# Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network

Authors: Zizhen Zhang; Yougang Wang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Hopf bifurcation for an SEIRS-V model with delays on the transmission of worms in a wireless sensor network is investigated. We focus on existence of the Hopf bifurcation by regarding the diverse delay as a bifurcation parameter. The results show that propagation of worms in the wireless sensor network can be controlled when the delay is suitably small under some certain conditions. Then, we study properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Finally, we give a numerical example to support the theoretical results.


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## Body

## 1. Introduction

In recent years, wireless sensor networks have received extensive attention due to their vast potential in many application environments. However, security of wireless sensor networks still remains one of the most critical challenges because sensor nodes are often placed in a hostile or dangerous environment [1]. Many epidemiological models [2–6] have been proposed to study and predict the spread of viruses in wireless networks motivated by the pioneering work of Murray [7] and Kephart and White [8, 9]. In [10], Mishra and Keshri proposed the following SEIRS-V model to describe the propagation of worms in a wireless sensor network:(1)dStdt=A-βStIt-ζ+pSt+δRt+ηVt,dEtdt=βStIt-ζ+αEt,dItdt=αEt-ζ+ε+γIt,dRtdt=γIt-ζ+δRt,dVtdt=pSt-ζ+ηVt,where S(t), E(t), I(t), R(t), and V(t) denote the number of susceptible, exposed (infected, but not infectious), infectious, recovered, and vaccinated sensor nodes at time t, respectively. A, p, α, β, γ, δ, η, ε, and ζ are the positive parameters of system (1) and for the specific meanings of them one can refer to [10]. Considering the time delays in system (1), Zhang and Si [11] proposed the following delayed SEIRS-V system:(2)dStdt=A-βStIt-ζ+pSt+δRt-τ2+ηVt-τ2,dEtdt=βStIt-ζ+αEt,dItdt=αEt-ζ+εIt-γIt-τ1,dRtdt=γIt-τ1-ζRt-δRt-τ2,dVtdt=pSt-ζVt-ηVt-τ2,where τ1≥0 is the time delay due to the period that antivirus software uses to clean worms in the infected nodes; τ2≥0 is the time delay due to the temporary immunity period of the recovered and the vaccinated nodes. For the convenience of analysis, Zhang and Si [11] let τ1=τ2 and considered the following system:(3)dStdt=A-βStIt-ζ+pSt+δRt-τ+ηVt-τ,dEtdt=βStIt-ζ+αEt,dItdt=αEt-ζ+εIt-γIt-τ,dRtdt=γIt-τ-ζRt-δRt-τ,dVtdt=pSt-ζVt-ηVt-τ.Zhang and Si [11] investigated existence and properties of the Hopf bifurcation of system (3).It should be pointed out that one of the significant features of worms in networks is its latent characteristic. Therefore, there exists a certain period before the exposed nodes develop themselves into the infectious ones. In addition, as far as we know, there have been some papers that deal with research of Hopf bifurcation of dynamical systems with multiple delays in recent years [12–15]. In [12], Xu et al. studied Hopf bifurcation of a ring of five neurons with delays. In [14], Bianca et al. investigated Hopf bifurcation of an economic growth model with two delays. Considering that there is a latent period of worms in the exposed nodes in system (3), we study the following system with delays:(4)dStdt=A-βStIt-ζ+pSt+δRt-τ2+ηVt-τ2,dEtdt=βStIt-αEt-τ1-ζEt,dItdt=αEt-τ1-ζ+εIt-γIt-τ2,dRtdt=γIt-τ2-ζRt-δRt-τ2,dVtdt=pSt-ζVt-ηVt-τ2,where τ1 is the time delay due to the latent period of worms in the exposed nodes and τ2 is the time delay due to the period that the antivirus software uses to clean worms in the infected nodes and that due to the temporary immunity period of the recovered and the vaccinated nodes.The structure of this paper is as follows. In Section2, we obtain sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation of system (4). In Section 3, we deal with the properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Some numerical simulations are carried out in Section 4 with the aim of verifying the obtained analytic results. Finally, conclusions and future work are summarized.


## 2. Hopf Bifurcation Analysis

By a direct computation, we know that ifR0=αβA(ζ+η)+pη(ζ+α)(ζ+ε+γ)/(ζ+p)(ζ+α)(ζ+η)(ζ+ε+γ)>1, then system (4) has a unique positive equilibrium D∗(S∗,E∗,I∗,R∗,V∗) in which(5)S∗=ζ+αζ+ε+γαβ,E∗=ζ+ε+γαI∗,R∗=γζ+δI∗,V∗=pζ+αζ+ε+γαβζ+η,I∗=αβAζ+δζ+η+pηζ+αζ+δζ+ε+γ-ζ+pζ+αζ+δζ+ηζ+ε+γβζ+αζ+δζ+ηζ+ε+γ-αβδγζ+η.The characteristic equation of system (4) at D∗ is(6)λ5+A4λ4+A3λ3+A2λ2+A1λ+A0+B4λ4+B3λ3+B2λ2+B1λ+B0e-λτ1+C4λ4+C3λ3+C2λ2+C1λ+C0e-λτ2+D3λ3+D2λ2+D1λ+D0e-2λτ2+E3λ3+E2λ2+E1λ+E0e-λτ1+τ2+F2λ2+F1λ+F0e-λτ1+2τ2+G2λ2+G1λ+G0e-3λτ2+H1λ+H0e-λτ1+3τ2=0,where(7)A0=-a11a22a33a44a55,A1=a11a22a33a44+a11a22a55a33+a44+a33a44a55a11+a22,A2=-a55a11a22+a33a44+a11+a22a33+a44-a11a22a33+a44+a33a44a11+a22,A3=a11a22+a33a44+a11+a22a33+a44+a55a11+a22+a33+a44,A4=-a11+a22+a33+a44+a55,B0=a44a55b32a11a23-a13a21-a11a22a33a44b22,B1=b22a11a22a33+a44+a33a44a11+a22+b32a44+a55a13a21-a23a11-a23a44a55,B2=a23b32a44+a55+b32a11a23-a13a21-b22a11a22+a33a44+a11+a22a33+a44,B3=b22a11+a22+a33+a44-a23b32,B4=-b22,C0=-a11a22a33a44b55+a55b44-a11a22a44a55b33,C1=b33a11a22a44+a55+a44a55a11+a22+a11a22+a11a33+a22a33a44b55+a55b44+a11a22a33b44+b55,C2=-a11a22+a11a33+a22a33b44+b55-a44b55+a55b44a11+a22+a33-b33a11a22+a44a55+a11+a22a44+a55,C3=b33a11+a22+a44+a55+a44b55+a55b44+a11+a22+a33b44+b55,C4=-b22+b44+b55,D0=-a11a22a33b44b55+a44b33b55+a55b33b44,D1=b33b44a11a22+a11a55+a22a55+b33b55a11a22+a11a44+a22a44+b44b55a11a22+a11a33+a22a33,D2=-b33b44a11+a22+a55-b33b55a11+a22+a44-b44b55a11+a22+a33,D3=b33b44+b33b55+b44b55,E0=b32a44b55+a55b44a11a23-a13a21-a11b22a44a55b33+a33a44b55+a33a55b44,E1=a11a33b22b44+b55+b22b33a11a44+a11a55+a44a55+a11b22+a33b22-a23b32a44b55+a55b44+b32b44+b55a13a21-a11a23,E2=a23b32b44+b55-b22b33a11+a44+a55-b22a44b55+a55b44+b44+b55a11+a33,E3=b22b33+b44+b55,F0=a21a55b14b32b43+b32b44b55a11a23-a13a21-a11b22a33b44b55+a44b33b55+a55b33b44,F1=b22b33b55a11+a44+b22b44b55a11+a33-a21b14b32b43+a23a32b44b55,F2=-b22b33b44+b33b55+b44b55,G0=-a11a22b33b44b55,G1=b33b44b55a11+a22,G2=-b33b44b55,H0=a21b14b32b43b55-a11b22b33b44b55,H1=b22b33b44b55,with(8)a11=-βI∗+ζ+p,a13=-βS∗,a21=βI∗,a22=-ζ,a23=βS∗,a33=-ζ+ε,a44=-ζ,a51=p,a55=-ζ,b14=δ,b15=η,b22=-α,b32=α,b33=-γ,b43=γ,b44=-δ,b55=-η.Case 1.
Whenτ1=τ2=0, (6) becomes(9)λ5+A14λ4+A13λ3+A12λ2+A11λ+A10=0,where(10)A10=A0+B0+C0+D0+E0+F0+G0+H0,A11=A1+B1+C1+D1+E1+F1+G1+H1,A12=A2+B2+C2+D2+E2+F2+G2,A13=A3+B3+C3+D3+E3,A14=A4+B4+C4.
It is easy to get thatA14=p+α+γ+δ+ε+η+5ζ+βI∗>0. Thus, we have the following condition: (H1)
If (11) holds, D∗ is locally asymptotically stable:(11)det2=A141A12A13>0,det3=A1410A12A13A140A11A12>0,det4=A14100A12A13A141A10A11A12A1300A10A11>0,det5=A141000A12A13A1410A10A11A12A13A1400A10A11A120000A10>0.Case 2.
Whenτ1>0, τ2=0, (6) becomes(12)λ5+A24λ4+A23λ3+A22λ2+A21λ+A20+B24λ4+B23λ3+B22λ2+B21λ+B20e-λτ1=0,where(13)A20=A0+C0+D0+G0,A21=A1+C1+D1+G1,A22=A2+C2+D2+G2,A23=A3+C3+D3,A24=A4+C4,B20=B0+E0+F0+H0,B21=B1+E1+F1+H1,B22=B2+E2+F2,B23=B3+E3,B24=B4.Let λ=iω1 (ω1>0) be a root of (12). Then,(14)B21ω1-B23ω13sin⁡τ1ω1+B24ω14-B22ω12+B20cos⁡τ1ω1=A22ω12-A24ω14-A20,B21ω1-B23ω13cos⁡τ1ω1-B24ω14-B22ω12+B20sin⁡τ1ω1=A23ω13-ω15-A21ω1.Thus, we have(15)ω110+s24ω18+s23ω16+s22ω14+s21ω1+s20=0,where(16)s20=A202-B202,s21=A212-B212-2A20A22+2B20B22,s22=A222-B222+2A20A24-2A21A23-2B20B24+2B21B23,s23=A232-B232+2A21-2A22A24+2B22B24,s24=A242-B242-2A23.Let ω12=v1; then, (15) becomes(17)v15+s24v14+s23v13+s22v12+s21v1+s20=0.According to the analysis of roots of (17) in [16], we assume that(H21)
Equation (17) has at least one positive root.Then, there exists a positive root v10 for (17). Thus, we obtain ω10=v10. Further,(18)τ10=1ω10×arccos⁡p28ω108+p26ω106+p24ω104+p22ω102+p20q28ω108+q26ω106+q284ω104+q22ω102+q20,where(19)q20=B202,q22=B212-2B22,q24=B222+2B20B24-B21B23,q26=B232-2B22B24,q28=B242,p20=-A20B20,p22=A20B22+A22B20-A21B21,p24=A21B23-A20B24-A22B22+A23B21-A24B20,p26=A22B24+A24B22-A23B23-B21,p28=B23-A24B24.Differentiating (12) regarding τ1, we have(20)dλdτ1-1=-5λ4+4A24λ3+3A23λ2+2A22λ+A21λλ5+A24λ4+A23λ3+A22λ2+A21λ+A20+4B24λ3+3B23λ2+2B22λ+B21λB24λ4+B23λ3+B22λ2+B21λ+B20-τ1λ.Then,(21)Redλdτ1λ=iω10-1=f21′v1∗B21ω10-B23ω1032+B24ω104-B22ω102+B202,where(22)f21v1=v15+s24v14+s23v13+s22v12+s21v1+s20and v1∗=ω102.Thus, we can conclude that if(H22)
f21′(v1∗)≠0 holds, then Re[dλ/dτ1]λ=iω10-1≠0.Summarizing the analysis above, we have the following.Theorem 1.
For system (4), let τ10 be specified by (18). If the conditions (H21)-(H22) are satisfied, then D∗(S∗,E∗,I∗,R∗,V∗) is asymptotically stable when τ1∈[0,τ10) and a Hopf bifurcation occurs at D∗(S∗,E∗,I∗,R∗,V∗) when τ1=τ10.Case 3.
Whenτ1=0 and τ2>0, (6) becomes(23)λ5+A34λ4+A33λ3+A32λ2+A31λ+A30+B34λ4+B33λ3+B32λ2+B31λ+B30e-λτ2+C33λ3+C32λ2+C31λ+C30e-2λτ2+D32λ2+D31λ+D30e-3λτ2=0.Multiplying by eλτ2 on both sides of (23), we have(24)B34λ4+B33λ3+B32λ2+B31λ+B30+λ5+A34λ4+A33λ3+A32λ2+A31λ+A30eλτ2+C33λ3+C32λ2+C31λ+C30e-λτ2+D32λ2+D31λ+D30e-2λτ2=0.Let λ=iω2 (ω2>0) be the root of (24). Then,(25)g31ω2cos⁡τ2ω2-g32ω2sin⁡τ2ω2+g33ω2=h31ω2sin⁡2τ2ω2+h32ω2cos⁡2τ2ω2,g34ω2sin⁡τ2ω2+g35ω2cos⁡τ2ω2+g36ω2=h31ω2cos⁡2τ2ω2-h32ω2sin⁡2τ2ω2,where(26)g31ω2=A34ω24-A32+C32ω22+A30+C30,g32ω2=ω25-A33-C33ω23+A31-C31ω2,g33ω2=B34ω24-B32ω22+B30,g34ω2=A34ω24-A32-C32ω22+A30-C30,g35ω2=ω25-A33+C33ω23+A31+C31ω2,g36ω2=B31ω2-B33ω23,h31ω2=-D31ω2,h32ω2=D32ω22-D30.According to the analysis in [11], we obtain the expressions of cos⁡τ2ω2 and sin⁡τ2ω2 when sin⁡τ2ω2=1-cos2τ2ω2 and we denote f31(ω2)=cos⁡τ2ω2 and f32(ω2)=sin⁡τ2ω2. Then, we obtain a function regarding ω2:(27)f312ω2+f322ω2=1.Next, we suppose that(H31)
Equation (27) has at least one positive root.Then, there exists ω210>0 which makes (24) have roots ±iω210. For ω210,(28)τ210=1ω210arccos⁡f31ω210.Similarly, one can also obtain expressions ofcos⁡τ2ω2 and sin⁡τ2ω2 for sin⁡τ2ω2=-1-cos2τ2ω2 and we denote f33(ω2)=cos⁡τ2ω2 and f34(ω2)=sin⁡τ2ω2. Then,(29)f332ω2+f342ω2=1.Obviously, if (29) has a root ω220>0, then (24) has roots ±iω220. For ω220,(30)τ220=1ω220arccos⁡f33ω220.Let(31)τ20=min⁡τ210,τ220,and ±iω20 are the roots of (24) when τ2=τ20. Differentiating (24) regarding τ2, one can obtain(32)dλdτ2-1=p30λ+p31λeλτ2+p32λe-λτ2+p33λe-2λτ2q30λ-q31λeλτ2-τ2λ,with(33)p30λ=4B34λ3+3B33λ2+2B32λ+B31,p31λ=5λ4+4A34λ3+3A33λ2+2A32λ+A31,p32λ=3C33λ2+2C32λ+C31,p33λ=2D32λ+D31,q30λ=C33λ4+C32+2D32λ3+C31+2D31λ2+C30+2D30λ,q31λ=λ6+A34λ5+A33λ4+A32λ3+A31λ2+A30λ.Define(34)Redλdτ2λ=iω20-1=P3RQ3R+P3IQ3IQ3R2+Q3I2.Obviously, if the condition(H32)
P3RQ3R+P3IQ3I≠0 holds, then Re[dλ/dτ2]λ=iω20-1≠0.Therefore, we can otain the following according to the analysis above.Theorem 2.
For system (4), let τ20 be specified by (31). If the conditions (H31)-(H32) are satisfied, then D∗(S∗,E∗,I∗,R∗,V∗) is asymptotically stable when τ2∈[0,τ20) and a Hopf bifurcation occurs at D∗(S∗,E∗,I∗,R∗,V∗) when τ2=τ20.Case 4 (τ1>0, τ2>0, and τ2∈(0,τ20)).
Letλ=iω1∗ (ω1∗>0) be the root of (6); then, one can obtain(35)g41ω1∗sin⁡τ1ω1∗+g42ω1∗cos⁡τ1ω1∗=h41ω1∗,g41ω1∗cos⁡τ1ω1∗-g42ω1∗sin⁡τ1ω1∗=h42ω1∗,where(36)g41ω1∗=B1ω1∗-B3ω1∗3+E1ω1∗-E3ω1∗3cos⁡τ2ω1∗-E0-E2ω1∗2sin⁡τ2ω1∗+F1ω1∗cos⁡2τ2ω1∗-F0-F2ω1∗2sin⁡2τ2ω1∗+H1ω1∗cos⁡3τ2ω1∗-H0sin⁡3τ2ω1∗,g42ω1∗=B4ω1∗4-B2ω1∗2+B0+E1ω1∗-E3ω1∗3sin⁡τ2ω1∗+E0-E2ω1∗2cos⁡τ2ω1∗+F1ω1∗sin⁡2τ2ω1∗+F0-F2ω1∗2cos⁡2τ2ω1∗+H1ω1∗sin⁡3τ2ω1∗+H0cos⁡3τ2ω1∗,h41ω1∗=A2ω1∗2-A4ω1∗4-A0+C3ω1∗3-C1ω1∗sin⁡τ2ω1∗+C2ω1∗2-C4ω1∗4-C0cos⁡τ2ω1∗+D3ω1∗3-D1ω1∗sin⁡2τ2ω1∗+D2ω1∗2-D0cos⁡2τ2ω1∗-G1ω1∗sin⁡3τ2ω1∗+G2ω1∗2-G0cos⁡3τ2ω1∗,h42ω1∗=A3ω1∗3-ω1∗5-A1ω1∗+C3ω1∗3-C1ω1∗cos⁡τ2ω1∗-C2ω1∗2-C4ω1∗4-C0sin⁡τ2ω1∗+D3ω1∗3-D1ω1∗cos⁡2τ2ω1∗-D2ω1∗2-D0sin⁡2τ2ω1∗-G1ω1∗cos⁡3τ2ω1∗-G2ω1∗2-G0sin⁡3τ2ω1∗.Thus, we can obtain(37)f40ω1∗+2f41ω1∗cos⁡τ2ω1∗+2f42ω1∗sin⁡τ2ω1∗+2f43ω1∗cos⁡2τ2ω1∗+2f44ω1∗sin⁡2τ2ω1∗+2f45ω1∗cos⁡3τ2ω1∗+2f46ω1∗sin⁡3τ2ω1∗=0,where(38)f40ω1∗=ω1∗10+A42-B42+C42-2A3ω1∗8+A32-B32+C32+D32-E32-2A2A4+2B2B4-2C2C4+2A1ω1∗6+A22-B22+C22+D22-E22-F22+G22+2A0A4-2A1A3-2B0B4+2B1B3+2C0C4-2C1C3-2D1D3+2E1E3ω1∗4+A12-B12+C12+D12-E12+G12-H12-2A0A2-2C0C2-2D0D2+2B0B2+2E0E2+2F0F2-2G0G2ω1∗2+A02+C02+D02+G02-B02-E02-F02-H02,f41ω1∗=A4C4-C3ω1∗8+A3C3-A2C4-A4C2-B3E3+B4E2+C3D3-C4D2+C1ω1∗6+A0C4-A1C3+A2C2-A3C1+A4C0-C1D3-C3D1+D2G2-D3G1+C2D2+C4D0+B1E3-B2E2+B3E1-B4E0+E3F1-E2F2ω1∗4+A1C1-A0C2-A2C0+C1D1-C0D2-C2D0+D1G1-D0G2-D2G0-B1E1+B2E0+B0E2-E1F1+E0F2+E2F0-F1H1+F2H0ω1∗2-B0E0-E0F0-F0H0,f42ω1∗=-C4ω1∗9+A3C4-A4C3+B4E3-C4D3+C2ω1∗7+A2C3+A4C1-A3C2-A1C4-B2E3+B3E2-B4E1+C2D3+C4D1-C3D2-D3G2-E3F2-C0ω1∗5+A1C2+A3C0-A2C1-A0C3+B0E3-B1E2+B2E1-B3E0-C0D3+C1D2-C2D1+D2G1+C3D0-D1G2-D3G0-E1F2+E2F1+F2H1-E3F0ω1∗3+A0C1-A1C0-B0E1-B1E0+C0D1-D0G1-C1D0-D1G0-E0F1+E1F0-F0H0+F1H0ω1∗,f43ω1∗=-D3ω1∗8+A3D3-A4D2+B4F2-C4G2+D1ω1∗6+A2D2-A3D1+A4D0-A1D3-B2F2+B3F1-B4F0-C3G1+C4G0+C2G2-E3H1ω1∗4+A1D1-A0D2-A2D0+B0F2+B2F0-B1F1-C0G2+C1G1-C2G0-E1H1+E2H0ω1∗2+A0D0+C0G0-B0F0-E0H0,f44ω1∗=D2-A4D3ω1∗7+A2D3-A3D2+A4D1+B3F2-B4F1-C3G2+C4G1-D0ω1∗5+A1D2-A2D1+A3D0-A0D3-B1F2+B2F1-B3F0-C2G1+C1G2+C3G0+E2H1-E3H0ω1∗3+A0D1-A1D0+C0G1-C1G0+B1F0-B0F1-E0H1+E1H0ω1∗,f45ω1∗=G1-A4G2ω1∗6+A2G2-A3G1+A4G0+B3H1-B4H0ω1∗4+A1G1-A0G2-A2G0-B1H1+B2H0ω1∗2+A0G0-B0H0,f46ω1∗=G2ω1∗7+A4G1-A3G2-B4H1-G0ω1∗5+A1G2-A2G1+A3G0+B2H1-B3H0ω1∗3+A0G1-A1G0-B0H1+B1H0ω1∗.Suppose that(H41)
Equation (37) has at least one positive root.Then, there exists a positive root ω10∗ for (37) and (6) has roots ±iω10∗. For ω10∗, we have(39)τ10∗=1ω10∗×arccos⁡g41ω10∗×h42ω10∗+g42ω10∗×h41ω10∗g412ω10∗+g422ω10∗.Differentiating (6) with respect to τ1, one can obtain(40)dλdτ1-1=P4∗λQ4∗λ-τ1λ,where(41)P4∗λ=p40λ+p41λe-λτ1+p42λe-λτ2+p43λe-2λτ2+p44λe-λτ1+τ2+p45λe-λτ1+2τ2+p46λe-3λτ2+p47λe-λτ1+3τ2,Q4∗λ=q41λe-λτ1+q42λe-λτ1+τ2+q43λe-λτ1+2τ2+q44λe-λτ1+3τ2,with(42)p40λ=5λ4+4A4λ3+3A3λ2+2A2λ+A1,p41λ=4B4λ3+3B3λ2+2B2λ+B1,p42λ=-τ2C4λ4+4C4-τ2C3λ3+3C3-τ2C2λ2+2C2-τ2C1λ+C1-τ2C0,p43λ=-2τ2D3λ3+3D3-2τ2D2λ2+2D2-2τ2D1λ+D1-2τ2D0,p44λ=-τ2E3λ3+3E3-τ2E2λ2+2E2-τ2E1λ+E1-τ2E0,p45λ=-2τ2F2λ2+2F2-τ2F1λ+F1-2τ2F0,p46λ=-3τ2G2λ2+2G2-3τ2G1λ+G1-3τ2G0,p47λ=-2τ2H2λ+H1-3τ2H0,q41λ=B4λ5+B3λ4+B2λ3+B1λ2+B0λ,q42λ=E3λ4+E2λ3+E1λ2+E0λ,q43λ=F2λ3+F1λ2+F0λ,q44λ=H1λ2+H0λ.Define(43)Redλdτ1λ=iω10∗-1=P4RQ4R+P4IQ4IQ4R2+Q4I2.Obviously, if the condition(H42)
P4RQ4R+P4IQ4I≠0 holds, then Re[dλ/dτ1]λ=iω10∗-1≠0.Thus, we have the following based on the analysis above.Theorem 3.
For system (4), let τ10∗ be specified by (39) and τ2∈(0,τ20). If the conditions (H41)-(H42) are satisfied, then D∗(S∗,E∗,I∗,R∗,V∗) is asymptotically stable when τ1∈[0,τ10∗) and a Hopf bifurcation occurs at D∗(S∗,E∗,I∗,R∗,V∗) when τ1=τ10∗.Case 5 (τ1>0, τ2>0, and τ1∈(0,τ10)).
Multiplying byeλτ2, (6) becomes(44)C4λ4+C3λ3+C2λ2+C1λ+C0+E3λ3+E2λ2+E1λ+E0e-λτ1+λ5+A4λ4+A3λ3+A2λ2+A1λ+A0eλτ2+D3λ3+D2λ2+D1λ+D0e-λτ2+B4λ4+B3λ3+B2λ2+B1λ+B0eλτ2-τ1+F2λ2+F1λ+F0e-λτ1+τ2.Let λ=iω2∗ (ω2∗>0) be a root of (44). Then,(45)g51ω2∗cos⁡τ2ω2∗-g52ω2∗sin⁡τ2ω2∗+g53ω2∗=h51ω2∗sin⁡2τ2ω2∗+h52ω2∗cos⁡2τ2ω2∗,g54ω2∗sin⁡τ2ω2∗+g55ω2∗cos⁡τ2ω2∗+g56ω2∗=h51ω2∗cos⁡2τ2ω2∗-h52ω2∗sin⁡2τ2ω2∗,where(46)g51ω2∗=A4ω2∗4-A2+D2ω2∗2+A0+D0+B4ω2∗4-B2+F2ω2∗2+B0+F0cos⁡τ1ω2∗+B1+F1ω2∗-B3ω2∗3sin⁡τ1ω2∗,g52ω2∗=ω2∗5-A3-D3ω2∗3+A1-D1ω2∗+B2-F2ω2∗2-B4ω2∗4-B0+F0sin⁡τ1ω2∗+B1-F1ω2∗-B3ω2∗3cos⁡τ1ω2∗,g53ω2∗=C4ω2∗4-C2ω2∗2+C0+E1ω2∗-E3ω2∗3sin⁡τ1ω2∗+E0-E2ω2∗2cos⁡τ1ω2∗,g54ω2∗=A4ω2∗4-A2-D2ω2∗2+A0-D0+B4ω2∗4-B2-F2ω2∗2+B0-F0cos⁡τ1ω2∗+B1-F1ω2∗-B3ω2∗3sin⁡τ1ω2∗,g55ω2∗=ω2∗5-A3+D3ω2∗3+A1+D1ω2∗+B2+F2ω2∗2-B4ω2∗2-B0-F0sin⁡τ1ω2∗+B1+F1ω2∗-B3ω2∗3cos⁡τ1ω2∗,g56ω2∗=C1ω2∗-C3ω2∗3+E1ω2∗-E3ω2∗3cos⁡τ1ω2∗-E0-E2ω2∗2sin⁡τ1ω2∗,h51ω2∗=G1ω2∗+H0sin⁡τ1ω2∗-H1ω2∗cos⁡τ1ω2∗,h52ω2∗=G0-G2ω2∗2-H0cos⁡τ1ω2∗-H1ω2∗sin⁡τ1ω2∗.Similar to Case3, we can obtain the expressions of cos⁡τ2ω2∗ and sin⁡τ2ω2∗ when sin⁡τ2ω2∗=1-cos2τ2ω2∗ and we denote f51(ω2∗)=cos⁡τ2ω2∗ and f52(ω2∗)=sin⁡τ2ω2∗. Then, one can obtain(47)f512ω2∗+f522ω2∗=1.In order to give the main results in this paper, we suppose that(H51)
Equation (47) has at least one positive root.If the condition(H51) holds, then there exists ω21∗>0 such that (6) has a pair of purely imaginary roots ±iω21∗. For ω21∗, we have(48)τ21∗=1ω21∗×arccos⁡f51ω21∗.Similarly, one can obtain expressions ofcos⁡τ2ω2∗ and sin⁡τ2ω2∗ when sin⁡τ2ω2∗=-1-cos2τ2ω2∗ and we denote f53(ω2∗)=cos⁡τ2ω2∗ and f54(ω2∗)=sin⁡τ2ω2∗. Then, one can obtain(49)f532ω2∗+f542ω2∗=1.If (49) has one positive root ω22∗, then (6) has roots ±iω22∗. For ω22∗, we have(50)τ22∗=1ω22∗×arccos⁡f53ω22∗.Let(51)τ20∗=min⁡τ21∗,τ22∗and ±iω20∗ be the roots of (6) with τ2=τ20∗. Differentiating (6) with respect to τ2, we get(52)dλdτ2-1=P5∗λQ5∗λ-τ2λ,where(53)P5∗λ=p50λ+p51λe-λτ1+p52λe-λτ2+p53λe-2λτ2+p54λe-λτ1+τ2+p55λe-λτ1+2τ2+p56λe-3λτ2+p57λe-λτ1+3τ2,Q5∗λ=q51λe-λτ2+q52λe-λτ1+τ2+q53λe-2λτ2+q54λe-λτ1+τ2+q55λe-3λτ2+q56λe-λτ1+3τ2,with(54)p50λ=5λ4+4A4λ3+3A3λ2+2A2λ+A1,p51λ=-τ1B4λ4+4B4-τ1-B3λ3+3B3-τ1B2λ2+2B2-τ1B1+B1-τ1B0,p52λ=4C4λ3+3C3λ2+2C2λ+C1,p53λ=3D3λ2+2D2λ+D1,p54λ=-τ1E3λ3+3E3-τ1E2λ2+2E2-τ1E1λ+E1-τ1E0,p55λ=-τ1F2λ2+2F2-τ1F1λ+F1-τ1F0,p56λ=2G2λ+G1,p57λ=-τ1H1λ+H1-τ1-H0,q51λ=C4λ5+C3λ4+C2λ3+C1λ2+C0λ,q52λ=E3λ4+E2λ3+E1λ2+E0λ,q53λ=2D3λ4+D2λ3+D1λ2+D0λ,q54λ=2F2λ3+F1λ2+F0λ,q55λ=3G2λ3+G1λ2+G0λ,q56λ=3H1λ2+H0λ.Define(55)Redλdτ2λ=iω20∗-1=P5RQ5R+P5IQ5IQ5R2+Q5I2.Obviously, if the condition(H52)
P5RQ5R+P5IQ5I≠0 holds, then Re[dλ/dτ2]λ=iω20∗-1≠0.Thus, we have the following based on the analysis above.Theorem 4.
For system (4), let τ20∗ be specified by (51) and τ1∈(0,τ10). If the conditions (H51)-(H52) are satisfied, then D∗(S∗,E∗,I∗,R∗,V∗) is asymptotically stable when τ2∈[0,τ20∗) and a Hopf bifurcation occurs at D∗(S∗,E∗,I∗,R∗,V∗) when τ2=τ20∗.


## 3. Properties of the Hopf Bifurcation

We will investigate properties of the Hopf bifurcation of system (4) with respect to τ1 and τ2∈(0,τ20) in this section. Let τ2∗<τ10∗ with τ2∗∈(0,τ20) and τ1=τ10∗+μ, μ∈R. By the transformation u1(t)=S(t)-S∗, u2(t)=E(t)-E∗, u3(t)=I(t)-I∗, u4(t)=R(t)-R∗, u5(t)=V(t)-V∗, and t→(t/τ1), system (4) can be written as(56)u˙t=Lμut+Fμ,ut,where ut=(u1(t),u2(t),u3(t),u4(t),u5(t))T∈C([-1,0],R5), and(57)Lμϕ=τ0+μAmaxϕ0+Bmax1ϕ-τ2∗τ10∗+Bmax2ϕ-1,Fμ,ϕ=τ10∗+μ-βϕ10ϕ30βϕ10ϕ30000,where(58)Amax=a110a1300a21a22a23000a32a3300000a440a51000a55,Bmax1=000b14b150000000b330000b43b4400000b55,Bmax2=000000b220000b320000000000000.According to the Riesz representation theorem, we know that there exists a5×5 matrix function η(θ,μ):[-1,0]→R5×5 such that(59)Lμϕ=∫-10dηθ,μϕθ,ϕ∈C.In fact, we choose(60)
η
θ
,
μ
=
τ
10
∗
+
μ
A
m
a
x
+
B
m
a
x
1
+
B
m
a
x
2
,
θ
=
0
,
τ
10
∗
+
μ
B
m
a
x
1
+
B
m
a
x
2
,
θ
∈
-
τ
2
∗
τ
10
∗
,
0
,
τ
10
∗
+
μ
B
m
a
x
2
,
θ
∈
-
1
,
-
τ
2
∗
τ
10
∗
,
0
,
θ
=
-
1
.For ϕ∈C([-1,0],R5), we define(61)
A
μ
ϕ
=
d
ϕ
θ
d
θ
,
-
1
≤
θ
<
0
,
∫
-
1
0
d
η
θ
,
μ
ϕ
θ
,
θ
=
0
,
R
μ
ϕ
=
0
,
-
1
≤
θ
<
0
,
F
μ
,
ϕ
,
θ
=
0
.Then, system (56) is equivalent to(62)u˙t=Aμut+Rμut,where ut(θ)=u(t+θ) for θ∈[-1,0].Then, we defineA∗:(63)
A
∗
φ
=
-
d
φ
s
d
s
,
0
<
s
≤
1
,
∫
-
1
0
d
η
T
s
,
0
φ
-
s
,
s
=
0
,and a bilinear form(64)φs,ϕθ=φ¯0ϕ0-∫θ=-10∫ξ=0θφ¯ξ-θdηθϕξdξ,where η(θ)=η(θ,0).Letρ(θ)=(1,ρ2,ρ3,ρ4,ρ5)Teiω10∗τ10∗θ be the eigenvector of A(0) with +iω10∗τ10∗ and let ρ∗(s)=1/K(1,ρ2∗,ρ3∗,ρ4∗,ρ5∗)eiω10∗τ10∗s be the eigenvector of A∗(0) with -iω10∗τ10∗. Then, we obtain(65)ρ2=a21iω10∗-a33-b33e-iτ2∗ω10∗iω10∗-a22-b22e-iτ10∗ω10∗iω10∗-a33-b33e-τ2∗ω10∗-a23b32e-iτ10∗ω10∗,ρ3=b32e-iτ10∗ω10∗iω10∗-a33-b33e-τ2∗ω10∗ρ2,ρ4=b43e-iτ2∗ω10∗iω10∗-a44-b44e-iτ2∗ω10∗ρ3,ρ5=a51ω10∗-a55-b55e-τ2∗ω10∗,ρ2∗=-iω10∗+a11a21+a51b15eiτ2∗ω10∗a21ω10∗+a55+b55eiτ2∗ω10∗,ρ3∗=iω10∗+a11iω10∗+a22+b22e-iτ10∗ω10∗a21b32e-iτ10∗ω10∗-a51b15eiτ2∗ω10∗iω10∗+a22+b22e-iτ10∗ω10∗a21b32e-iτ10∗ω10∗iω10∗+a55+b55eiτ2∗ω10∗,ρ4∗=-b14eiτ2∗ω10∗iω10∗+a44+b44eiτ2∗ω10∗,ρ5∗=-b15e-iτ2∗ω10∗iω140∗+a55+b55e-iτ2∗ω10∗.From (64), we obtain(66)ρ∗,ρ=1K¯1+ρ2ρ¯2∗+ρ3ρ¯3∗+ρ4ρ¯4∗+ρ5ρ¯5∗+τ10∗e-iτ10∗ω10∗ρ2b22ρ2∗¯+b32ρ3∗¯+τ2∗e-iτ2∗ω10∗ρ3b33ρ¯3∗+b43ρ¯4∗+ρ4b14+b44ρ¯4∗+ρ5b15+b55ρ¯5∗.Then, we can choose(67)K¯=1+ρ2ρ¯2∗+ρ3ρ¯3∗+ρ4ρ¯4∗+ρ5ρ¯5∗+τ10∗e-iτ10∗ω10∗ρ2b22ρ2∗¯+b32ρ3∗¯+τ2∗e-iτ2∗ω10∗ρ3b33ρ¯3∗+b43ρ¯4∗+ρ4b14+b44ρ¯4∗+ρ5b15+b55ρ¯5∗.such that 〈ρ∗,ρ〉=1 and 〈ρ∗,ρ¯〉=0.Following the same algorithms introduced in [17] and the similar computation used in [18–22], we obtain(68)g20=2βτ10∗K¯ρ3ρ¯2∗-1,g11=βτ10∗K¯ρ3+ρ¯3ρ¯2∗-1,g02=2βτ10∗K¯ρ¯3ρ¯2∗-1,g21=2βτ10∗K¯ρ¯2∗-1W1110ρ3+12W2010ρ¯3+W1130+12W2030,with(69)W20θ=ig20ρ0τ10∗ω10∗eiτ10∗ω10∗θ+ig¯02ρ¯03τ10∗ω10∗e-iτ10∗ω10∗θ+E1e2iτ10∗ω10∗θ,W11θ=-ig11ρ0τ10∗ω10∗eiτ10∗ω10∗θ+ig¯11ρ¯0τ10∗ω10∗e-iτ10∗ω10∗θ+E2,E1=2a11′0-a13-b14e-2iτ2∗ω10∗-b15e-2iτ2∗ω10∗-a21a22′-a23000-a32-b32e-2iτ10∗ω10∗a33′0000-b43e-2iτ2∗ω10∗a44′0-a51000a55′-1-βρ3βρ3000,E2=-a110a13b14b15a21a22+b22a23000a32+b32a33+b330000b43a44+b440a51000a55+b55-1-βρ3+ρ¯3βρ3+ρ¯3000,where(70)a11′=2iω10∗-a11,a22′=2iω10∗-a22-b22e-2iω10∗τ10∗,a33′=2iω10∗-a33-b33e-2iω10∗τ2∗,a44′=2iω10∗-a44-b44e-2iω10∗τ2∗,a55′=2iω10∗-a55-b55e-2iω10∗τ2∗.Consequently, we can compute the following values:(71)C10=i2τ10∗ω10∗g11g20-2g112-g0223+g212μ2=-ReC10Reλ′τ10∗,β2=2ReC10,T2=-Im⁡C10+μ2Im⁡λ′τ10∗τ10∗ω10∗.According to the analysis about properties of the Hopf bifurcation in [17], we have the following for system (4).Theorem 5.
Letμ2, β2, and T2 be specified by (71). μ2 determines the direction of the Hopf bifurcation (supercritical if μ2>0, subcritical if μ2<0); β2 determines the stability of the bifurcating periodic solutions (stable if β<0, unstable if β>0); T2>0 determines the period of the bifurcating periodic solutions (increasing if T2>0, decreasing if T2<0).


## 4. A Numerical Example

In this section we give a numerical example of system (4) to validate the analysis above. By extracting the same values from [11], we get the following system:(72)dStdt=2-0.3StIt-0.323St+0.3Rt-τ2+0.06Vt-τ2,dEtdt=0.3StIt-0.003Et-0.25Et-τ1,dItdt=0.25Et-τ1-0.073It-0.4It-τ2,dRtdt=0.4It-τ2-0.003Rt-0.3Rt-τ2,dVtdt=0.32St-0.003Vt-0.06Vt-τ2.It is easy to getR0=4.8750>1 and system (72) has a unique positive equilibrium D∗(1.5956,44.7772,23.6666,31.2430,8.1046). Also we get that the condition (H1) holds.Firstly, we can obtainω10=0.2162 and τ10=7.9833 by some computations. According to Theorem 1, we can conclude that the positive equilibrium D∗(1.5956,44.7772,23.6666,31.2430,8.1046) is asymptotically stable for τ1<τ10=7.9833. In this case, the propagation of worms can be predicted and controlled easily. However, once the value of τ1 is above τ10=7.9833, a Hopf bifurcation will occur which implies that the propagation of worms will be out of control. This property can be illustrated by the bifurcation diagram with respect to τ1 in Figure 1. Similarly, we have ω20=0.4883 and τ20=2.4273. The corresponding bifurcation diagram with respect to τ2 is as shown in Figure 2.Figure 1
Bifurcation regardingτ1 with τ2=0.Figure 2
Bifurcation regardingτ2 with τ1=0.Secondly, we may obtainω10∗=1.8164 and τ10∗=3.5477 for τ1>0 and τ2=1.5∈(0,τ20) by computing. It follows from Theorem 3 that when τ1=τ10∗=3.5477, a Hopf bifurcation occurs, which can be shown by the bifurcation diagram regarding τ1 and τ2=1.5 in Figure 3. Moreover, we have C1(0)=-1.2298+3.8550i, μ2=455.4815>0, β2=-2.4596<0, and T2=-22.5006<0. Thus, It follows from Theorem 5 that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable and decreasing. Since the bifurcating periodic solutions are stable, the numbers of every class of nodes in system (72) may coexist in an oscillatory mode, which is not welcome in networks.Figure 3
Bifurcation regardingτ1 with τ2=1.5.Lastly, by computing, we obtainω20∗=2.0979 and τ20∗=1.5536 for τ2>0 and τ1=3.5∈(0,τ10). By Theorem 4, we know that the positive equilibrium D∗(1.5956,44.7772,23.6666,31.2430,8.1046) is asymptotically stable for τ2<τ20∗, which makes it possible for the propagation of worms to be predicted and controlled by taking effective measures. However, the positive equilibrium D∗(1.5956,44.7772,23.6666,31.2430,8.1046) loses stability and a Hopf bifurcation occurs, when τ2 passes through τ20∗=1.5536. In this case, it is difficult to take measures to control the propagation of worms. The Hopf bifurcation phenomenon can be illustrated by Figure 4.Figure 4
Bifurcation regardingτ2 with τ1=3.5.


## 5. Conclusions

In the present paper, the generalization of the delayed SEIRS-V model describing worms spreading in a wireless sensor network investigated in [11] by inserting the latent period delay in the exposed sensor nodes has been considered. We find that not only τ1 but also τ2 can influence stability of system (4) and make system (4) undergo a Hopf bifurcation under some certain conditions. It has been shown that characteristics of the propagation of worms in system (4) can be easily predicted and eliminated when the value of delay is below the corresponding critical value and the propagation of worms in system (4) may be out of control once the delay passes through the corresponding critical value. Thus, we can conclude that the worms propagation can be controlled by postponing occurrence of a Hopf bifurcation. Subsequently, we use the normal form approach theory and center manifold theory introduced in [17] to deal with properties of the Hopf bifurcation when τ1>0 and τ2∈(0,τ20).According to the numerical example, we can conclude that the dynamics of system (4) is more complicated than that of the system considered in [11]. However, it should be pointed out that we assume that the exposed sensor nodes can not infect other nodes, which is not consistent with reality, because the exposed nodes can also infect other nodes through vulnerability seeking or other methods. Consequently, it is more realistic to investigate dynamics of the proposed model in this paper with graded infection rate. We leave this as our future work.



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*Source: 9898726-2017-01-18.xml*