Delay Induced Hopf Bifurcation of an Epidemic Model with Graded Infection Rates for Internet Worms

A delayed SEIQRS worm propagation model with different infection rates for the exposed computers and the infectious computers is investigated in this paper. The results are given in terms of the local stability and Hopf bifurcation. Sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained by using eigenvalue method and choosing the delay as the bifurcation parameter. In particular, the direction and the stability of the Hopf bifurcation are investigated by means of the normal form theory and center manifold theorem. Finally, a numerical example is also presented to support the obtained theoretical results.


Introduction
In the wake of developments in computer technology and communication technology, there is a rapid increase in computer viruses which has brought about huge financial losses [1][2][3].Therefore, it is extremely urgent to analyze and protect computers against viruses.In order to understand the spread law of computer viruses over the Internet and in view of the high similarity between computer viruses and biological viruses, many computer virus propagation models have been developed and analyzed.For example, see [4][5][6][7][8][9][10][11][12][13] and the cited references therein.
All the models above assume that the infected computer has no infectivity.This is inconsistent with the fact that an infected computer which is in latency can also infect other computers through file copying or file downloading.Based on this, Yang et al. proposed some models [14][15][16][17], by taking into account the fact that a computer immediately possesses infectivity once it is infected.However, these models make an assumption that the exposed computers and the infectious computers have the same infectivity.This is not consistent with the reality, because the infection rate of the exposed computers is less than that of the infectious ones.Thus, Wang et al. [18] proposed the following SEIQRS model with graded infection rates for Internet worms: where (), (), (), (), and () denote the numbers of the susceptible, the exposed, the infectious, the quarantined, and the recovered computers at time  in the Internet, respectively.∏ is the recruitment of the susceptible computers;  1 and  2 are the contact rates of the exposed computers and the infectious computers, respectively;  is the natural death rate of the computers;  1 and  2 are the death rates of the exposed computers and the infectious computers due to the attack of worms, respectively;  is the quarantined rate of the infectious computers;  −1 1 is the average cured time; , ,  2 , , and  are the state transition rates.Wang et al. [18] investigated local and global stability of system (1).
It should be pointed out that Wang et al. [18] neglect the fact that it needs a period to clean the worms in the exposed, the infectious, and the quarantined computers for the antivirus software.Time delays cause a stable equilibrium to become unstable and cause Hopf bifurcation phenomenon for a dynamical system.The occurrence of Hopf bifurcation means that the state of worm prevalence changes from an equilibrium point to a limit cycle, which is not welcomed in networks.Therefore, it is meaningful to investigate the effect of time delays on stability of dynamical systems.
As far as we know, there have been some researches on Hopf bifurcation of delayed computer virus models.For example, Feng et al. [19] studied the Hopf bifurcation of a delayed SIRS viral infection model in computer networks by taking the time delay due to the latent and temporary immune period as the bifurcation parameter.Dong et al. [9] proposed a delayed SEIR computer virus model with multistate antivirus and studied the Hopf bifurcation of the model by choosing the delay where the infectious nodes use antivirus software to clean the viruses as the bifurcation parameter.Zhang and Bi [20] investigated the Hopf bifurcation of a delayed computer virus propagation model with infectivity in latent period.For some other research works on the Hopf bifurcation of computer virus models one can refer to [21][22][23][24].Specially, Zhang et al. [24] studied the existence and properties of the Hopf bifurcation of a computer virus model with antidote in vulnerable system by regarding the time delay due to the period that the infected computers use to reinstall system as a bifurcation parameter.Motivated by the work above and considering the effect of the time delay on system (1), we consider the following delayed worm propagation model: where  is the time delay due to the period that the antivirus software uses to clean the worms in the exposed, the infectious, and the quarantined computers.The object of this paper is to study the existence and properties of the Hopf bifurcation of system (2).The remainder of this paper is organized as follows.The existence of a Hopf bifurcation is discussed by choosing the delay as the bifurcation parameter in Section 2. Properties of the Hopf bifurcation are studied by means of the normal form theory and the center manifold theorem.An example together with its numerical simulations is also presented in order to illustrate the effectiveness of our obtained theoretical results.
Multiplying   on both sides of (4), one can get For  > 0, we assume that  =  ( > 0) is the root of (13); then where Then, with Since sin  = ± √ 1 − cos 2 , (16) can be transformed into It equals where Based on the discussion about the distribution of the root of ( 19) in [25,26], we obtain the expression of cos , say cos  =  1 () .
For  > 0, we obtain  0 = 1.2030,  0 = 7.5508, and   ( 0 ) = 0.9208 − 0.2975 by some complicated computations using Matlab software package.Based on Theorem 1, we know that the endemic equilibrium  * is asymptotically stable when  ∈ [0, 7.5508).In this case, propagation of the worms can be controlled easily and this is exhibited by numerical simulation shown in Figure 1.However, once the magnitude of the time delay  passes through the critical value  0 = 7.5508,  * will lose its stability and a Hopf bifurcation occurs.In this case, the worm propagation will become unstable and propagation of the worms will be out of control, which can be illustrated in Figure 2. The bifurcation phenomenon can be also exhibited by the bifurcation diagram shown in Figure 3.It follows from (46) that  1 (0) = −0.9606− 0.2731,  2 = 1.0432 > 0,  2 = −1.9212< 0, and  2 = 0.0642.Thus, according to Theorem 2, the bifurcation at  0 = 7.5508 is supercritical and stable, and the period of the Hopf bifurcation increases.

Conclusions and Further Developments
A delayed SEIQRS model with graded infection rates for Internet worms is proposed in this paper based on the model in the literature [18].Compared with the models investigated in the previous literatures, the exposed and the infectious computers in the model considered in our paper have different infection rates.Also, we incorporate the time delay due to the period that the antivirus software uses to clean the worms in the exposed, the infectious, and the quarantined computers.Therefore, the model considered in our paper is more general.
We mainly investigate the effect of the time delay  on the stability of the model.Based on the numerical simulations presented in our paper, we found that when the value of the delay  is below the critical value  0 , the model is locally asymptotically stable, which means that the number of the five classes computers in system (2) will be in an ideal steady state and we can control prevalence of the worms in (2) easily.However, once the value of the delay  is above  0 , a Hopf bifurcation occurs.This phenomenon suggests that the numbers of computers of the five classes in system (2) will fluctuate periodically in a range.This is not helpful to control the prevalence of the worms.Hence, we should control the phenomenon by combining some bifurcation control strategies and other relative features of virus prevalence, for example, topological structures of networks, which is our future work.
Thus, our further research directions include the possibility of linking the results obtained with the model proposed in the present paper with the results coming from the networks theory.Specifically, the interest focuses on the possibility of gaining a deep understanding of the impact of the network topology on the viral prevalence.