Stability and Hopf Bifurcation of a Delayed Epidemic Model of Computer Virus with Impact of Antivirus Software

In this paper, we investigate an SLBRS computer virusmodel with time delay and impact of antivirus software.The proposedmodel considers the entering rates of all computers since every computer can enter or leave the Internet easily. It has been observed that there is a stability switch and the system becomes unstable due to the effect of the time delay. Conditions under which the system remains locally stable and Hopf bifurcation occurs are found. Sufficient conditions for global stability of endemic equilibrium are derived by constructing a Lyapunov function. Formulae for the direction, stability, and period of the bifurcating periodic solutions are conducted with the aid of the normal form theory and center manifold theorem. Numerical simulations are carried out to analyze the effect of some of the parameters in the system on the dynamic behavior of the system.


Introduction
Computer viruses are programs created to carry out activities in a computer without consent of its owner.They not only disrupt the normal functionalities of computer system and damage data files in the computer, but also cause heavy economic losses and tremendous social impacts [1][2][3].In recent years, mathematical modeling enjoys popularity in both analyzing and controlling computer viruses based on the similarity between computer viruses and biological viruses.A few works proposing SIR models have appeared in the literatures [4][5][6].In [4], Amador studied a stochastic SIRA epidemic model for computer viruses and analyzed the quasistationary distribution, the extinction time, and the number of infections in order to understand the spreading mechanism of computer viruses.In [5], Ozturk and Gulsu proposed an approximate solution to a modified SIR computer virus model by using the shifted Chebyshev collocation method.In [6], Khanh studied the stability and approximate iterative solutions of an SIR computer virus model with antidotal component.
Considering the latent period of computer viruses, some models with latency are proposed by some scholars [1,[7][8][9][10].In [7], Yang investigated global stability of a VEISV network worm attack model by using the Li-Muldowney geometric approach.In [8], Keshri et al. proposed a reduced SEIR scalefree network model and studied its stability.In [9], Hosseini et al. formulated a discrete-time SEIRS model of computer virus propagation in scale-free networks and analyzed the local and global stability of the model.In [1], Guillen et al. proposed an improved SEIRS worm model with considering accurate positions for dysfunctional hosts and their replacements in state transition.In [10], Ren and Xu investigated an SEIR-KS computer virus propagation model based on the kill signals.They studied the local and global stability of the model by applying Routh-Hurwitz criterion and Lyapunov functional method.There are also some other models considering the latency of computer viruses with quarantine [11][12][13][14] and vaccination [15][16][17][18][19].
However, as stated in [20], those above models with the exposed compartment neglect the fact that a computer can infect other computers through file copying or file downloading.Therefore, to overcome the above-mentioned defect, computer virus models with infectivity in latent period have received much attention in recent years [21][22][23][24][25][26].Unfortunately, most of these models still have some defects.On the one hand, they ignore the effect of time delay.As is known, there are some time delays of one type or another Rate of latent and breaking computers reinstall the operating system in the transmission process of computer viruses due to latent period, temporary immunity period, or other reasons.On the other hand, only the susceptible computers are regarded as the entering computers, but every computer can enter or leave the Internet easily in reality.Finally, they neglect the effect of antivirus software, especially the effect of antivirus software on the susceptible computers.Based on the discussion above, we investigated a delayed SLBRS computer virus model with impact of antivirus software based on the following model proposed in [27]: where (), (), (), and () denote the numbers of susceptible, latent, breaking, and recovered computers at time , respectively.More parameters are listed in Table 1 as follows.
Considering the temporary immune period of the recovered computers, we incorporate the time delay due to the temporary immunity period into system (1) and obtain the following delayed model: where  is the time delay due to the temporary immunity period.
The remainder of the paper is structured as follows.In Section 2, conditions for local stability of the endemic equilibrium and the existence of Hopf bifurcation are performed.Section 3 deals with global stability of the endemic equilibrium.Section 4 is devoted to establishing the formulae to determine the direction, stability, and period of the bifurcating periodic solutions.Some numerical simulations are presented to illustrate the theoretical results in Section 5. We end the paper with a brief conclusion in Section 6.
(i) Effect of the recovered rate ( 1 ): in Figures 7(a)-7(d), we can see that the numbers of susceptible and recovered computers increase; nevertheless, the numbers of latent and breaking computers decrease, when the number of  1 increases.And the system changes its behavior from limit cycle to stable focus as we increase the value of  1 , from 0.1 to 0.3, which can be shown as in Figure 8.
(ii) Effect of the rate of latent and breaking computers reinstall the operating system ( 2 ): in the same manner,     we can see from Figures 9(a)-9(d) that the numbers of susceptible and latent computers increase and the number of breaking computers decreases, when the number of  2 increases.But it does not affect the number of recovered computers, which can be also seen from the expression of  * in Section 2. Also, we observe that  2 does not affect the dynamics of the system; it remains at limit cycle when we choose  = 3.6755.This property can be illustrated by Figure 10.
(iii) Effect of the entering rates ( 1 ,  2 ,  3 ,  4 ): as is shown in Figures 11-14, the numbers of all computers increase when the numbers of  1 and  4 increase.However, the      number of susceptible computers decreases and the numbers of latent, breaking, and recovered computers increase, when the numbers of  2 and  3 increase.In addition, we find that the entering rates does not affect the dynamics of the systems.

Conclusions
In this paper, a delayed SLBRS computer virus model is presented by incorporating the time delay due to the temporary immunity period of the recovered computers based on the model proposed in [27].Compared with the model in [27], we mainly consider the effect of the time delay on its dynamic behavior.Compared with other computer virus models, we assume that every computer can enter the Internet, which is consistent with the reality.Further, we also consider the effect of antivirus software on the susceptible computers in the presented model.Thus, the computer virus model proposed in our paper is more general.
It has been shown that the endemic equilibrium  * ( * ,  * ,  * ,  * ) is locally asymptotically stable when  ∈ [0, 0 ) under some certain conditions.In this case, the propagation of the computer virus in system (2) can be controlled easily.Once the value of the time delay passes through  0 ,  * ( * ,  * ,  * ,  * ) loses its stability and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from  * ( * ,  * ,  * ,  * ).In this case, the numbers of the four classes computers in system (2) will oscillate in the vicinity of  * ( * ,  * ,  * ,  * ).Namely, the propagation of the computer virus will be out of control.Therefore, the results obtained in the present paper can help us to gain insight into the spreading process of computer viruses.Also, sufficient conditions for global stability of the endemic equilibrium are derived by constructing a suitable Lyapunov function.Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and center manifold theorem.Numerical simulations are presented to verify the analytical predictions.In addition, it has been observed in our simulations that the recovered rate  1 can change the dynamics of the system from limit cycle to stable focus as its value increases.Thus, it is strongly recommended that users of computers connected to Internet should periodically run antivirus software of the newest version.From the point of this view, we can conclude that the results of the proposed model in our paper can be used to evaluate the effectiveness of antivirus software.In addition, the numbers of latent and breaking computers decrease, when the reinstalling of the operating system rate increases.Thus, it can be concluded that users should reinstall operating system if necessary.Finally, the numbers of latent and breaking computer will also increase, when the values of entering rates of all computers  1 ,  2 ,  3 , and  4 increase.Therefore, the manager of a network should control the number of computers connected to the network properly.
Of course, when we pursue a low level of infections, we should also consider the cost of the measures we carry out.In addition, it should be pointed out that the model investigated in the literature [27] and our present paper assumes that the latent computers and the breaking computers have the same infection rate .In the near future, we will investigate the optimal control problem of the following general system (63) so as to achieve a low level of infections at a low cost by using the method introduced in [32 where  1 and  2 are the infection rate of the latent computers and the breaking computers, respectively.

Figure 7 :
Figure 7: Time plots of , , , and  for different  1 at  = 3.2575.Rest of the parameters are taken as given in the text.

Figure 8 :Figure 9 :
Figure 8: Dynamic behavior of system (61): projection on L-B-R with  = 3.6755 >  0 = 3.4685 for different  1 .Rest of the parameters are taken as given in the text.

Figure 10 :
Figure 10: Dynamic behavior of system (61): projection on L-B-R with  = 3.6755 >  0 = 3.4685 for different  2 .Rest of the parameters are taken as given in the text.

Figure 11 :
Figure 11: Time plots of , , , and  for different  1 at  = 3.2575.Rest of the parameters are taken as given in the text.

Figure 12 :
Figure 12: Time plots of , , , and  for different  2 at  = 3.2575.Rest of the parameters are taken as given in the text.

Figure 13 :
Figure 13: Time plots of , , , and  for different  3 at  = 3.2575.Rest of the parameters are taken as given in the text.

Figure 14 :
Figure 14: Time plots of , , , and  for different  3 at  = 3.2575.Rest of the parameters are taken as given in the text.

Table 1 :
Parameters and their meanings in this paper.