# Hopf Bifurcation Analysis for a Computer Virus Model with Two Delays

Authors: Zizhen Zhang; Huizhong Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with a computer virus model with two delays. Its dynamics are studied in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the
positive equilibrium and existence of the local Hopf bifurcation are obtained by regarding the possible combinations of the two delays as a bifurcation parameter. Furthermore, explicit formulae for determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are obtained by using the normal form method and center manifold theory. Finally, some numerical simulations are presented to support the theoretical results.


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## Body

## 1. Introduction

Since the pioneering work of Kephart and White [1, 2], many classical epidemic models such as SIR [3–5], SIRS [6–8], SEIR [9, 10], and SEIRS [11, 12], SEIQV [13] have been used to describe the spread of a computer virus in computer network due to the high similarity between computer viruses and biological viruses. In [9], Yuan and Chen proposed the following SEIR model:
(1)dS(t)dt=μN-v(t)S(t)-ρSRS(t)-μS(t),dE(t)dt=v(t)S(t)-mEαE(t),dI(t)dt=αE(t)-(γ+μ)I(t),dR(t)dt=ρSRS(t)+ρERE(t)+γI(t)-μR(t),
where S(t), E(t), I(t), and R(t) denote the numbers of nodes at time t in states susceptible, exposed, infectious, and recovered, respectively. Yuan and Chen [9] studied the behaviors of virus propagation with the presence of antivirus countermeasures by analyzing the equilibrium stability of system (1).It is well known that time delays of one or other reasons can cause a stable equilibrium to become unstable and make a system bifurcate periodic solutions and dynamical systems with delay have been studied by many scholars [14–23]. Starting from this point and considering that the antivirus software may use a period to clean the viruses in a computer, Dong et al. [10] proposed the following model with delay:
(2)dS(t)dt=μN-βI(t)S(t)-(ρSR+μ)S(t),dE(t)dt=βI(t)S(t)-(α+μ)E(t)-ρERE(t-τ),dI(t)dt=αE(t)-(γ+μ)I(t),dR(t)dt=ρSRS(t)+ρERE(t-τ)+γI(t-τ)-μR(t),
where N is the total number of computers in a network. μ describes the impact of quarantine or replacement. ρSR describes the impact of implementing real-time immunization. ρER describes the impact of cleaning the virus and immunizing the computers. β is the transmission coefficient. α and γ are the state transition rates. And τ is the period which a computer uses antivirus software to clean viruses. Dong et al. [10] discussed the local stability and existence of local Hopf bifurcation of system (2). Properties of the Hopf bifurcation were also investigated in [10].However, Yuan and Chen [9] Dong et al. [10] supposed that the recovered computers have a permanent immunization period and can no longer be infected. This is not consistent with real situation. In order to overcome limitation and considering that the recovered computers may be infected again after a temporary immunity period, we investigate the following system with two delays in this paper:
(3)dS(t)dt=μN-βI(t)S(t)-(ρSR+μ)S(t)+ɛR(t-τ2),dE(t)dt=βI(t)S(t)-(α+μ)E(t)-ρERE(t-τ1),dI(t)dt=αE(t)-(γ+μ)I(t),dR(t)dt=ρSRS(t)+ρERE(t-τ1)+γI(t-τ1)-μR(t)-ɛR(t-τ2),
where τ1 is the period that a computer uses antivirus software to clean viruses and τ2 is the temporary immunity period after which a recovered computer may be infected again. ɛ is the transition rate from R to S.This paper is organized as follows. In Section2, local stability of the positive equilibrium and the existence of local Hopf bifurcation are discussed. In Section 3, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. In order to illustrate the validity of the theoretical analysis, some numerical simulations are presented in Section 4. Some main conclusions are drawn in Section 5.


## 2. Stability and Existence of Local Hopf Bifurcation

In this section, we will study the stability of positive equilibrium and the existence of Hopf bifurcation. It is not difficult to verify if(4)R0=αβμN(μ+ɛ)+ɛρSR(μ+γ)(α+μ+ρER)(μ+ɛ)(μ+γ)(μ+ρSR)(α+μ+ρER)>1
system (3) has a unique positive equilibrium D*(S*,E*,I*,R*), where
(5)S*=(μ+γ)(α+μ+ρER)αβ,I*=(αβμN(μ+ɛ)+ɛρSR(μ+γ)(α+μ+ρER)-(μ+ɛ)(μ+γ)(μ+ρSR)(α+μ+ρER)αβμN(μ+ɛ))×(αβμ(μ+ɛ+γ)+βμ(μ+γ)(μ+ɛ+ρER))-1,E*=μ+γαI*,R*=ρSR(μ+γ)(α+μ+ρER)+[αβγ+βρER(μ+γ)]I*αβ(μ+ɛ).R0 is called the basic reproduction number.Linearizing system (3) at the positive equilibrium D* yields the following linear system:
(6)dS(t)dt=a11S(t)+a13I(t)+c14R(t-τ2),dE(t)dt=a21S(t)+a22E(t)+a23I(t)+b22E(t-τ1),dI(t)dt=a32E(t)+a33I(t),dR(t)dt=a41S(t)+a44R(t)+b42E(t-τ1)+b43I(t-τ1)+c44R(t-τ2),
where
(7)a11=-(μ+ρSR+βI*),a13=-βS*,a21=βI*,a22=-(α+μ),a23=βS*,a32=α,a33=-(μ+γ),a41=ρSR,a44=-μ,b22=-ρER,b42=ρER,b43=γ,c14=ɛ,c44=-ɛ.
Thus, the characteristic equation of system (3) at D* is
(8)λ4+A3λ3+A2λ2+A1λ+A0+(B3λ3+B2λ2+B1λ+B0)e-λτ1+(C3λ3+C2λ2+C1λ+C0)e-λτ2+(D2λ2+D1λ+D0)e-λ(τ1+τ2)=0,
where
(9)A0=a11a44(a22a33-a23a32)+a13a21a32a44,A1=a23a32(a11+a44)-a11a22(a33+a44)-a13a21a32-a33a44(a11+a22),A2=(a11+a22)(a33+a44)+a11a22+a33a44-a23a32,A3=-(a11+a22+a33+a44),B0=a11a33a44b22,B1=-a11a33b22-a44b22(a11+a33),B2=b22(a11+a33+a44),B3=-b22,C0=(a22a33-a23a32)(a11c44-a41c14)+a13a21a32c44,C1=a41c14(a22+a33)+c44(a23a32-a11a22)-a33a44(a11+a22),C2=c44(a11+a22+a33)-a41c14,C3=-c44,D2=b22c44,D1=c14(a41b22-a21b42)-b22c44(a11+a33).D0=a33(a11b22c44+a21b42c14)-c14(a21a32b43+a33a41b22).Case 1 (τ1=τ2=0).
Equation (8) reduces to
(10)λ4+A13λ3+A12λ2+A11λ+A10=0,
where
(11)A10=A0+B0+C0+D0,A11=A1+B1+C1+D1,A12=A2+B2+C2+D2,A13=A3+B3+C3.
By the Routh-Hurwitz criterion, sufficient conditions for all roots of (10) to have a negative real part are given in the following form:
(12)Det1=A13>0,Det2=|A131A11A12|>0,Det3=|A1310A11A12A130A10A11|>0,Det4=|A13100A11A12A1310A10A11A12000A10|>0.
Thus, if condition(H1) (12) holds, then the positive equilibrium D* is locally asymptotically stable in the absence of delay.Case 2 (τ1>0,τ2=0).
Whenτ1>0, τ2=0, (8) becomes the following form:
(13)λ4+A23λ3+A22λ2+A21λ+A20+(B23λ3+B22λ2+B21λ+B20)e-λτ1=0,
where
(14)A23=A3+C3,A22=A2+C2,A21=A1+C1,A20=A0+C0,B23=B3,B22=B2+D2,B21=B1+D1,B20=B0+D0.
Let λ=iω1 (ω1>0) be the root of (13). Then, we can get
(15)(B21ω1-B23ω13)sinτ1ω1+(B20-B22ω12)cosτ1ω1=A22ω12-ω14-A20,(B21ω1-B23ω13)cosτ1ω1-(B20-B22ω12)sinτ1ω1=A23ω13-A21ω1,
which follows that
(16)ω18+e23ω16+e22ω14+e21ω12+e20=0,
where
(17)e20=A202-B202,e21=A212-B212-2A20A22+2B20B22,e22=A222-B222-2A21A23+2A20+2B21B23,e23=A232-B232-2A22.
Let ω12=v1; then, (16) becomes
(18)v14+e23v13+e22v12+e21v1+e20=0.
Discussion about the roots of (18) is similar to that in [15]. Therefore, we have the following lemma.Lemma 1.
Let(19)f1(v1)=v14+e23v13+e22v12+e21v1+e20.p=12e22-316e232,q=132e233-18e22e23+e21,α1=(q2)3+(p3)3,β1=-12+32i,y1=-q2+α13+-q2-α13,y2=β1-q2+α13+β12-q2-α13,y3=β12-q2+α13+β1-q2-α13,v1i=yi-3e234,i=1,2,3.
Then, for (18), one has the following:(i)
ife20<0, (18) has at least one positive root;(ii)
ife20≥0 and α1≥0, (18) has positive roots if and only if v11>0 and f1(v11)<0;(iii)
ife20≥0 and α1<0, (18) has positive roots if and only if there exists at least one v1*∈{v11,v12,v13} such that v1*>0 and f1(v1*)≤0.
In what follows, we assume that(H21): the coefficients in f1(v1) satisfy one of the following conditions in (a)–(c):(a)
e20<0;(b)
e20≥0, α1≥0, v11>0, and f1(v11)<0;(c)
e20≥0, α1<0, and there exists at least one v1*∈{v11,v12,v13} such that v1*>0 and f1(v1*)≤0.
If(H21) holds, we know that (16) has at least a positive root ω10 such that (13) has a pair of purely imaginary root ±iω10. The corresponding critical value of the delay is
(20)τ1k=1ω10arccosp6ω106+p4ω104+p2ω102+p0q6ω106+q4ω104+q2ω102+q0+2kπω10,k=0,1,2,…,
where
(21)p0=-A20B20,p2=A20B22-A21B21+A22B20,p4=A21B23-A22B22+A23B21-B20,p6=B22-A23B23,q0=B202,q2=B212-2B20B22,q4=B222-2B21B23,q6=B232.
Differentiating both sides of (13) regarding τ1, we can obtain
(22)[dλdτ1]-1=-4λ3+3A23λ2+2A22λ+A21λ5+A23λ4+A22λ3+A21λ2+A20λ+3B23λ2+2B22λ+B21B23λ4+B22λ3+B21λ2+B20λ-τ1λ.
Thus,
(23)Re[dλdτ]τ1=τ10-1=(4ω106+3(A232-2A22)ω104+2(A222+2A20-2A21A23)ω102+A212-2A20A22)×(ω108+(A232-2A22)ω106+(A222+2A20-2A21A23)ω104+(A212-2A20A22)ω102+A202)-1-(3B232ω104+2(B222-2B21B23)ω102+B212-2B20B22(B222-2B21B23))×(B232ω106+(B222-2B21B23)ω104+(B212-2B20B22)ω102+B202)-1.
From (16), we can get
(24)Re[dλdτ]τ1=τ10-1=f1′(v1*)×(B232ω106+(B222-2B21B23)ω104+(B212-2B20B22)ω102+B202)-1,
where v1*=ω102.
Thus, if condition(H22)f1′(v1*)≠0 holds, then the transversality condition is satisfied. According to the Hopf bifurcation theorem in [24], we have the following results.Theorem 2.
Suppose that the conditions(H21)-(H22) hold. Consider(i)
the positive equilibriumD*(S*,E*,I*,R*) of system (3) is asymptotically stable for τ1∈[0,τ10);(ii)
system (3) undergoes a Hopf bifurcation at the positive equilibrium D*(S*,E*,I*,R*) when τ1=τ10 and a family of periodic solutions bifurcate from the positive equilibrium D*(S*,E*,I*,R*) near τ1=τ10.Case 3 (τ1=0,τ2>0).
Whenτ1=0, τ2>0, (8) becomes the following form:
(25)λ4+A33λ3+A32λ2+A31λ+A30+(C33λ3+C32λ2+C31λ+C30)e-λτ2=0,
where
(26)A33=A3+B3,A32=A2+B2,A31=A1+B1,A30=A0+B0,C33=C33,C32=C2+D2,C31=C1+D1,C30=C0+D0.
Let λ=iω2 (ω2>0) be the root of (25), and we have
(27)(C31ω2-C33ω23)sinτ2ω2+(C30-C32ω22)cosτ2ω2=A32ω22-ω24-A30,(C31ω2-C33ω23)cosτ2ω2-(C30-C32ω22)sinτ2ω2=A33ω23-A31ω2,
from which, we can obtain
(28)ω28+e33ω26+e32ω24+e31ω22+e30=0,
where
(29)e30=A302-C302,e31=A312-C312-2A30A32+2C20C32,e32=A322-B322-2A31A33+2A30+2C31C33,e33=A332-C332-2A32.
Let ω22=v2, then (28) becomes
(30)v24+e33v23+e32v22+e31v2+e30=0.
Define
(31)f2(v2)=v24+e33v23+e32v22+e31v2+e30,m=12e32-316e332,n=132e333-18e32e33+e31,α2=(n2)3+(m3)3,β2=-12+32i,z1=-n2+α23+-n2-α23,z2=β2-n2+α23+β22-n2-α23,z3=β22-n2+α23+β2-n2-α23,v2i=zi-3e334,i=1,2,3.
Next, we assume that (H31): the coefficients in f2(v2) satisfy one of the following conditions in (a′)–(c′):(a′)
e30<0;(b′)
e30≥0, α2≥0, v21>0, and f2(v21)<0;(c′)
e30≥0, α2<0, and there exists at least one v2*∈{v21,v22,v23} such that v2*>0 and f2(v2*)≤0.According to Lemma 1, we know that if the condition (H31) holds, then (28) has at least a positive root ω20 such that (25) has a pair of purely imaginary roots ±iω20. The corresponding critical value of the delay is
(32)τ2k=1ω20arccosm6ω206+m4ω204+m2ω202+m0n6ω206+n4ω204+n2ω202+n0+2kπω20,k=0,1,2,…,
where
(33)m0=-A30C30,m2=A30C32-A31C31+A32C30,m4=A31C33-A32C32+A33C31-C30,m6=C32-A33C33,n0=C302,n2=C312-2C30C32,n4=C322-2C31C33,n6=C332.
As in Case 2, we know that if f2′(v2*)≠0 where v2*=ω202, then Re[dλ/dτ2]τ2=τ20-1≠0. That is, the transversality condition is satisfied if the condition (H32)f2′(v2*)≠0 holds. Thus, according to the Hopf bifurcation theorem in [24], we have the following results.Theorem 3.
Suppose the conditions(H31)-(H32) hold.(i)
The positive equilibriumD*(S*,E*,I*,R*) of system (3) is asymptotically stable for τ2∈[0,τ20).(ii)
System (3) undergoes a Hopf bifurcation at the positive equilibrium D*(S*,E*,I*,R*) when τ2=τ20 and a family of periodic solutions bifurcate from the positive equilibrium D*(S*,E*,I*,R*) near τ2=τ20.Case 4 (τ1=τ2=τ>0).
Substitutingτ1=τ2=τ>0 into (8), then (8) becomes
(34)λ4+A43λ3+A42λ2+A41λ+A40+(B43λ3+B42λ2+B41λ+B40)e-λτ+(D42λ2+D41λ+D40)e-2λτ=0,
where
(35)A43=A3,A42=A2,A41=A1,A40=A0,B43=B3+C3,B42=B2+C2,B41=B1+C1,B40=B0+C0,D42=D2,D41=D1,D40=D0.
Multiplying by eλτ, (34) becomes
(36)B43λ3+B42λ2+B41λ+B40×(λ4+A43λ3+A42λ2+A41λ+A40)eλτ+(D42λ2+D41λ+D40)e-λτ=0.
Let λ=iω (ω>0) be the root of (36); consider
(37)(ω4-(A42+D42)ω2+A40+D40)cosτω+(A43ω3-(A41-D41)ω)sinτω=B32ω2-B40,(ω4-(A42-D42)ω2+A40-D40)sinτω-(A43ω3-(A41+D41)ω)cosτω=B43ω3-B41ω,
which follows that
(38)sinτω=g7ω7+g5ω5+g3ω3+g1ωω8+h6ω6+h4ω4+h2ω2+h0,cosτω=g6ω6+g4ω4+g2ω2+g0ω8+h6ω6+h4ω4+h2ω2+h0,
where
(39)g0=B40(D40-A40),g1=B40(A41+D41)-B41(A40+D40),g2=B40(A42-D42)-B41(A41-D41)+B42(A40-D40),g3=B41(A42+D42)-B42(A41+D41)+B43(A40+D40)-A43B40,g4=B42(D42-A42)+B43(A41-D41)+A43B41-B40,g5=A42B43-B41-B43(A42+D42),g6=B42-A43B43,g7=B43,h0=A402-D402,h2=A412-D412-2A40A42+2D40D42,h4=A422-D422-2A41A43+2A40,h6=A432-2A42.
As is known, sin2τω+cos2τω=1. Thus, we have
(40)ω16+e47ω14+e46ω12+e45ω10+e44ω8+e43ω6+e42ω4+e41ω2+e40=0,
where
(41)e40=h02-g02,e41=2h0h2-2g0g2-g12,e42=h22-g22+2h0h4-2g0g4-2g1g3,e43=2h0h6+2h2h4-2g0g6-2g1g5-2g2g4-g33,e44=h42-g42+2h0+2h2h6-2g1g7-2g2g6-2g3g5,e45=2h2+2h4h6-2g3g7-2g4g6-g52,e46=h62-g62+2h4-2g5g7,e47=2h6-g72.
Let ω2=v, then (40) becomes
(42)v8+e47v7+e46v6+e45v5+e44v4+e43v3+e42v2+e41v+e40=0.
In order to give the main results in this paper, we make the following assumption.(
H
41
) (42) has at least one positive real root.Suppose that condition(H41) holds. Without loss of generality, we assume that (42) has eight positive real roots, which are denoted as v1,v2,…,v8, respectively. Then (40) has eight positive real roots ωk=vk. For every fixed ωk, the corresponding critical value of time delay is
(43)τk(j)=1ωkarccosg6ωk6+g4ωk4+g2ωk2+g0ωk8+h6ωk6+h4ωk4+h2ωk2+h0+2jπωk,k=1,2,…;j=0,1,2,….
Define
(44)τ0=min{τk(0)∣k=1,2,…,8},ω0=ωk|τ=τ0.
Taking the derivative of with respect to (36), it is easy to obtain
(45)[dλdτ]-1=-(-(D42λ3+D41λ2+D40λ)e-λτ)-1(3B43λ2+2B42λ+B41+(2D42λ+D41)e-λτ+(4λ3+3A43λ2+2A42λ+A41)eλτ)×((λ5+A43λ4+A42λ3+A41λ2+A40λ)eλτ-(D42λ3+D41λ2+D40λ)e-λτ)-1)-τλ.
Thus, we have
(46)Re[dλdτ]τ=τ0-1=-PRQR+PIQIQR2+QI2,
where
(47)PR=(A41+D41-3A43ω02)cosτ0ω0+2(2ω03-A42ω0+D42ω0)sinτ0ω0-3B43ω02+B41,PI=(A41-D41-3A43ω02)sinτ0ω0-2(2ω03-A42ω0-D42ω0)cosτ0ω0+2B42ω0,QR=(ω05-(A42+D42)ω03+(A40+D40)ω0)sinτ0ω0+(A43ω04-(A41+D41)ω02)cosτ0ω0,QI=(ω05-(A42+D42)ω03+(A40+D40)ω0)cosτ0ω0-(A43ω04-(A41+D41)ω02)sinτ0ω0.
Obviously, if condition PRQR+PIQI≠0 holds, then Re[dλ/dτ]τ=τ0-1≠0. Thus, by the Hopf bifurcation theorem in [24], we have the following results.Theorem 4.
Suppose that conditions(H41)-(H42) hold.(i)
The positive equilibriumD*(S*,E*,I*,R*) of system (3) is asymptotically stable for τ∈[0,τ0).(ii)
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.Case 5 (τ1>0, τ2∈(0,τ20)).
We consider (8) with τ2 in its stable interval, regarding τ1 as a parameter.
Letλ=iω1* (ω1*>0) be the root of (8); we obtain
(48)c0(ω1*)+2c1(ω1*)cosτ2ω1*+2c2(ω1*)sinτ2ω1*=0,
where
(49)c0(ω1*)=ω1*8+(A32-B32+C32-2A2)ω1*6+(A22-B22+C22-D22+2A0-2A1A3+2B1B3-2C1C3{A22})ω1*4+(A12-B12+C12-D12-2A-0A2+2B0B2-2C0C2+2D0D2{D12})ω1*2+A02-B02+C02-D02,c1(ω1*)=(A3C3-C2)ω1*6+(A2C2+B3D1-B2D2+C0-A1C3-A3C1)×ω1*4+(A1C1-A0C2-A2C0+B0D2+B2D0-B1D1)ω1*2+A0C0+B0D0,c2(ω1*)=-C3ω1*7+(A2C3-A3C2+B3D2+C1)ω1*5+(A1C2+A3C0-A0C3-A2C1-B1D2+B2D1-B3D0)ω1*3×(A0C1-A1C0-B0D1+B1D0)ω1*.
We assume that (H51) (48) has at least finite positive roots. And we denote the positive roots of (48) as ω11*, ω12*, and ω1k*. For every fixed ω1i*, the corresponding critical value of time delay is
(50)τ1i*(j)=1ω1i*arccosΔ1Δ4+Δ2Δ3Δ12+Δ22+2jπω1i*,i=1,2,k;j=0,1,2,…,
where
(51)Δ1=D1ω1i*cosτ2ω1i*+(D2ω1i*2-D0)sinτ2ω1i*-B3ω1i*3+B1ω1i*,Δ2=D1ω1i*sinτ2ω1i*-(D2ω1i*2-D0)cosτ2ω1i*-B2ω1i*2+B0,Δ3=(C3ω1i*3-C1ω1i*)sinτ2ω1i*+(C2ω1i*2-C0)cosτ2ω1i*+A2ω1i*2-ω1i*4-A0,Δ4=(C3ω1i*3-C1ω1i*)cosτ2ω1i*+(C2ω1i*2-C0)sinτ2ω1i*+A3ω1i*3-A1ω1i*.
Let
(52)τ1*=min{τ1i*(0)∣i=1,2,…,k},ω1*=ω1i*|τ1=τ1*.
Next, we make the following assumption: (H51)Re[dλ/dτ]τ1=τ1*-1≠0. Thus, by the Hopf bifurcation theorem in [24], we have the following results.Theorem 5.
Suppose that conditions(H51)-(H52) hold and τ2∈(0,τ20).(i)
The positive equilibriumD*(S*,E*,I*,R*) of system (3) is asymptotically stable for τ1∈[0,τ1*).(ii)
System (3) undergoes a Hopf bifurcation at the positive equilibrium D*(S*,E*,I*,R*) when τ1=τ1* and a family of periodic solutions bifurcate from the positive equilibrium D*(S*,E*,I*,R*) near τ1=τ1*.


## 3. Direction and Stability of the Hopf Bifurcation

In this section, we will investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions w.r.t.τ1 for τ2∈(0,τ20) by using the normal form theory and the center manifold argument in [24]. We assume that τ2*<τ1* where τ2*∈(0,τ20).Letu1(t)=S(t)-S*, u2(t)=E(t)-E*, u3(t)=I(t)-I*, u4(t)=R(t)-R*, τ1=τ1*+μ,μ∈R, and normalize t→(t/τ1). Then system (3) can be transformed into the following form:
(53)u˙(t)=Lμut+F(μ,ut),
where ut=(u1(t),u2(t),u3(t),u4(t))T∈C=C([-1,0],R4) and
(54)Lμϕ=(τ1*+μ)(A′ϕ(0)+C′ϕ(-τ2*τ1*)+B′ϕ(-1)),F(μ,ϕ)=(τ1*+μ)(-βϕ1(0)ϕ3(0)βϕ1(0)ϕ3(0)00),
where
(55)A′=(a110a130a21a22a2300a32a330a4100a44),B′=(00000b220000000b42b430),C′=(000c1400000000000c44).By the Riesz representation theorem, there exists a4×4 matrix function η(θ,μ):[-1,0]→R4 whose elements are of bounded variation such that
(56)Lμϕ=∫-10‍dη(θ,μ)ϕ(θ),ϕ∈C.
In fact, we choose
(57)η(θ,μ)={(τ1*+μ)(A′+B′+C′),θ=0,(τ1*+μ)(B′+C′),θ∈[-τ2*τ1*,0),(τ1*+μ)B′,θ∈(-1,-τ2*τ1*),0,θ=1.
For ϕ∈C([-1,0],R4), we define
(58)A(μ)ϕ={dϕ(θ)dθ,-1≤θ<0,∫-10‍dη(θ,μ)ϕ(θ),θ=0,R(μ)ϕ={0,-1≤θ<0,F(μ,ϕ),θ=0.
Then system (53) can be transformed into the following operator equation:
(59)u˙(t)=A(μ)ut+R(μ)ut.
The adjoint operator A* of A is defined by
(60)A*(φ)={-dφ(s)ds,0<s≤1,∫-10‍dηT(s,0)φ(-s),s=0,
associated with a bilinear form(61)〈φ(s),ϕ(θ)〉=φ¯(0)ϕ(0)-∫θ=-10‍∫ξ=0θ‍φ¯(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0).Letq(θ)=(1,q2,q3,q4)Teiτ1*ω1*θ be the eigenvector of A(0) corresponding to +iτ1*ω1*, and let q*(s)=D(1,q2*,q3*,q4*)eiτ1*ω1*s be the eigenvector of A*(0) corresponding to -iτ1*ω1*. From the definition of A(0) and A*(0), we can get
(62)q2=a21(iω1*-a33)(iω1*-a33)(iω1*-a22-b22e-τ1*ω1*)-a23a32,q3=a21a32(iω1*-a33)(iω1*-a22-b22e-τ1*ω1*)-a23a32,q4=iω1*-a11c14e-iτ2*ω1*-a13a21a32eiτ2*ω1*c14(iω1*-a33)(iω1*-a22-b22e-τ1*ω1*)-a23a32c14,q2*=a41c14eiτ2*ω1*a21(iω1*+a44+c44eiτ2*ω1*)-iω1*+a11a21,q3*=-(iω1*+a22+b22eiτ1*ω1*)q2*+b42eiτ1*ω1*q4*a32,q4*=-c14eiτ2*ω1*iω1*+a44+c44eiτ2*ω1*.
From (61), we can get
(63)〈q*,q〉=D¯[∫1+q2q¯2*+q3q¯3*+q4q¯4*(q2(b22q¯2*+b42q¯4*)+b43q3q¯4*)+τ2*e-iτ2*ω1*q4(c14+c44q¯4*)+τ1*e-iτ1*ω1*(q2(b22q¯2*+b42q¯4*)+b43q3q¯4*)∫].
Then, we choose
(64)D¯=[(q2(b22q¯2*+b42q¯4*)+b43q3q¯4*)1+q2q¯2*+q3q¯3*+q4q¯4*+τ2*e-iτ2*ω1*q4(c14+c44q¯4*)+τ1*e-iτ1*ω1*(q2(b22q¯2*+b42q¯4*)+b43q3q¯4*)]-1,
such that 〈q*,q〉=1, 〈q*,q¯〉=0.Next, we can get the coefficients which can be used to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by following the algorithms introduced in [24]:
(65)g20=2βτ1*D¯q3(q¯2*-1),g02=2βτ1*D¯q¯3(q¯2*-1),g11=bt1*D¯(q3+q¯3)(q¯2*-1),g21=2βτ1*D¯(q¯2*-1)(W11(1)(0)q3+12W20(1)(0)q¯3+W11(3)(0)+12W20(3)(0)),
with
(66)W20(θ)=ig20q(0)τ1*ω1*eiτ1*ω1*θ+ig¯02q¯(0)3τ1*ω1*e-iτ1*ω1*θ+E1e2iτ1*ω1*θ,W11(θ)=-ig11q(0)τ1*ω1*eiτ1*ω1*θ+ig¯11q¯(0)τ1*ω1*e-iτ1*ω1*θ+E2,
where E1 and E2 can be determined by the following equations respectively(67)E1=2(2iω1*-a110-a13-c14e-2iτ2*ω1*-a212iω1*-a22-b22e-2iτ1*ω1*-a2300-a322iω1*-a330-a41-b42e-2iτ1*ω1*-b43e-2iτ1*ω1*2iω1*-a44-c44e-2iτ2*ω1*)-1(E1(1)E1(2)00),E2=-(a110a13c14a21a22+b22a2300a32a330a41b42b43a44+c44)-1(E2(1)E2(2)00),with(68)E1(1)=-βq3,E1(2)=βq3,E2(1)=-β(q3+q¯3),E2(2)=β(q3+q¯3).Therefore, we can calculate the following values:(69)C1(0)=i2τ1*ω1*(g11g20-2|g11|2-|g02|23)+g212,μ¯=-Re{C1(0)}Re{λ′(τ1*)},β¯=2Re{C1(0)},T¯=-Im{C1(0)}+μ¯Im{λ′(τ1*)}τ1*ω1*.
Based on the discussion above, we can obtain the following results.Theorem 6.
For system (3), if μ¯>0 (μ¯<0), the Hopf bifurcation is supercritical (subcritical). If β¯<0 (β¯>0) the bifurcating periodic solutions are stable (unstable). If T¯>0 (T¯<0), the period of the bifurcating periodic solutions increases (decreases).


## 4. Numerical Simulation

In this section, we present some numerical simulations to verify the theoretical analysis in Sections2 and 3. Let N=1000, α=0.1, β=0.01, γ=0.08, ɛ=0.15, μ=0.01, ρSR=0.2, ρER=0.2. Then, we can get a particular case of system (3):
(70)dS(t)dt=10-0.01I(t)S(t)-0.21S(t)+0.15R(t-τ2),dE(t)dt=0.01I(t)S(t)-0.11E(t)-0.2E(t-τ1),dI(t)dt=0.1E(t)-0.09I(t),dR(t)dt=0.2S(t)+0.2E(t-τ1)+0.08I(t-τ1)-0.01R(t)-0.15R(t-τ2).Then, we can getR0=2.5996>1 and the unique positive equilibrium D*(27.9,239.2915,265.8794,466.9290) of system (70). Further, we have Det1=3.4288>0, Det2=5.4964>0, Det3=0.6898>0, Det4=9.6572e-004>0. That is, condition (H1) holds.Forτ1>0, τ2=0, by some complicated computation, we obtain ω10=0.6102, τ10=17.2530. By Theorem 2, we know that when τ1∈[0,17.2530) the positive equilibrium D* is asymptotically stable which can be illustrated by Figures 1 and 2. However, if we let τ1=20.05>17.2530=τ10, the positive equilibrium D* becomes unstable and a Hopf bifurcation occurs and a branch of periodic solutions bifurcates from the positive equilibrium D*. This property can be shown in Figures 3 and 4. Similarly, we have ω20=0.1181, τ20=52.4927 for τ1=0, τ2>0. The corresponding wave forms and plots are shown in Figures 5, 6, 7, and 8.The track of the statesS, E, I, and R for τ1=15.5000<17.2530=τ10.
(a)
(b)
(c)
(d)Figure 2
The phase plot of the statesS, E, and R for τ1=15.5000<17.2530=τ10.The track of the statesS, E, I, and R for τ1=20.0500>17.2530=τ10.
(a)
(b)
(c)
(d)Figure 4
The phase plot of the statesS, E, and R for τ1=20.0500>17.2530=τ10.The track of the statesS, E, I, and R for τ2=45.5000<52.4927=τ20.
(a)
(b)
(c)
(d)Figure 6
The phase plot of the statesS, E, and R for τ2=45.5000<52.4927=τ20.The track of the statesS, E, I, and R for τ2=67.5000>52.4927=τ20.
(a)
(b)
(c)
(d)Figure 8
The phase plot of the statesS, E, and R for τ2=67.5000>52.4927=τ20.Forτ1=τ2=τ>0, we get ω0=2.0670, τ0=5.1094. According to Theorem 4, we can conclude that when τ∈[0,5.1094), the positive equilibrium D* is asymptotically stable, which can be seen from Figures 9 and 10. If τ=5.23>5.1094=τ0; then, the positive equilibrium D* becomes unstable, and a branch of bifurcating periodic solutions occurs, which can be shown in Figures 11 and 12.The track of the statesS, E, I, and R for τ=4.7500<5.1094=τ0.
(a)
(b)
(c)
(d)Figure 10
The phase plot of the statesS, E, and R for τ=4.7500<5.1094=τ0.The track of the statesS, E, I, and R for τ=5.2300>5.1094=τ0.
(a)
(b)
(c)
(d)Figure 12
The phase plot of the statesS, E, and R for τ=5.2300>5.1094=τ0.Lastly, we haveω1*=1.6115, τ1*=7.8533 for τ1>0, τ2=3∈(0,τ20). The corresponding wave forms and phase plots are shown in Figures 13, 14, 15, and 16. In addition, we have λ′(τ1*)=0.3938+0.1113i, C1(0)=-0.2079-0.6067i, μ¯=0.5279, β¯=-0.4158<0, T¯=0.0433>0. Thus, by Theorem 6, we can conclude that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the periodic solutions increases.The track of the statesS, E, I, and R for τ1=7.2000<7.8533=τ1* and τ2*=3.
(a)
(b)
(c)
(d)Figure 14
The phase plot of the statesR, S, and E for τ1=7.2000<7.8533=τ1* and τ2*=3.The track of the statesS, E, I, and R for τ1=8.4500>7.8533=τ1* and τ2*=3.
(a)
(b)
(c)
(d)Figure 16
The phase plot of the statesR, S, and E for τ1=8.4500>7.8533=τ1* and τ2*=3.


## 5. Conclusions

Based on the SEIR model considered in the literature [10], we propose an SEIRS model with two delays for the propagation of computer viruses in networks. The effects of the two delays on the dynamics of the model are investigated. It is found that the two delays may lead to local Hopf bifurcation and make the model unstable under some certain conditions. When the corresponding delay is suitable and small, the positive equilibrium is asymptotically stable. In this case, the propagation of computer viruses can be predicted and controlled. However, a local Hopf bifurcation occurs and a branch of periodic solutions bifurcates from the positive equilibrium when the corresponding delay passes though a critical value. In such conditions, propagation of the computer viruses is out of control. This phenomenon is not welcome in networks. Therefore, In order to control and even eliminate the propagation of computer viruses, the two delays τ1 and τ2 in the model should remain less than the corresponding critical value. Furthermore, by using the normal form theory and center manifold theorem, the properties of the Hopf bifurcation such as direction and stability are determined. Some numerical simulations are also included to testify the theoretical analysis.



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*Source: 560804-2013-11-13.xml*