# Stability and Hopf Bifurcation Analysis for a Computer Virus Propagation Model with Two Delays and Vaccination

Authors: Zizhen Zhang; Yougang Wang; Dianjie Bi; Luca Guerrini
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

A further generalization of an SEIQRS-V (susceptible-exposed-infectious-quarantined-recovered-susceptible with vaccination) computer virus propagation model is the main topic of the present paper. This paper specifically analyzes effects on the asymptotic dynamics of the computer virus propagation model when two time delays are introduced. Sufficient conditions for the asymptotic stability and existence of the Hopf bifurcation are established by regarding different combination of the two delays as the bifurcation parameter. Moreover, explicit formulas that determine the stability, direction, and period of the bifurcating periodic solutions are obtained with the help of the normal form theory and center manifold theorem. Finally, numerical simulations are employed for supporting the obtained analytical results.


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## Body

## 1. Introduction

Computer viruses, including conventional viruses and network worms, can propagate among computers with no human awareness and popularization of Internet has been the major propagation channel of viruses [1, 2]. The past few decades have witnessed the great financial losses caused by computer viruses. Therefore, it is of considerable importance to investigate the laws describing propagation of computer viruses in order to provide some help with preventing computer viruses. For that purpose and in view of the fact that propagation of computer viruses among computers resembles that of biological viruses among a population, many dynamical models describing propagation of computer viruses across the Internet have been established by the scholars at home and abroad, such as conventional models [3–8], stochastic models [9–12], and delayed models [13–18]. There are also some other computer virus models [19–21] combined with network theory to investigate the impact of the network topology, the patch forwarding, and the network eigenvalue on the viral prevalence.As is known, vaccination is regarded as one of the most effective measures of preventing computer viruses and the awareness that there exist many infected computers would enhance the probability that the user of a susceptible computer will make his computer vaccinated [22, 23]. However, the mentioned models above neglect the influence of vaccination strategy on the propagation of computer viruses. Recently, considering the importance of vaccination, Kumar et al. [24] proposed the following SEIQRS-V computer virus propagation model:(1)dStdt=A-βStIt-dSt-ρSt+θRt+χVt,dEtdt=βStIt-dEt-γEt,dItdt=γEt-dIt-αIt-δIt-ηIt,dQtdt=δIt-dQt-αQt-εQt,dRtdt=εQt-dRt-θRt+ηIt,dVtdt=ρSt-dVt-χVt,where S(t), E(t), I(t), Q(t), R(t), and V(t) denote the numbers of the uninfected computers, the exposed computers, the infected computers, the quarantined computers, recovered computers, and vaccinated computers at time t, respectively. A is the birth rate of new computers in the network; d is the death rate of the computers due to the reason other than the attack of viruses; α is the death rate of computers due to the attack of viruses; β is the contact rate of the uninfected computers; ρ, θ, χ, γ, δ, η, and ε are the transition rates between the states in system (1).Obviously, system (1) neglects the delays in the procedure of viruses’ propagation and it is investigated under the assumption that the transition between the states is instantaneous. This is not reasonable with reality. For example, it needs a period to clean the viruses in the infected and quarantined computers for antivirus software and there is usually a temporary immunity period for the recovered and the vaccinated computers because of the effect of the antivirus software. In addition, a stability switch occurs even when an ignored delay is small for a dynamical system. Based on this, we introduce two delays into system (1) and get the following delayed system: (2)dStdt=A-βStIt-dSt-ρSt+θRt-τ2+χVt-τ2,dEtdt=βStIt-dEt-γEt,dItdt=γEt-dIt-αIt-δIt-ηIt-τ1,dQtdt=δIt-dQt-αQt-εQt-τ1,dRtdt=εQt-τ1-dRt-θRt-τ2+ηIt-τ1,dVtdt=ρSt-dVt-χVt-τ2,where τ1 is the time delay due to the period that antivirus software uses to clean the viruses in the infected and quarantined computers and τ2 is the time delay due to the temporary immunity period of the recovered and the vaccinated computers.To the best of our knowledge, until now, there is no good analysis on system (2). Therefore, it is meaningful to analyze the proposed system with two delays.The rest of this paper is organized as follows. In the next section, we analyze the threshold of Hopf bifurcation of system (2) by regarding different combination of the two delays as the bifurcation parameter. In Section 3, by means of the normal form theory and center manifold theorem, direction and stability of the Hopf bifurcation for τ1>0 and τ2>0 are investigated. Simulation results of system (2) are shown in Section 4. Finally, we finish the paper with conclusions in Section 5.


## 2. Analysis of Hopf Bifurcation

By direct computation, we know that ifAR0(d+χ)>d2+(ρ+χ)d and β(d+θ)(d+α+ε)>R0θεδ+R0θη(d+α+ε), then system (2) has a unique viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗), where (3)S∗=d+γd+α+δ+ηβγ=1R0,E∗=d+α+δ+ηγI∗,R∗=εδ+ηd+α+εd+θd+α+εI∗,V∗=ρd+χR0,Q∗=δd+α+εI∗,I∗=d+θd+α+εd2+ρ+χd-AR0d+χd+χR0θεδ+d+α+εR0θη-βd-βθ,R0=βγd+γd+α+δ+η.The linearized section of system (2) at P∗(S∗,E∗,I∗,Q∗,R∗,V∗) is as follows: (4)dStdt=a1St+a2It+c1Rt-τ2+c2Vt-τ2,dEtdt=a3St+a4Et+a5It,dItdt=a6Et-a7It+b1It-τ1,dQtdt=a8It+a9Qt+b2Qt-τ1,dRtdt=a10Rt+b3It-τ1+b4Qt-τ1+c3Rt-τ2,dVtdt=a11St+a12Vt+c4Vt-τ2,where (5)a1=-βI∗+d+ρ,a2=-βS∗,a3=βI∗,a4=-d+γ,a5=βS∗,a6=γ,a7=-d+α+δ,a8=δ,a9=-d+α,a10=-d,a11=ρ,a12=-d,b1=-η,b2=-ε,b3=η,b4=ε,c1=θ,c2=χ,c3=-θ,c4=-χ.Then, the characteristic equation for system (4) can be obtained: (6)λ6+A5λ5+A4λ4+A3λ3+A2λ2+A1λ+A0+B5λ5+B4λ4+B3λ3+B2λ2+B1λ+B0e-λτ1+C5λ5+C4λ4+C3λ3+C2λ2+C1λ+C0e-λτ2+D4λ4+D3λ3+D2λ2+D1λ+D0e-λτ1+τ2+E3λ3+E2λ2+E1λ+E0e-λτ1+2τ2+F4λ4+F3λ3+F2λ2+F1λ+F0e-2λτ1+G4λ4+G3λ3+G2λ2+G1λ+G0e-2λτ2+H3λ3+H2λ2+H1λ+H0e-λ2τ1+τ2+I2λ2+I1λ+I0e-2λτ1+τ2=0,with (7)A0=a9a12a1a4a7a10+a2a3a6a10-a1a5a6a10-a3a6b3c1,A1=a5a6a1a9a10+a12+a10a12a1+a9+a3a6b3c1a9+a12-a2a3a6a9a10+a9a12+a10a12-a1a4a7a9a10+a12-a10a12a1a4a7+a9+a7a9a1+a4,A2=a2a3a6a9+a10+a12+a1a4a7a9-a3a6b3c1-a5a6a1a9+a10a12+a1+a9a10+a12+a10a12a1a4+a7a9+a1+a4a7+a9+a10+a12a1a4a7+a9+a7a9a1+a4,A3=a5a6a1+a9+a10+a12-a2a3a6-a1a4a7+a9-a7a9a1+a4-a10a12a1+a4+a7+a9-a10+a12a1a4+a7a9+a1+a4a7+a9,A4=a1a4+a7a9+a10a12-a5a6+a1+a4a7+a9+a10+a12a1+a4+a7+a9,A5=-a1+a4+a7+a9+a10+a12,B0=a1a4a10a12a7b2+a9b1-a6a10a12b2a1a5+a2a3,B1=a5a6b2a1a10+a1a12+a10a12-a2a3a6b2a10+a12-a7b2+a9b1a1a4a10+a12+a10a12a1+a4B1-a1a4a10a12b1+b2B2=a7b2+a9b1a1a4+a10a12+a1+a4a10+a12+a2a3a6b2+b1+b2a1a4a10+a12+a10a12a1+a4,B3=a5a6b2-a7b2+a9b1a1+a4+a10+a12-b1+b2a1a4+a10a12+a1+a4a10+a12,B4=a7b2+a9b1+b1+b2a1+a4+a10+a12,B5=-b1+b2,C0=a1a4a7a9a10c4+a12c3+a9a10a11c2a4a7-a5a6+a6a9a10c4+a12c3a2a3-a1a5,C1=a9c3+c4a1a5a6-a2a3a6-a1a4a7+a5a6a11c2a9+a10+a6a1a5-a2a3+a5a9a10c4+a12c3-a11c2a4a7a9+a10+a9a10a4+a7,C2=c3+c4a1a4a7+a9+a7a9a1+a4+a10c4+a12c3a1a4+a7a9+a1+a4a7+a9-a5a6a10c4+a12c3-a11c2+a6c3+c4a2a3-a1a5-a5a9+a11c2a4a7+a9a10+a4+a7a9+a10,C3=a5a6c3+c4-a11c2a4+a7+a9+a10-a10c4+a12c3a1+a4+a7+a9-c3+c4a1a4+a7a9+a1+a4a7+a9,C4=a10c4+a11c2+a12c3+c3+c4a1+a4+a7+a9,C5=-c3+c4,D0=a4a1a10c4+a1a12c3+a10a11c2a7b2+a9b1+a6a3a8a12b4c4-a5a10a11b2c2-a6b2a1a5-a2a3a10c4+a12c3,D1=c3+c4a5b2a1a5-a2a3-a1a4a7b2+a9b1+a6a5a11b2c2-a3a8b4c4+a5a6b2a10c4+a12c3-a11c2a4a10b1+b2+a4+a10a7b2+a9b1-a10c4+a12c3a1a4b1+b2+a1+a4a7b2+a9b1,D2=a10c4+a12c3a7b2+a9b1+b1+b2a1+a4+c3+c4a1a4b1+b2+a1+a4a7b2+a9b1D2+a11c2a7b2+a9b1+a4+a10b1+b2-a5a6b2c3+c4,D3=-c3+c4a7b2+a9b1+b1+b2a1+a4-b1+b2a10c4+a11c2+a12c3,D4=b1+b2c3+c4,E0=a4c3a7b2+a9b1a1c4+a11c2+a6b2c3c4a2a3-a1a5-a6a3a9b3c1c4+a5a11b2c2c3,E1=a6c4a5b2c3+a3b3c1-a11c2c3b1a4+a9+b2a4+a7-c3c4a1a4b1+b2+a1+a4a7b2+a9b1,E2=c3c4a7b2+a9b1+a1+a4b1+b2+a11c2c3b1+b2,E3=-c3c4b1+b2,F0=a1a4a10a12b1b2,F1=-b1b2a1a4a10+a12+a10a12a1+a4,F2=b1b2a1a4+a10a12+a1+a4a10+a12,F3=-b1b2a1+a4+a10+a12,F4=b1b2,G0=a9c3c4a1a4a7-a1a5a6+a2a3a6+a4a7a9a11c2c3,G1=a11c2c3a4a7+a4a9+a7a9+a5a6c3c4a1+a9-c3c4a1a4a7+a9+a7a9a1+a4,G2=c3c4a1a4+a7a9+a1+a4a7+a9+a11c2c3a4+a7+a9-a5a6c3c4,G3=c3c4a1+a4+a7+a9+a11c2c3,G4=c3c4,H0=a4b1b2a1a10c4+a12c3+a10a11c2-a3a6a12b2b3c1,H1=a3a6b2b3c1-b1b2a1a4c3+c4+a11c2a4+a10-b1b2a1+a4a10c4+a12c3,H2=b1b2a10c4+a11c2+a12c3+a1+a4c3+c4,H3=-b1b2c3+c4,I0=a4b1b2c3a11c2+a1c4-a3a6b2b3c1c4,I1=b1b2c3c4a1+a4+a11b1b2c2c3,I2=b1b2c3c4.Case 1 (τ1=τ2=0).
Whenτ1=τ2=0, (6) becomes (8)λ6+A15λ5+A14λ4+A13λ3+A12λ2+A11λ+A10=0,where (9)A10=A0+B0+C0+D0+E0+F0+G0+H0+I0,A11=A1+B1+C1+D1+E1+F1+G1+H1+I1,A12=A2+B2+C2+D2+E2+F2+G2+H2+I2,A13=A3+B3+C3+D3+E3+F3+G3+H3,A14=A4+B4+C4+D4+E4+F4+G4,A15=A5+B5+C5.
Clearly,D1=A15=βI∗+ρ+γ+δ+η+ε+θ+χ+2α+6d>0. Thus, if condition (H1) (see (10)) holds, then system (2) without delay is locally asymptotically stable: (10)D2=det⁡A151A13A14>0,D3=det⁡A1510A13A14A15A11A12A13>0,D4=det⁡A05100A03A14A151A01A12A13A140A10A11A12>0,D5=det⁡A151000A13A14A1510A11A12A13A14A150A10A11A12A13000A10A11>0,D6=A10>0.Case 2 (τ1>0; τ2=0).
Equation (6) equals (11)λ6+A25λ5+A24λ4+A23λ3+A22λ2+A21λ+A20+B25λ5+B24λ4+B23λ3+B22λ2+B21λ+B20e-λτ1+F24λ4+F23λ3+F22λ2+F21λ+F20e-2λτ1=0,where (12)A20=A0+C0+G0,A21=A1+C1+G1,A22=A2+C2+G2,A23=A3+C3+G3,A24=A4+C4+G4,A25=A5+G5,B20=B0+D0+E0,B21=B1+D1+E1,B22=B2+D2+E2,B23=B3+D3+E3,B24=B4+D4+E4,B25=B5.Multiplying eλτ1 on left and right of (11), one has (13)B25λ5+B24λ4+B23λ3+B22λ2+B21λ+B20+λ6+A25λ5+A24λ4+A23λ3+A22λ2+A21λ+A20eλτ1+F24λ4+F23λ3+F22λ2+F21λ+F20e-λτ1=0.Assume that λ=iω1(ω1>0) is the root of (13): (14)L21ω1cos⁡τ1ω1-L22ω1sin⁡τ1ω1=L23ω1,L24ω1sin⁡τ1ω1+L25ω1cos⁡τ1ω1=L26ω1,with (15)L21ω1=A24+F24ω14-ω16-A22+F22ω12+A20+F20,L22ω1=A25ω15-A23-F23ω13+A21-F21ω1,L23ω1=B22ω12-B24ω14-B20,L24ω1=A24-F24ω14-ω16-A22-F22ω12+A20-F20,L25ω1=A25ω15-A23+F23ω13+A21+F21ω1,L26ω1=B23ω13-B25ω15-B21ω1.Thus, one can obtain the expressions of cos⁡τ1ω1 and sin⁡τ1ω1 as follows: (16)cos⁡τ1ω1=L22ω1×L26ω1+L23ω1×L24ω1L21ω1×L24ω1+L22ω1×L25ω1,sin⁡τ1ω1=L21ω1×L26ω1-L23ω1×L25ω1L21ω1×L24ω1+L22ω1×L25ω1.Then, we can get (17)cos2τ1ω1+sin2τ1ω1=1.Suppose that (H21) (see (17)) has at least one positive root.
If condition(H21) holds, then there exists ω10>0 such that (13) has a pair of purely imaginary roots ±iω10. For ω10, (18)τ10=1ω10×arccos⁡L22ω10×L26ω10+L23ω10×L24ω10L21ω10×L24ω10+L22ω10×L25ω10.Differentiating (13) with respect to τ1, one has (19)dλdτ1-1=F21λF22λ-τ1λ,where (20)F21λ=5B25λ4+4B24λ4+3B23λ2+2B22λ+B21+6λ5+5A25λ4+4A24λ3+3A23λ2+2A22λ+A21eλτ1+4F24λ3+3F23λ2+2F22λ+F21e-λτ1,F22λ=4F24λ4+3F23λ3+2F22λ2+F21λe-λτ1-λ7+A25λ6+A24λ5+A23λ4+A22λ3+A21λ2+A20λeλτ1.Thus, (21)Re⁡dλdτ1λ=iω10-1=G2R×H2R+G2I×H2IH2R2+H2I2,with (22)G2R=5B25ω104-3B23ω102+B21+5A25ω104-3A23+F23ω102+A21+F21cos⁡τ10ω10-6ω105-4A24-F24ω103+2A22-F22ω10sin⁡τ10ω10,G2I=2B22ω10-4B24ω103+5A25ω104-3A23-F23ω102+A21-F21sin⁡τ10ω10+6ω105-4A24+F24ω103+2A22+F22ω10cos⁡τ10ω10,H2R=F21+A20ω10+A24ω105-ω107-3F23+A22ω103sin⁡τ10ω10+4F24-A23ω104-2F22-A21ω102-A25ω106cos⁡τ10ω10,H2I=F21-A20ω10-A24ω105+ω107-3F23-A22ω103cos⁡τ10ω10-4F24+A23ω104-2F22-A21ω102-A25ω106sin⁡τ10ω10.
Thus, if condition(H22)G2R×H2R+G2I×H2I≠0 holds, then Re⁡[dλ/dτ1]λ=iω10≠0. Based on the Hopf bifurcation theorem in [25], we have the following results.Theorem 1.
Suppose that conditions(H1), (H21), and (H22) hold for system (2). The viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) is locally asymptotically stable when τ1∈[0,τ10) and a Hopf bifurcation occurs at the viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) when τ1=τ10.Case 3 (τ1=0; τ2>0).
Equation (6) becomes(23)λ6+A35λ5+A34λ4+A33λ3+A32λ2+A31λ+A30+C35λ5+C34λ4+C33λ3+C32λ2+C31λ+C30e-λτ2+G34λ4+G33λ3+G32λ2+G31λ+G30e-2λτ2=0,where(24)A30=A0+B0+F0,A31=A1+B1+F1,A32=A2+B2+F2,A33=A3+B3+F3,A34=A4+B4+F4,A35=A5+B5,C30=C0+D0+H0,C31=C1+D1+H1,C32=C2+D2+H2,C33=C3+D3+H3,C34=C4+D4,C35=C5,G30=E0+G0+I0,G31=E1+G1+I1,G32=E2+G2+I2,G33=E3+G3,G34=G4.Multiplying eλτ2 on left and right of (23), one has (25)C35λ5+C34λ4+C33λ3+C32λ2+C31λ+C30+λ6+A35λ5+A34λ4+A33λ3+A32λ2+A31λ+A30eλτ2+G34λ4+G33λ3+G32λ2+G31λ+G30e-λτ2=0.Let λ=iω2(ω2>0) be the root of (25):(26)L31ω2cos⁡τ2ω2-L32ω2sin⁡τ2ω2=L33ω2,L34ω2sin⁡τ2ω2+L35ω2cos⁡τ2ω2=L36ω2,with(27)L31ω2=A34+G34ω24-ω26-A32+G32ω22+A30+G30,L32ω2=A35ω25-A33-G33ω23+A31-G31ω2,L33ω2=C32ω22-C34ω24-C30,L34ω2=A34-G34ω24-ω26-A32-G32ω22+A30-G30,L35ω2=A35ω25-A33+G33ω23+A31+G31ω2,L36ω2=C33ω23-C35ω25-C31ω2.Then,(28)cos⁡τ2ω2=L32ω2×L36ω2+L33ω2×L34ω2L31ω2×L34ω2+L32ω2×L35ω2,sin⁡τ2ω2=L31ω2×L36ω2-L33ω2×L35ω2L31ω2×L34ω2+L32ω2×L35ω2.And the equation following equation regarding τ2 can be obtained: (29)cos2τ2ω2+sin2τ2ω2=1.Suppose that (H31) (see (29)) has at least one positive root.
If condition(H31) holds, then there exists ω20>0 such that (25) has a pair of purely imaginary roots ±iω20. For ω20, (30)τ20=1ω20×arccos⁡L32ω20×L36ω20+L33ω20×L34ω20L31ω20×L34ω20+L32ω20×L35ω20.Differentiate both sides of (25) with respect to τ2. Then, (31)dλdτ2-1=F31λF32λ-τ2λ,where(32)F31λ=5C35λ4+4C34λ4+3C33λ2+2C32λ+C31+6λ5+5A35λ4+4A34λ3+3A33λ2+2A32λ+A31eλτ2+4G34λ3+3G33λ2+2G32λ+G31e-λτ2,F32λ=4G34λ4+3G33λ3+2G32λ2+G31λe-λτ2-λ7+A35λ6+A34λ5+A33λ4+A32λ3+A31λ2+A30λeλτ2.Thus,(33)Re⁡dλdτ2λ=iω20-1=G3R×H3R+G3I×H3IH3R2+H3I2,with(34)G3R=5C35ω204-C33ω202+C31+5A35ω204-3A33+G33ω202+A31+G31cos⁡τ20ω20-6ω205-4A34+G34ω203+2A32+G32ω20sin⁡τ20ω20,G3I=2C22ω20-4C24ω203+5A35ω204-3A33+G33ω202+A31-G31sin⁡τ20ω20+6ω205-4A34+G34ω203+2A32+G32ω20cos⁡τ20ω20,H3R=G31+A30ω20+A34ω205-ω207-3G33+A32ω203sin⁡τ20ω20+4G34-A33ω204-2G32-A31ω202-A35ω206cos⁡τ20ω20,H3I=G31-A30ω20-A34ω205+ω207-3G33-A32ω203cos⁡τ20ω20+4G34+A33ω204-2G32+A31ω202-A35ω206sin⁡τ20ω20.
Similar to Case2, we know that if condition (H32)G3R×H3R+G3I×H3I≠0 holds, then Re[dλ/dτ2]λ=iω20≠0. In conclusion, we have the following results.Theorem 2.
Suppose that conditions(H1), (H31), and (H32) hold for system (2). The viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) is locally asymptotically stable when τ2∈[0,τ20) and a Hopf bifurcation occurs at the viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) when τ2=τ20.Case 4 (τ1>0; τ2∈(0,τ20)).
Regardingτ1 as the bifurcation parameter when τ2∈(0,τ20), multiplying by eλτ1, (6) becomes (35)B5λ5+B4λ4+B3λ3+B2λ2+B1λ+B0+D4λ4+D3λ3+D2λ2+D1λ+D0e-λτ2+E3λ3+E2λ2+E1λ+E0e-2λτ2+λ6+A5λ5+A4λ4+A3λ3+A2λ2+A1λ+A0eλτ1+C5λ5+C4λ4+C3λ3+C2λ2+C1λ+C0eλτ1-τ2+F4λ4+F3λ3+F2λ2+F1λ+F0e-λτ1+G4λ4+G3λ3+G2λ2+G1λ+G0eλτ1-2τ2+H3λ3+H2λ2+H1λ+H0e-λτ1+τ2+I2λ2+I1λ+I0e-λτ1+2τ2.
Letλ=iω1∗(ω1∗>0) be the root of (35), and for the convenience we still denote ω1∗ as ω1; then, (36)L41ω1cos⁡τ1ω1-L42ω1sin⁡τ1ω1=L43ω1,L44ω1sin⁡τ1ω1+L45ω1cos⁡τ1ω1=L46ω1,where(37)L41ω1=A4+F4ω14-ω16-A2+F2ω12+A0+F0+C4ω14-C2+H2ω12+C0+H0cos⁡τ2ω1+C5ω15-C3+H3ω13+C1+H1ω1sin⁡τ2ω1+G4ω14-G2+I2ω12+G0+I0cos⁡2τ2ω1+G1+I1ω1-G3ω13sin⁡2τ2ω1,L42ω1=A5ω15-A3-F3ω13+A1-F1ω1+C5ω15-C3-H3ω13+C1-H1ω1cos⁡τ2ω1-C4ω14-C2-H2ω12+C0-H0sin⁡τ2ω1+G1-I1ω1-G3ω13cos⁡2τ2ω1-G4ω14-G2-I2ω12+G0-I0sin⁡2τ2ω1,L43ω1=B2ω12-B4ω14-B0+D3ω13-D1ω1sin⁡τ2ω1+D2ω12-D4ω14-D0cos⁡τ2ω1+E3ω13-E1ω1sin⁡τ2ω1+E2ω12-E0cos⁡τ2ω1,L44ω1=A4-F4ω14-ω16-A2-F2ω12+A0-F0+C4ω14-C2-H2ω12+C0-H0cos⁡τ2ω1+C5ω15-C3-H3ω13+C1-H1ω1sin⁡τ2ω1+G4ω14-G2-I2ω12+G0-I0cos⁡2τ2ω1+G1-I1ω1-G3ω13sin⁡2τ2ω1,L45ω1=A5ω15-A3+F3ω13+A1+F1ω1+C5ω15-C3+H3ω13+C1+H1ω1cos⁡τ2ω1-C4ω14-C2+H2ω12+C0+H0sin⁡τ2ω1+G1+I1ω1-G3ω13cos⁡2τ2ω1-G4ω14-G2+I2ω12+G0+I0sin⁡2τ2ω1,L46ω1=B3ω13-B3ω13-B1ω1+D3ω13-D1ω1cos⁡τ2ω1-D2ω12-D4ω14-D0sin⁡τ2ω1+E3ω13-E1ω1cos⁡τ2ω1-E2ω12-E0sin⁡τ2ω1.Thus,(38)cos⁡τ1ω1=L42ω1×L46ω1+L43ω1×L44ω1L41ω1×L44ω1+L42ω1×L45ω1,sin⁡τ1ω1=L41ω1×L46ω1-L43ω1×L45ω1L41ω1×L44ω1+L42ω1×L45ω1.Then, we get(39)cos2τ1ω1+sin2τ1ω1=1.Suppose that (H41) (see (39)) has at least one positive root.
If(H41) holds, then there exists ω10>0 such that (35) has a pair of purely imaginary roots ±iω10∗. For ω10∗,(40)τ10∗=1ω10∗×arccos⁡L42ω10∗×L46ω10∗+L43ω10∗×L44ω10∗L41ω10∗×L44ω10∗+L42ω10∗×L45ω10∗.Differentiating both sides of (25) with respect to τ2,(41)dλdτ1-1=F41λF42λ-τ1λ,where(42)F41λ=5B5λ4+4B4λ3+3B3λ2+2B2λ+B1+4D4-τ2D3λ3-τ2D4λ4+3D3-τ2D2λ2+2D2-D1λ+D1-τ2D0e-λτ2+6λ5+5A5λ4+4A4λ3+3A3λ2+2A2λ+A1eλτ1+3E3-2τ2E2λ2-2τ2E3λ3+2E2-τ2E1λ+E1-2τ2E0e-2λτ2+5C5-τ2C4λ4-τ2C5λ5+4C4-τ2C3λ3+3C3-τ2C2λ2+2C2-τ2C1λ+C1-τ2C0eλτ1-τ2+4F4λ3+3F3λ2+2F2λ+F1e-λτ1+3H3-τ2H2λ2-τ2H3λ3+2H2-τ2H1λ+H1-τ2H0e-λτ1+τ2+4G4-2τ2G3λ3-2τ2G4λ4+3G3-τ2G2λ2+2G2-τ2G1λ+G1-2τ2G0eλτ1-2τ2+2I2-τ2I1λ-2τ2I2λ2+I1-2τ2I0e-λτ1+2τ2,F42λ=F4λ5+F3λ4+F2λ3+F1λ2+F0λe-λτ1+H3λ4+H2λ3+H1λ2+H0λe-λτ1+τ2+I2λ3+I1λ2+I0λe-λτ1+2τ2-λ7+A5λ6+A4λ5+A3λ4+A2λ3+A1λ2+A0λeλτ1-C5λ6+C4λ5+C3λ4+C2λ3+C1λ2+C0λeλτ1-τ2-G4λ5+G3λ4+G2λ3+G1λ2+G0λeλτ1-2τ2.
Define(43)Re⁡dλdτ1λ=iω10∗-1=G4R×H4R+G4I×H4IH4R2+H4I2.
Similar to Case2, we know that if condition (H42)G4R×H4R+G4I×H4I≠0 holds, then Re⁡[dλ/dτ1]λ=iω10∗≠0. Thus, we have the following results.Theorem 3.
Letτ2∈(0,τ20) and suppose that conditions (H1), (H41), and (H42) hold for system (2). The viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) is locally asymptotically stable when τ1∈[0,τ10∗) and a Hopf bifurcation occurs at the viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) when τ1=τ10∗.Case 5 (τ1∈(0,τ10); τ2>0).
Regardingτ2 as the bifurcation parameter when τ1∈(0,τ10), multiplying by eλτ2, (6) becomes (44)C5λ5+C4λ4+C3λ3+C2λ2+C1λ+C0+D4λ4+D3λ3+D2λ2+D1λ+D0e-λτ1+H3λ3+H2λ2+H1λ+H0e-2λτ1+G4λ4+G3λ3+G2λ2+G1λ+G0e-λτ2+λ6+A5λ5+A4λ4+A3λ3+A2λ2+A1λ+A0eλτ2+B5λ5+B4λ4+B3λ3+B2λ2+B1λ+B0eλτ2-τ1+E3λ3+E2λ2+E1λ+E0e-λτ2+τ1+F4λ4+F3λ3+F2λ2+F1λ+F0eλτ2-2τ1+I2λ2+I1λ+I0e-λτ2+2τ1.
Letλ=iω2∗(ω2∗>0) be the root of (44), and for the convenience we still denote ω2∗ as ω2; then, (45)L51ω2cos⁡τ2ω2-L52ω1sin⁡τ2ω2=L53ω2,L54ω2sin⁡τ2ω2+L55ω1cos⁡τ2ω2=L56ω2,where(46)L51ω2=A4+G4ω24-ω26-A2+G2ω22+A0+G0+B4ω24-B2+E2ω22+B0+E0cos⁡τ1ω2+B5ω25-B3+E3ω23+B1-E1ω2sin⁡τ1ω2+F4ω24-F2+I2ω22+F0+I0cos⁡2τ1ω2+F1+I1ω2-F3ω23sin⁡2τ1ω2,L52ω2=A5ω25-A3-G3ω23+A1-G1ω2+B2-E2ω22-B4ω24+E0-B0sin⁡τ1ω2+B5ω25-B3-E3ω23+B1-E1ω2cos⁡τ1ω2-F4ω24-F2-I2ω22+F0-I0sin⁡2τ1ω2+F1-I1ω2-F3ω23cos⁡2τ1ω2,L53ω2=C2ω22-C4ω24-C0+D3ω23-D1ω2sin⁡τ1ω2+D2ω22-D4ω24-D0cos⁡τ1ω2+H3ω23-H1ω2sin⁡2τ1ω2+H2ω22-H0cos⁡2τ1ω2,L54ω2=A4-G4ω24-ω26-A2-G2ω22+A0-G0+B4ω24-B2-E2ω22+B0-E0cos⁡τ1ω2+B5ω25-B3-E3ω23+B1-E1ω2sin⁡τ1ω2+F4ω24-F2-I2ω22+F0-I0cos⁡2τ1ω2+F1-I1ω2-F3ω23sin⁡2τ1ω2,L55ω2=A5ω25-A3+G3ω23+A1+G1ω2-B4ω24-B2+E2ω22+E0+B0sin⁡τ1ω2+B5ω25-B3+E3ω23+B1+E1ω2cos⁡τ1ω2-F4ω24-F2+I2ω22+F0-I0sin⁡2τ1ω2+F1+I1ω2-F3ω23cos⁡2τ1ω2,L56ω2=C3ω23-C5ω25-C1ω2+D3ω23-D-1ω2cos⁡τ1ω2-D2ω22-D4ω24-D0sin⁡τ1ω2+H3ω23-H1ω2cos⁡2τ1ω2-H2ω22-H0sin⁡2τ1ω2.Thus,(47)cos⁡τ2ω2=L52ω2×L56ω2+L53ω2×L54ω2L51ω2×L54ω2+L52ω2×L55ω2,sin⁡τ2ω2=L51ω2×L56ω2-L53ω2×L55ω2L51ω2×L54ω2+L52ω2×L55ω2.Then, we get (48)cos2τ2ω2+sin2τ2ω2=1.
If(H51) holds, then there exists ω20∗>0 such that (35) has a pair of purely imaginary roots ±iω20∗. For ω20∗,(49)τ20∗=1ω20∗×arccos⁡L52ω20∗×L56ω20∗+L53ω20∗×L54ω20∗L51ω20∗×L54ω20∗+L52ω20∗×L55ω20∗.Differentiating (25) with respect to τ2, one can get (50)dλdτ2-1=F51λF52λ-τ2λ,where(51)F51λ=5C5λ4+4C4λ3+3C3λ2+2C2λ+C1+4D4-τ1D3λ3-τ1D4λ4+3D3-τ1D2λ2+2D2-τ1D1λ+D1-τ1D0e-λτ1+3H3-2τ1H2λ2-2τ1H3λ3+2H2-τ1H1λ+H1-2τ1H0e-2λτ1+4G4λ3+3G3λ2+2G2λ+G1e-λτ2+6λ5+5A5λ4+4A4λ3+3A3λ2+2A2λ+A1eλτ2+5B5-τ1B4λ4-τ1B5λ5+4B4-τ1B3λ3+3B3-τ1B2λ2+2B2-τ1B1λ+B1-τ1B0eλτ2-τ1+3E3-τ1E2λ2-τ1E3λ3+2E2-τ1E1λ+E1-τ1E0e-λτ2+τ1+4F4-2τ1F3λ3-2τ1F4λ4+3F3-2τ1F2λ2+2F2-2τ1F1λ+F1-2τ1F0eλτ2-2τ1+2I2-τ1I1λ-2τ1I2λ2+I1-2τ1I0eλτ2+2τ1,F52λ=G4λ5+G3λ4+G2λ3+G1λ2+G0λe-λτ2+E3λ4+E2λ3+E1λ2+E0λe-λτ2+τ1+I2λ3+I1λ2+I0λe-λτ2+2τ1-F4λ5+F3λ4+F2λ3+F1λ2+F0λeλτ2-2τ1-B5λ6+B4λ5+B3λ4+B2λ3+B1λ2+B0λeλτ2-τ1-λ7+A5λ6+A4λ5+A3λ4+A2λ3+A1λ2+A0λeλτ2.Thus,(52)Re⁡dλdτ2λ=iω20∗-1=G3R×H3R+G3I×H3IH3R2+H3I2.
Therefore, we know that if condition(H52)G5R×H5R+G5I×H5I≠0 holds, then Re⁡[dλ/dτ2]λ=iω20∗≠0. Then, we have the following results.Theorem 4.
Letτ1∈(0,τ10) and suppose that conditions (H1), (H51), and (H52) hold for system (2). The viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) is locally asymptotically stable when τ2∈[0,τ20∗) and a Hopf bifurcation occurs at the viral equilibrium P∗(S∗,E∗,I∗,Q∗,R∗,V∗) when τ2=τ20∗.


## 3. Properties of the Hopf Bifurcation

In this section, we shall investigate direction and stability of the Hopf bifurcation under the case whereτ1∈(0,τ10) and τ2>0. Set u1(t)=S(t)-S∗, u2(t)=E(t)-E∗, u3(t)=I(t)-I∗, u4(t)=Q(t)-Q∗, u5(t)=R(t)-R∗, u6(t)=V(t)-V∗, and t→(t/τ2). For convenience, we assume that τ1∗∈(0,τ10)<τ20∗ throughout this section. Then, system (2) becomes functional differential equations in C=C([-1,0],R6): (53)u˙t=Lμut+Fμ,ut,with (54)Lμϕ=τ20∗+μAϕ0+Bϕ-τ1∗τ20∗+Cϕ-1,Fμ,ϕ=τ20∗+μ-βϕ10ϕ30βϕ10ϕ300000,where (55)A=a10a2000a3a4a50000a6a700000a8a9000000a100a110000a12,B=00000000000000b1000000b20000b3b400000000,C=0000c1c2000000000000000000000c30000000c4.Based on the Riesz representation theorem, there exists a 6×6 function η(θ,μ):[-1,0]→R6×6 such that (56)Lμϕ=∫-10dηθ,μϕθ,ϕ∈C.In fact, we choose (57)ηθ,μ=τ20∗+μA+B+B,θ=0,τ20∗+μB+C,θ∈-τ1∗τ20∗,0,τ20∗+μB,θ∈-1,-τ1∗τ20∗,0,θ=-1.For ϕ∈C([-1,0],R6), we define (58)Aμϕ=dϕθdθ,-1≤θ<0,∫-10dηθ,μϕθ,θ=0,Rμϕ=0,-1≤θ<0,Fμ,ϕ,θ=0.Then, system (53) becomes (59)u˙t=Aμut+Rμut,where ut(θ)=u(t+θ) for θ∈[-1,0].DefineA∗ as follows: (60)A∗φ=-dφsds,0<s≤1,∫-10dηTs,0φ-s,s=0,and a bilinear form (61)φs,ϕθ=φ-0ϕ0-∫θ=-10∫ξ=0θφ-ξ-θdηθϕξdξ,where η(θ)=η(θ,0).Letq(θ)=(1,q2,q3,q4,q5,q6)Teiω20∗τ20∗θ be the eigenvector of A(0) with +iω20∗τ20∗ and let q∗(s)=D(1,q2∗,q3∗,q4∗,q5∗)eiω20∗τ20∗s be the eigenvector of A∗(0) with -iω20∗τ20∗. Then, according to the definition of A(0) and A∗(0), we obtain (62)q2=a3+a5q3iω20∗-a3,q3=a3a6iω20∗-a3iω20∗-a7-b1e-iτ1∗ω20∗-a5a6,q4=a8q3iω20-a9-b2e-iτ1∗ω20∗,q5=iω20∗-a1-a2q3-c2e-iτ20∗ω20∗c1e-iτ20∗ω20∗,q6=iω20∗-a11a12+c4e-iτ20∗ω20∗,q2∗=-iω20∗+a1a3,q3∗=iω20∗+a1iω20∗+a4a3a6,q4∗=-iω20∗+a7+b1eiτ1∗ω20∗q3a8+a2a3+iω20∗+a1a5a3a8+b3iω20∗+a9+b2eiτ1∗ω20∗a8b4,q5=-iω20∗+a9+b2eiτ1∗ω20∗q4b4eiτ1∗ω20∗,q6∗=-c2eiτ20∗ω20∗iω20∗+a12+c4eiτ20∗ω20∗.In addition, from (61), we have (63)q∗s,qθ=q-0q0-∫-10∫ξ=0θq-∗ξ-θdηθqξdξ=D-1,q-2∗,q-3∗,q-4∗,q-5∗,q-6∗1,q2,q3,q4,q5,q6T-∫-10∫ξ=0θD-1,q-2∗,q-3∗,q-4∗,q-5∗,q-6∗e-iτ1∗ω20∗ξ-θdηθ1,q2,q3,q4,q5,q6Teiτ1∗ω20∗ξdξ-∫-10∫ξ=0θD-1,q-2∗,q-3∗,q-4∗,q-5∗,q-6∗e-iτ20∗ω20∗ξ-θdηθ1,q2,q3,q4,q5,q6Teiτ20∗ω20∗ξdξ=D-1+q2q-2∗+q3q-3∗+q4q-4∗+q5q-5∗+q6q-6∗+τ1∗eiτ1∗ω20∗∫-101,q-2∗,q-3∗,q-4∗,q-5∗,q-6∗B1,q2,q3,q4,q5,q6T+τ1∗eiτ2∗ω20∗∫-101,q-2∗,q-3∗,q-4∗,q-5∗,q-6∗C1,q2,q3,q4,q5,q6T=D-1+q2q-2∗+q3q-3∗+q4q-4∗+q5q-5∗+q6q-6∗+τ1∗e-iτ1∗ω20∗b3b1q-3∗+b3q-5∗+b4b2q-4∗+b4q-5∗+τ20∗e-iτ20∗ω20∗c1q-5∗+c2q-6∗+c3q5q-5∗+c4q6q-6∗.Thus, we can choose (64)D-=1+q2q-2∗+q3q-3∗+q4q-4∗+q5q-5∗+q6q-6∗+τ1∗e-iτ1∗ω20∗b3b1q-3∗+b3q-5∗+b4b2q-4∗+b4q-5∗+τ20∗e-iτ20∗ω20∗c1q-5∗+c2q-6∗+c3q5q-5∗+c4q6q-6∗-1,such that 〈q∗,q〉=1, 〈q∗,q-〉=0.Then, using the algorithms from Hassard et al. [25] and the similar computation process in [26–29], we obtain (65)g20=2βτ20∗D-q3q-2∗-1,g11=βτ20∗D-Re⁡q3q-2∗-1,g02=2βτ20∗D-q-3q-2∗-1,g21=2βτ20∗D-q-2∗-1W1110q3+12W2010q-3+W1130+12W2030,with (66)W20θ=ig20q0τ20∗ω20∗eiτ20∗ω20∗θ+ig-02q-03τ20∗ω20∗e-iτ20∗ω20∗θ+E1e2iτ20∗ω20∗θ,W11θ=-ig11q0τ20∗ω20∗eiτ20∗ω20∗θ+ig-11q-0τ20∗ω20∗e-iτ20∗ω20∗θ+E2,E1=2a1′0-a20-c1e-2iτ20∗ω20∗-c2e-2iτ20∗ω20∗-a3a2′-a50000-a6a3′00000-a8a4′0000-b3e-2iτ1∗ω20∗-b4e-2iτ1∗ω20∗a5′0-a110000a6′-1×-βq3βq30000,E2=-a10a20c1c2a3a4a50000a6a7+b100000a8a9+b20000b3b4a10+c30a110000a12+c4-1×-βRe⁡q3βRe⁡q30000,where(67)a1′=2iω20∗-a1,a2′=2iω20∗-a4,a3′=2iω20∗-a7-b1e-2iτ1∗ω20∗,a4′=2iω20∗-a9-b2e-2iτ1∗ω20∗,a5′=2iω20∗-a10-c3e-2iτ20∗ω20∗,a6′=2iω20∗-a12-c4e-2iτ20∗ω20∗.Then, we can get the following coefficients:(68)C10=i2τ20∗ω20∗g11g20-2g112-g0223+g212,μ2=-Re⁡C10Re⁡λ′τ20∗,ρ2=2Re⁡C10,T2=-Im⁡C10+μ2Im⁡λ′τ20∗τ20∗ω20∗.Thus, we have the following results.Theorem 5.
The sign ofμ2 determines direction of the Hopf bifurcation: if μ2>0 (μ2<0), then the Hopf bifurcation is supercritical (subcritical); the sign of ρ2 determines stability of the bifurcating periodic solutions: if ρ2<0 (ρ2>0), then the bifurcating periodic solutions are stable (unstable); the sign of T2 determines period of the bifurcating solutions: if T2>0 (T2<0), then the period of the bifurcating periodic solutions increases (decreases).


## 4. Numerical Simulation

In this section, we present some numerical results of system (2) in order to validate the analytical predictions obtained in Sections 2 and 3. We choose a set of parameters as follows: A=100, β=0.009, d=0.05, ρ=0.65, θ=0.05, χ=0.55, γ=0.45, α=0.035, δ=0.1, η=0.35, and ε=0.07, and consider the following special case of (2): (69)dStdt=100-0.009StIt-0.05St-0.65St+0.05Rt-τ2+0.55Vt-τ2,dEtdt=0.009StIt-0.05Et-0.45Et,dItdt=0.45Et-0.05It-0.035It-0.1It-0.35It-τ1,dQtdt=δIt-0.05Qt-0.035Qt-0.07Qt-τ1,dRtdt=0.07Qt-τ1-0.05Rt-0.05Rt-τ2+0.35It-τ1,dVtdt=0.65St-0.05Vt-0.55Vt-τ2,from which we can get the unique viral equilibrium P∗(66.0494,277.7978,233.6617,150.7495, 923.3406, 71.7439). It can be easily verified that condition (H1) is satisfied when τ1=τ2=0.By computation, we haveω10=0.8554 and τ10=4.1056. Then, we get λ′(τ10)=2.3686+i1.0212. Thus, we know that conditions (H21) and (H22) hold. We can conclude that all roots that cross the imaginary axis at iω10 cross from left to right as τ1 increases by the theory in [22]. According to Theorem 1, P∗(66.0494,277.7978,233.6617,150.7495,923.3406, 71.7439) is asymptotically stable when τ1∈(0,τ10). This property can be illustrated by Figures 1 and 2. In this case, spreading law of the computer viruses can be predicted and the viruses can be controlled and eliminated. However, once the value of τ1 passes through the critical value τ10, P∗(66.0494,277.7978,233.6617,150.7495,923.3406, 71.7439) loses its stability and a Hopf bifurcation occurs, which can be shown in Figures 3 and 4. The occurrence of a Hopf bifurcation means that the state of computer viruses propagation changes from the viral equilibrium point to a limit cycle. This makes spreading of the computer viruses be out of control.Figure 1
The projection of the phase portrait of system (69) in (S,E,V)-space with τ1=3.605.Figure 2
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ1=3.605.Figure 3
The projection of the phase portrait of system (69) in (S,E,V)-space with τ1=4.60.Figure 4
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ1=4.60.Similarly, we have the following:ω20=1.8255 and τ20=3.7424 when τ1=0 and τ2>0; ω10∗=0.9665 and τ10∗=3.1862 when τ1>0 and τ2=2.25∈(0,τ20); ω20∗=2.4217 and τ20∗=3.0254 when τ2>0 and τ1=2.45∈(0,τ10). The corresponding phase plots are shown in Figures 5–8, Figures 9–12, and Figures 13–16, respectively. In addition, for τ2>0 and τ1=2.45∈(0,τ10), we have C1(0)=-17.2982+i13.5056 and λ′(τ20∗)=0.3796+i2.0581 by some complex computation. Based on (68), we get μ2=45.5692>0, ρ2=-34.5964<0, and T2=-14.6441<0. Therefore, the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the bifurcating periodic solutions decreases.Figure 5
The projection of the phase portrait of system (69) in (S,E,V)-space with τ2=3.65.Figure 6
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ2=3.65.Figure 7
The projection of the phase portrait of system (69) in (S,E,V)-space with τ2=3.805.Figure 8
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ2=3.805.Figure 9
The projection of the phase portrait of system (69) in (S,E,V)-space with τ1=2.86 and τ2=2.25∈(0,τ20).Figure 10
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ1=2.86 and τ2=2.25∈(0,τ20).Figure 11
The projection of the phase portrait of system (69) in (S,E,V)-space with τ1=3.574 and τ2=2.25∈(0,τ20).Figure 12
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ1=3.574 and τ2=2.25∈(0,τ20).Figure 13
The projection of the phase portrait of system (69) in (S,E,V)-space with τ2=2.862 and τ1=2.45∈(0,τ10).Figure 14
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ2=2.862 and τ1=2.45∈(0,τ10).Figure 15
The projection of the phase portrait of system (69) in (S,E,V)-space with τ2=3.225 and τ1=2.45∈(0,τ10).Figure 16
The projection of the phase portrait of system (69) in (I,Q,R)-space with τ2=3.225 and τ1=2.45∈(0,τ10).According to the numerical simulation results, we know that the time delay should remain less than the corresponding threshold in order to control and predict the viruses’ propagation by decreasing the period that antivirus software uses to clean the computer viruses and the temporary immunity period of the recovered and the vaccinated computers. To this end, we can adjust the parameters of our proposed model in real-world networks, such as timely updating the antivirus software on computers, properly controlling the number of computers attached to the network, and timely disconnecting computers from the network when the connections are unnecessary. Of course, in the next step, we also need to collect large amount of relevant data and estimate the parameters involved in our proposed model through statistical analysis in real-world networks. Namely, we have to adjust the parameters in the model so as to control viruses’ propagation effectively if it is necessary.


## 5. Conclusions

It is definitely an interesting work to consider the effect of delays on dynamical systems, because a stability switch occurs even when an ignored delay is small for a dynamical system. Based on this fact, we introduce the time delay due to the period that antivirus software uses to clean the computer viruses in the infectious and quarantined computers (τ1) and the time delay due to the temporary immunity period of the recovered and the vaccinated computers (τ2) into the SEIQRS-V computer virus propagation model considered in [21]. We obtain some conditions for local stability and Hopf bifurcation occurring by analyzing distribution of roots of the associated characteristic equation.By computation, there exists a corresponding critical value of the time delay below which system (2) is stable and above which system (2) is unstable. When the system is stable, the characteristics of the propagation of computer viruses can be easily predicted and then the computer viruses can get eliminated. Otherwise, the propagation of the computer viruses is out of control. Therefore, stability of the computer virus propagation system must be guaranteed in practice. In addition, we find that the effect of τ2 on system (2) is marked compared with τ1, because the critical value of τ2 is much smaller when we only consider it. At last, we have also derived the explicit formula which can determine direction and stability of the Hopf bifurcation under the case where τ1∈(0,τ10) and τ2>0.



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*Source: 3536125-2017-03-20.xml*