# Dynamics of a Computer Virus Propagation Model with Delays and Graded Infection Rate

Authors: Zizhen Zhang; Limin Song
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

A four-compartment computer virus propagation model with two delays and graded infection rate is investigated in this paper. The critical values where a Hopf bifurcation occurs are obtained by analyzing the distribution of eigenvalues of the corresponding characteristic equation. In succession, direction and stability of the Hopf bifurcation when the two delays are not equal are determined by using normal form theory and center manifold theorem. Finally, some numerical simulations are also carried out to justify the obtained theoretical results.


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## Body

## 1. Introduction

In recent years, with the fast development and popularization of computer technologies and network, Internet has offered numerous functionalities and facilities to the world. Meanwhile, Internet has also become a powerful mechanism for propagating computer viruses. Computer viruses are computer programs which have serious effects on individual and corporate computer systems in the network, such as modifying data and formatting disks [1, 2].In order to analyze the propagation laws of computer viruses in the network, many epidemiological models have been borrowed to depict the spread of computer viruses because of the high similarity between the computer viruses and the biological viruses [3–5]. In [6–11], Mishra et al. proposed SIRS computer virus models in different forms. Yuan and Chen presented the SEIR computer virus propagation model in [12] and they studied the stability of the model. Based on the the work in [12], Dong et al. proposed the SEIR computer virus model with time delay in [13] and they investigated the Hopf bifurcation of the model. There are also some other different computer virus models which have been proposed by other scholars in recent years and one can refer to [14–18]. However, all the computer virus models above which incorporate the latent status of the viruses assume that the latent computers have no infection ability. This is not consistent with the reality, because an infected computer which is in latency can also infect other computers through file copying or file downloading. Based on this fact, Yang et al. established a computer virus propagation model with graded infection rate in [19]:(1)dStdt=μ-β1StLt-β2StAt+αRt-μSt,dLtdt=β1StLt+β2StAt-εLt-μLt,dAtdt=εLt-γAt-μAt,dRtdt=γAt-αRt-μRt,where S(t), L(t), A(t), and R(t) are the percentages of susceptible computers, latent computers, active computers, and recovered computers on the Internet, at time t, respectively. μ is the rate at which external computers are connected to the Internet and it is also the rate at which internal computers are disconnected from the Internet; β1 is the infected rate of the susceptible computers by the latent computers; β2 is the infected rate of the susceptible computers by the active computers; α is the rate at which the recovered computers become susceptibly virus-free again; ε is the rate at which the latent computers break out; and γ is the rate at which the active computers are cured by the antivirus software.As pointed out in [9], one of the typical features of computer viruses is their latent characteristic. Therefore, they need a period to become active computers for the latent ones. Likewise, the antivirus software needs a period to clean the viruses in the active computers. Based on this and motivated by the work about the dynamical system with delay in [20–24], we incorporate two delays into system (1) and obtain the following delayed computer virus model:(2)dStdt=μ-β1StLt-β2StAt+αRt-μSt,dLtdt=β1StLt+β2StAt-εLt-τ1-μLt,dAtdt=εLt-τ1-γAt-τ2-μAt,dRtdt=γAt-τ2-αRt-μRt,where τ1 is the latent period of the computer viruses and τ2 is the period that the antivirus software needs to clean the viruses in the active computers.The rest of this paper is organized as follows. In Section2, we present the existence of the viral equilibrium and conditions for the local stability of the viral equilibrium and existence of the Hopf bifurcation are derived. Direction and stability of the Hopf bifurcation are studied in Section 3 and some numerical simulations are performed in Section 4 to justify the obtained theoretical findings by taking some relevant values of the parameters in system (2) and using the Matlab software package. Finally, we end this paper with concluding remarks in Section 5.


## 2. Existence of Local Hopf Bifurcation

By a simple computation, we know that ifε+μγ+μ/β1(γ+μ)+β2ε<1 and (ε+μ)(γ+μ)/ε>αγ/α+μ, then system (2) has a unique viral equilibrium E∗(S∗,L∗,A∗,R∗), where(3)S∗=γ+με+μβ1γ+μ+β2ε,L∗=γ+μεA∗,R∗=γα+μA∗,A∗=A1∗A2∗,A1∗=μ-με+μγ+μβ1γ+μ+β2ε,A2∗=ε+μγ+με-αγα+μ.LetS¯(t)=S(t)-S∗, L¯(t)=L(t)-L∗, A¯(t)=A(t)-A∗, R¯(t)=R(t)-R∗. Dropping the bars, system (2) becomes(4)dStdt=a1St+a2Lt+a3At+a4Rt-β1StLt-β2StAt,dLtdt=a5St+a6Lt+a7At+b1Lt-τ1+β1StLt+β2StAt,dAtdt=a8At+b2Lt-τ1+c1At-τ2,dRtdt=a9Rt+c2At-τ2,where(5)a1=-β1L∗+β2A∗+μ,a2=-β1S∗,a3=-β2S∗,a4=α,a5=β1L∗+β2A∗,b1=-ε,a6=β1S∗-μ,a7=β2S∗,a8=-μ,b2=ε,c1=-γ,a9=-α+μR∗,c2=γ.The linear system of system (4) is(6)dStdt=a1St+a2Lt+a3At+a4Rt,dLtdt=a5St+a6Lt+a7At+b1Lt-τ1,dAtdt=a8At+b2Lt-τ1+c1At-τ2,dRtdt=a9Rt+c2At-τ2.The corresponding characteristic equation is(7)λ4+a03λ3+a02λ2+a01λ+a00+b03λ3+b02λ2+b01λ+b00e-λτ1+c03λ3+c02λ2+c01λ+c00e-λτ2+d02λ2+d01λ+d00e-λτ1+τ2=0,where(8)a00=a1a6-a2a5a8a9,a01=a2a5-a1a6a8+a9,a02=a1a6-a2a5+a8a9+a1+a6a8+a9,a03=-a1+a6+a8+a9,b00=a1a8a9b1+a3a5-a1a7a9b2,b01=a7b2a1+a9-b1a1a8+a1a9+a8a9,b02=b1a1+a8+a9-a7b2,b03=-b1,c00=a1a6-a2a5a9c1,c01=c1a2a5-a1a6-a1a9-a6a9,c02=c1a1+a6+a9,c03=-c1,d00=a1a9b1c1+a4a5b2c2,d01=-b1c1a1+a9,d02=b1c1.Case 1 (τ1=τ2=0).
Forτ1=τ2=0, (7) becomes(9)λ4+a13λ3+a12λ2+a11λ+a10=0,where(10)a10=a00+b00+c00+d00,a11=a01+b01+c01+d01,a12=a02+b02+c02+d02,a13=a03+b03+c03.
Thus, according to the Routh-Hurwithz theorem, we know that if conditions(H11)a10>0, a13>0, and a13a12>a11 hold, then viral equilibrium E∗(S∗,L∗,A∗,R∗) of system (2) without delay is locally asymptotically stable.Case 2 (τ1>0, τ2=0).
Forτ1>0 and τ2=0, we can get the following from (7):(11)λ4+a23λ3+a22λ2+a21λ+a20+b23λ3+b22λ2+b21λ+b20e-λτ1=0,where(12)a20=a00+c00,a21=a01+c01,a22=a02+c02,a23=a03+c03,b20=b00+d00,b21=b01+d01,b22=b02+d02,b23=b03.We assume that λ=iω1(ω1>0) is a root of (11). Then,(13)b21ω1-b23ω13sin⁡τ1ω1+b20-b22ω12cos⁡τ1ω1=a22ω12-ω14-a20,b21ω1-b23ω13cos⁡τ1ω1-b20-b22ω12sin⁡τ1ω1=a23ω13-a21ω1,which implies that(14)ω18+g23ω16+g22ω14+g21ω12+g20=0,with(15)g20=a202-b202,g21=a212-b212-2a20a22+2b20b22,g22=a222-b222+2b21b23-2a21a23+2a20,g23=a232-b232-2a22.Let ω12=v1; then (14) becomes(16)v14+g23v13+g22v12+g21v1+g20=0.Discussion about distribution of roots for (16) is similar to that in [25]. Therefore, we directly assume that (H21) (16) has at least one positive equilibrium v10.If(H21) holds, we know that (14) has at least one positive root ω10=v10 such that (11) has a pair of purely imaginary roots ±iω10. For ω10,(17)τ10=1ω10arccos⁡h21ω10h22ω10, where(18)h21ω10=b22-a23b23ω106+a21b23+a23b21-a22b22-b20ω104+a20b22+a22b20-a21b21ω102-a20b20,h22ω10=b232ω106+b222-2b21b23ω104+b212-2b20b22ω102+b202. Differentiating both sides of (11) with respect to τ1, one can obtain(19)dλdτ1-1=-4λ3+3a23λ2+2a22λ+a21λλ4+a23λ3+a22λ2+a21λ+a20+3b23λ2+2b22λ+b21λb23λ3+b22λ2+b21λ+b20-τ1λ. Thus,(20)Re⁡dλdτ1τ1=τ10-1=f1′v1∗h22ω10, where v1∗=ω102 and f1(v1)=v14+g23v13+g22v12+g21v1+g20. Therefore, if condition (H22)f1′(v1∗)≠0 holds, then Re[dλ/dτ1]τ1=τ10-1≠0. Based on the discussion above and according to the Hopf bifurcation theorem in [26], we obtain the following.Theorem 1.
If conditionsH21-(H22) hold, then(i)
viral equilibriumE∗(S∗,L∗,A∗,R∗) of system (2) is locally asymptotically stable for τ1∈[0,τ10);(ii)
system (2) undergoes a Hopf bifurcation at viral equilibrium E∗(S∗,L∗,A∗,R∗) when τ1=τ10 and a family of periodic solutions bifurcate from E∗(S∗,L∗,A∗,R∗).Case 3 (τ1=0,τ2>0).
Forτ1=0 and τ2>0, (7) becomes(21)λ4+a33λ3+a32λ2+a31λ+a30+c33λ3+c32λ2+c31λ+c30e-λτ2=0,where(22)a30=a00+b00,a31=a01+b01,a32=a02+b02,a33=a03+b03,c30=c00+d00,c31=c01+d01,c32=c02+d02,c33=c03.Let λ=iω2(ω2>0) be a root of (21). Then,(23)c31ω2-c33ω23sin⁡τ2ω2+c30-c32ω22cos⁡τ2ω2=a32ω22-ω24-a30,c31ω2-c33ω23cos⁡τ2ω2-c30-c32ω22sin⁡τ2ω2=a33ω23-a31ω2.It follows that(24)ω28+g33ω26+g32ω24+g31ω22+g30=0,with(25)g30=a302-c302,g31=a312-c312-2a30a32+2c30b32,g32=a322-c322+2c31c33-2a31a33+2a30,g33=a332-b332-2a32.Let ω22=v2; then, we have(26)v24+g33v23+g32v22+g31v2+g30=0.Similar to Case2, we make the following assumption. (H31) (26) has at least one positive root v20. If condition (H31) holds, then there exists ω20=v20 such that (21) has a pair of purely imaginary roots ±iω20. For ω20,(27)τ20=1ω20arccos⁡h31ω20h32ω20, where(28)h31ω20=c32-a33c33ω206+a31c33-a32c32+a33c31-c30ω204+a30c32-a31c31+a32c30ω202-a30c30,h32ω20=c332ω206+b322-2c31c33ω204+c312-2c30c32ω202+c302. In addition, we have(29)dλdτ1-1=-4λ3+3a33λ2+2a32λ+a31λλ4+a33λ3+a32λ2+a31λ+a30+3c33λ2+2c32λ+c31λc33λ3+c32λ2+c31λ+c30-τ2λ. Further,(30)Re⁡dλdτ2τ2=τ20-1=f2′v2∗h32ω20, where v2∗=ω202 and f2(v2)=v24+g33v23+g32v22+g31v2+g30. Therefore, if condition (H32)f2′(v2∗)≠0 holds, then Re[dλ/dτ2]τ2=τ20-1≠0. Based on the discussion above and according to the Hopf bifurcation theorem in [26], we obtain the following.Theorem 2.
If conditionsH31-(H32) hold, then(i)
viral equilibriumE∗(S∗,L∗,A∗,R∗) of system (2) is locally asymptotically stable for τ2∈[0,τ20);(ii)
system (2) undergoes a Hopf bifurcation at viral equilibrium E∗(S∗,L∗,A∗,R∗) when τ2=τ20 and a family of periodic solutions bifurcate from E∗(S∗,L∗,A∗,R∗).Case 4 (τ1=τ2=τ>0).
Forτ1=τ2=τ>0, we have(31)λ4+a43λ3+a42λ2+a41λ+a40+b43λ3+b42λ2+b41λ+b40e-λτ+d42λ2+d41λ+d40e-2λτ=0,with(32)a40=a00,a41=a01,a42=a02,a43=a03,b40=b00+c00,b41=b01+c01,b42=b02+c02,b43=b03+c03,d40=d00,d41=d01,d42=d02. Multiplying by eλτ, (31) becomes the following:(33)b43λ3+b42λ2+b41λ+b40+λ4+a43λ3+a42λ2+a41λ+a40eλτ+d42λ2+d41λ+d40e-λτ=0.Let λ=iω(ω>0) be a root of (37); it is easy to get(34)g41ωcos⁡τω-g42ωsin⁡τω=g43ω,g44ωsin⁡τω+g45ωcos⁡τω=g46ω,where(35)g41ω=ω4-a42+d42ω2+a40+d40,g42ω=a41-d41ω-a43ω3,g43ω=b42ω2-b40,g44ω=ω4-a42-d42ω2+a40-d40,g45ω=a41+d41ω-a43ω3,g46ω=b43ω3-b41ω.It leads to(36)cos⁡τω=g42ω×g46ω+g43ω×g44ωg41ω×g44ω+g42ω×g45ω,sin⁡τω=g41ω×g46ω-g43ω×g45ωg41ω×g44ω+g42ω×g45ω. Thus, we can get the following equation with respect to ω:(37)cos2⁡τω+sin2⁡τω=1.Next, we make the following assumption.(H41) (37) has at least one positive root ω0. Then, for ω0, we have(38)τ0=1ω0arccos⁡g42ω0×g46ω0+g43ω0×g44ω0g41ω0×g44ω0+g42ω0×g45ω0. Taking the derivative of λ with respect to τ, we obtain(39)dλdτ-1=-g47λg48λ-τλ, where(40)g47λ=3b43λ2+2b42λ+b41+4λ3+3a43λ2+2a42λ+a41eλτ+2d42λ+d41e-λτ,g48λ=λ5+a43λ4+a42λ3+a41λ2+a40λeλτ-d42λ3+d41λ2+d40λe-λτ. Then, we can get that(41)Re⁡dλdττ=τ0-1=-h41ω0×h43ω0+h42ω0×h44ω0h432ω0+h442ω0, where(42)h41ω0=a41+d41-3a43ω02cos⁡τ0ω0-2a42-d42ω0-2ω03sin⁡τ0ω0+b41-3b43ω02,h42ω0=a41-d41-3a43ω02sin⁡τ0ω0+2a42+d42ω0-2ω03cos⁡τ0ω0+2b42ω0,h43ω0=a43ω04-a41+d41ω02cos⁡τ0ω0-ω05-a42+d42ω03+a40-d40ω0sin⁡τ0ω0,h44ω0=a43ω04-a41-d41ω02sin⁡τ0ω0+ω05-a42+d42ω03+a40+d40ω0cos⁡τ0ω0.We assume that(H42)h41(ω0)×h43(ω0)+h42(ω0)×h44(ω0)≠0. Clearly, if condition (H42) holds, then we can conclude that Re[dλ/dτ]τ=τ0-1≠0. Therefore, according to the Hopf bifurcation theorem in [26], we obtain the following.Theorem 3.
If conditionsH41-(H42) hold, then(i)
viral equilibriumE∗(S∗,L∗,A∗,R∗) of system (2) is locally asymptotically stable for τ∈[0,τ0);(ii)
system (2) undergoes a Hopf bifurcation at viral equilibrium E∗(S∗,L∗,A∗,R∗) when τ=τ0 and a family of periodic solutions bifurcate from E∗(S∗,L∗,A∗,R∗).Case 5 (τ1>0, τ2∈(0,τ20)).
Letλ=iω1′ be the root of (7). Then,(43)g51ω1′sin⁡τ1ω1′+g52ω1′cos⁡τ1ω1′=g53ω1′,g51ω1′cos⁡τ1ω1′-g52ω1′sin⁡τ1ω1′=g54ω1′,where(44)g51ω1′=b01ω1′-b03ω1′3+d01ω1′cos⁡τ2ω1′-d00-d02ω1′2sin⁡τ2ω1′,g52ω1′=b00-b02ω1′2+d01ω1′sin⁡τ2ω1′+d00-d02ω1′2cos⁡τ2ω1′,g53ω1′=a02ω1′2-ω1′4-a00-c01ω1′-c03ω1′3sin⁡τ2ω1′-c00-c02ω1′2cos⁡τ2ω1′,g54ω1′=a03ω1′3-a01ω1′-c01ω1′-c03ω1′3cos⁡τ2ω1′+c00-c02ω1′2sin⁡τ2ω1′.Thus, one can get the following equation with respect to ω1′:(45)g512ω1′+g522ω1′=g532ω1′+g542ω1′.Similar to Case4, we assume that (H51) (45) has at least one positive root ω10′. Then, for ω10′, we have(46)τ10′=1ω10′arccos⁡g51ω10′×g54ω10′+g52ω10′×g53ω10′g512ω10′+g522ω10′. Differentiating (7) with respect to τ1, we get(47)dλdτ-1=g55λg56λ-τ1λ, with(48)g55λ=4λ3+3a03λ2+2a02λ+a01+3b03λ2+2b02λ+b01e-λτ1+3c03-τ2c02λ2-τ2c03λ3+2c02-τ2c01λ+c01-τ2c00e-λτ2+2d02-τ2d01λ-τ2d02λ2+d01-τ2d00e-λτ1+τ2,g56λ=c03λ4+c02λ3+c01λ2+c00λe-λτ2+d02λ3+d01λ2+d00λe-λτ1+τ2.Then, we obtain(49)Re⁡dλdτ1τ1=τ10′-1=h51ω10′×h53ω10′+h52ω10′×h54ω10′h532ω10′+h542ω10′, where(50)h51ω10′=2b02ω10′+2d02-τ2d01cos⁡τ2ω10′-τ2ω10′2+d01-τ2d00sin⁡τ2ω10′sin⁡τ10′ω10′+b01-3b03ω10′2+2d02-τ2d01sin⁡τ2ω10′+τ2ω10′2+d01-τ2d00cos⁡τ2ω10′cos⁡τ10′ω10′+2c02-τ2c01ω10′+τ2c03ω10′3sin⁡τ2ω10′+c01-τ2c00-3c03-τ2c02ω10′2cos⁡τ2ω10′+a01-3a03ω10′2,h52ω10′=2b02ω10′+2d02-τ2d01cos⁡τ2ω10′-τ2ω10′2+d01-τ2d00sin⁡τ2ω10′cos⁡τ10′ω10′-b01-3b03ω10′2+2d02-τ2d01sin⁡τ2ω10′+τ2ω10′2+d01-τ2d00cos⁡τ2ω10′sin⁡τ10′ω10′+2c02-τ2c01ω10′+τ2c03ω10′3cos⁡τ2ω10′-c01-τ2c00-3c03-τ2c02ω10′2sin⁡τ2ω10′+a02ω10′-4ω10′3,h53ω10′=c00ω10′-c02ω10′3+d00ω10′-d02ω10′3cos⁡τ10′ω10′+d01ω10′2sin⁡τ10′ω10′sin⁡τ2ω10′+c03ω10′4-c01ω10′2+d00ω10′-d02ω10′3sin⁡τ10′ω10′-d01ω10′2cos⁡τ10′ω10′cos⁡τ2ω10′,h54ω10′=c00ω10′-c02ω10′3+d00ω10′-d02ω10′3cos⁡τ10′ω10′+d01ω10′2sin⁡τ10′ω10′cos⁡τ2ω10′-c03ω10′4-c01ω10′2+d00ω10′-d02ω10′3sin⁡τ10′ω10′-d01ω10′2cos⁡τ10′ω10′sin⁡τ2′ω10′.We assume that(H52)h51(ω10′)×h53(ω10′)+h52(ω10′)×h54(ω10′)≠0. Thus, we know that Re[dλ/dτ1]τ1=τ10′-1≠0, if condition (H52) holds. Therefore, according to the Hopf bifurcation theorem in [26], we obtain the following.Theorem 4.
If conditionsH51-(H52) hold and τ2∈(0,τ20), then(i)
viral equilibriumE∗(S∗,L∗,A∗,R∗) of system (2) is locally asymptotically stable for τ1∈[0,τ10′);(ii)
system (2) undergoes a Hopf bifurcation at viral equilibrium E∗(S∗,L∗,A∗,R∗) when τ1=τ10′ and a family of periodic solutions bifurcate from E∗(S∗,L∗,A∗,R∗).


## 3. Properties of the Hopf Bifurcation

In this section, we shall investigate direction of the Hopf bifurcation and stability of the bifurcating periodic solution of system (2) when τ1=τ10′ and τ2=τ20′∈(0,τ20) by using the center manifold theorem and the normal form theory which has been developed by Hassard et al. [26].Letτ1=τ10′+μ, u1=S(τ1t), u2=L(τ1t), u3=A(τ1t), u4=R(τ1t), μ∈R. Then, μ=0 is the Hopf bifurcation value of system (2) and system (2) can be rewritten as(51)u˙t=Lμut+Fμ,ut,where u(t)=u1,u2,u3,u4T∈C=C-1,0,R4 and Lμ: C→R4 and F:R×C→R4 are given, respectively, by(52)Lμϕ=τ10′+μA∗ϕ0+C∗ϕ-τ20′τ10′+B∗ϕ-1, with(53)A∗=a1a2a3a4a5a6a7000a80000a9,B∗=00000b1000b2000000,C∗=0000000000c1000c20,F1=-β1ϕ10ϕ20-β2ϕ10ϕ30,F2=β1ϕ10ϕ20+β2ϕ10ϕ30,ϕ=ϕ1,ϕ2,ϕ3,ϕ4T∈C-1,0,R4. Using Riesz representation theorem, there exists matrix η(θ,μ):[-1,0]→R4 such that(54)Lμϕ=∫-10dηθ,μϕθ,ϕ∈C-1,0,R4. In fact, choose(55)ηθ,μ=τ10′+μA∗+B∗+C∗,θ=0,τ10′+μB′+C∗,θ∈-τ20′τ10′,0,τ10′+μB∗,θ∈-1,-τ20′τ10′,0,θ=-1. For ϕ∈C([-1,0],R4), define(56)Aμϕ=dϕθdθ,-1≤θ<0,∫-10dηθ,μϕθ,θ=0,Rμϕ=0,-1≤θ<0,Fμ,ϕ,θ=0.Then, system (51) can be rewritten in the following form:(57)u˙t=Aμut+Rμut.For ϕ∈C([-1,0],R4), φ∈C1([-1,0],(R4)∗), (58)A∗φ=-dφsds,0<s≤1,∫-10dηTs,0φ-s,s=0, and bilinear form(59)φ,ϕ=φ¯0ϕ0-∫θ=-10∫ξ=0θφ¯ξ-θdηθϕξdξ,are defined, where η(θ)=η(θ,0), A=A(0), and A∗ are adjoint operators.Based on the discussion above, one can conclude that±iω10′τ10′ are common eigenvalues of A(0) and A∗. The eigenvectors of A(0) and A∗ can be calculated corresponding to +iω10′τ10′ and -iω10′τ10′, respectively. Let q(θ)=1,q2,q3,q4Teiω10′τ10′θ be the eigenvector of A(0) corresponding to +iω10′τ10′ and q∗(θ)=K1,q2∗,q3∗,q4∗eiω10′τ10′s be the eigenvector of A∗ corresponding to -iω10′τ10′. By some complex computations, we obtain(60)q2=a5q20,q3=b2e-iτ10′ω10′q2iω10′-a8-c1e-iτ20′ω10′,q4=c2e-iτ20′ω10′q3iω10′-a9,q2∗=-iω10′+a1a5,q3∗=a2+iω10′+a6+b1eiτ10′ω10′q2∗b2eiτ10′ω10′,q4∗=-a3+a7q2∗+iω10′+a8+c1eiτ20′ω10′q3∗c2eiτ20′ω10′,with(61)q20=iω10′-a6-b1e-iτ10′ω10′-a7b2e-iτ10′ω10′iω10′-a8-c1e-iτ20′ω10′. From (59), we get(62)K¯=1+q2q¯2∗+q3q¯3∗+q4q¯4∗+τ10′e-iω10′τ10′q2b1q¯2∗+b2q¯3∗+τ20′e-iω10′τ10′q3c1q¯3∗+c2q¯4∗-1, such that 〈q∗,q〉=1. In what follows, we can obtain the coefficients by using the method introduced in [26]:(63)g20=2τ10′K¯q¯2∗-1β1q2+β2q3,g11=τ10′K¯q¯2∗-1β1Re⁡q2+β2Re⁡q3,g02=2τ10′K¯q¯2∗-1β1q¯2+β2q¯3,g21=2τ10′K¯q¯2∗-1β1W1110q2+12W2010q¯2+W1120+12W2020+β2W1110q3+12W2010q¯3+W1130+12W2030, with(64)W20θ=ig20q0τ10′ω10′eiτ10′ω10′θ+ig¯02q¯03τ10′ω10′e-iτ10′ω10′θ+E1e2iτ10′ω10′θ,W11θ=-ig11q0τ10′ω10′eiτ10′ω10′θ+ig¯11q¯0τ10′ω10′e-iτ10′ω10′θ+E2, where(65)E1=2a1∗-a2-a3-a4-a5a6∗a700-b2e-iτ10′ω10′a8∗000-c2e-iτ20′ω10′a9∗-1-β1q2-β2q3β1q2+β2q300,E2=-a1a2a3a4a5a6+b1a700b2a8+c1000c2a9-1-β1Re⁡q2-β2Re⁡q3β1Re⁡q2+β2Re⁡q300, with(66)a1∗=2iω10′-a1,a6∗=2iω10′-a6-b1e-iτ10′ω10′,a8∗=2iω10′-a8-c1e-iτ20′ω10′,a9∗=2iω10′-a9. Thus, we can compute the following coefficients:(67)C10=i2ω10′τ10′g11g20-2g112-g0223+g212,μ2=-Re⁡C10Re⁡λ′τ10′,β2=2Re⁡C10,T2=-Im⁡C10+μ2Im⁡λ′τ10′ω10′τ10′.In conclusion, we have the following results in this section.Theorem 5.
For system (2),(i)
the direction of the Hopf bifurcation is determined byμ2: if μ2>0, the Hopf bifurcation is supercritical; if μ2<0, the Hopf bifurcation is subcritical;(ii)
the stability of the bifurcating periodic solutions is determined byβ2: if β2<0, the bifurcating periodic solutions are stable; if β2>0, the bifurcating periodic solutions are unstable;(iii)
the period of the bifurcating periodic solution is determined byT2: if T2>0, the period of the bifurcating periodic solutions increases; if T2<0, the period of the bifurcating periodic solutions decreases.


## 4. Numerical Simulations

In this section, we present some numerical simulation results of system (2) to illustrate our theoretical results. We choose a set of parameters as follows: μ=0.001, β1=0.1, β2=0.15, α=0.05, ε=0.05, and γ=0.02. Then, we get the following system:(68)dStdt=0.001-0.1StLt-0.15StAt+0.05Rt-0.001St,dLtdt=0.1StLt+0.15StAt-0.05Lt-τ1-0.001Lt,dAtdt=0.05Lt-τ1-0.02At-τ2-0.001At,dRtdt=0.02At-τ2-0.05Rt-0.001Rt.It is easy to verify that(ε+μ)(γ+μ)/β1γ+μ+β2ε<1 and (ε+μ)(γ+μ)/ε>αγ/α+μ, which ensures the fact that system (68) has a unique viral equilibrium E∗(0.1116,0.2073,0.4936,0.1936).Forτ1=τ2=0, by direct computation by Matlab 7.0, we can get a10>0, a13>0, and a13a12>a11, which means that viral equilibrium E∗(0.1116,0.2073,0.4936,0.1936) is locally asymptotically stable.Forτ1>0,τ2=0, we obtain that (16) has a unique positive root v10=0.0013 and (14) has a unique positive root ω10=0.0363. Further, we get the critical value of delay τ10=28.4522 and f1′(v1∗)=0.0612>0. Thus, we can see that the conditions in Theorem 1 hold true. It follows that viral equilibrium E∗(0.1116,0.2073,0.4936,0.1936) is locally asymptotically stable for τ1∈[0,28.4522) and system (68) undergoes a Hopf bifurcation at viral equilibrium E∗(0.1116,0.2073,0.4936,0.1936) when τ1=28.4522. This property can be depicted in Figures 1–4. Similarly, we can also obtain ω20=0.2756, τ20=95.9606 for τ1=0,τ2>0 and ω0=0.7114, τ0=24.2508 for τ1=τ2=τ>0, respectively. The corresponding phase plots are depicted in Figures 5–8 and Figures 9–12, respectively.Figure 1
The phase plot of statesS∗, A∗, and R∗ with τ1=22.36<28.4522=τ10.Figure 2
The phase plot of statesS∗, L∗, and A∗ with τ1=22.36<28.4522=τ10.Figure 3
The phase plot of statesS∗, A∗, and R∗ with τ1=29.49>28.4522=τ10.Figure 4
The phase plot of statesS∗, L∗, and A∗ with τ1=29.49>28.4522=τ10.Figure 5
The phase plot of statesS∗, A∗, and R∗ with τ2=58.48<95.9606=τ20.Figure 6
The phase plot of statesS∗, L∗, and A∗ with τ2=58.48<95.9606=τ20.Figure 7
The phase plot of statesS∗, A∗, and R∗ with τ2=110.96>95.9606=τ20.Figure 8
The phase plot of statesS∗, L∗, and A∗ with τ2=110.96>95.9606=τ20.Figure 9
The phase plot of statesS∗, A∗, and R∗ with τ=21.35<24.2508=τ0.Figure 10
The phase plot of statesS∗, L∗, and A∗ with τ=21.35<24.2508=τ0.Figure 11
The phase plot of statesS∗, A∗, and R∗ with τ=25.47>24.2508=τ0.Figure 12
The phase plot of the statesS∗, L∗, and A∗ with τ=25.47>24.2508=τ0.Forτ1>0 and τ2=65.25∈(0,τ20), we obtain ω10′=0.0089, τ10′=18.6255. The simulation results can be seen in Figures 13–16. In addition, we obtain C1(0)=-0.2107+0.0062i, λ′(τ10′)=0.0031-0.1086i by some complex computations. Further, we have μ2=67.9677>0, β2=-0.4214<0, T2=44.4907>0. Therefore, we can conclude that the Hopf bifurcation is supercritical; the bifurcating periodic solutions are stable; and the period of the bifurcating periodic solutions increases.Figure 13
The phase plot of statesS∗, A∗, and R∗ with τ1=8.65<18.6255=τ10′ and τ2=65.25.Figure 14
The phase plot of statesS∗, L∗, and A∗ with τ1=8.65<18.6255=τ10′ and τ2=65.25.Figure 15
The phase plot of statesS∗, A∗, and R∗ with τ1=22.4057>18.6255=τ10′ and τ2=65.25.Figure 16
The phase plot of statesS∗, L∗, and A∗ with τ1=22.4057>18.6255=τ10′ and τ2=65.25.


## 5. Conclusions

In the present paper, an improved model for propagation of computer virus propagation model in the network is introduced and studied by incorporating the delay due to the latent period of the computer viruses and the delay due to the period that the antivirus software needs to clean the viruses in the active computers into the model proposed in [19]. We mainly investigate effect of the two delays on the model.By choosing different combination of the two delays as a bifurcation parameter, it has been found that both the two delays can change the stability of the viral equilibrium of the model under some conditions. When the value of the delay is below corresponding critical value, the model is locally asymptotically stable which indicates that the law of propagation of the computer viruses in system (2) can be predicted. However, when the value of the delay is above the corresponding critical value, a Hopf bifurcation occurs and a family of periodic solutions bifurcate from the viral equilibrium, which suggests that the percentages of susceptible, latent, active, and recovered computers in system (2) will fluctuate periodically in a range. This is not helpful in predicting the law of propagation of the computer viruses. Therefore, we should control the occurrence of the Hopf bifurcation by using some bifurcation control strategies and we leave this as our near future work. Furthermore, the properties of the Hopf bifurcation when τ1>0 and τ2∈(0,τ20) have been investigated in detail. Finally, some numerical simulations are also included to support the theoretical results obtained in the paper.



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