Research Article Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network

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1 e Scientific World Journal Volume 4, Article ID 944, 4 pages Research Article Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network Zizhen Zhang, and Huizhong Yang School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 333, China Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 4, China Correspondence should be addressed to Huizhong Yang; yhz@jiangnan.edu.cn Received 6 June 4; Accepted 3 July 4; Published 3 July 4 Academic Editor: Luca Guerrini Copyright 4 Z. Zhang and H. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An SEIQRS model for the transmission of malicious objects in computer network with two delays is investigated in this paper. We show that possible combination of the two delays can affect the stability of the model and make the model bifurcate periodic solutions under some certain conditions. For further investigation, properties of the periodic solutions are studied by using the normal form method and center manifold theory. Finally, some numerical simulations are given to justify the theoretical results.. Introduction Computer viruses in network have posed a major threat to our work and life with the rapid popularization of the Internet. Many virus propagation models [ 4] havebeen proposed to understand the way that computer viruses propagateafterkephartandwhite[5] proposehefirst epidemiological model of computer viruses. In [], Thommes andcoatesproposedamodifiedversionoftheseimodel to predict the virus propagation in a network. In [3], Wen andzhongstudiedansirmodelonbipartitenetworks and they proved the existence and the asymptotic stability of the endemic equilibrium by applying the theory of the multigroup model. In [4], Mishra and Jha proposed the following SEIQRS model to describe the transmission of malicious objects in computer network by introducing a new compartment quarantine into the SEIRS model proposed in []: ds (t) =A β I (t) d +ηr(t), de (t) =β I (t) (d+μ)e(t), di (t) =μe(t) (d+α+γ+δ)i(t), dq (t) dr (t) =δi(t) (d+α+ε) Q (t), =γi(t) +εq(t) (d+η)r(t),, E(t), I(t), Q(t), and R(t) denote the sizes of nodes in the states susceptible, exposed, infectious, quarantined, and recovered at time t, respectively.a istherateatwhich new computers are attached to the network. d is the rate at which computers are disconnected to the network. α is the crashing rate of computers due to the attack of malicious objects. β is the transmission rate. μ, γ, δ, ε, andη are the state transition rates. As is known, an infected computer becomes a recovered one by using antimalicious software and the recovered computer has a temporary immunity, and computer virus models with delay have been studied by many scholars [6 ]. In [6], Ren et al. investigated local and global stability of a delayed viral infection model in computer virus propagation model. In [8], Dong et al. proposed a delayed SEIR computer virus model and studied the problem of Hopf bifurcation of the model by regarding the delay as a bifurcating parameter. Motivated by the work above, Liu [] incorporated the time ()

2 The Scientific World Journal delay due to the temporary immunity period into system () and proposed the following SEIQRS model with time delay: ds (t) de (t) di (t) dq (t) dr (t) =A β I (t) d +ηr(t τ), =β I (t) (d+μ)e(t), =μe(t) (d+α+γ+δ)i(t), =δi(t) (d+α+ε) Q (t), =γi(t) +εq(t) dr(t) ηr(t τ), τ isthetimedelayduetothetemporaryimmunity period. However, we know that an infected computer needs a period to clean viruses by antivirus software and then becomes a recovered one. Therefore, there is a time delay before the infected computers develop themselves into the recovered ones. And there have been some papers that deal with the research of Hopf bifurcation of dynamical system with multiple delays [3 8]. In [3], Xu and He considered a two-neuron network with resonant bilinear terms and two delays. They studied the problem of Hopf bifurcation by regarding the sum of the two delays as a bifurcation parameter. In [6], Meng et al. studied the Hopf bifurcation of a three-species system with two delays by regarding possible combination of the two delays as a bifurcation parameter. Motivated by the work above, we consider the following SEIQRS computer virus model with two delays in the present paper: ds (t) de (t) di (t) dq (t) dr (t) =A β I (t) d +ηr(t τ ), =β I (t) (d+μ)e(t), =μe(t) (d+α+δ) I (t) γi(t τ ), =δi(t) (d+α) Q (t) εq(t τ ), () =γi(t τ )+εq(t τ ) dr(t) ηr(t τ ), (3) we give a numerical example to support the theoretical results in the paper.. Local Stability and Existence of Local Hopf Bifurcation By a simple computation, it is easy to get that if R = Aμβ/d(d+μ)(α+δ+γ+d) >,thensystem(3)hasaunique positive equilibrium P (S,E,I,Q,R ), S = I = (d + μ) (d + α + δ + γ) μβ, E = (d+α+δ+γ)i, μ (d + η) (d+α+ε)(aμβ d (d + μ) (d + α + δ + γ)), β Q δi = d+α+ε, R = γi d+η + εδi (d + η) (d+α+ε), (4) and R isthebasicreproductionnumber.itiseasytogetthe linearization of system (3)atP (S,E,I,Q,R ): ds (t) =a S (t) +a 3 I (t) +b 5 R(t τ ), de (t) =a S (t) +a E (t) +a 3 I (t), di (t) =a 3 E (t) +a 33 I (t) +c 33 I(t τ ), (5) dq (t) =a 43 I (t) +a 44 Q (t) +c 44 Q(t τ ), dr (t) =a 55 R (t) +b 55 R(t τ )+c 53 I(t τ ) +c 54 Q(t τ ), a = (βi +d), a 3 = βs, a =βi, a = (d+μ), a 3 =βs, a 3 =μ, a 33 = (d+α+δ), a 43 =δ, a 44 = (d+α), a 55 = d, b 5 =η, b 55 = η, c 33 = γ, c 44 = ε, c 53 =γ, c 54 =ε. Thus, the characteristic equation of system (5) is (6) τ is the time delay due to the temporary immunity period and τ is the time delay due to the period that the infected computer uses to clean viruses by antivirus software. The main purpose of this paper is to investigate the effects of the two delays on system (3) and the remainder of this paper is organized as follows. Sufficient conditions for local stability and existence of local Hopf bifurcation are obtained by analyzing the distribution of the roots of the associated characteristic equation in Section. PropertiesoftheHopf bifurcation are further investigated by using the normal form method and center manifold theory in Section 3.In Section 4, λ 5 +m 4 λ 4 +m 3 λ 3 +m λ +m λ+m +(n 4 λ 4 +n 3 λ 3 +n λ +n λ+n )e λτ +(p 4 λ 4 +p 3 λ 3 +p λ +p λ+p )e λτ +(q 3 λ 3 +q λ +q λ+q )e λτ +(r 3 λ 3 +r λ +r λ+r )e λ(τ +τ ) +(s λ +s λ+s )e λ(τ +τ ) =, (7)

3 The Scientific World Journal 3 m =(a a 3 a 3 a a a 33 a 3 a a 3 )a 44 a 55, m =(a 44 +a 55 )(a 3 a a 3 a a 3 a 3 ) a 3 a 3 a 44 a 55 +a a a 33 a 44 +a a a 55 (a 33 +a 44 ) +a 33 a 44 a 55 (a +a ), m =a 3 a 3 (a +a 44 +a 55 ) a 55 (a a +a 33 a 44 ) a 3 a a 3 a a (a 33 +a 44 ) a 33 a 44 (a +a ) a 55 (a +a )(a 33 +a 44 ), m 3 =a a +a 33 a 44 +(a +a )(a 33 +a 44 ) a 3 a 3 +a 55 (a +a +a 33 +a 44 ), m 4 = (a +a +a 33 +a 44 +a 55 ), n =(a a 3 a 3 a a a 33 a 3 a a 3 )a 44 b 55, n =a a b 55 (a 33 +a 44 )+a 33 a 44 b 55 (a +a ) +a 3 a a 3 b 55 a 3 a 3 b 55 (a +a 44 ), n =(a 3 a 3 a a a 33 a 44 )b 55 (a +a )(a 33 +a 44 )b 55, n 3 =b 55 (a +a +a 33 +a 44 ), n 4 = b 55, p =(a a 3 a 3 a )a 3 a 55 c 44 (a 44 c 33 +a 33 c 44 )a a a 55, p =a a c 33 (a 44 +a 55 )+a 44 a 55 c 33 (a +a ) +a a c 44 (a 33 +a 55 ) a 3 a 3 c 44 (a +a 55 ) +a 33 a 55 c 44 (a +a ) a 3 a a 3 c 44, p =a 3 a 3 c 44 c 33 (a +a )(a 44 +a 55 ) c 33 (a a +a 44 a 55 ) c 44 (a +a )(a 33 +a 55 ) c 44 (a a +a 33 a 55 ), p 3 =c 33 (a +a +a 44 +a 55 )+c 44 (a +a +a 33 +a 55 ), p 4 = (c 33 +c 44 ), q =a a a 55 c 33 c 44, q 3 =c 33 c 44, q =(a a +a a 55 +a a 55 )c 33 c 44, q = (a +a +a 55 )c 33 c 44, r =a a 3 b 5 (a 44 c 53 a 43 c 54 ) a a b 55 (a 33 c 44 +a 44 c 33 ) a 3 b 55 c 44 (a a 3 +a 3 a ), r = (a +a +a 33 )b 55 c 44 (a +a +a 44 )b 55 c 33, r 3 =(c 33 +c 44 )b 55, s =a a 3 b 5 c 44 c 53 a a b 55 c 33 c 44, s =(a +a )b 55 c 33 c 44, s = b 55 c 33 c 44. Case (τ =τ =). When τ =τ =,(7)becomes λ 5 +A 4 λ 4 +A 3 λ 3 +A λ +A λ+a =, A =m +n +p +q +r +s, A =m +n +p +q +r +s, A =m +n +p +q +r +s, A 3 =m 3 +n 3 +p 3 +q 3 +r 3, A 4 =m 4 +n 4 +p 4 =βi +5d+α+μ+δ+ε+γ+η>. Let det =A 4.Obviously,det >. Therefore, if the condition (H ):() holds, then the positive equilibrium P (S, E,I,Q,R ) of system (3) is locally asymptotically stable without delay. Consider det = det 3 = det 4 = A 4 A A >, 3 A 4 A A 3 A 4 >, A A A 4 A A 3 A 4 A A A A 3 A A >, A 4 A A 3 A 4 det 5 = A A A A 3 A 4 >. A A A A (8) (9) () Case (τ >, τ = ). When τ =,(7) becomesthe following form: λ 5 +A 4 λ 4 +A 3 λ 3 +A λ +A λ+a r =(a a +a a 44 +a a 44 )b 55 c 33 a c 3 b 5 c 53 +(a a +a a 33 +a a 33 )b 55 c 44 +a 3 a 3 b 55 c 44, +(B 4 λ 4 +B 3 λ 3 +B λ +B λ+b )e λτ =, ()

4 4 The Scientific World Journal A =m +p +q, A =m +p +q, A =m +p +q, A 3 =m 3 +p 3 +q 3, A 4 =m 4 +p 4, B =n +r +s, B =n +r +s, B =n +r +s, B 3 =n 3 +r 3, B 4 =n 4. Let λ=iω (ω >)bearootof(). Then, we obtain () p = A B, p =A B A B +A B, p 4 =A B 3 +A 3 B A B 4 A B A 4 B, p 6 =A B 4 +A 4 B A 3 B 3 B, p 8 =B 3 A 4 B 4, q =B, q =B B B, q 4 =B +B B 4 B B 3, q 6 =B 3 B B 4, q 8 =B 4. (8) (B ω B 3 ω 3 ) sin τ ω +(B 4 ω 4 B ω +B ) cos τ ω =A ω A 4ω 4 A, (B ω B 3 ω 3 ) cos τ ω (B 4 ω 4 B ω +B ) sin τ ω = ω 5 +A 3ω 3 A ω. It follows that with (3) ω +c 4ω 8 +c 3ω 6 +c ω 4 +c ω +c =, (4) c =A B, c =A B A A +B B, c =A B +A A 4 A A 3 B B 4 +B B 3, c 3 =A 3 +A A A 4 B 3 +B B 4, c 4 =A 4 B 4 A 3. Let ω = V,then(4)becomes (5) V 5 +c 4V 4 +c 3V 3 +c V +c V +c =. (6) If all the parameters of system (3) aregiven,onecanget all the rootsof (6) by the software package Matlab. In order to give the main results in this paper, we make the following assumption. (H ) (6) has at least one positive real root. If the condition (H ) holds, then there exists a V such that () has a pair of purely imaginary roots ±iω =±i V. For ω, the corresponding critical value of time delay is Taking the derivative of λ with respect to τ in (), one can obtain [ dλ ] dτ Thus, 5λ 4 +4A = 4 λ 3 +3A 3 λ +A λ+a λ(λ 5 +A 4 λ 4 +A 3 λ 3 +A λ +A λ+a ) + Re [ dλ ] dτ 4B 4 λ 3 +3B 3 λ +B λ+b λ(b 4 λ 4 +B 3 λ 3 +B λ +B λ+b ) τ λ. λ=iω (9) f = (V ) (B ω B 3 ω 3 ) +(B 4 ω 4 B ω +B ), () f (V )=V 5 +c 4V 4 +c 3V 3 +c V +c V +c and V =ω. Obviously, if the condition (H )f (V ) =holds, then Re [dλ/dτ ] λ=iω =. According to the Hopf bifurcation theoremin [9], we have the following results for system (3). Theorem. For system (3), if the conditions (H )-(H ) hold, then the positive equilibrium P (S,E,I,Q,R ) of system (3) is asymptotically stable for τ [,τ ) and system (3) undergoes a Hopf bifurcation at the positive equilibrium P (S,E,I,Q,R ) when τ =τ. Case 3 (τ =, τ >). When τ =,(7)becomes λ 5 +A 34 λ 4 +A 33 λ 3 +A 3 λ +A 3 λ+a 3 τ = arccos p 8ω 8 +p 6ω 6 +p 4ω 4 +p ω +p ω q 8 ω 8 +q 6ω 6 +q 4ω 4 +q ω +q, (7) +(B 34 λ 4 +B 33 λ 3 +B 3 λ +B 3 λ+b )e λτ +(C 33 λ 3 +C 3 λ +C 3 )e λτ =, ()

5 The Scientific World Journal 5 A 3 =m +n, A 3 =m +n, A 3 =m +n, A 33 =m 3 +n 3, A 34 =m 4 +n 4, B 3 =p +r, B 3 =p +r, B 3 =p +r, B 33 =p 3 +r 3, B 34 =p 4, C 3 =q +s, C 3 =q +s, C 3 =q +s, C 33 =q 3. () Multiplying e λτ on both sides of (), we have B 34 λ 4 +B 33 λ 3 +B 3 λ +B 3 λ+b +(λ 5 +A 34 λ 4 +A 33 λ 3 +A 3 λ +A 3 λ+a 3 )e λτ p 3 =A 3 (C 3 B 3 ), p 3 =(A 3 +C 3 )B 3 (A 3 +C 3 )B 3, p 3 =B 3 (C 3 A 3 )+A 3 B 3 +A 3 B 3 B 3 C 3 B 3 C 3, p 33 =B 3 (A 3 +C 3 )+B 33 (A 3 +C 3 ) B 3 (A 33 +C 33 ) B 3 (A 3 +C 3 ), p 34 =B 3 (A 33 C 33 )+B 3 (C 3 A 3 )+B 33 (A 3 C 3 ) +B 34 (C 3 A 3 ) A 34 B 3, p 35 =B 3 (A 33 +C 33 )+B 34 (A 3 +C 3 ) B 33 (A 3 +C 3 ) A 34 B 3, p 36 =A 34 B 3 +A 3 B 34 A 33 B 33 B 3 +B 33 C 33 B 34 C 3, +(C 33 λ 3 +C 3 λ +C 3 )e λτ =. Let λ=iω (ω >)betherootof(3), then we obtain M 3 sin τ ω +M 3 cos τ ω =M 33, M 34 cos τ ω M 35 sin τ ω =M 36, M 3 =A 34 ω 4 +(C 3 A 3 )ω +A 3 C 3, M 3 =ω 5 (A 33 +C 33 )ω 3 +(A 3 +C 3 )ω, M 33 =B 33 ω 3 B 3ω, M 34 =A 34 ω 4 (A 3 +C 3 )ω +A 3 +C 3, M 35 =ω 5 (A 33 C 33 )ω 3 +(A 3 C 3 )ω, M 36 =B 3 ω B 34ω 4 B 3. Then, we obtain (3) (4) (5) p 37 =A 34 B 33 B 3 (A 33 +C 33 )B 34, p 38 =B 33 A 34 C 34, q 3 =A 3 C 3, q 3 =A 3 C 3 A 3A 3 +C 3 C 3, q 34 =A 3 C 3 A 3A 34 +C 3 C 33 A 3 A 33, q 36 =A 33 C 33 +A 3 A 3 A 34, q 38 =A 34 A 33. (7) Then, we obtain ω +c 39ω 8 +c 38ω 6 +c 37ω 4 +c 36ω +c 35ω +c 34 ω 8 +c 33ω 6 +c 3ω 4 +c 3ω +c 3 =, c 3 =q 3 p 3, c 3 =q 3 q 3 p 3 p 3 p 3, c 3 =q 3 p 3 +q 3q 34 p 3 p 34 p 3 p 33, c 33 =q 3 q 36 +q 3 q 34 p 3 p 36 p 3 p 35 p 3 p 34 p 33, c 34 =q 34 p 34 +q 3q 38 +q 3 q 36 p 3 p 38 p 3 p 37 p 3 p 36 p 33 p 35, c 35 =q 3 q 38 p 3 p 38 p 33 p 37 p 34 p 36 p 35, (8) cos τ ω = sin τ ω = p 38 ω 8 +p 36ω 6 +p 34ω 4 +p 3ω +p 3 ω +q 38ω 8 +q 36ω 6 +q 34ω 4 +q 3ω +q, 3 p 37 ω 7 +p 35ω 5 +p 33ω 3 +p 3ω +q 38ω 8 +q 36ω 6 +q 34ω 4 +q 3ω +q, 3 (6) ω c 36 =q 36 p 36 +q 34q 38 +q 3 p 34 p 38 p 35 p 37, c 37 =q 34 +q 36 q 38 p 36 p 38 p 37, c 38 =q 38 p 38 +q 36, c 39 =q 38. (9)

6 6 The Scientific World Journal Let ω = V,then(8)becomes V +c 39V 9 +c 38V 8 +c 37V 7 +c 36V 6 +c 35V 5 +c 34 V 4 +c 33V 3 +c 3V +c 3V +c 3 =. (3) Q 3R =((A 33 +C 33 )ω 4 ω6 (A 3 +C 3 )ω ) cos τ ω (A 34 ω 5 +(A 3 C 3 )ω 3 +(A 3 C 3 )ω ) sin τ ω, Similar as in Case, we make the following assumption. (H 3 ) (3) has at least one positive real root. If the condition (H 3 ) holds, then there exists a V such that (3) hasapair of purely imaginary roots ±iω = ±i V.Forω,the corresponding critical value of time delay is τ = arccos ω p 38 ω 8 +p 36ω 6 +p 34ω 4 +p 3ω +p 3 +q 38ω 8 +q 36ω 6 +q 34ω 4 +q 3ω +q. 3 (3) ω Differentiating two sides of (3)withrespecttoτ,wehave [ dλ ] = g 3 (λ) +g 3 (λ) e λτ +g 33 (λ) e λτ τ dτ g 34 (λ) e λτ g35 (λ) e λτ λ, (3) g 3 (λ) =4B 34 λ 3 +3B 33 λ +B 3 λ+b 3, g 3 (λ) =5λ 4 +4A 34 λ 3 +3A 33 λ +A 3 λ+a 3, g 33 (λ) =3C 33 λ +C 3 λ+c 3, g 34 (λ) =C 33 λ 4 +C 3 λ 3 +C 3 λ +C 3 λ, g 35 (λ) =λ 6 +A 34 λ 5 +A 33 λ 4 +A 3 λ 3 +A 3 λ +A 3 λ. (33) Q 3I =((A 33 C 33 )ω 4 ω6 (A 3 C 3 )ω ) sin τ ω +(A 34 ω 5 (A 3 +C 3 )ω 3 +(A 3 +C 3 )ω ) cos τ ω. (35) Obviously, if the condition (H 3 )P 3R Q 3R +P 3I Q 3I = holds, then Re [dλ/dτ ] λ=iω =. According to the Hopf bifurcationtheoremin [9], we have the following results for system (3). Theorem. For system (3), if the conditions (H 3 )-(H 3 ) hold, then the positive equilibrium P (S,E,I,Q,R ) of system (3) is asymptotically stable for τ [,τ ) and system (3) undergoes a Hopf bifurcation at the positive equilibrium P (S,E,I,Q,R ) when τ =τ. Case 4 (τ >, τ >, τ (,τ )). We consider system (3) under the condition that τ is in its stable interval and τ is a bifurcation parameter. Let λ=iω (ω >)betherootof(7), then we obtain M 4 sin τ ω +M 4 cos τ ω =M 43, (36) M 4 cos τ ω M 4 sin τ ω =M 44, M 4 =n ω n 3 ω 3 +(r ω r 3 ω 3 ) cos τ ω (r r ω ) sin τ ω +s ω cos τ ω (s s ω ) sin τ ω, M 4 =n 4 ω 4 n ω +n +(r ω r 3 ω 3 ) sin τ ω Thus, Re [ dλ ] dτ λ=iω = P 3RQ 3R +P 3IQ 3I Q3R, (34) +Q 3I +(r r ω ) cos τ ω +s ω sin τ ω +(s s ω ) cos τ ω, M 43 =(p 3 ω 3 p ω ) sin τ ω (p 4 ω 4 p ω +p ) cos τ ω +(q 3 ω 3 q ω ) sin τ ω P 3R =(5ω 4 3(A 33 +C 33 )5ω +A 3 +C 3 ) cos τ ω +(4A 34 ω 3 +(C 3 A 3 )ω ) sin τ ω +B 3 3B 33 ω, P 3I = (5ω 4 3(A 33 C 33 )5ω +A 3 C 3 ) sin τ ω (4A 34 ω 3 (A 3 +C 3 )ω ) cos τ ω +B 3 ω 4B 34 ω 3, +(q ω q ) cos τ ω +m ω m 4ω 4 m, M 44 =(p 3 ω 3 p ω ) cos τ ω +(p 4 ω 4 p ω +p ) sin τ ω +(q 3 ω 3 q ω ) cos τ ω (q ω q ) sin τ ω +m 3 ω 3 ω5 m ω. (37)

7 The Scientific World Journal 7 Then, we can obtain f 4 (ω )+f 4 (ω ) cos τ ω +f 4 (ω ) sin τ ω +f 43 (ω ) cos τ ω +f 44 (ω ) sin τ ω +f 45 (ω ) cos τ ω cos τ ω f 44 (ω )=q ω 7 +(n 3s m 3 q q )ω 5 +(m q +m 3 q n s n 3 s )ω 3 +(n s m q )ω, f 45 (ω )=(p 3 q 3 p 4 q )ω 6 +(p q +p 4 q p q 3 p 3 q +r s r 3 s )ω 4 +f 46 (ω ) cos τ ω sin τ ω +f 47 (ω ) sin τ ω sin τ ω +f 48 (ω ) sin τ ω cos τ ω =, f 4 (ω )=ω +(m 4 n 4 +p 4 m 3)ω 8 +(m 3 n 3 +p 3 +q 3 r 3 +m m m 4 +n n 4 p p 4 )ω 6 +(m n +p +q r s +m m 4 (38) +(p q p q p q r s +r s r s )ω +p q +r s, f 46 (ω )= p 4 q 3 ω 7 +(p q 3 p 3 q +p 4 q +r 3 s )ω 5 +(p q p q 3 p q p 3 q r s +r s r 3 s )ω 3 +(p q p q r s r s )ω, f 47 (ω )=(p 3 q 3 p 4 q )ω 6 +(p q p q 3 p 3 q p 4 q r s +r 3 s )ω 4 +(p q p q p q +r s r s +r s )ω +p q r s, m m 3 +n n 4 +n n 3 p p 3 q q 3 +r r 3 )ω 4 +(m n +p +q r s m m +n n p p q q +r r +s s )ω +m n +p +q r s, f 4 (ω )=(m 4 p 4 p 3 )ω 8 +(m 3 p 3 +p m p 4 m 4 p n 3 r 3 +n 4 r )ω 6 +(m p 4 m p 3 +m p m 3 p +m 4 p +n r 3 n r +n 3 r n 4 r )ω 4 +(m p m p m p +n r n r +n r )ω +m p n r, f 4 (ω )= p 4 ω 9 +(m 3p 4 m 4 p 3 +n 4 r 3 +p )ω 7 +(m p 3 m p 4 m 3 p +m 4 p n r 3 +n 3 r n 4 r p )ω 5 +(m p m p 3 m p +m 3 p +n r 3 n r +n r n 3 r )ω 3 +(m p m p n r +n r )ω, f 43 (ω )= q 3 ω 8 +(m 3q 3 m 4 q +q )ω 6 +(m q m q 3 m 3 q +m 4 q +n 3 s )ω 4 +(m q m q m q n s )ω, f 48 (ω )=p 4 q 3 ω 7 +(p 3q p q 3 p 4 q r 3 s )ω 5 +(p q 3 p q +p q p 3 q +r s r s r 3 s )ω 3 +(p q +p q +r s r s )ω. (39) In order to give the main results in this paper, we make the following assumption. (H 4 ) (38) has at least one positive real root. If the conditions (H 4 ) hold, then there exists a ω such that (7) has a pair of purely imaginary roots ±iω.forω,the corresponding critical value of time delay is τ = ω arccos M 4M 44 +M 4 M 43 M4. (4) +M 4 ω =ω Differentiating two sides of (7)withrespecttoτ,wehave [ dλ ] =(g dτ 4 (λ) +g 4 (λ) e λτ +g 43 (λ) e λτ +g 44 (λ) e λ(τ +τ ) +g 45 (λ) e λτ +g 46 (λ) e λ(τ +τ ) ) (g 47 (λ) e λτ +g 48 (λ) e λ(τ +τ ) τ λ, +g 49 (λ)e λ(τ +τ ) ) (4)

8 8 The Scientific World Journal g 4 (λ) =5λ 4 +4m 4 λ 3 +3m 3 λ +m λ+m, g 4 (λ) =4n 4 λ 3 +3n 3 λ +n λ+n, g 43 (λ) = τ p 4 λ 4 +(4p 4 τ p 3 )λ 3 +(3p 3 τ p )λ +(p +τ p )λ+p τ p, g 44 (λ) = τ q 3 3 +(3q 3 τ q ) +(q τ p )λ +q τ q, g 45 (λ) = τ s λ +(s τ s )λ+s τ s, g 46 (λ) =3r 3 λ +r λ+r, g 47 (λ) =n 4 λ 5 +n 3 λ 4 +n λ 3 +n λ +n λ, g 48 (λ) =r 3 λ 4 +r λ 3 +r λ +r λ, g 49 (λ) =s λ 3 +s λ +s λ. (4) P 4I =m ω 4m 4(ω )3 +r ω cos τ ω + ((3r 3 (ω ) r ) sin τ ω +(s τ s )ω cos τ ω (τ s (ω ) +s τ s ) sin τ ω ) cos τ ω + (3n 3 (ω ) n r ω sin τ ω (r 3r 3 (ω ) ) cos τ ω (s τ s )ω sin τ ω (τ s (ω ) +s τ s ) cos τ ω ) sin τ ω +((p τ p )ω (4p 4 τ p 3 )(ω )3 ) cos τ ω ((τ p 3p 3 )(ω ) τ p 4 (ω )4 +p τ p ) sin τ ω +(τ q 3 (ω )3 +(q τ q )ω ) cos τ ω ((τ q 3q 3 )(ω ) +q τ q ) sin τ ω, (44) Thus, Q 4R =(n 4 (ω )5 n (ω )3 +n ω Re [ dλ ] dτ λ=iω = P 4RQ 4R +P 4I Q 4I Q4R, (43) +Q 4I +(r ω r (ω )3 ) cos τ ω (r 3 (ω )4 r (ω ) ) sin τ ω +(s ω s (ω )3 ) cos τ ω +s (ω ) sin τ ω ) sin τ ω P 4R =5(ω )4 3m 3 (ω ) +m +(n ω 4n 4(ω )3 +r ω cos τ ω + (3r 3 (ω ) r ) sin τ ω +(s τ s )ω cos τ ω (τ s (ω ) +s τ s ) sin τ ω ) sin τ ω +(n 3n 3 (ω ) +r ω sin τ ω +(r 3r 3 (ω ) ) cos τ ω +(s τ s )ω sin τ ω +(τ s (ω ) +s τ s ) cos τ ω ) cos τ ω +((p τ p )ω (4p 4 τ p 3 )(ω )3 ) sin τ ω +((τ p 3p 3 )(ω ) τ p 4 (ω )4 +p τ p ) cos τ ω +(n 3 (ω )4 n (ω ) +(r ω r (ω )3 ) sin τ ω +(r 3 (ω )4 r (ω ) ) cos τ ω +(s ω s (ω )3 ) sin τ ω s (ω ) cos τ ω ) cos τ ω, Q 4I =(n 4 (ω )5 n (ω )3 +n ω +(r ω r (ω )3 ) cos τ ω (r 3 (ω )4 r (ω ) ) sin τ ω +(s ω s (ω )3 ) cos τ ω +s (ω ) sin τ ω ) cos τ ω (n 3 (ω )4 n (ω ) +(r ω r (ω )3 ) sin τ ω +(r 3 (ω )4 r (ω ) ) cos τ ω +(s ω s (ω )3 ) sin τ ω +(τ q 3 (ω )3 +(q τ q )ω ) sin τ ω +((τ q 3q 3 )(ω ) +q τ q ) cos τ ω, s (ω ) cos τ ω ) sin τ. (45)

9 The Scientific World Journal 9 Obviously, if the condition (H 4 )P 4R Q 4R +P 4I Q 4I = holds, then Re [dλ/dτ ] λ=iω =. According to the Hopf bifurcationtheoremin [9], we have the following results for system (3). Theorem 3. For system (3), if the conditions (H 4 )-(H 4 ) hold and τ (,τ ), then the positive equilibrium P (S,E,I,Q,R ) of system (3) is asymptotically stable for τ [,τ ) and system (3) undergoes a Hopf bifurcation at the positive equilibrium P (S,E,I,Q,R ) when τ =τ. 3. Direction and Stability of the Hopf Bifurcation In this section, we determine the properties of the Hopf bifurcation of system (3) withrespecttoτ for τ (,τ ). Throughout this section, we assume that τ < τ, τ (,τ ). Let τ =τ +μ, μ Rso that μ=is the Hopf bifurcation value of system (3).Rescalingthetimedelaybyt (t/τ ). Let u (t) = S, u (t) = E(t) E, u 3 (t) = I(t) I, u 4 (t) = Q(t) Q, u 5 (t) = R(t) R,thensystem(3)canbe written as a PDE in C = C([, ], R 5 ): u (t) =L μ u t +F(μ,u t ), (46) and L μ :C R 5, F:R C R 5 are given, respectively, by L μ φ=(τ +μ)(a φ () +C ( τ τ )+B φ ( )), with F(μ,φ)=(τ +μ) ( ( βφ () φ 3 () βφ () φ 3 () ), ) a a 3 a a a 3 A = ( a 3 ), a 43 a 44 ( a 55 ) b 5 B = ( ), ( b 55 ) C = ( c 33 ). c 44 ( c 53 c 54 ) (47) (48) By the Riesz representation theorem, there exists a 5 5 matrix function η(θ, μ) : [, ] R 5 whose elements are of bounded variation such that In fact, we choose L μ φ= dη (θ, μ) φ (θ), φ C. (49) (τ +μ)(a +B +C ), θ=, (τ { +μ)(b +C ), θ [ τ τ,), η (θ, μ) = (τ +μ)b, θ (, τ ), { {, θ =. For φ C([, ], R 5 ), we define dφ (θ), θ <, { dθ A(μ)φ= { dη (θ, μ) φ (θ), θ =, { R(μ)φ={, F(μ,φ), θ<, θ=. τ (5) (5) Then system (46) can be transformed into the following operator equation u (t) =A(μ) u t +R(μ) u t. (5) For φ C([, ], (R 5 ) ), we define the adjoint operator A of A dφ (s), <s, A { ds (φ) = { dη T (s, ) φ ( s), s =, { associated with a bilinear form φ (s),φ(θ) =φ () φ () η(θ) = η(θ, ). θ= θ φ (ξ θ) dη (θ) φ (ξ) dξ, ξ= (53) (54)

10 The Scientific World Journal Let ρ(θ) = (, ρ,ρ 3,ρ 4,ρ 5 ) T e iω τ θ be the eigenvector of A corresponding to +iω τ and let ρ (s) = D(, ρ, ρ 3,ρ 4,ρ 5 )eiω τ s be the eigenvector of A corresponding to iω τ. From the definition of A() and A () and by a simple computation, we obtain Let D=[+ρ ρ +ρ 3ρ 3 +ρ 4ρ 4 +ρ 5ρ 5 +τ e iω τ ρ 5 +τ e iω τ (ρ 3 (c 33ρ 3 +c 53 ρ 5 )+ρ 4 (c 44ρ 4 +c 54 ρ 5 )) ]. (57) ρ 3 = ρ = a +a 3 ρ 3 iω a, a a 3 (iω a ) (iω c 33e iω τ ) a 3 a 3, a ρ 4 = 43 ρ 3 iω a, 44 c 44 e iω τ ρ 5 = iω a a 3 ρ 3, b 5 e iω τ ρ = iω +a a, ρ 3 = (iω +a )ρ a 3, From (54), we have ρ (s),ρ(θ) ρ 4 = c 54 e iω τ ρ 5 iω +a, 44 +c 44 e iω τ ρ 5 = b 5 e iω τ iω +a. 55 +b 55 e iω τ = D[+ρ ρ +ρ 3ρ 3 +ρ 4ρ 4 +ρ 5ρ 5 +τ e iω τ ρ 5 +τ e iω τ (ρ 3 (c 33ρ 3 +c 53 ρ 5 )+ρ 4 (c 44ρ 4 +c 54 ρ 5 ))]. (55) (56) Then, ρ,ρ =, ρ, ρ =. Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in[9] and using a computation process similar to that in []: with g =βτ D(ρ )ρ() () ρ (3) (), g =βτ D(ρ )(ρ() () ρ (3) () +ρ () () ρ (3) ()), g =βτ D(ρ )ρ() () ρ (3) (), g =βτ D(ρ ) (W () () ρ(3) () + W() () ρ(3) () +W (3) () ρ() () + W(3) () ρ() ()), W (θ) = ig ρ () τ ω +E e iτ ω θ, W (θ) = ig ρ () τ ω e iτ ω θ + ig ρ () 3τ e iτ ω θ ω e iτ ω + ig ρ () e iτ ω θ +E, τ ω (58) (59) E and E can be determined by the following equations, respectively: ( a a 3 b 5 e iω τ E = ( a a a 3 ( a 3 a 33 a 43 a 44 c 53 e iω τ c 54 e iω τ a 55 a a 3 b 5 a a a 3 E = ( a 3 c 33 ) a 43 a 44 +c 44 ( c 53 c 54 a 55 +b 55 ) ) ) E () E () ( ), ( ) E () E () ( ), ( ) (6)

11 The Scientific World Journal E(t) I(t) Figure : The phase plot of the states S, E, andi for τ = 7.85 < = τ Q(t).5..4 R(t) Figure:ThephaseplotofthestatesS, Q, andr for τ = 7.85 < = τ..6.8 with a =iω a, a =iω a, a 33 =iω c 33e iω τ, a 44 =iω a 44 c 44 e iω τ, a 55 =iω a 55 b 55 e iω τ, E () = βρ () () ρ (3) (), E () =βρ () () ρ (3) (), E () = β(ρ () () ρ (3) () +ρ () () ρ (3) ()), E () =β(ρ () () ρ (3) () +ρ () () ρ (3) ()). Then, we can get the following coefficients: C () = i τ ω (g g g g )+ g 3, (6) E(t) I(t) Figure 3: The phase plot of the states S, E, andi for τ = 9.85 > = τ. 4. Numerical Simulation In this section, we present some numerical simulations to verify the theoretical results in Sections and 3.LetA =.33, β =.75, d =., η =., μ =.3, α =., γ =.8, δ =.38, ε =.3. Then, we get the following particular case of system (3):.8 μ = Re {C ()} Re {λ (τ )}, β =Re {C ()}, T = Im {C ()}+μ Im {λ (τ )} τ. ω In conclusion, we have the following results. (6) Theorem 4. For system (3), ifμ > (μ < ), the Hopf bifurcation is supercritical (subcritical). If β < (β > ) the bifurcating periodic solutions are stable (unstable). If T > (T < ), the period of the bifurcating periodic solutions increases (decreases). ds (t) =.33.75S (t) I (t). +.R (t τ ), de (t) =.75S (t) I (t).4e(t), di (t) =.3E (t).68i (t).8i (t τ ), dq (t) =.38I (t).3q (t).3q(t τ ), dr (t) =.8I (t τ ) +.3Q (t τ ).R (t),.r (t τ ). (63)

12 The Scientific World Journal Q(t) R(t) Figure 4: The phase plot of the states S, Q, andr for τ = 9.85 > = τ Q(t).5 R(t) Figure 6: The phase plot of the states S, Q, andr for τ = 7.75 < 8.8 = τ. E(t) I(t)...5 Figure 5: The phase plot of the states S, E, andi for τ = 7.75 < 8.8 = τ. It is easy to verify that R =.584 >. Then,weget theuniquepositiveequilibriump (.589,.5656,.973,.5,.433) of system (63). Further, we can obtain det =.48 >, det = >, det 3 =..74 >, det 4 =. >, anddet 5 =.6 >. Thatis,the condition (H ) holds. For τ >, τ =. By some complex computation, we obtain ω =.3397, τ = , andf (V ) =.3 >. Thatis,theconditions(H ) and (H ) hold. According to Theorem, wecanconcludethatwhenτ [,τ ), the positive equilibrium P (.589,.5656,.973,.5,.433) of system (63) isasymptoticallystable.however, when the value of τ passes through the critical value τ, the positive equilibrium P (.589,.5656,.973,.5,.433) of system (63) willloseitsstabilityandahopf bifurcation occurs at the positive equilibrium of system (63). This property can be illustrated by Figures 4.Ascanbeseen from Figures -, ifwechooseτ = 7.85 < τ,itiseasyto see from Figures - that the positive equilibrium P (.589,.5656,.973,.5,.433) of system (63) isasymptot- ically stable. However, if we choose τ = 9.85 > τ, then the positive equilibrium P (.589,.5656,.973,.5,.433) loses its stability and a Hopf bifurcation occurs,whichcanbeillustratedbyfigures3-4.similarly,we E(t) I(t).5 Figure 7: The phase plot of the states S, E, andi for τ = 9.37 > 8.8 = τ. have ω =.769, τ = 8.8 and P 3R Q 3R +P 3I Q 3I =.39 >. Namely, the conditions (H 3 ) and (H 3 ) hold. The corresponding phase plots are shown in Figures 5, 6, 7, and8. For τ >, τ > and τ = 5.5 (,τ ).We obtain ω = 3.659, τ = by some complex computations. The corresponding phase plots are shown in Figures 9. As illustrated by Figures 9-, whenτ = 5.5 (, τ ),thepositiveequilibriump (.589,.5656,.973,.5,.433) of system (63)isasymptoticallystable. However,ascanbeseenfromFigures-, thepositive equilibrium P (.589,.5656,.973,.5,.433) of system (63) becomes unstable and a Hopf bifurcation occurs at P (.589,.5656,.973,.5,.433) when τ = 6.5 > τ. This property is consistent with Theorem 3. In addition, we have λ (τ ) = i, C () = i. Thus, we have μ = >, β =.666 <, T =.7 <.FromTheorem 4,wecanconcludethatthe Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable, and the period of the periodic solutions decreases..5

13 The Scientific World Journal E(t).5.5 Q(t).5 R(t).4. I(t) Figure8:ThephaseplotofthestatesS, Q, andr for τ = 9.37 > 8.8 = τ. Figure : The phase plot of the states S, E, andi for τ = 6.5 > = τ and τ = 5.5 (, τ ) E(t) I(t) Q(t).5 R(t) Figure 9: The phase plot of the states S, E, andi for τ = 5.5 < = τ and τ = 5.5 (, τ ). Figure : The phase plot of the states S, Q, andr for τ = 6.5 > = τ and τ = 5.5 (, τ ) Q(t).5 R(t) Figure : The phase plot of the states S, Q, andr for τ = 5.5 < = τ and τ = 5.5 (, τ ). 5. Conclusions This paper is concerned with a delayed SEIQRS model for the transmission of malicious objects in computer network. Compared with the literature [], we consider not only the time delay due to the temporary immunity period but also the time delay due to the period that the infected computer uses to clean viruses by antivirus software. That is, the system we considered in this paper is more general than that in the literature []. By considering the possible combination of the two delays as a bifurcation parameter, we find that when the delay is below the corresponding critical value, the positive equilibrium of system (3) is locally asymptotically stable. However, when the delay passes through the corresponding critical value, the positive equilibrium of system (3) loses its stability and system (3) undergoes a Hopf bifurcation, which is not welcomed in networks. Furthermore, direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are determined by using the normal form method and center manifold theory. Numerical simulations are presented to illustrate the theoretical analysis and results. Since the occurrence of the Hopf bifurcation is not welcomed in networks, we should control the Hopf bifurcation by some bifurcation control strategies such as the state feedback and parameter perturbation and so on. This is a further problem, which can be studied in the future. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

14 4 The Scientific World Journal Acknowledgments The authors are grateful to the anonymous referees and the editor for their valuable comments and suggestions on the paper. This work was supported by the National Natural Science Foundation of China (6737), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ4A5). References [] R. W. Thommes and M. J. Coates, Modeling virus propagation in peer-to-peer networks, in Proceedings of the 5th International Conference on Information, Communications and Signal Processing, pp , Bangkok, Thailand, December 5. [] B. K. Mishra and D. K. Saini, SEIRS epidemic model with delay for transmission of malicious objects in computer network, Applied Mathematics and Computation,vol.88,no.,pp , 7. [3] L. Wen and J. Zhong, Global asymptotic stability and a property of the SIS model on bipartite networks, Nonlinear Analysis: Real World Applications, vol.3,no.,pp ,. [4] B. K. Mishra and N. Jha, SEIQRS model for the transmission of malicious objects in computer network, Applied Mathematical Modelling,vol.34,no.3,pp.7 75,. [5] J.O.KephartandS.R.White, Directed-graphepidemiological models of computer viruses, in Proceedings of the IEEE ComputerSocietySymposiumonResearchinSecurityandPrivacy, pp , May 99. [6] J.G.Ren,X.F.Yang,L.X.Yang,Y.Xu,andF.Yang, Adelayed computer virus propagation model and its dynamics, Chaos, Solitons & Fractals,vol.45,no.,pp.74 79,. [7] L.Feng,X.Liao,H.Li,andQ.Han, Hopfbifurcationanalysis of a delayed viral infection model in computer networks, Mathematical and Computer Modelling,vol.56,no.7-8,pp.67 79,. [8] T. Dong, X. Liao, and H. Li, Stability and Hopf bifurcation in a computer virus model with multistate antivirus, Abstract and Applied Analysis, vol., Article ID 84987, 6 pages,. [9] Z. Zhang and H. Yang, Stability and Hopf bifurcation in a delayed SEIRS worm model in computer network, Mathematical Problems in Engineering,vol.3,ArticleID3974,9pages, 3. [] Y. Yao, W. Xiang, A. Qu, G. Yu, and F. Gao, Hopf bifurcation in an SEIDQV worm propagation model with quarantine strategy, Discrete Dynamics in Nature and Society, vol., Article ID 34868, 8 pages,. [] Y. Yao, N. Zhang, W. L. Xiang, G. Yu, and F. X. Gao, Modeling and analysis of bifurcation in a delayed worm propagation model, Applied Mathematics, vol. 3, ArticleID 97369, pages, 3. [] J. Liu, Hopf bifurcation in a delayed SEIQRS model for the transmission of malicious objects in computer network, Applied Mathematics, vol. 4, Article ID 4998, 8 pages, 4. [3] C.J.XuandX.F.He, Stabilityandbifurcationanalysisinaclass of two-neuron networks with resonant bilinear terms, Abstract and Applied Analysis, vol., Article ID 69763, pages,. [4] S. Gakkhar and A. Singh, Complex dynamics in a prey predator system with multiple delays, Communications in Nonlinear Science and Numerical Simulation, vol.7,no.,pp.94 99,. [5] J. Liu, C. W. Sun, and Y. M. Li, Stability and Hopf bifurcation analysis for a Gause-type predator-prey system with multiple delays, Abstract and Applied Analysis, vol.3,articleid , pages, 3. [6] X. Meng, H. Huo, and H. Xiang, Hopf bifurcation in a threespecies system with delays, Applied Mathematics and Computing, vol. 35, no. -, pp ,. [7] Z. Zhang, H. Yang, and J. Liu, Stability and Hopf bifurcation in a modified HOLling-Tanner predator-prey system with multiple delays, Abstract and Applied Analysis,vol.,Article ID 36484, 9 pages,. [8] X. Meng, H. Huo, X. B. Zhang, and H. Xiang, Stability and Hopf bifurcation in a three-species system with feedback delays, Nonlinear Dynamics, vol. 64, no. 4, pp ,. [9] B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 98. [] C. Bianca, M. Ferrara, and L. Guerrini, The Cai model with time delay: existence of periodic solutions and asymptotic analysis, Applied Mathematics & Information Sciences,vol.7,no.,pp. 7,3.

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