Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network

An SEIQRS model for the transmission of malicious objects in computer network with two delays is investigated in this paper. We show that possible combination of the two delays can affect the stability of the model and make the model bifurcate periodic solutions under some certain conditions. For further investigation, properties of the periodic solutions are studied by using the normal form method and center manifold theory. Finally, some numerical simulations are given to justify the theoretical results.


Introduction
Computer viruses in network have posed a major threat to our work and life with the rapid popularization of the Internet. Many virus propagation models [1][2][3][4] have been proposed to understand the way that computer viruses propagate after Kephart and White [5] proposed the first epidemiological model of computer viruses. In [1], Thommes and Coates proposed a modified version of the SEI model to predict the virus propagation in a network. In [3], Wen and Zhong studied an SIR model on bipartite networks and they proved the existence and the asymptotic stability of the endemic equilibrium by applying the theory of the multigroup model. In [4], Mishra and Jha proposed the following SEIQRS model to describe the transmission of malicious objects in computer network by introducing a new compartment quarantine into the SEIRS model proposed in [2]: where ( ), ( ), ( ), ( ), and ( ) denote the sizes of nodes in the states susceptible, exposed, infectious, quarantined, and recovered at time , respectively. is the rate at which new computers are attached to the network. is the rate at which computers are disconnected to the network. is the crashing rate of computers due to the attack of malicious objects. is the transmission rate. , , , , and are the state transition rates. As is known, an infected computer becomes a recovered one by using antimalicious software and the recovered computer has a temporary immunity, and computer virus models with delay have been studied by many scholars [6][7][8][9][10][11][12]. In [6], Ren et al. investigated local and global stability of a delayed viral infection model in computer virus propagation model. In [8], Dong et al. proposed a delayed SEIR computer virus model and studied the problem of Hopf bifurcation of the model by regarding the delay as a bifurcating parameter. Motivated by the work above, Liu [12] incorporated the time 2 The Scientific World Journal delay due to the temporary immunity period into system (1) and proposed the following SEIQRS model with time delay: where ≥ 0 is the time delay due to the temporary immunity period. However, we know that an infected computer needs a period to clean viruses by antivirus software and then becomes a recovered one. Therefore, there is a time delay before the infected computers develop themselves into the recovered ones. And there have been some papers that deal with the research of Hopf bifurcation of dynamical system with multiple delays [13][14][15][16][17][18]. In [13], Xu and He considered a two-neuron network with resonant bilinear terms and two delays. They studied the problem of Hopf bifurcation by regarding the sum of the two delays as a bifurcation parameter. In [16], Meng et al. studied the Hopf bifurcation of a three-species system with two delays by regarding possible combination of the two delays as a bifurcation parameter. Motivated by the work above, we consider the following SEIQRS computer virus model with two delays in the present paper: where 1 is the time delay due to the temporary immunity period and 2 is the time delay due to the period that the infected computer uses to clean viruses by antivirus software.
The main purpose of this paper is to investigate the effects of the two delays on system (3) and the remainder of this paper is organized as follows. Sufficient conditions for local stability and existence of local Hopf bifurcation are obtained by analyzing the distribution of the roots of the associated characteristic equation in Section 2. Properties of the Hopf bifurcation are further investigated by using the normal form method and center manifold theory in Section 3. In Section 4, we give a numerical example to support the theoretical results in the paper.
( 21 ) (16) has at least one positive real root. If the condition ( 21 ) holds, then there exists a V 10 such that (11) has a pair of purely imaginary roots ± 10 = ± √ V 10 .
( 41 ) (38) has at least one positive real root. If the conditions ( 41 ) hold, then there exists a * 10 such that (7) has a pair of purely imaginary roots ± * 10 . For * 10 , the corresponding critical value of time delay is * Differentiating two sides of (7) with respect to 1 , we have Thus, Re   [19], we have the following results for system (3).
Then, ⟨ * , ⟩ = 1, ⟨ * , ⟩ = 0. Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in [19] and using a computation process similar to that in [20]: where 1 and 2 can be determined by the following equations, respectively: Then, we can get the following coefficients: In conclusion, we have the following results.

Conclusions
This paper is concerned with a delayed SEIQRS model for the transmission of malicious objects in computer network. Compared with the literature [12], we consider not only the time delay due to the temporary immunity period but also the time delay due to the period that the infected computer uses to clean viruses by antivirus software. That is, the system we considered in this paper is more general than that in the literature [12]. By considering the possible combination of the two delays as a bifurcation parameter, we find that when the delay is below the corresponding critical value, the positive equilibrium of system (3) is locally asymptotically stable. However, when the delay passes through the corresponding critical value, the positive equilibrium of system (3) loses its stability and system (3) undergoes a Hopf bifurcation, which is not welcomed in networks. Furthermore, direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are determined by using the normal form method and center manifold theory. Numerical simulations are presented to illustrate the theoretical analysis and results.
Since the occurrence of the Hopf bifurcation is not welcomed in networks, we should control the Hopf bifurcation by some bifurcation control strategies such as the state feedback and parameter perturbation and so on. This is a further problem, which can be studied in the future.