Qualitative analysis for a delayed epidemic model with latent and breaking-out over the Internet

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1 Zhang and Wang Advances in Difference Equations (17) 17:31 DOI /s R E S E A R C H Open Access Qualitative analysis for a delayed epidemic model with latent and breaking-out over the Internet Zizhen Zhang * and Yougang Wang * Correspondence: zzzhaida@163.com School of Management Science and Engineering Anhui University of Finance and Economics Bengbu 333 China Abstract We generalize a delayed computer virus model known as the SLBQRS model in a computer network by introducing the time delay due to the period that the antivirus software uses to clean viruses in the breaking out computers and the quarantined computers. By choosing the delay as the parameter we prove the existence of a Hopf bifurcation as the delay crosses a critical value. Moreover we study properties of the Hopf bifurcation by applying the center manifold theorem and the normal form theory. Finally we carry out numerical simulations to support the obtained theoretical conclusions. Keywords: computer virus; Hopf bifurcation; stability; SLBQRS model 1 Introduction With the rapid development of the communication technology and network applications the Internet has become an important platform for people sharing news and ideas [1]. Meanwhile the Internet has become a powerful mechanism for propagating malicious computer virus programs such as worms Trojans horses and so on. What is more serious enormous financial losses and social panic have also been caused by these computer viruses []. Therefore it is urgent to understand the behavior of computer viruses and to pose effective measures of controlling their spread across the Internet. In recent years many models have been established to investigate spread of computer viruses since the pioneering work of Kephart and White [3]. They investigated the computer virus spreading in the Internet by using the SIS epidemic model inspired by the compelling similarities between computer viruses and their biological counterparts. The SIR classical epidemic model and some other different epidemic models are used to investigate the spread of malware in the Internet [4 7]. The authorsof [8 11] proposed different SEIR models that assume that the recovered computers in networks have a permanent immunization period. SEIRS models and SEIQRS models have also been developed and studied in [1 15]. However all the models mentioned overlook the fact that a computer can infect other computers immediately after it gets infected [16]. This is not consistent with realization. Because of that an infected computer has infectivity during its latent period. Very recently Kumar et al. [17] proposed the following computer virus model with The Author(s) 17. This article is distributed under the terms of the Creative Commons Attribution 4. International License ( which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original author(s) and the source provide a link to the Creative Commons license and indicate if changes were made.

2 Zhangand WangAdvances in Difference Equations (17) 17:31 Page of 13 graded infection rate based on the work by Yang et al. [18]: ds(t) dl(t) db(t) dq(t) dr(t) = μ βs(t)(l(t)+b(t))+εr(t) μs(t) = βs(t)(l(t)+ B(t)) (μ + α)l(t) = αl(t) (μ + γ + η + σ )B(t) = γ B(t) (μ + σ + δ)q(t) = δq(t) (μ + ε)r(t)+ ηb(t) (1) where S(t) L(t) B(t) Q(t) and R(t) denote the percentages of uninfected computers that have no immunity infected computers that are latent infected computers that are breaking out infected computers that are quarantined and recovered computers that have temporary immunity at time t respectively;μ istherateatwhichexternalcomputersare connected to the Internet and it is also the rate at which the internal computers are disconnected from the Internet; σ is the crashing rate of the computers due to the attack of computer viruses; and α β γ ε ηandδ are the state transition rates of system (1). It is well known that it needs some time to clean the computer viruses in the infected computers that are breaking out and in the infected computers that are quarantined for the anti-virus softwares. That is system (1) neglects the delay due to the period that the anti-virus software needs to clean the viruses. To the best of our knowledge until now thereisnoanalysisonthedynamicsofsystem(1) with time delay. Motivated by this fact we consider the following delayed system: ds(t) dl(t) db(t) dq(t) dr(t) = μ βs(t)(l(t)+b(t))+εr(t) μs(t) = βs(t)(l(t)+ B(t)) (μ + α)l(t) = αl(t) (μ + γ + η + σ )B(t) = γ B(t) (μ + σ + δ)q(t) = δq(t τ) (μ + ε)r(t)+ηb(t τ) () where τ is the delay due to the period that the antivirus software needs to clean the viruses. This paper mainly focuses on the effect of time delay on system (). The subsequent materials of this paper are organized as follows. In Section weprove the local stability of the viral equilibrium and the existence of a Hopf bifurcation. Section 3 is devoted to investigations in direction of the Hopf bifurcation and stability of the bifurcating periodic solutions. Then we validate the obtained results by using simulations in Section 4. We summarize this work in Section 5. Stability of viral equilibrium and existence of Hopf bifurcation According to the analysis in [19] we can obtain that system ()hasauniqueviralequilib- rium E (S L B Q R ) if the basic reproduction number R = S = Q = (α + μ)(γ + μ + η + σ ) β(α + γ + μ + η + σ ) = 1 R L = γ + μ + η + σ B α γ μ + σ + δ B R = δγ + η(μ + σ + δ) (μ + ε)(μ + σ + δ) B β(α+γ +μ+η+σ ) (α+μ)(γ +μ+η+σ ) >1where

3 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 3 of 13 B = μα(μ + σ + δ)(μ + ε)(1 R ) R αγδε +(μ + σ + δ)[r αηε β(μ + ε)(μ + α + γ + η + σ )]. The linearized system of system () at the viral equilibrium E (S L B Q R )is ds(t) dl(t) db(t) dq(t) dr(t) = a 11 S(t)+a 1 L(t)+a 13 B(t)+a 15 R(t) = a 1 S(t)+a L(t)+a 3 B(t) = a 3 L(t)+a 33 B(t) = a 43 B(t)+a 44 Q(t) = a 55 R(t)+b 53 Q(t τ)+b 54 B(t τ) (3) where a 11 = β(l + B ) μ a 1 = βs a 13 = βs a 15 = ε a 1 = β(l + B ) a = βs (μ + α) a 3 = βs a 3 = α a 33 = μ + γ + σ + η a 43 = γ a 44 = (μ + σ + δ) a 55 = (μ + ε) b 53 = η b 54 = δ. Thecharacteristicequationofsystem() ate (S L B Q R ) can be obtained as follows: λ 5 + p 4 λ 4 + p 3 λ 3 + p λ + p 1 λ + p +(q 1 λ + q )e λτ = (4) with p = a 44 a 55 [ a33 (a 1 a 1 a 11 a )+a 3 (a 11 a 3 a 13 a 1 ) ] p 1 = a 55 [ a11 a (a 33 + a 44 )+a 33 a 44 (a 11 + a ) ] + a 11 a a 33 a 44 + a 13 a 1 a 3 (a 44 + a 55 ) a 3 a 3 [ a11 (a 44 + a 55 )+a 44 a 55 ] a 1 a 1 [ a33 (a 44 + a 55 )+a 44 a 55 ] p = a 3 a 3 (a 11 + a 44 + a 55 )+a 1 a 1 (a 33 + a 44 + a 55 ) a 13 a 1 a 3 a 11 a (a 33 + a 44 ) a 33 a 44 (a 11 + a ) a 55 [ a11 a + a 33 a 44 +(a 11 + a )(a 33 + a 44 ) ] p 3 = a 11 a + a 33 a 44 + a 55 (a 11 + a + a 33 + a 44 ) +(a 11 + a )(a 33 + a 44 ) a 1 a 1 a 3 a 3 p 4 = (a 11 + a + a 33 + a 44 + a 55 ) q = a 15 a 3 (a 44 b 53 a 43 b 54 ) q 1 = a 15 a 3 b 53. When τ =equation(4)becomes λ 5 + p 4 λ 4 + p 3 λ 3 + p λ +(p 1 + q 1 )λ + p + q =. (5)

4 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 4 of 13 Suppose that the following inequalities are satisfied which we refer to as condition (H 1 ): det 1 = p 4 > (6) ( ) p 4 1 det = > (7) p p 3 p 4 1 det 3 = p p 3 p 4 > (8) p 1 + q 1 p p 4 1 p p 3 p 4 1 det 4 = > (9) p + q p 1 + q 1 p 1 p 3 p + q p 1 + q 1 p 4 1 p p 3 p 4 1 det 5 = p + q p 1 + q 1 p p 3 p 4 >. (1) p + q p 1 + q 1 p p + q Then E (S L B Q R ) is locally asymptotically stable. For τ >letλ = iω (ω > ) be a root of equation (4). Then q 1 ω sin τω+ q cos τω= p ω p 4 ω 4 p q 1 ω cos τω q sin τω= p 3 ω 3 ω 5 p 1 ω. (11) It follows that ω 1 + e 4 ω 8 + e 3 ω 6 + e ω 4 + e 1 ω + p = (1) where e = p q e 1 = p 1 p p q 1 e = p +p p 4 p 1 p 3 e 3 = p 3 +p 1 p p 4 e 4 = p 4 p 3. Letting ω = vequation(1)becomes v 5 + e 4 v 4 + e 3 v 3 + e v + e 1 v + e =. (13) The discussion of the distribution of positive real roots of equation (13) is similar to that in []. Obviously if e <thenequation(13) has at least one positive root. In the following we discuss the distribution of the roots of equation (13)ase. Denote h(v)=v 5 + e 4 v 4 + e 3 v 3 + e v + e 1 v + e. (14)

5 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 5 of 13 Then h (v)=5v 4 +4e 4 v 3 +3e 3 v +e v + e 1. (15) Define 5v 4 +4e 4 v 3 +3e 3 v +e v + e 1 =. (16) Let v = z e 4 5. Then equation (16)becomes z 4 + c z + c 1 z + c = (17) with c = 3 65 e e 4 e 3 5 e 4e e 1 c 1 = 8 15 e e 4e e c = 6 5 e e 3. If c 1 = then we can obtain the four roots of equation (17) as follows: c + c + z 1 = z = c c z 3 = z 4 = with = c 4c.Thusv i = z i e 4 5 have the following result. (i =134)aretherootsofequation(16). Then we Lemma 1 Suppose that e and c 1 =. (i) If < then equation (13) has no positive real roots; (ii) If c and c > then equation (13) has no positive real roots; (iii) If (i) and (ii) are not satisfied then equation (13) has positive real roots if and only if there exists at least one v {v 1 v v 3 v 4 } such that v >and h(v ). Next we assume that c 1. Consider the resolvent of equation (17) that is Denote ( s c ) 1 4(s c ) 4 c = (18) s 3 c s 4c s 1 +4c c c 1 =. (19) α 1 = 1 3 c 4c β 1 = 7 c c c c 1

6 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 6 of 13 1 = 1 7 α β 1 σ 1 = i. By the Cardan formulaequation (17) has the following three roots: s 1 = 3 α α c 3 3 s = σ 1 α σ1 3 α c 3 s 3 = σ1 3 α σ 1 α c 3. Let s = s 1 c.thenequation(17)isequivalentto [ ] z 4 + s z + s 4 (s c )z c 1 z + s 4 c =. () If s > c thenequation()is ( z + s ) ( s c z After factorization we obtain and Denote z + s c z z s c z + = s c + c 1 s c ) =. (1) c 1 + s = () s c c 1 + s =. (3) s c c 1 c 1 3 = s c. s c s c Then we can obtain the roots of equation (17): z 1 = s c + z = s c s c + 3 s c 3 z 3 = z 4 =. Then v i = z i e 4 5 result. (i =134)aretherootsofequation(16).Thuswehavethefollowing Lemma Suppose that e c 1 and s > c. (i) If <and 3 < then equation (13) has no positive real roots; (ii) If (i) is not satisfied then equation (13) has positive real roots if and only if there exists at least one v {v 1 v v 3 v 4 } such that v >and h(v ).

7 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 7 of 13 Finally if s < c thenequation()is ( z + s ) ( c s z Denote v = c 1 c s c 1 (c s ) e 4 5. Hence we have the following result. ) =. (4) Lemma 3 Suppose that e c 1 and s < c. Then equation (13) has positive real c roots if and only if 1 + s 4(c s ) =and v >h( v). In conclusion we can obtain that equation (13) has at one positive real root if the coefficients in equation (13) satisfy one of the conditions in (H ): (H ) (a) e <;(b)e c 1 =andc <or c > and there exists at least one v {v 1 v v 3 v 4 } such that v >and h(v ) ; (c)e c 1 s > c or 3 and there exists at least one v {v 1 v v 3 v 4 } such that v >and h(v ) ; c (d) e c 1 s < c 1 + s 4(c s ) =and v > h( v). Suppose that the coefficients in equation (13) satisfy one of the conditions in (H ). Then equation (13) has at least one positive real root v such that equation (4) hasapairof purely imaginary roots ±iω = ±i v.forω wehave τ = 1 ω arccos { q1 ω 6 +(p 3q 1 p 4 q )ω 4 +(p q p 1 q 1 )ω p q q + q 1 ω Differentiating equation (4)withrespecttoτ and using equation (4) we get }. (5) [ ] dλ 1 = 5λ4 +4p 4 λ 3 +3p 3 λ +p λ + p 1 dτ λ(λ 5 + p 4 λ 4 + p 3 λ 3 + p λ + p 1 λ + p ) + q 1 λ(q 1 λ + q ) τ λ. (6) Substituting λ = iω into equation (6) and taking its real part we have [ dλ Re dτ ] 1 = h (v ) τ=τ q + q 1 ω where v = ω. Obviously if condition (H 3 ) h (ω ) holds then Re[ dλ dτ ] 1 τ=τ. Thus we can conclude that if condition (H 3 ) holds then the transversality condition is satisfied. According to the Hopf bifurcation theorem in [19]we have the following: Theorem 1 For system () if conditions (H 1 )-(H 3 ) hold then the viral equilibrium E (S L B Q R ) is asymptotically stable for τ [ τ ). A Hopf bifurcation occurs at the viral equilibrium E (S L B Q R ) when τ = τ and a family of periodic solutions bifurcate from the viral equilibrium E (S L B Q R ) near τ = τ.

8 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 8 of 13 3 Stability of the bifurcating periodic solutions Let u 1 (t) =S(t) S u (t) =L(t) L u 3 (t) =B(t) B u 4 (t) =Q(t) Q u 5 (t) =R(t) R τ = τ + μ μ R. By the transformation t = t/τ system () becomes the following functional differential equation in C = C([ 1 ] R 5 ): u(t)=l μ u t + F(μ u t ) (7) where u t =(u 1 (t) u (t) u 3 (t) u 4 (t) u 5 (t)) T =(S(t) L(t) B(t) Q(t) R(t)) T R 5 u t (θ)=u(t + θ) CandL μ : C R 5 and F(μ u t ) R 5 are defined by a 11 a 1 a 13 a 15 a 1 a L μ φ =(τ + μ) a 3 a 33 φ() + (τ + μ) φ( 1) a 43 a 44 a 55 b 53 b 54 and β(φ 1 ()φ () + φ 1 ()φ 3 ()) β(φ 1 ()φ () + φ 1 ()φ 3 ()) F(μ φ)=. By the Riesz representation theorem there is a bounded-variation function η(θ μ) in θ [ 1 ] such that L μ φ = dη(θ μ)φ(θ) forφ C. (8) 1 In fact we choose a 11 a 1 a 13 a 15 a 1 a η(θ μ)=(τ + μ) a 3 a 33 δ(θ) a 43 a 44 a 55 +(τ + μ) δ(θ +1) b 53 b 54 where δ(θ) is the Dirac delta function. For φ C([ 1 ] R 5 ) we set dφ(θ) 1 θ < dθ A(μ)φ = dη(θ μ)φ(θ) θ = 1

9 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 9 of 13 and 1 θ < R(μ)φ = F(μ φ) θ =. Then system (7)canberewrittenas u(t)=a(μ)u t + R(μ)u t. (9) For ϕ C 1 ([ 1]) (R 5 ) wedefinetheoperator A dϕ(s) <s 1 ds (ϕ)= 1 dηt (s)ϕ( s) s = and the bilinear inner form ϕ(s) φ(θ) = ϕ()φ() θ θ= 1 ξ= ϕ(ξ θ) dη(θ)φ(ξ) dξ (3) where η(θ)=η(θ). Then it is easy to see that A() and A () are adjoint operators and that ±iω are the eigenvalues of A() and also the eigenvalues of A (). Let ρ(θ)=(1ρ ρ 3 ρ 4 ρ 5 ) T e iω τ θ and ρ (s)=(1ρ ρ 3 ρ 4 ρ 5 )eiω τ s be the eigenvectors of A() and A () corresponding to +iω τ respectively. Then it is not difficult to show that ρ = iω a 33 a 1 a 3 ρ 3 ρ 3 = a 3 (iω a )(iω a 33 ) a 3 a 3 ρ 4 = a 43ρ 3 iω a 44 ρ 5 = b 53ρ 3 + b 54 ρ 4 (iω a 33 )e iτ ω ρ = iω + a 11 a 1 ρ 3 = a 1 +(iω + a )ρ a 3 ρ 4 = a 15 b 54 e iτ ω (iω + a 44 )(iω + a 55 ) ρ 5 = a 15 iω + a 55. Using equation (3) we can obtain ρ (s) ρ(θ) = D [ 1+ρ ρ + ρ 3 ρ 3 + ρ 4 ρ 4 + ρ 5 ρ 5 + τ e iτ ω ρ 5 (b 53ρ 3 + b 54 ρ 4 ) ]. Then we choose D = [ 1+ρ ρ + ρ 3 ρ 3 + ρ 4 ρ 4 + ρ 5 ρ 5 + τ e iτ ω ρ 5 (b 53ρ 3 + b 54 ρ 4 ) ] 1 such that ρ ρ =1and ρ ρ =.

10 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 1 of 13 Following the method introduced in[19] and using a similar computation process in [1 3] we can get the following coefficients: g =τ Dβ ( ρ 1) (ρ + ρ 3 ) g 11 = τ Dβ ( ρ 1)( Re{ρ } + Re{ρ 3 } ) g =τ Dβ ( ρ 1) ( ρ + ρ 3 ) g 1 =βτ D ( ρ 1)( W (1) 11 ()ρ + 1 W (1) () ρ + W () 11 () + 1 W () () + W (1) 11 ()ρ W (1) () ρ 3 + W (3) 11 () + 1 W (3) () ) with W (θ)= ig ρ() e iτ ω θ + iḡ ρ() e iτ ω θ + E 1 e iτ ω θ τ ω 3τ ω W 11 (θ)= ig 11ρ() e iτ ω θ + iḡ 11 ρ() e iτ ω θ + E τ ω τ ω where E 1 and E can be determined by the following equations: iω a 11 a 1 a 13 a 15 a 1 iω a a 3 a 3 iω a 33 e iτ ω a 43 iω a 44 b 53 e iτ ω b 54 e iτ ω iω a 55 β(ρ + ρ 3 ) β(ρ + ρ 3 ) = a 11 a 1 a 13 a 15 β(re{ρ } + Re{ρ 3 }) a 1 a a 3 β(re{ρ } + Re{ρ 3 }) a 3 a 33 E =. a 43 a 44 b 53 b 54 a 55 Then we can get the following coefficients: i C 1 () = (g 11 g g 11 g ) + g 1 τ ω 3 μ = Re{C 1()} Re{λ (τ )} β =Re { C 1 () } T = Im{C 1()} + μ Im{λ (τ )} τ ω. (31) E 1

11 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 11 of 13 In conclusion we have the following results. Theorem Let E (S L B Q R ) be the viral equilibrium of system (). (i) TheHopfbifurcationattheviralequilibriumE (S L B Q R ) is supercritical if μ >and subcritical if μ <; (ii) The bifurcating periodic solutions are stable if β <and unstable if β >; (iii) The period of the bifurcating periodic solutions increases if T >and decreases if T <. 4 Numerical simulations This section is concerned with some numerical simulations of system () to illustrate the results obtained in Sections and 3. Wefixtheparametersetμ =.β =.3ε =.3 α =.3μ =.35γ =.1η =.4σ =.δ =.1andletτ vary. With these parameter values we get the following particular case of system (): ds(t) dl(t) db(t) dq(t) dr(t) =..3S(t)(L(t)+B(t)) +.3R(t).S(t) =.3βS(t)(L(t)+B(t)).3L(t) =.3L(t).7B(t) =.1B(t).3Q(t) =.1Q(t τ).3r(t)+.4b(t τ). (3) By direct computation we get R = Then we obtain the unique viral equilibrium E ( ). Further we can verify that the conditions for the occurrence of a Hopf bifurcation are satisfied. By complex computations we obtain ω =.5499andτ = FromTheorem1 we easily see that E ( ) is asymptotically stable for τ [ ) whichisillustratedbyfigure1. AHopfbifurcationoccurswhenτ = τ = E ( ) loses its stability and a family of periodic solutions bifurcate from E ( ). Therefore the bifurcating periodic solutions exist at least for the values of τ larger than the critical value τ.this propertycanbeseenfromfigure. Further we obtain λ (τ )= i and C 1 () = i by some complex computations. Then we can get β = 1.893<μ =1.7575>andT =.3931 >. According to Theorem we can conclude that the Hopf bifurcation is supercritical the bifurcating periodic solutions are stable and the period of the bifurcating periodic solutions increases. Since the bifurcating periodic solutions are stable we know that the five kinds of computers in system (3) may coexist in an oscillatory mode from the viewpoint of biology. In this case it is difficult to control the propagation of computer virus described in system (3). 5 Conclusions A delayed SLBQRS computer virus model is proposed based on the model studied in [17]. Compared with the model studied in [17] the model in this paper incorporates the time delay due to the period that the antivirus software uses to clean the viruses in the breaking out computers and the quarantined computers. So our delayed SLBQRS computer virus model is more general.

12 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 1 of 13 Figure 1 E is locally asymptotically stable when τ = < τ = with initial value Figure P loses its stability when τ = > τ = with initial value We mainly investigate the effect of the time delay on the model. We prove that there exists a critical value τ of the time delay below which the model is stable and above which the model will lose its stability. Practically speaking propagation of the computer viruses can be controlled easily when the model is stable. However it will be out of control when the model is unstable. That is the time delay in system () can affect propagation of the computer viruses. Furthermore properties of the Hopf bifurcation at the critical value τ are also investigated. It should be pointed out that the delayed model in this paper only considers the time delay due to the period that the antivirus software uses to clean the viruses. There are also some other types of delay in system () such as latent delay and delay due to a temporary immunity period of the recovered computers. We will investigate dynamics of system () with multiple delays in the near future. Competing interests The authors declare that they have no competing interests.

13 Zhangand WangAdvances in Difference Equations (17) 17:31 Page 13 of 13 Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements The authors would like to thank the editor and anonymous referees for their constructive suggestions on improving the presentation of the paper. This work was supported by Natural Science Foundation of the Higher Education Institutions of Anhui Province (No. KJ15A144) and Anhui Provincial Natural Science Foundation (Nos QF QF145). Received: 9 October 16 Accepted: December 16 References 1. Kumar M Mishar BK Panda TC: Effect of quarantine and vaccination on infectious nodes in computer network. Int. J. Comput. Netw. Appl (15). Szor P: The Art of Computer Virus Research and Defense. Addison-Wesley Reading (5) 3. Kephart JO White SR: Directed-graph epidemiological models of computer viruses. In: Proc IEEE Comput. Society Symp. Res. Secur. Privacy pp (1991) 4. Piqueria JRC: A modified epidemiological model for computer viruses. Appl. Math. Comput (9) 5. Piqueira JRC Cesar FB: Dynamical models for computer virus propagation. Math. Probl. Eng. 8 Article ID9456 (8) 6. Keeling MJ Eames KTD: Network and epidemic models. J. R. Soc. Interface (5) 7. Piqueira JRC Navarro BF Monteiro LHA: Epidemiological models applied to virus in computer network. J. Comput. Sci (5) 8. Yuan YH Chen G: Network virus epidemic model with the point-to-group information propagation. Appl. Math. Comput (8) 9. Dong T Liao XF Li HQ: Stability and Hopf bifurcation in a computer virus model with multistate antivirus. Abstr. Appl. Anal. 1 Article ID (1) 1. Peng M He X Huang JJ Dong T: Modeling computer virus and its dynamics. Math. Probl. Eng. 13ArticleID (13) 11. PengMMouHJ:Anovelcomputervirusmodelanditsstability.J.Netw (14) 1. Mishra BK Saini DK: SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl. Math. Comput (7) 13. Zhang ZZ Yang HZ: Hopf bifurcation analysis for a computer virus model with two delays. Abstr. Appl. Anal. 13 Article ID 5684 (13) 14. Mishra BK Pandey SK: Dynamic model of worms with vertical transmission in computer network. Appl. Math. Comput (11) 15. Mishra BK Jha N: SEIQRS model for the transmission of malicious objects in computer network. Appl. Math. Model (1) 16. Yang LX Yang XF: A new epidemic model of computer viruses. Commun. Nonlinear Sci. Numer. Simul (14) 17. Kumar M Mishar BK Panda TC: Stability analysis of a quarantined epidemic model with latent and breaking-out over the Internet. Int. J. Hybrid Inf. Technol (15) 18. Yang M Zhang Z Li Q Zhang G: An SLBRS model with vertical transmission of computer virus over the Internet. Discrete Dyn. Nat. Soc. 1 Article ID (1) 19. Hassard BD Kazarinoff ND Wan YH: Theory and Applications of Hopf Bifurcation. Cambridge University Press Cambridge (1981). Zhang TL Jiang HJ Teng ZD: On the distribution of the roots of a fifth degree exponential polynomial with application to a delayed neural network model. Neurocomputing (9) 1. Bianca C Ferrara M Guerrini L: The cai model with time delay: existence of periodic solutions and asymptotic analysis. Appl. Math. Inf. Sci (13). Bianca C Ferrara M Guerrini L: The time delays effects on the qualitative behavior of an economic growth model. Abstr. Appl. Anal. 13Article ID 9114 (13) 3. Bianca C Guerrini L: Existence of limit cycles in the Solow model with delayed-logistic population growth. Sci. World J. 14 Article ID 786 (14)