Stability and Hopf Bifurcation Analysis for a Computer Virus Propagation Model with Two Delays and Vaccination

Abstract

A further generalization of an SEIQRS-V (susceptible-exposed-infectious-quarantined-recovered-susceptible with vaccination) computer virus propagation model is the main topic of the present paper. This paper specifically analyzes effects on the asymptotic dynamics of the computer virus propagation model when two time delays are introduced. Sufficient conditions for the asymptotic stability and existence of the Hopf bifurcation are established by regarding different combination of the two delays as the bifurcation parameter. Moreover, explicit formulas that determine the stability, direction, and period of the bifurcating periodic solutions are obtained with the help of the normal form theory and center manifold theorem. Finally, numerical simulations are employed for supporting the obtained analytical results.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Zizhen Zhang 1 and Yougang Wang 1 and Dianjie Bi 1 and Luca Guerrini 2

Academic Editor:Lu-Xing Yang

1, School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China

2, Department of Management, Marche Polytechnic University, Piazza Martelli 8, 60121 Ancona, Italy

Received 5 January 2017; Accepted 23 February 2017; 20 March 2017

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.

1. Introduction

Computer viruses, including conventional viruses and network worms, can propagate among computers with no human awareness and popularization of Internet has been the major propagation channel of viruses [1, 2]. The past few decades have witnessed the great financial losses caused by computer viruses. Therefore, it is of considerable importance to investigate the laws describing propagation of computer viruses in order to provide some help with preventing computer viruses. For that purpose and in view of the fact that propagation of computer viruses among computers resembles that of biological viruses among a population, many dynamical models describing propagation of computer viruses across the Internet have been established by the scholars at home and abroad, such as conventional models [3-8], stochastic models [9-12], and delayed models [13-18]. There are also some other computer virus models [19-21] combined with network theory to investigate the impact of the network topology, the patch forwarding, and the network eigenvalue on the viral prevalence.

As is known, vaccination is regarded as one of the most effective measures of preventing computer viruses and the awareness that there exist many infected computers would enhance the probability that the user of a susceptible computer will make his computer vaccinated [22, 23]. However, the mentioned models above neglect the influence of vaccination strategy on the propagation of computer viruses. Recently, considering the importance of vaccination, Kumar et al. [24] proposed the following SEIQRS-V computer virus propagation model: [figure omitted; refer to PDF] where S(t), E(t), I(t), Q(t), R(t), and V(t) denote the numbers of the uninfected computers, the exposed computers, the infected computers, the quarantined computers, recovered computers, and vaccinated computers at time t, respectively. A is the birth rate of new computers in the network; d is the death rate of the computers due to the reason other than the attack of viruses; α is the death rate of computers due to the attack of viruses; β is the contact rate of the uninfected computers; ρ, θ, χ, γ, δ, η, and [straight epsilon] are the transition rates between the states in system (1).

Obviously, system (1) neglects the delays in the procedure of viruses' propagation and it is investigated under the assumption that the transition between the states is instantaneous. This is not reasonable with reality. For example, it needs a period to clean the viruses in the infected and quarantined computers for antivirus software and there is usually a temporary immunity period for the recovered and the vaccinated computers because of the effect of the antivirus software. In addition, a stability switch occurs even when an ignored delay is small for a dynamical system. Based on this, we introduce two delays into system (1) and get the following delayed system: [figure omitted; refer to PDF] where _τ1 is the time delay due to the period that antivirus software uses to clean the viruses in the infected and quarantined computers and _τ2 is the time delay due to the temporary immunity period of the recovered and the vaccinated computers.

To the best of our knowledge, until now, there is no good analysis on system (2). Therefore, it is meaningful to analyze the proposed system with two delays.

The rest of this paper is organized as follows. In the next section, we analyze the threshold of Hopf bifurcation of system (2) by regarding different combination of the two delays as the bifurcation parameter. In Section 3, by means of the normal form theory and center manifold theorem, direction and stability of the Hopf bifurcation for _τ1 >0 and _τ2 >0 are investigated. Simulation results of system (2) are shown in Section 4. Finally, we finish the paper with conclusions in Section 5.

2. Analysis of Hopf Bifurcation

By direct computation, we know that if A_R0 (d+χ)>^d2 +(ρ+χ)d and β(d+θ)(d+α+[straight epsilon])>_R0 θ[straight epsilon]δ+_R0 θη(d+α+[straight epsilon]), then system (2) has a unique viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ), where [figure omitted; refer to PDF] The linearized section of system (2) at _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then, the characteristic equation for system (4) can be obtained: [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

Case 1 (_τ1 =_τ2 =0).

When _τ1 =_τ2 =0, (6) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Clearly, _D1 =_A15 =β_{I[low *]} +ρ+γ+δ+η+[straight epsilon]+θ+χ+2α+6d>0. Thus, if condition (_H1 ) (see (10)) holds, then system (2) without delay is locally asymptotically stable: [figure omitted; refer to PDF]

Case 2 (_τ1 >0; _τ2 =0).

Equation (6) equals [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Multiplying ^eλ_τ1 on left and right of (11), one has [figure omitted; refer to PDF] Assume that λ=i_ω1 (_ω1 >0) is the root of (13): [figure omitted; refer to PDF] with [figure omitted; refer to PDF] Thus, one can obtain the expressions of cos[...]_τ1ω1 and sin[...]_τ1ω1 as follows: [figure omitted; refer to PDF] Then, we can get [figure omitted; refer to PDF] Suppose that (_H21 ) (see (17)) has at least one positive root.

If condition (_H21 ) holds, then there exists _ω10 >0 such that (13) has a pair of purely imaginary roots ±i_ω10 . For _ω10 , [figure omitted; refer to PDF] Differentiating (13) with respect to _τ1 , one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

Thus, if condition (_H22 ) _G2R ×_H2R +_G2I ×_H2I ≠0 holds, then Re[...][dλ/d_τ1]λ=iω10 ≠0. Based on the Hopf bifurcation theorem in [25], we have the following results.

Theorem 1.

Suppose that conditions (_H1 ), (_H21 ), and (_H22 ) hold for system (2). The viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) is locally asymptotically stable when _τ1 ∈[0,_τ10 ) and a Hopf bifurcation occurs at the viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) when _τ1 =_τ10 .

Case 3 (_τ1 =0; _τ2 >0).

Equation (6) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Multiplying ^eλ_τ2 on left and right of (23), one has [figure omitted; refer to PDF] Let λ=i_ω2 (_ω2 >0) be the root of (25): [figure omitted; refer to PDF] with [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] And the equation following equation regarding _τ2 can be obtained: [figure omitted; refer to PDF] Suppose that (_H31 ) (see (29)) has at least one positive root.

If condition (_H31 ) holds, then there exists _ω20 >0 such that (25) has a pair of purely imaginary roots ±i_ω20 . For _ω20 , [figure omitted; refer to PDF] Differentiate both sides of (25) with respect to _τ2 . Then, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

Similar to Case 2, we know that if condition (_H32 ) _G3R ×_H3R +_G3I ×_H3I ≠0 holds, then Re [dλ/d_τ2]λ=iω20 ≠0. In conclusion, we have the following results.

Theorem 2.

Suppose that conditions (_H1 ), (_H31 ), and (_H32 ) hold for system (2). The viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) is locally asymptotically stable when _τ2 ∈[0,_τ20 ) and a Hopf bifurcation occurs at the viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) when _τ2 =_τ20 .

Case 4 (_τ1 >0; _τ2 ∈(0,_τ20 )).

Regarding _τ1 as the bifurcation parameter when _τ2 ∈(0,_τ20 ), multiplying by ^eλ_τ1 , (6) becomes [figure omitted; refer to PDF]

Let λ=i^{ω1[low *]} (^{ω1[low *]} >0) be the root of (35), and for the convenience we still denote ^{ω1[low *]} as _ω1 ; then, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Then, we get [figure omitted; refer to PDF] Suppose that (_H41 ) (see (39)) has at least one positive root.

If (_H41 ) holds, then there exists _ω10 >0 such that (35) has a pair of purely imaginary roots ±i^{ω10[low *]} . For ^{ω10[low *]} , [figure omitted; refer to PDF] Differentiating both sides of (25) with respect to _τ2 , [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Define [figure omitted; refer to PDF]

Similar to Case 2, we know that if condition (_H42 ) _G4R ×_H4R +_G4I ×_H4I ≠0 holds, then Re[...][dλ/d_{τ1]λ=i^{ω10[low *]}} ≠0. Thus, we have the following results.

Theorem 3.

Let _τ2 ∈(0,_τ20 ) and suppose that conditions (_H1 ), (_H41 ), and (_H42 ) hold for system (2). The viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) is locally asymptotically stable when _τ1 ∈[0,^{τ10[low *]} ) and a Hopf bifurcation occurs at the viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) when _τ1 =^{τ10[low *]} .

Case 5 (_τ1 ∈(0,_τ10 ); _τ2 >0).

Regarding _τ2 as the bifurcation parameter when _τ1 ∈(0,_τ10 ), multiplying by ^eλ_τ2 , (6) becomes [figure omitted; refer to PDF]

Let λ=i^{ω2[low *]} (^{ω2[low *]} >0) be the root of (44), and for the convenience we still denote ^{ω2[low *]} as _ω2 ; then, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Then, we get [figure omitted; refer to PDF]

If (_H51 ) holds, then there exists ^{ω20[low *]} >0 such that (35) has a pair of purely imaginary roots ±i^{ω20[low *]} . For ^{ω20[low *]} , [figure omitted; refer to PDF] Differentiating (25) with respect to _τ2 , one can get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF]

Therefore, we know that if condition (_H52 ) _G5R ×_H5R +_G5I ×_H5I ≠0 holds, then Re[...][dλ/d_{τ2]λ=i^{ω20[low *]}} ≠0. Then, we have the following results.

Theorem 4.

Let _τ1 ∈(0,_τ10 ) and suppose that conditions (_H1 ), (_H51 ), and (_H52 ) hold for system (2). The viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) is locally asymptotically stable when _τ2 ∈[0,^{τ20[low *]} ) and a Hopf bifurcation occurs at the viral equilibrium _{P[low *]} (_{S[low *]} ,_{E[low *]} ,_{I[low *]} ,_{Q[low *]} ,_{R[low *]} ,_{V[low *]} ) when _τ2 =^{τ20[low *]} .

3. Properties of the Hopf Bifurcation

In this section, we shall investigate direction and stability of the Hopf bifurcation under the case where _τ1 ∈(0,_τ10 ) and _τ2 >0. Set _u1 (t)=S(t)-_{S[low *]} , _u2 (t)=E(t)-_{E[low *]} , _u3 (t)=I(t)-_{I[low *]} , _u4 (t)=Q(t)-_{Q[low *]} , _u5 (t)=R(t)-_{R[low *]} , _u6 (t)=V(t)-_{V[low *]} , and t[arrow right](t/_τ2 ). For convenience, we assume that ^{τ1[low *]} ∈(0,_τ10 )<^{τ20[low *]} throughout this section. Then, system (2) becomes functional differential equations in C=C([-1,0],^R6 ): [figure omitted; refer to PDF] with [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Based on the Riesz representation theorem, there exists a 6×6 function η(θ,μ):[-1,0][arrow right]^R6×6 such that [figure omitted; refer to PDF] In fact, we choose [figure omitted; refer to PDF] For [varphi]∈C([-1,0],^R6 ), we define [figure omitted; refer to PDF] Then, system (53) becomes [figure omitted; refer to PDF] where _ut (θ)=u(t+θ) for θ∈[-1,0].

Define ^{A[low *]} as follows: [figure omitted; refer to PDF] and a bilinear form [figure omitted; refer to PDF] where η(θ)=η(θ,0).

Let q(θ)=(1,_q2 ,_q3 ,_q4 ,_q5 ,_q6^{)Teiω20[low *]τ20[low *] θ} be the eigenvector of A(0) with +i^{ω20[low *]τ20[low *]} and let ^{q[low *]} (s)=D(1,^{q2[low *]} ,^{q3[low *]} ,^{q4[low *]} ,^{q5[low *]} )^{eiω20[low *]τ20[low *] s} be the eigenvector of ^{A[low *]} (0) with -i^{ω20[low *]τ20[low *]} . Then, according to the definition of A(0) and ^{A[low *]} (0), we obtain [figure omitted; refer to PDF] In addition, from (61), we have [figure omitted; refer to PDF] Thus, we can choose [figure omitted; refer to PDF] such that [...]^{q[low *]} ,q[...]=1, [...]^{q[low *]} ,q-[...]=0.

Then, using the algorithms from Hassard et al. [25] and the similar computation process in [26-29], we obtain [figure omitted; refer to PDF] with [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Then, we can get the following coefficients: [figure omitted; refer to PDF] Thus, we have the following results.

Theorem 5.

The sign of _μ2 determines direction of the Hopf bifurcation: if _μ2 >0 (_μ2 <0), then the Hopf bifurcation is supercritical (subcritical); the sign of _ρ2 determines stability of the bifurcating periodic solutions: if _ρ2 <0 (_ρ2 >0), then the bifurcating periodic solutions are stable (unstable); the sign of _T2 determines period of the bifurcating solutions: if _T2 >0 (_T2 <0), then the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Simulation

In this section, we present some numerical results of system (2) in order to validate the analytical predictions obtained in Sections 2 and 3. We choose a set of parameters as follows: A=100, β=0.009, d=0.05, ρ=0.65, θ=0.05, χ=0.55, γ=0.45, α=0.035, δ=0.1, η=0.35, and [straight epsilon]=0.07, and consider the following special case of (2): [figure omitted; refer to PDF] from which we can get the unique viral equilibrium _{P[low *]} (66.0494,277.7978,233.6617,150.7495, 923.3406, 71.7439). It can be easily verified that condition (_H1 ) is satisfied when _τ1 =_τ2 =0.

By computation, we have _ω10 =0.8554 and _τ10 =4.1056. Then, we get ^{λ[variant prime]} (_τ10 )=2.3686+i1.0212. Thus, we know that conditions (_H21 ) and (_H22 ) hold. We can conclude that all roots that cross the imaginary axis at i_ω10 cross from left to right as _τ1 increases by the theory in [22]. According to Theorem 1, _{P[low *]} (66.0494,277.7978,233.6617,150.7495,923.3406, 71.7439) is asymptotically stable when _τ1 ∈(0,_τ10 ). This property can be illustrated by Figures 1 and 2. In this case, spreading law of the computer viruses can be predicted and the viruses can be controlled and eliminated. However, once the value of _τ1 passes through the critical value _τ10 , _{P[low *]} (66.0494,277.7978,233.6617,150.7495,923.3406, 71.7439) loses its stability and a Hopf bifurcation occurs, which can be shown in Figures 3 and 4. The occurrence of a Hopf bifurcation means that the state of computer viruses propagation changes from the viral equilibrium point to a limit cycle. This makes spreading of the computer viruses be out of control.

Figure 1: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ1 =3.605.

[figure omitted; refer to PDF]

Figure 2: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ1 =3.605.

[figure omitted; refer to PDF]

Figure 3: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ1 =4.60.

[figure omitted; refer to PDF]

Figure 4: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ1 =4.60.

[figure omitted; refer to PDF]

Similarly, we have the following: _ω20 =1.8255 and _τ20 =3.7424 when _τ1 =0 and _τ2 >0; ^{ω10[low *]} =0.9665 and ^{τ10[low *]} =3.1862 when _τ1 >0 and _τ2 =2.25∈(0,_τ20 ); ^{ω20[low *]} =2.4217 and ^{τ20[low *]} =3.0254 when _τ2 >0 and _τ1 =2.45∈(0,_τ10 ). The corresponding phase plots are shown in Figures 5-8, Figures 9-12, and Figures 13-16, respectively. In addition, for _τ2 >0 and _τ1 =2.45∈(0,_τ10 ), we have _C1 (0)=-17.2982+i13.5056 and ^{λ[variant prime]} (^{τ20[low *]} )=0.3796+i2.0581 by some complex computation. Based on (68), we get _μ2 =45.5692>0, _ρ2 =-34.5964<0, and _T2 =-14.6441<0. Therefore, the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the bifurcating periodic solutions decreases.

Figure 5: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ2 =3.65.

[figure omitted; refer to PDF]

Figure 6: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ2 =3.65.

[figure omitted; refer to PDF]

Figure 7: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ2 =3.805.

[figure omitted; refer to PDF]

Figure 8: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ2 =3.805.

[figure omitted; refer to PDF]

Figure 9: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ1 =2.86 and _τ2 =2.25∈(0,_τ20 ).

[figure omitted; refer to PDF]

Figure 10: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ1 =2.86 and _τ2 =2.25∈(0,_τ20 ).

[figure omitted; refer to PDF]

Figure 11: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ1 =3.574 and _τ2 =2.25∈(0,_τ20 ).

[figure omitted; refer to PDF]

Figure 12: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ1 =3.574 and _τ2 =2.25∈(0,_τ20 ).

[figure omitted; refer to PDF]

Figure 13: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ2 =2.862 and _τ1 =2.45∈(0,_τ10 ).

[figure omitted; refer to PDF]

Figure 14: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ2 =2.862 and _τ1 =2.45∈(0,_τ10 ).

[figure omitted; refer to PDF]

Figure 15: The projection of the phase portrait of system (69) in (S,E,V)-space with _τ2 =3.225 and _τ1 =2.45∈(0,_τ10 ).

[figure omitted; refer to PDF]

Figure 16: The projection of the phase portrait of system (69) in (I,Q,R)-space with _τ2 =3.225 and _τ1 =2.45∈(0,_τ10 ).

[figure omitted; refer to PDF]

According to the numerical simulation results, we know that the time delay should remain less than the corresponding threshold in order to control and predict the viruses' propagation by decreasing the period that antivirus software uses to clean the computer viruses and the temporary immunity period of the recovered and the vaccinated computers. To this end, we can adjust the parameters of our proposed model in real-world networks, such as timely updating the antivirus software on computers, properly controlling the number of computers attached to the network, and timely disconnecting computers from the network when the connections are unnecessary. Of course, in the next step, we also need to collect large amount of relevant data and estimate the parameters involved in our proposed model through statistical analysis in real-world networks. Namely, we have to adjust the parameters in the model so as to control viruses' propagation effectively if it is necessary.

5. Conclusions

It is definitely an interesting work to consider the effect of delays on dynamical systems, because a stability switch occurs even when an ignored delay is small for a dynamical system. Based on this fact, we introduce the time delay due to the period that antivirus software uses to clean the computer viruses in the infectious and quarantined computers (_τ1 ) and the time delay due to the temporary immunity period of the recovered and the vaccinated computers (_τ2 ) into the SEIQRS-V computer virus propagation model considered in [21]. We obtain some conditions for local stability and Hopf bifurcation occurring by analyzing distribution of roots of the associated characteristic equation.

By computation, there exists a corresponding critical value of the time delay below which system (2) is stable and above which system (2) is unstable. When the system is stable, the characteristics of the propagation of computer viruses can be easily predicted and then the computer viruses can get eliminated. Otherwise, the propagation of the computer viruses is out of control. Therefore, stability of the computer virus propagation system must be guaranteed in practice. In addition, we find that the effect of _τ2 on system (2) is marked compared with _τ1 , because the critical value of _τ2 is much smaller when we only consider it. At last, we have also derived the explicit formula which can determine direction and stability of the Hopf bifurcation under the case where _τ1 ∈(0,_τ10 ) and _τ2 >0.

Acknowledgments

This work was supported by Natural Science Foundation of Anhui Province (nos. 1608085QF151, 1608085QF145, and 1708085MA17) and Natural Science Foundation of the Higher Education Institutions of Anhui Province (nos. KJ2014A006 and KJ2015A144).

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