#### Abstract

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.

#### 1. 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 [1–4] 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 [6–12]. 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 delay due to the temporary immunity period into system (1) and proposed the following SEIQRS model with time delay: where 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 [13–18]. 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 is the time delay due to the temporary immunity period and 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.

#### 2. Local Stability and Existence of Local Hopf Bifurcation

By a simple computation, it is easy to get that if , then system (3) has a unique positive equilibrium , where and is the basic reproduction number. It is easy to get the linearization of system (3) at : where Thus, the characteristic equation of system (5) is where

*Case 1 (). *When , (7) becomes

Let . Obviously, . Therefore, if the condition : (10) holds, then the positive equilibrium of system (3) is locally asymptotically stable without delay. Consider

*Case 2 (, ). *When , (7) becomes the following form:
where
Let () be a root of (11). Then, we obtain
It follows that
with
Let , then (14) becomes

If all the parameters of system (3) are given, one can get all the roots of (16) by the software package Matlab. In order to give the main results in this paper, we make the following assumption.

(16) has at least one positive real root.

If the condition holds, then there exists a such that (11) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is
where
Taking the derivative of with respect to in (11), one can obtain
Thus,
where and .

Obviously, if the condition holds, then . According to the Hopf bifurcation theorem in [19], we have the following results for system (3).

Theorem 1. *For system (3), if the conditions - hold, then the positive equilibrium of system (3) is asymptotically stable for and system (3) undergoes a Hopf bifurcation at the positive equilibrium when .*

*Case 3 (, ). *When , (7) becomes
where
Multiplying on both sides of (21), we have
Let () be the root of (23), then we obtain
where
Then, we obtain
where
Then, we obtain
where
Let , then (28) becomes

Similar as in Case 2, we make the following assumption. (30) has at least one positive real root. If the condition holds, then there exists a such that (23) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is
Differentiating two sides of (23) with respect to , we have
where
Thus,
where

Obviously, if the condition holds, then . According to the Hopf bifurcation theorem in [19], we have the following results for system (3).

Theorem 2. *For system (3), if the conditions - hold, then the positive equilibrium of system (3) is asymptotically stable for and system (3) undergoes a Hopf bifurcation at the positive equilibrium when .*

*Case 4 (, , ). *We consider system (3) under the condition that is in its stable interval and is a bifurcation parameter.

Let () be the root of (7), then we obtain
where
Then, we can obtain
where
In order to give the main results in this paper, we make the following assumption.

(38) has at least one positive real root. If the conditions hold, then there exists a such that (7) has a pair of purely imaginary roots . For , the corresponding critical value of time delay is
Differentiating two sides of (7) with respect to , we have
where
Thus,
where