Stability and Hopf Bifurcation for a Delayed Computer Virus Model with Antidote in Vulnerable System

A delayed computer virus model with antidote in vulnerable system is investigated. Local stability of the endemic equilibrium and existence of Hopf bifurcation are discussed by analyzing the associated characteristic equation. Further, direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are presented to show consistency with the obtained results.


Introduction
Applications based on computer networks are becoming more and more popular in our daily life.While bringing convenience to us, computer networks are exposed to various threats [1,2].Therefore, it is urgent to explore the spreading law of computer viruses in networks.To this end, many dynamical models describing propagation of computer viruses have been established by scholars at home and abroad.Particularly the classic epidemic models, such as SIRS [3][4][5] model, SEIRS model [6], and SEIQRS model [7,8], are used to investigate the spreading law of computer viruses due to the common feature between the computer virus and the biological virus.Some computer virus models with infectivity in both seizing and latent computers have been also proposed by Yang et al. [9][10][11][12][13].
Recently, Khanh and Huy [14] proposed the following computer virus model with different antidote rates of nodes in vulnerable system considering the immunizations ways and the vulnerabilities of the operating system: where (), (), (), and () are the sizes of susceptible, exposed, infectious, and recovered nodes at time , respectively;  is the constant recruitment of the susceptible nodes;  is the same rate at which every node in the states (), (), (), and () disconnects from the network;  is the constant rate at which every susceptible node acquires temporary immunity due to antidote and Khanh and Huy [14] assume that  <  taking system vulnerability into account; , , , and  are the other state transition rates of system (1).Khanh and Huy [14] studied stability of system (1) and suggested some effective strategies for eliminating viruses.It is well known that time delays of one type or another have been incorporated into computer virus models due to latent period of virus, temporary immunization period of nodes, or other reasons.Computer virus models with time delay have been investigated extensively in recent years [15][16][17][18].Time delays can play a complicated role in the dynamics of dynamical systems, especially that they can cause Hopf bifurcation phenomenon of the models.In reality, 2 Journal of Control Science and Engineering the occurrence of Hopf bifurcation means that the state of computer virus spreading changes from an equilibrium point to a limit cycle, which is not welcomed in networks.Motivated by the work above and considering that it needs a period to reinstall system for the infected nodes in the network, we propose the following system with time delay: where  is the time delay due to the period that the infected nodes use to reinstall system.The rest of the paper is organized as follows.Section 2 obtains the basic reproduction number of system (2) and discusses local stability of the endemic equilibrium and existence of Hopf bifurcation; Section 3 determines direction of the Hopf bifurcation and stability of the bifurcated periodic solutions; Section 4 covers numerical analysis and simulations.Finally, Section 5 summarizes this work.

Existence of Hopf Bifurcation
and  0 is the basic reproduction number of system (2).
Based on the leading matrix of system ( which derives that where When  = 0, (5) takes the following form: According to the expressions of   and   ( = 0, 1, 2, 3), we can get with  = / and ,  are specified by ( 6) and ( 7), respectively.According to the assumption  <  considering system vulnerability, we know that  0 +  0 > 0,  1 +  1 > 0,  2 +  2 > 0, and  3 +  3 > 0. In addition, one can be conclude that based on the analysis by Khanh and Huy in [14] when  0 > 1.Therefore, we can conclude that the endemic equilibrium  * ( * ,  * ,  * ,  * ) is locally stable for  0 > 1 according to the Routh-Hurwitz criterion.The time delay is always positive in our physical system.It follows that if instability occurs for a particular value of the delay  > 0, a characteristic root of ( 5) must intersect the imaginary axis according to Corollary 2.4 in [19].Assume that (5) has a purely imaginary root  ( > 0).Then, the following identity must be true which yields the following equation: where Denote  2 = V; (13) becomes Define Thus, Set Let  = V + 3 3 /4.Then, (18) becomes where Denote Discussion about the distribution of the roots of ( 15) is similar to that in [20].Thus, we have the following lemma.Lemma 1.For (15), one has the following: (15) has at least one positive root.

has positive roots if and only
(iii) If  0 ≥ 0 and  < 0, (15) has positive roots if and only if there exists at least one In what follows, we assume that ( 1 )and the coefficients in (V) satisfy one of the following conditions in (a)-(c).
(c)  0 ≥ 0 and  < 0, and there exists at least one Suppose that the condition ( 1 ) holds; then (15) has at least one positive root V 0 such that (13) has a positive root  0 = √ V 0 .Further, we can get Differentiating both sides of ( 5) with respect to  yields Further, we have where According to the Hopf bifurcation theorem in [21], we can obtain the following results.

Numerical Simulations
In this section, numerical simulations are presented by taking partial parameters from numerical simulations in [14]  ( By virtue of the chosen values of parameters, we can get  0 = 1.8182 and the unique endemic equilibrium  * (1.5714, 0.6712, 0.0224, 0.5427).Further, we can verify that the conditions indicated in Theorem 2 are satisfied.In this case we can obtain  0 = 0.5209,  0 = 5.3442, and   ( 2 0 ) = 1.9227 ̸ = 0. Therefore,  * (1.5714, 0.6712, 0.0224, 0.5427) is asymptotically stable when 0 <  <  0 = 5.3442 according to Theorem 2, which can be shown as the numerical simulation in Figure 1.In this case, propagation of the computer viruses can be controlled easily.However, system (68) undergoes a  Hopf bifurcation at  * (1.5714, 0.6712, 0.0224, 0.5427) when  passes through the critical value  0 .This property can be illustrated by the numerical simulation in Figure 2 and propagation of the computer viruses will be out of control.In addition, according to (67), we have  2 = 0.8048 > 0,  2 = −1.2498< 0, and  2 = −0.7991< 0. Thus, we know that the direction of the Hopf bifurcation at  0 is supercritical; the period of the bifurcating periodic solutions decreases; and the bifurcating periodic solutions are stable.From the viewpoint of biology, if the periodic solutions bifurcating from the Hopf bifurcation are stable, then the susceptible, exposed, infectious, and recovered nodes in system (68) may coexist in an oscillatory mode.This phenomenon is not welcome in a real network.We regret to say that only supercritical Hopf bifurcation is identified in our numerical case study.However, it should be pointed out that subcritical Hopf bifurcations are quite usual in dynamical systems with time delay, and the existence of an unstable periodic solution makes the dynamics even more intricate, which has been discussed in population dynamics in the literature [22].
It should be also pointed out that we know that onset of the Hopf bifurcation can be postponed by properly increasing values of the parameters , , and  based on numerical simulations.In reality, cure rate of the exposed node  and cure rate of the infectious nodes can be properly enhanced by updating of antivirus software on nodes; disconnecting rate of every node  can be properly increased by the managers of a real network.Therefore, propagation of the computer viruses in system (2) can be controlled by timely updating of antivirus software on nodes and timely disconnecting nodes from a real network when the connections are unnecessary.

Conclusions
A delayed computer virus model is proposed based on the literature [14] considering that the outbreak of computer virus usually lags.By theoretical analysis, the critical value of the delay  0 where the Hopf bifurcation occurs is obtained and our results show that the time delay plays an important role in the stability of the considered computer virus model in the present paper.
When the value of the delay  <  0 , the endemic equilibrium of the model is asymptotically stable and the propagation of the computer virus is divinable.However, when the value of the delay  >  0 , the endemic equilibrium of the model will lose its stability and a Hopf bifurcation occurs.In this case, propagation of the computer viruses will be out of control.Therefore, we should postpone occurrence of the Hopf bifurcation by taking some effective measures, such as timely updating of antivirus software on nodes and timely disconnecting nodes from the network when the connections are unnecessary.In addition, properties of the Hopf bifurcation are also investigated by using the normal form theory and center manifold theorem.
Of course, various factors in a real network can affect propagation of the computer viruses.Among them, time delay is important and the present paper thus concentrates on analyzing it.Other network impacts of computer viruses such as network topology, network eigenvalue, and patch forwarding cannot be neglected, and they will be a major emphasis of our future research.Also, stability of the endemic equilibrium and existence of the periodic solutions are investigated only at the critical value  0 .As stated in the literature [22], it is an open question yet whether there exists stable endemic equilibrium for even larger time delay.This is the other interesting future research direction for us.