1+C1,A12=A2+B2+C2,A13=A3+B3.

By the Routh-Hurwitz criterion, the sufficient conditions for all roots of (11) to have a negative real part given in the following form: (13)D1=A13>0,D2=det(A13    1A11    A12)>0,D3=det(A13    1    0A11    A12    A130    A10    A11)>0,(14)D4=det(A13    1    0    0A11    A12    A13    10    A10    A11    A120    0    0    A10)>0.

Thus, if the condition (H1) holds, which means that (13) and (14) are satisfied, the positive equilibrium D* is locally asymptotically stable in the absence of delay.

For τ>0, let λ=iω(ω>0) be a root of (10). Then, we can get (15)(ω4-(A2+C2)ω2+A0+C0)cosτω-((A1-C1)ω-A3ω3)sinτω=B2ω2-B0,(ω4-(A2-C2)ω2+A0-C0)sinτω+((A1+C1)ω-A3ω3)cosτω=B3ω3-B1ω. Then, we can get (16)sinτω=p7ω7+p5ω5+p3ω3+p1ωω8+q6ω6+q4ω4+q2ω2+q0,cosτω=p6ω6+p4ω4+p2ω2+p0ω8+q6ω6+q4ω4+q2ω2+q0, where (17)p0=B0(C0-A0),p1=(A1+C1)B0-(A0+C0)B1,p2=(A2-C2)B0+(A0-C0)B2+(C1-A1)B1,p3=(A0+C0)B3-(A1+C1)B2+(A2+C2)B1-A3B0,p4=(A1-C1)B3-(A2-C2)B2+A3B1-B0,p5=A3B2-B1-(A2+C2)B3,p6=B2-A3B3,p7=B3,q0=A02-C02,q2=A12-C12-2A0A2+2C0C2,q4=A22-C22+2A0-2A1A3,q6=A32-2A2.

It is well known that sin2τω+cos2τω=1. Thus, we have (18)ω16+e7ω14+e6ω12+e5ω10+e4ω8+e3ω6+e2ω4+e1ω2+e0=0, where (19)e0=q02-p02,e1=2q0q2-2p0p2-p12,e2=q22+2q0q4-p22-2p0p4-2p1p3,e3=2q0q6+2q2q4-p32-2p0p6-2p1p5-2p2p4,e4=q42+2q0+2q2q6-p42-2p1p7-2p2p6-2p3p5,e5=2q2+2q4q6-2p3p7-2p4p6-p52,e6=q62-p62+2q4-2p5p7,e7=2q6-p72.

Let ω2=v, and then (18) becomes (20)v8+e7v7+e6v6+e5v5+e4v4+e3v3+e2v2+e1v+e0=0.

In order to give the main results in this paper, we made the following assumption.

( H 2 ) Equation (20) has at least one positive real root.

Suppose that the condition (H2) holds. Without loss of generality, we suppose that (20) has eight positive real roots, which are denoted as v1,v2,,v8, respectively. Then, (18) has eight positive roots ωk=vk,k=1,2,,8. For every fixed ωk, the corresponding critical value of time delay is (21)τk(j)=1ωkarccosp6ω6+p4ωk4+p2ωk2+p0ω8+q6ωk6+q4ωk4+q2ωk2+q0+2jπωk,22222222222222222k=1,2,,8,j=0,1,2,.

Let (22)τ0=min{τk(0)},k{1,2,,8},ω0=ωk|τ=τ0.

Taking the derivative of with respect to (10), it is easy to obtain (23)[dλdτ]-1=(3B3λ2+2B2λ+B1+(2C2λ+C1)e-λτ+(4λ3+3A3λ2+2A2λ+A1)eλτ)×((C2λ3+C1λ2+C0λ)e-λτ-(λ5+A3λ4+A2λ3+A1λ2+A0λ)eλτ))-1-τλ. Then, we have (24)Re[dλdτ]τ=τ0-1=PRQR+PIQIQR2+QI2,PR=(A1+C1-3A3ω02)cosτ0ω0+2(2ω03-(A2-C2)ω0)sinτ0ω0-3B3ω02+B1,PI=(A1-C1-3A3ω02)sinτ0ω0+2(ω03-(A2+C2)ω0)cosτ0ω0+2B2ω0,QR=(A1ω02-A3ω04)cosτ0ω0+(ω05-A2ω03+A0ω0)sinτ0ω0,QI=(A1ω02-A3ω04)sinτ0ω0-(ω05-A2ω03+A0ω0)cosτ0ω0.

Obviously, if the condition (H3)PRQR+PIQI0 holds, then Re[dλ/dτ]τ=τ0-10. Thus, according to the Hopf bifurcation theorem in , we have the following results.

Theorem 1.

Suppose that the conditions (H1)–(H3) hold. The positive equilibrium D*(S*,E*,I*,R*) of system (3) is asymptotically stable for τ[0,τ0). System (3) undergoes a Hopf bifurcation at the positive equilibrium D*(S*,E*,I*,R*) when τ=τ0 and a family of periodic solutions bifurcate from the positive equilibrium D*(S*,E*,I*,R*) near τ=τ0.

3. Properties of the Hopf Bifurcation

In the previous section, we have obtained the conditions for the Hopf bifurcation to occur when τ=τ0. In this section, we will study properties of the Hopf bifurcation such as direction and stability by using the normal form theory and the center manifold theorem in .

Let u1(t)=S(t)-S*, u2(t)=E(t)-E*, u3(t)=I(t)-I*, u4(t)=R(t)-R*, and τ=τ0+μ,μR and normalize t(t/τ). Then system (3) can be transformed into the following form: (25)u˙(t)=Lμut+F(μ,ut), where ut=(u1(t),u2(t),u3(t),u4(t))TC=C([-1,0],R4) and Lμ:CR4 and F:R×CR4 are given, respectively, by (26)Lμϕ=(τ0+μ)(a11a12a130a21a22a2300a32a330000a44)ϕ(0)-(τ0+μ)(000b14000000b33000b43b44)ϕ(-1),F(μ,ϕ)=(τ0+μ)(-βϕ1(0)ϕ3(0)βϕ1(0)ϕ3(0)00).

By the Riesz representation theorem, there exists a 4  ×  4 matrix function η(θ,μ):[-1,0]R4 whose elements are of bounded variation such that (27)Lμϕ=-10dη(θ,μ)ϕ(θ),ϕC.

In fact, we choose (28)η(θ,μ)=(τ0+μ)(a11a12a130a21a22a2300a32a330000a44)δ(θ)-(τ0+μ)(000b14000000b33000b43b44)δ(θ+1), where δ is the Dirac delta function.

For ϕC([-1,0],R4), we define (29)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 (25) can be transformed into the following operator equation: (30)u˙(t)=A(μ)ut+R(μ)ut.

The adjoint operator A* of A is defined by (31)A*(φ)={-dφ(s)ds,0<s1-10dηT(s,0)φ(-s),s=0, associated with a bilinear form: (32)φ(s),ϕ(θ)=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 corresponding to iω0τ0 and let ρ*(s)=D(1,ρ2*,ρ3*,ρ4*)eiω0τ0s be the eigenvector of A* corresponding to -iω0τ0. From the definition of A(0) and A*(0), we can get (33)ρ2=-iω0-a33-b33e-iτ0ω0(iω0-a22)(iω0-a33-b33e-iτ0ω0)-a23a32,ρ3=a21a32(iω0-a22)(iω0-a33-b33e-iτ0ω0),ρ4=b43e-τ0ω0iω0-a44+b33e-τ0ω0ρ3,ρ2*=-a11iω0+a21,r3*=-(iω0(a12-a11)+a12a21-a11a22a32(iω0+a21),ρ4*=b14eiτ0ω0iω0+a44+b44eiτ0ω0.

From (32), we have (34)ρ*,ρ=D-[1+ρ2ρ-2*+ρ3ρ-3*+ρ4ρ-4*+τ0e-iτ0ω0×(ρ3(b33ρ-3*+b43ρ-4*)+ρ4(b14+b44ρ-4*))τ0e-iτ0ω0].

Then, we choose (35)D-=[1+ρ2ρ-2*+ρ3ρ-3*+ρ4ρ-4*+τ0e-iτ0ω0×(ρ3(b33ρ-3*+b43ρ-4*)+ρ4(b14+b44ρ-4*))τ0e-iτ0ω0]-1 such that q*,q=1, q*,q-=0.

Next, we can obtain the coefficients which will be used to determine the properties of the Hopf bifurcation by following the algorithms introduced in  and using a computation process as in : (36)g20=2βτ0D-ρ3(ρ-2*-1),g02=2βτ0D-ρ-3(ρ-2*-1),g11=βτ0D-(r3+r-3)(r-2*-1)),g21=2βτ0D-(ρ-2*-1)(W11(1)(0)ρ3+12W20(1)(0)ρ-3+W11(3)(0)+12W20(3)(0)), with (37)W20(θ)=ig20q(0)τ0ω0eiτ0ω0θ+ig-02q-(0)3τ0ω0e-iτ0ω0θ+E1e2iτ0ω0θ,W11(θ)=-ig11q(0)τ0ω0eiτ0ω0θ+ig-11q-(0)τ0ω0e-iτ0ω0θ+E2, where E1 and E2 can be determined by the following equations, respectively:(38)E1=2(2iω0-a11-a12-a13-b14e-2iω0τ0-a212iω0-a22-a2300-a322iω0-a33-b33e-2iω0τ0000-b43e-2iω0τ02iω0-a44-b44e-2iω0τ0)-1(E1(1)E1(2)00),E2=-(a11a12a13b14a21a22a2300a32a33+b33000  b43a44+b44)-1(E2(1)E2(2)00),with (39)E1(1)=-βρ3,E1(2)=βρ3,E2(1)=-β(ρ3+ρ-3),E2(2)=β(ρ3+ρ-3).

Therefore, we can calculate the following values: (40)C1(0)=i2τ0ω0(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re{C1(0)}Re{λ(τ0)},β2=2Re{C1(0)},T2=-Im{C1(0)}+μ2Im{λ(τ0)}τ0ω0. Based on the discussion above, we can obtain the following results.

Theorem 2.

For system (3), if μ2>0(μ2<0), the Hopf bifurcation is supercritical (subcritical). If β2<0(β2>0), the bifurcating periodic solutions are stable (unstable). If T2>0(T2<0) the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Simulation

In this section, we present a numerical example to verify the theoretical analysis in Sections 2 and 3. Let b=0.7, d=0.06, p=0.02, q=0.01, β=0.1, ε=0.2, γ=0.05, η=0.01, and ζ=0.1. We can get a particular case of system (3): (41)dS(t)dt=0.7-0.1I(t)S(t)-0.014E(t)-0.007I(t)-0.06S(t)+0.1R(t-τ),dE(t)dt=0.1I(t)S(t)+0.014E(t)+0.007I(t)-0.2E(t)-0.06E(t),dI(t)dt=0.2E(t)-0.05I(t-τ)-0.06I(t)-0.01I(t),dR(t)dt=0.05I(t-τ)-0.06R(t)-0.1R(t-τ).

Then, we can get that R0=7.5774>1, (ε+d)(d+γ+η)=0.0312>pb(d+γ+η)+qbε=0.0031, and system (41) has a unique positive equilibrium D*(1.4050,1.5422,4.9350, and 2.9610). Further, we have D1=1.0795>0, D2=0.3360>0, D3=0.0141>0, and D4=2.8200e-004>0. That is, condition (H1) holds. Thus, we can obtain ω0=0.1245, τ0=13.6109, and λ(τ0)=0.0073+0.0299i. Therefore, from Theorem 1, the positive equilibrium D* is asymptotically stable when τ[0,13.6109) and unstable when τ>13.6109. As can be seen from Figures 1, 2, and 3, when τ=12.5[0,13.6109), the positive equilibrium D* is asymptotically stable. However, when τ=14.95>13.6109, the positive equilibrium D* will lose its stability, a Hopf bifurcation occurs, and a family of periodic solutions bifurcate from the positive equilibrium D*. This property can be illustrated by Figures 4, 5, and 6. In addition, from (40), we can get C1(0)=-0.00904+0.6833i, μ2=12.3836>0, β2=-0.1808<0, and T2=-0.6217<0. Thus, according to Theorem 2, we know that the Hopf bifurcation is supercritical. The bifurcating periodic solutions are stable and the period of the periodic solutions decreases.

The track of the states S, E, I, and R for τ=12.5000<13.6109=τ0.

The phase plot of the states S, E, and I for τ=12.5000<13.6109=τ0.

The phase plot of the states E, I, and R for τ=12.5000<13.6109=τ0.

The track of the states S, E, I, and R for τ=14.9500>13.6109=τ0.

The phase plot of the states S, E, and I for τ=14.9500>13.6109=τ0.

The phase plot of the states E, I, and R for τ=14.9500>13.6109=τ0.

5. Conclusions

In this paper, we incorporate the time delay due to the period the antivirus software has to use to clean the worms in one node and the temporary immunity period of the recovered nodes into the model considered in the literature  and get a delayed SEIRS epidemic model for the transmission of worms in computer network through vertical transmission. The effects of the time delay on the dynamics of the model are investigated. It is found that the time delay can play a complicated role on the model by analyzing the distribution of the roots of the associated characteristic equation. When the time delay is suitable small, the positive equilibrium is asymptotically stable. However, a local Hopf bifurcation occurs and a branch of periodic solutions bifurcates from the positive equilibrium when the delay passes through the critical value τ0. Furthermore, the properties of the Hopf bifurcation such as direction and stability are determined by using the normal form theory and center manifold theorem. In order to verify the theoretical analysis, a numerical example is also included.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61273070), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institution, Major Science Foundation Subject for the Education Department of Anhui Province under Project no. ZD200905, and a Project funded by the Ministry of Education (12YJA630136).

Kephart J. O. White S. R. Measuring and modeling computer virus prevalence Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy May 1993 2 15 2-s2.0-0027150413 Kephart J. O. White S. R. Directed-graph epidemiological models of computer viruses Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy May 1991 343 358 2-s2.0-0026156688 Thommes R. W. Coates M. J. Modeling virus propagation in peer-to-peer networks Proceedings of the IEEE International Conference on Information, Communications and Signal Processing December 2005 981 985 2-s2.0-34147092603 Zou C. C. Towsley D. Gong W. On the performance of Internet worm scanning strategies Performance Evaluation 2006 63 7 700 723 2-s2.0-33646150900 10.1016/j.peva.2005.07.032 Yuan H. Chen G. Network virus-epidemic model with the point-to-group information propagation Applied Mathematics and Computation 2008 206 1 357 367 10.1016/j.amc.2008.09.025 MR2474981 ZBL1162.68404 Mishra B. K. Pandey S. K. Dynamic model of worms with vertical transmission in computer network Applied Mathematics and Computation 2011 217 21 8438 8446 10.1016/j.amc.2011.03.041 MR2802253 ZBL1219.68080 Wierman J. C. Marchette D. J. Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction Computational Statistics and Data Analysis 2004 45 1 3 23 10.1016/S0167-9473(03)00113-0 MR2041255 ZBL05373937 Kafai Y. B. White S. Understanding virtual epidemics: children's folk conceptions of a computer virus Journal of Science Education and Technology 2008 17 6 523 529 2-s2.0-56749178276 10.1007/s10956-008-9102-x Wang F. W. Zhang Y. K. Wang C. G. Ma J. Moon S. Stability analysis of a SEIQV epidemic model for rapid spreading worms Computers and Security 2010 29 4 410 418 2-s2.0-77951206095 10.1016/j.cose.2009.10.002 Piqueira J. R. C. Araujo V. O. A modified epidemiological model for computer viruses Applied Mathematics and Computation 2009 213 2 355 360 10.1016/j.amc.2009.03.023 MR2536658 ZBL1185.68133 Yao Y. Xie X. W. Guo H. Yu G. Gao F. X. Tong X. J. Hopf bifurcation in an Internet worm propagation model with time delay in quarantine Mathematical and Computer Modelling 2013 57 2635 2646 2-s2.0-79960561776 10.1016/j.mcm.2011.06.044 Ren J. G. Yang X. F. Yang L. X. Xu Y. Yang F. A delayed computer virus propagation model and its dynamics Chaos, Solitons and Fractals 2012 45 1 74 79 10.1016/j.chaos.2011.10.003 MR2863589 ZBL06196114 Bai H. Y. Zhai Y. H. Hopf bifurcation analysis for the model of the chemostat with one species of organism Abstract and Applied Analysis 2012 2013 7 10.1155/2013/829045 829045 MR3045058 ZBL1272.92040 Feng L. Liao X. Li H. Han Q. Hopf bifurcation analysis of a delayed viral infection model in computer networks Mathematical and Computer Modelling 2012 56 7-8 167 179 10.1016/j.mcm.2011.12.010 MR2947124 ZBL1255.34071 Zhu G. Wei J. Stability and Hopf bifurcation analysis of coupled optoelectronic feedback loops Abstract and Applied Analysis 2013 2013 11 10.1155/2013/829045 918943 MR3035277 ZBL1271.93124 Wang B. Shi P. Karimi H. R. Lim C. C. Observer-based sliding mode control for stabilization of a dynamic system with delayed output feedback Mathematical Problems in Engineering 2013 2013 6 537414 10.1155/2013/537414 Zhang T. L. Liu J. L. Teng Z. D. Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure Nonlinear Analysis. Real World Applications 2010 11 1 293 306 10.1016/j.nonrwa.2008.10.059 MR2570549 ZBL1195.34130 Wang R. Li J. Adaptive neural control for a class of outputs time-delay nonlinear systems Mathematical Problems in Engineering 2012 2012 16 10.1155/2012/852161 852161 MR2983806 ZBL1264.93110 Zhu G. Wei J. J. Synchronized Hopf bifurcation analysis in a delay-coupled semiconductor lasers system Journal of Applied Mathematics 2012 2012 20 10.1155/2012/257635 257635 MR2979448 ZBL1251.78012 Zhu W. L. Ruan X. F. Qin Y. Zhuang J. Exponential stability of stochastic nonlinear dynamical price system with delay Mathematical Problems in Engineering 2013 2013 9 10.1155/2013/168169 168169 MR3064553 Hassard B. D. Kazarinoff N. D. Wan Y. H. Theory and Applications of Hopf Bifurcation 1981 Cambridge, UK Cambridge University Press Xu C. J. He X. F. Stability and bifurcation analysis in a class of two-neuron networks with resonant bilinear terms Abstract and Applied Analysis 2011 2011 21 697630 10.1155/2011/697630 MR2802842 ZBL1218.37122