Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

Abstract

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

1. Introduction

Since the pioneering work of Kephart and White [1, 2], classical epidemiological models have been used to understand and predict virus propagations in computer network by many authors [3–10]. In [3], Thommes and Coates proposed a modified version of the SEI model to predict the virus propagation in a network. In [5], Yuan and Chen proposed an e-SEIR model and studied the behavior of virus propagation with the presence of antivirus countermeasures. Yuan and Chen [5] supposed that the recovered computers have a permanent immunization period and can no longer be infected. Considering that there is no permanent recovery from the virus till a node is attached to the computer network, Mishra and Pandey [6] proposed the following epidemic model for the transmission of worms with vertical transmission: 𝑑𝑆(𝑡)(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)(𝑡),(𝑡)𝑑𝑡=𝑏−𝛽𝐼−𝑝𝑏𝐸−𝑞𝑏𝐼−𝑑𝑆+𝜁𝑅𝑑𝐸(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)(𝑡),(𝑡)𝑑𝑡=𝛽𝐼+𝑝𝑏𝐸+𝑞𝑏𝐼−𝜀𝐸−𝑑𝐸𝑑𝐼(𝑡)(𝑡)(𝑡)(𝑡),(𝑡)𝑑𝑡=𝜀𝐸−𝛾𝐼−𝑑𝐼−𝜂𝐼𝑑𝑅(𝑡)(𝑡)(𝑡),𝑑𝑡=𝛾𝐼−𝑑𝑅−𝜁𝑅 (1) where 𝑆(𝑡), 𝐸(𝑡), 𝐼(𝑡), and 𝑅(𝑡) denote the numbers of nodes at time 𝑡 in susceptible, exposed, infectious and recovered states, respectively. 𝑏 is the recruitment rate of susceptible nodes to the computer network. 𝑑 is the crashing rate of nodes due to the reason other than the attack of worms. 𝑝 and 𝑞 are the fractions of the new nodes from the exposed and the infectious classes, respectively, that are introduced into the exposed class. 𝛽 is the transmission rate. 𝜀, 𝛾, and 𝜁 are the state transition rates. 𝜂 is the crashing rate of nodes due to the attack of worms. Mishra and Pandey [6] discussed the characteristic of the basic reproduction number and investigated the global stability of system (1) by constructing a new Liyapunov function.

It is well known that time delays can play a complicated role on the dynamics of a system. They can cause the stable equilibrium of a system to become unstable and make a system bifurcate periodic solutions. Dynamical systems with delay have been studied by many scholars [11–20]. Motivated by the works above and considering that the antivirus software may use a period to clean the worms in one node and that the recovered nodes may have a temporary immunity period due to the antivirus software, we consider the following delayed system in this paper: 𝑑𝑆(𝑡)(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)𝑑𝑡=𝑏−𝛽𝐼−𝑝𝑏𝐸−𝑞𝑏𝐼−𝑑𝑆+𝜁𝑅𝑡−𝜏₂,𝑑𝐸(𝑡)(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)(𝑡),(𝑡)𝑑𝑡=𝛽𝐼+𝑝𝑏𝐸+𝑞𝑏𝐼−𝜀𝐸−𝑑𝐸𝑑𝐼(𝑡)𝑑𝑡=𝜀𝐸−𝛾𝐼𝑡−𝜏₁(𝑡)(𝑡),(𝑡)−𝑑𝐼−𝜂𝐼𝑑𝑅𝑑𝑡=𝛾𝐼𝑡−𝜏₁−𝑑𝑅(𝑡)−𝜁𝑅𝑡−𝜏₂, (2) where 𝜏₁ is the period to clean the worms in one node and 𝜏₂ is the temporary immunity period. For the convenience of analysis, throughout this paper, we assume that the period to clean the worms in one node and the temporary immunity period are the same.

Let 𝜏₁=𝜏₂=𝜏, and then system (2) becomes the following form: 𝑑𝑆(𝑡)(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)(),(𝑡)𝑑𝑡=𝑏−𝛽𝐼−𝑝𝑏𝐸−𝑞𝑏𝐼−𝑑𝑆+𝜁𝑅𝑡−𝜏𝑑𝐸(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)(𝑡),(𝑡)𝑑𝑡=𝛽𝐼+𝑝𝑏𝐸+𝑞𝑏𝐼−𝜀𝐸−𝑑𝐸𝑑𝐼(𝑡)()(𝑡)(𝑡),(𝑡)𝑑𝑡=𝜀𝐸−𝛾𝐼𝑡−𝜏−𝑑𝐼−𝜂𝐼𝑑𝑅()(𝑡)().𝑑𝑡=𝛾𝐼𝑡−𝜏−𝑑𝑅−𝜁𝑅𝑡−𝜏 (3)

The main purpose of this paper is to investigate the effects of the time delay on the dynamics of system (3). We study the stability of the positive equilibrium of system (3) and find the critical value of the time delay where a Hopf bifurcation occurs. We also study the properties of the Hopf bifurcation such as direction and stability.

This paper is organized as follows. In Section 2, we investigate local stability of the positive equilibrium and obtain the sufficient conditions for the existence of local Hopf bifurcation. In Section 3, we determine the direction and the stability of the Hopf bifurcation by using the normal form theory and center manifold theorem. In order to testify the theoretical analysis, a numerical example is presented in Section 4.

2. Stability and Existence of Local Hopf Bifurcation

It is not difficult to verify that if 𝑅₀=𝑏𝛽𝜀+𝑏𝑑𝑞𝜀+𝑏𝑑𝑝𝑑+𝛾+𝜂𝑑()()𝑑+𝜀𝑑+𝛾+𝜂>1,𝑑+𝜀𝑑+𝛾+𝜂>𝑝𝑏𝑑+𝛾+𝜂+𝑞𝑏𝜀, (4) system (2) has a unique positive equilibrium 𝐷_∗(𝑆_∗,𝐸_∗,𝐼_∗,𝑅_∗), where 𝑆_∗=(𝑑+𝜀)𝑑+𝛾+𝜂−𝑝𝑏𝑑+𝛾+𝜂−𝑞𝑏𝜀,𝐼𝛽𝜀_∗=(()×𝛽()𝑑+𝜁)𝑏𝛽𝜀+𝑑𝑝𝑏𝑑+𝛾+𝜂+𝑞𝑏𝜀−𝑑+𝜀𝑑+𝛾+𝜂𝑑+𝜀)(𝑑+𝜁𝑑+𝛾+𝜂−𝛽𝛾𝜀𝜁⁻¹,𝐸_∗=𝑑+𝛾+𝜂𝜀𝐼_∗,𝑅_∗=𝛾𝐼𝑑+𝜁^∗. (5)

𝑅₀ is called the basic reproduction number.

The Jacobian matrix of system (3) about the positive equilibrium 𝐷_∗ is𝐽𝐷_∗=𝜆−𝑎₁₁−𝑎₁₂−𝑎₁₂−𝑏₁₄𝑒^−𝜆𝜏−𝑎₂₁𝜆−𝑎₂₂−𝑎₂₃00−𝑎₃₂𝜆−𝑎₃₃−𝑏₃₃𝑒^−𝜆𝜏000−𝑏₄₃𝑒^−𝜆𝜏𝜆−𝑎₄₄−𝑏₄₃𝑒^−𝜆𝜏, (6)where 𝑎₁₁=−𝑑+𝛽𝐼_∗,𝑎₁₂𝑎=−𝑝𝑏,₁₃=−𝑞𝑏+𝛽𝑆_∗,𝑎₂₁=𝛽𝐼_∗,𝑎₂₂=𝑝𝑏−𝑑−𝜀,𝑎₂₃=𝑞𝑏+𝛽𝑆_∗,𝑎₃₂=𝜀,𝑎₃₃,𝑎=−𝑑+𝜂₄₄=−𝑑,𝑏₁₄=𝜁,𝑏₃₃𝑏=−𝛾,₄₃=𝛾,𝑏₄₄=−𝜁. (7) Thus, the characteristic equation of system (2) at the positive equilibrium 𝐷_∗ is 𝜆⁴+𝐴₃𝜆³+𝐴₂𝜆²+𝐴₁𝜆+𝐴₀+𝐵₃𝜆³+𝐵₂𝜆²+𝐵₁𝜆+𝐵₀𝑒^−𝜆𝜏+𝐶₂𝜆²+𝐶₁𝜆+𝐶₀𝑒^−2𝜆𝜏=0, (8) where 𝐴₀=𝑎₁₁𝑎₄₄𝑎₂₂𝑎₃₃−𝑎₂₃𝑎₃₂+𝑎₂₁𝑎₄₄𝑎₁₃𝑎₃₂−𝑎₁₂𝑎₃₃,𝐴₁=𝑎₃₃+𝑎₄₄𝑎₁₂𝑎₂₁−𝑎₁₁𝑎₂₂+𝑎₂₃𝑎₃₂𝑎₁₁+𝑎₄₄−𝑎₃₃𝑎₄₄𝑎₁₁+𝑎₂₂+𝑎₁₃𝑎₂₁𝑎₃₂,𝐴₂=𝑎₁₁+𝑎₂₂𝑎₃₃+𝑎₄₄+𝑎₁₁𝑎₂₂+𝑎₃₃𝑎₄₄−𝑎₁₂𝑎₂₁−𝑎₂₃𝑎₃₂,𝐴₃=−𝑎₁₁+𝑎₂₂+𝑎₃₃+𝑎₄₄,𝐵₀=𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁𝑎₃₃𝑏₄₄+𝑎₄₄𝑏₃₃+𝑎₁₃𝑎₂₁−𝑎₁₁𝑎₂₃𝑎₃₂𝑏₄₄,𝐵₁=𝑏₃₃+𝑏₄₄𝑎₁₂𝑎₂₁−𝑎₁₁𝑎₂₂+𝑎₂₃𝑎₃₂𝑏₄₄−𝑎₁₁+𝑎₂₂𝑎₃₃𝑏₄₄+𝑎₄₄𝑏₃₃,𝐵₂=𝑎₁₁+𝑎₂₂𝑏₃₃+𝑏₄₄+𝑎₃₃𝑏₄₄+𝑎₄₄𝑏₃₃,𝐵₃=−𝑏₃₃+𝑏₄₄,𝐶₂=𝑏₃₃𝑏₄₄,𝐶₁=𝑏₃₃𝑏₄₄𝑎₁₁+𝑎₂₂,𝐶₀=𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁𝑏₃₃𝑏₄₄−𝑎₂₁𝑎₃₂𝑏₁₄𝑏₄₃. (9)

Multiplying 𝑒^𝜆𝜏 on both sides of (8), it is easy to get 𝐵₃𝜆³+𝐵₂𝜆²+𝐵₁𝜆+𝐵₀+𝜆⁴+𝐴₃𝜆³+𝐴₂𝜆²+𝐴₁𝜆+𝐴₀𝑒^𝜆𝜏+𝐶₂𝜆²+𝐶₁𝜆+𝐶₀𝑒^−𝜆𝜏=0. (10)

When 𝜏=0, (10) reduces to 𝜆⁴+𝐴₁₃𝜆³+𝐴₁₂𝜆²+𝐴₁₁𝜆+𝐴₁₀=0, (11) where 𝐴₁₀=𝐴₀+𝐵₀+𝐶₀,𝐴₁₁=𝐴₁+𝐵₁+𝐶₁,𝐴₁₂=𝐴₂+𝐵₂+𝐶₂,𝐴₁₃=𝐴₃+𝐵₃. (12)

By the Routh-Hurwitz criterion, the sufficient conditions for all roots of (11) to have a negative real part given in the following form: 𝐷₁=𝐴₁₃𝐷>0,₂𝐴=det¹³1𝐴₁₁𝐴₁₂𝐷>0,₃𝐴=det𝐴¹³10₁₁𝐴₁₂𝐴₁₃0𝐴₁₀𝐴₁₁𝐷>0,₄𝐴=det¹³𝐴100₁₁𝐴₁₂𝐴₁₃10𝐴₁₀𝐴₁₁𝐴₁₂000𝐴₁₀>0. (13)(14)

Thus, if the condition (𝐻₁) holds, which means that (13) and (14) are satisfied, the positive equilibrium 𝐷_∗ is locally asymptotically stable in the absence of delay.

For 𝜏>0, let 𝜆=𝑖𝜔(𝜔>0) be a root of (10). Then, we can get 𝜔⁴−𝐴₂+𝐶₂𝜔²+𝐴₀+𝐶₀−𝐴cos𝜏𝜔₁−𝐶₁𝜔−𝐴₃𝜔³sin𝜏𝜔=𝐵₂𝜔²−𝐵₀,𝜔⁴−𝐴₂−𝐶₂𝜔²+𝐴₀−𝐶₀+𝐴sin𝜏𝜔₁+𝐶₁𝜔−𝐴₃𝜔³cos𝜏𝜔=𝐵₃𝜔³−𝐵₁𝜔. (15) Then, we can get 𝑝sin𝜏𝜔=⁷𝜔⁷+𝑝₅𝜔⁵+𝑝₃𝜔³+𝑝₁𝜔𝜔⁸+𝑞₆𝜔⁶+𝑞₄𝜔⁴+𝑞₂𝜔²+𝑞₀,𝑝cos𝜏𝜔=⁶𝜔⁶+𝑝₄𝜔⁴+𝑝₂𝜔²+𝑝₀𝜔⁸+𝑞₆𝜔⁶+𝑞₄𝜔⁴+𝑞₂𝜔²+𝑞₀, (16) where 𝑝₀=𝐵₀𝐶₀−𝐴₀,𝑝₁=𝐴₁+𝐶₁𝐵₀−𝐴₀+𝐶₀𝐵₁,𝑝₂=𝐴₂−𝐶₂𝐵₀+𝐴₀−𝐶₀𝐵₂+𝐶₁−𝐴₁𝐵₁,𝑝₃=𝐴₀+𝐶₀𝐵₃−𝐴₁+𝐶₁𝐵₂+𝐴₂+𝐶₂𝐵₁−𝐴₃𝐵₀,𝑝₄=𝐴₁−𝐶₁𝐵₃−𝐴₂−𝐶₂𝐵₂+𝐴₃𝐵₁−𝐵₀,𝑝₅=𝐴₃𝐵₂−𝐵₁−𝐴₂+𝐶₂𝐵₃,𝑝₆=𝐵₂−𝐴₃𝐵₃,𝑝₇=𝐵₃,𝑞₀=𝐴²₀−𝐶²₀,𝑞₂=𝐴²₁−𝐶²₁−2𝐴₀𝐴₂+2𝐶₀𝐶₂,𝑞₄=𝐴²₂−𝐶²₂+2𝐴₀−2𝐴₁𝐴₃,𝑞₆=𝐴²₃−2𝐴₂. (17)

It is well known that sin²𝜏𝜔+cos²𝜏𝜔=1. Thus, we have 𝜔¹⁶+𝑒₇𝜔¹⁴+𝑒₆𝜔¹²+𝑒₅𝜔¹⁰+𝑒₄𝜔⁸+𝑒₃𝜔⁶+𝑒₂𝜔⁴+𝑒₁𝜔²+𝑒₀=0, (18) where 𝑒₀=𝑞²₀−𝑝²₀,𝑒₁=2𝑞₀𝑞₂−2𝑝₀𝑝₂−𝑝²₁,𝑒₂=𝑞²₂+2𝑞₀𝑞₄−𝑝²₂−2𝑝₀𝑝₄−2𝑝₁𝑝₃,𝑒₃=2𝑞₀𝑞₆+2𝑞₂𝑞₄−𝑝²₃−2𝑝₀𝑝₆−2𝑝₁𝑝₅−2𝑝₂𝑝₄,𝑒₄=𝑞²₄+2𝑞₀+2𝑞₂𝑞₆−𝑝²₄−2𝑝₁𝑝₇−2𝑝₂𝑝₆−2𝑝₃𝑝₅,𝑒₅=2𝑞₂+2𝑞₄𝑞₆−2𝑝₃𝑝₇−2𝑝₄𝑝₆−𝑝²₅,𝑒₆=𝑞²₆−𝑝²₆+2𝑞₄−2𝑝₅𝑝₇,𝑒₇=2𝑞₆−𝑝²₇. (19)

Let 𝜔²=𝑣, and then (18) becomes 𝑣⁸+𝑒₇𝑣⁷+𝑒₆𝑣⁶+𝑒₅𝑣⁵+𝑒₄𝑣⁴+𝑒₃𝑣³+𝑒₂𝑣²+𝑒₁𝑣+𝑒₀=0. (20)

In order to give the main results in this paper, we made the following assumption.

(𝐻₂) Equation (20) has at least one positive real root.

Suppose that the condition (𝐻₂) holds. Without loss of generality, we suppose that (20) has eight positive real roots, which are denoted as 𝑣₁,𝑣₂,…,𝑣₈, respectively. Then, (18) has eight positive roots 𝜔_𝑘=√𝑣_𝑘,𝑘=1,2,…,8. For every fixed 𝜔_𝑘, the corresponding critical value of time delay is 𝜏_𝑘^(𝑗)=1𝜔_𝑘𝑝arccos⁶𝜔⁶+𝑝₄𝜔⁴_𝑘+𝑝₂𝜔²_𝑘+𝑝₀𝜔⁸+𝑞₆𝜔⁶_𝑘+𝑞₄𝜔⁴_𝑘+𝑞₂𝜔²_𝑘+𝑞₀+2𝑗𝜋𝜔_𝑘,𝑘=1,2,…,8,𝑗=0,1,2,…. (21)

Let 𝜏₀=min𝜏_𝑘⁽⁰⁾{},𝜔,𝑘∈1,2,…,8₀=𝜔_{𝑘𝜏=𝜏0}. (22)

Taking the derivative of with respect to (10), it is easy to obtain 𝑑𝜆𝑑𝜏⁻¹=3𝐵₃𝜆²+2𝐵₂𝜆+𝐵₁+2𝐶₂𝜆+𝐶₁𝑒^−𝜆𝜏+4𝜆³+3𝐴₃𝜆²+2𝐴₂𝜆+𝐴₁𝑒^𝜆𝜏×𝐶₂𝜆³+𝐶₁𝜆²+𝐶₀𝜆𝑒^−𝜆𝜏−𝜆⁵+𝐴₃𝜆⁴+𝐴₂𝜆³+𝐴₁𝜆²+𝐴₀𝜆𝑒^𝜆𝜏−1−𝜏𝜆. (23) Then, we have Re𝑑𝜆𝑑𝜏⁻¹_𝜏=𝜏0=𝑃_𝑅𝑄_𝑅+𝑃_𝐼𝑄_𝐼𝑄²_𝑅+𝑄²_𝐼,𝑃_𝑅=𝐴₁+𝐶₁−3𝐴₃𝜔²₀cos𝜏₀𝜔₀+22𝜔³₀−𝐴₂−𝐶₂𝜔₀sin𝜏₀𝜔₀−3𝐵₃𝜔²₀+𝐵₁,𝑃_𝐼=𝐴₁−𝐶₁−3𝐴₃𝜔²₀sin𝜏₀𝜔₀+2𝜔³₀−𝐴₂+𝐶₂𝜔₀cos𝜏₀𝜔₀+2𝐵₂𝜔₀,𝑄_𝑅=𝐴₁𝜔²₀−𝐴₃𝜔⁴₀cos𝜏₀𝜔₀+𝜔⁵₀−𝐴₂𝜔³₀+𝐴₀𝜔₀sin𝜏₀𝜔₀,𝑄_𝐼=𝐴₁𝜔²₀−𝐴₃𝜔⁴₀sin𝜏₀𝜔₀−𝜔⁵₀−𝐴₂𝜔³₀+𝐴₀𝜔₀cos𝜏₀𝜔₀. (24)

Obviously, if the condition (𝐻₃)𝑃_𝑅𝑄_𝑅+𝑃_𝐼𝑄_𝐼≠0 holds, then Re[𝑑𝜆/𝑑𝜏]⁻¹_𝜏=𝜏0≠0. Thus, according to the Hopf bifurcation theorem in [21], we have the following results.

Theorem 1. Suppose that the conditions (H₁)–(H₃) hold. The positive equilibrium 𝐷_∗(𝑆_∗,𝐸_∗,𝐼_∗,𝑅_∗) of system (3) is asymptotically stable for 𝜏∈[0,𝜏₀). System (3) undergoes a Hopf bifurcation at the positive equilibrium 𝐷_∗(𝑆_∗,𝐸_∗,𝐼_∗,𝑅_∗) when 𝜏=𝜏₀ and a family of periodic solutions bifurcate from the positive equilibrium 𝐷_∗(𝑆_∗,𝐸_∗,𝐼_∗,𝑅_∗) near 𝜏=𝜏₀.

3. Properties of the Hopf Bifurcation

In the previous section, we have obtained the conditions for the Hopf bifurcation to occur when 𝜏=𝜏₀. In this section, we will study properties of the Hopf bifurcation such as direction and stability by using the normal form theory and the center manifold theorem in [21].

Let 𝑢₁(𝑡)=𝑆(𝑡)−𝑆_∗, 𝑢₂(𝑡)=𝐸(𝑡)−𝐸_∗, 𝑢₃(𝑡)=𝐼(𝑡)−𝐼_∗, 𝑢₄(𝑡)=𝑅(𝑡)−𝑅_∗, and 𝜏=𝜏₀+𝜇,𝜇∈𝑅 and normalize 𝑡→(𝑡/𝜏). Then system (3) can be transformed into the following form: 𝑢̇(𝑡)=𝐿_𝜇𝑢_𝑡+𝐹𝜇,𝑢_𝑡, (25) where 𝑢_𝑡=(𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡),𝑢₄(𝑡))^𝑇∈𝐶=𝐶([−1,0],𝑅⁴) and 𝐿_𝜇∶𝐶→𝑅⁴ and 𝐹∶𝑅×𝐶→𝑅⁴ are given, respectively, by 𝐿_𝜇𝜙=𝜏₀𝑎+𝜇¹¹𝑎₁₂𝑎₁₃0𝑎₂₁𝑎₂₂𝑎₂₃00𝑎₃₂𝑎₃₃0000𝑎₄₄𝜙(0)−𝜏₀+𝜇000𝑏₁₄000000𝑏₃₃000𝑏₄₃𝑏₄₄𝜙(),𝐹=𝜏−1𝜇,𝜙₀+𝜇−𝛽𝜙₁(0)𝜙₃(0)𝛽𝜙₁(0)𝜙₃(0)00. (26)

By the Riesz representation theorem, there exists a 4×4 matrix function 𝜂(𝜃,𝜇)∶[−1,0]→𝑅⁴ whose elements are of bounded variation such that 𝐿_𝜇𝜙=⁰₋₁𝑑𝜂𝜃,𝜇𝜙(𝜃),𝜙∈𝐶. (27)

In fact, we choose 𝜂𝜃,𝜇=𝜏₀𝑎+𝜇¹¹𝑎₁₂𝑎₁₃0𝑎₂₁𝑎₂₂𝑎₂₃00𝑎₃₂𝑎₃₃0000𝑎₄₄𝛿(𝜃)−𝜏₀+𝜇000𝑏₁₄000000𝑏₃₃000𝑏₄₃𝑏₄₄𝛿(𝜃+1), (28) where 𝛿 is the Dirac delta function.

For 𝜙∈𝐶([−1,0],𝑅⁴), we define 𝐴𝜇(𝜃)𝜙=𝑑𝜙∫𝑑𝜃,−1≤𝜃<0,₀₋₁𝜙(𝜃)𝑅𝜇𝐹𝑑𝜂𝜃,𝜇,𝜃=0,𝜙=0,−1≤𝜃<0,𝜇,𝜙,𝜃=0. (29) Then system (25) can be transformed into the following operator equation: 𝑢̇(𝑡)=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑢_𝑡. (30)

The adjoint operator 𝐴^∗ of 𝐴 is defined by 𝐴^∗𝜑=−𝑑𝜑(𝑠)∫𝑑𝑠,0<𝑠≤1₀₋₁𝑑𝜂^𝑇(𝑠,0)𝜑(−𝑠),𝑠=0, (31) associated with a bilinear form: 𝜑(𝑠)(𝜃=,𝜙)𝜑(0)𝜙(0)−⁰_𝜃=−1^𝜃_𝜉=0𝜑(𝜉−𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉, (32) where 𝜂(𝜃)=𝜂(𝜃,0).

Let 𝜌(𝜃)=(1,𝜌₂,𝜌₃,𝜌₄)^𝑇𝑒^{𝑖𝜔0𝜏0𝜃} be the eigenvector of 𝐴 corresponding to 𝑖𝜔₀𝜏₀ and let 𝜌^∗(𝑠)=𝐷(1,𝜌^∗₂,𝜌^∗₃,𝜌^∗₄)𝑒^{𝑖𝜔0𝜏0𝑠} be the eigenvector of 𝐴^∗ corresponding to −𝑖𝜔₀𝜏₀. From the definition of 𝐴(0) and 𝐴^∗(0), we can get 𝜌₂=−𝑖𝜔₀−𝑎₃₃−𝑏₃₃𝑒^{−𝑖𝜏0𝜔0}𝑖𝜔₀−𝑎₂₂𝑖𝜔₀−𝑎₃₃−𝑏₃₃𝑒^{−𝑖𝜏0𝜔0}−𝑎₂₃𝑎₃₂,𝜌₃=𝑎₂₁𝑎₃₂𝑖𝜔₀−𝑎₂₂𝑖𝜔₀−𝑎₃₃−𝑏₃₃𝑒^{−𝑖𝜏0𝜔0},𝜌₄=𝑏₄₃𝑒^{−𝜏0𝜔0}𝑖𝜔₀−𝑎₄₄+𝑏₃₃𝑒^{−𝜏0𝜔0}𝜌₃,𝜌^∗₂𝑎=−¹¹𝑖𝜔₀+𝑎₂₁,𝑟^∗₃=−𝑖𝜔₀𝑎₁₂−𝑎₁₁+𝑎₁₂𝑎₂₁−𝑎₁₁𝑎₂₂𝑎₃₂𝑖𝜔₀+𝑎₂₁,𝜌^∗₄=𝑏₁₄𝑒^{𝑖𝜏0𝜔0}𝑖𝜔₀+𝑎₄₄+𝑏₄₄𝑒^{𝑖𝜏0𝜔0}. (33)

From (32), we have 𝜌^∗,𝜌=𝐷1+𝜌₂𝜌^∗₂+𝜌₃𝜌^∗₃+𝜌₄𝜌^∗₄+𝜏₀𝑒^{−𝑖𝜏0𝜔0}×𝜌₃𝑏₃₃𝜌^∗₃+𝑏₄₃𝜌^∗₄+𝜌₄𝑏₁₄+𝑏₄₄𝜌^∗₄. (34)

Then, we choose 𝐷=1+𝜌₂𝜌^∗₂+𝜌₃𝜌^∗₃+𝜌₄𝜌^∗₄+𝜏₀𝑒^{−𝑖𝜏0𝜔0}×𝜌₃𝑏₃₃𝜌^∗₃+𝑏₄₃𝜌^∗₄+𝜌₄𝑏₁₄+𝑏₄₄𝜌^∗₄⁻¹ (35) such that ⟨𝑞^∗,𝑞⟩=1, ⟨𝑞^∗,𝑞⟩=0.

Next, we can obtain the coefficients which will be used to determine the properties of the Hopf bifurcation by following the algorithms introduced in [21] and using a computation process as in [22]: 𝑔₂₀=2𝛽𝜏₀𝐷𝜌₃𝜌^∗₂,𝑔−1₀₂=2𝛽𝜏₀𝐷𝜌₃𝜌^∗₂,𝑔−1₁₁=𝛽𝜏₀𝐷𝑟₃+𝑟₃𝑟^∗₂,𝑔−1₂₁=2𝛽𝜏₀𝐷𝜌^∗₂−1𝑊⁽¹⁾₁₁(0)𝜌₃+12𝑊⁽¹⁾₂₀(0)𝜌₃+𝑊⁽³⁾₁₁(0)+12𝑊⁽³⁾₂₀(0), (36) with 𝑊₂₀(𝜃)=𝑖𝑔₂₀𝑞(0)𝜏₀𝜔₀𝑒^{𝑖𝜏0𝜔0𝜃}+𝑖𝑔₀₂𝑞(0)3𝜏₀𝜔₀𝑒^{−𝑖𝜏0𝜔0𝜃}+𝐸₁𝑒^{2𝑖𝜏0𝜔0𝜃},𝑊₁₁(𝜃)=−𝑖𝑔₁₁𝑞(0)𝜏₀𝜔₀𝑒^{𝑖𝜏0𝜔0𝜃}+𝑖𝑔₁₁𝑞(0)𝜏₀𝜔₀𝑒^{−𝑖𝜏0𝜔0𝜃}+𝐸₂, (37) where 𝐸₁ and 𝐸₂ can be determined by the following equations, respectively:𝐸₁=22𝑖𝜔₀−𝑎₁₁−𝑎₁₂−𝑎₁₃−𝑏₁₄𝑒^{−2𝑖𝜔0𝜏0}−𝑎₂₁2𝑖𝜔₀−𝑎₂₂−𝑎₂₃00−𝑎₃₂2𝑖𝜔₀−𝑎₃₃−𝑏₃₃𝑒^{−2𝑖𝜔0𝜏0}000−𝑏₄₃𝑒^{−2𝑖𝜔0𝜏0}2𝑖𝜔₀−𝑎₄₄−𝑏₄₄𝑒^{−2𝑖𝜔0𝜏0−1}𝐸₁⁽¹⁾𝐸₁⁽²⁾00,𝐸₂𝑎=−¹¹𝑎₁₂𝑎₁₃𝑏₁₄𝑎₂₁𝑎₂₂𝑎₂₃00𝑎₃₂𝑎₃₃+𝑏₃₃000𝑏₄₃𝑎₄₄+𝑏₄₄⁻¹𝐸₂⁽¹⁾𝐸₂⁽²⁾00, (38)with 𝐸₁⁽¹⁾=−𝛽𝜌₃,𝐸₁⁽²⁾=𝛽𝜌₃,𝐸₂⁽¹⁾=−𝛽𝜌₃+𝜌₃,𝐸₂⁽²⁾=𝛽𝜌₃+𝜌₃. (39)

Therefore, we can calculate the following values: 𝐶₁(0)=𝑖2𝜏₀𝜔₀𝑔₁₁𝑔₂₀−2𝑔₁₁²−𝑔₀₂²3+𝑔₂₁2,𝜇₂𝐶=−Re₁(0)Re𝜆^′𝜏₀,𝛽₂=2Re𝐶₁(0,𝑇₂)𝐶=−Im₁(0)+𝜇₂Im𝜆^′𝜏₀𝜏₀𝜔₀. (40) Based on the discussion above, we can obtain the following results.

Theorem 2. For system (3), if 𝜇₂>0(𝜇₂<0), the Hopf bifurcation is supercritical (subcritical). If 𝛽₂<0(𝛽₂>0), the bifurcating periodic solutions are stable (unstable). If 𝑇₂>0(𝑇₂<0) the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Simulation

In this section, we present a numerical example to verify the theoretical analysis in Sections 2 and 3. Let 𝑏=0.7, 𝑑=0.06, 𝑝=0.02, 𝑞=0.01, 𝛽=0.1, 𝜀=0.2, 𝛾=0.05, 𝜂=0.01, and 𝜁=0.1. We can get a particular case of system (3): 𝑑𝑆(𝑡)(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)(),(𝑡)𝑑𝑡=0.7−0.1𝐼−0.014𝐸−0.007𝐼−0.06𝑆+0.1𝑅𝑡−𝜏𝑑𝐸(𝑡)𝑆(𝑡)(𝑡)(𝑡)(𝑡)(𝑡),(𝑡)𝑑𝑡=0.1𝐼+0.014𝐸+0.007𝐼−0.2𝐸−0.06𝐸𝑑𝐼(𝑡)()(𝑡)(𝑡),(𝑡)𝑑𝑡=0.2𝐸−0.05𝐼𝑡−𝜏−0.06𝐼−0.01𝐼𝑑𝑅()(𝑡)().𝑑𝑡=0.05𝐼𝑡−𝜏−0.06𝑅−0.1𝑅𝑡−𝜏 (41)

Then, we can get that 𝑅₀=7.5774>1, (𝜀+𝑑)(𝑑+𝛾+𝜂)=0.0312>𝑝𝑏(𝑑+𝛾+𝜂)+𝑞𝑏𝜀=0.0031, and system (41) has a unique positive equilibrium 𝐷_∗(1.4050,1.5422,4.9350, and 2.9610). Further, we have 𝐷₁=1.0795>0, 𝐷₂=0.3360>0, 𝐷₃=0.0141>0, and 𝐷₄=2.8200𝑒−004>0. That is, condition (𝐻₁) holds. Thus, we can obtain 𝜔₀=0.1245, 𝜏₀=13.6109, and 𝜆^′(𝜏₀)=0.0073+0.0299𝑖. Therefore, from Theorem 1, the positive equilibrium 𝐷_∗ is asymptotically stable when 𝜏∈[0,13.6109) and unstable when 𝜏>13.6109. As can be seen from Figures 1, 2, and 3, when 𝜏=12.5∈[0,13.6109), the positive equilibrium 𝐷_∗ is asymptotically stable. However, when 𝜏=14.95>13.6109, the positive equilibrium 𝐷_∗ will lose its stability, a Hopf bifurcation occurs, and a family of periodic solutions bifurcate from the positive equilibrium 𝐷_∗. This property can be illustrated by Figures 4, 5, and 6. In addition, from (40), we can get 𝐶₁(0)=−0.00904+0.6833𝑖, 𝜇₂=12.3836>0, 𝛽₂=−0.1808<0, and 𝑇₂=−0.6217<0. Thus, according to Theorem 2, we know that the Hopf bifurcation is supercritical. The bifurcating periodic solutions are stable and the period of the periodic solutions decreases.

Figure 1 

The track of the states 𝑆, 𝐸, 𝐼, and 𝑅 for 𝜏=12.5000<13.6109=𝜏0.

Figure 2 

The phase plot of the states 𝑆, 𝐸, and 𝐼 for 𝜏=12.5000<13.6109=𝜏0.

Figure 3 

The phase plot of the states 𝐸, 𝐼, and 𝑅 for 𝜏=12.5000<13.6109=𝜏0.

Figure 4 

The track of the states 𝑆, 𝐸, 𝐼, and 𝑅 for 𝜏=14.9500>13.6109=𝜏0.

Figure 5 

The phase plot of the states 𝑆, 𝐸, and 𝐼 for 𝜏=14.9500>13.6109=𝜏0.

Figure 6 

The phase plot of the states 𝐸, 𝐼, and 𝑅 for 𝜏=14.9500>13.6109=𝜏0.

5. Conclusions

In this paper, we incorporate the time delay due to the period the antivirus software has to use to clean the worms in one node and the temporary immunity period of the recovered nodes into the model considered in the literature [6] and get a delayed SEIRS epidemic model for the transmission of worms in computer network through vertical transmission. The effects of the time delay on the dynamics of the model are investigated. It is found that the time delay can play a complicated role on the model by analyzing the distribution of the roots of the associated characteristic equation. When the time delay is suitable small, the positive equilibrium is asymptotically stable. However, a local Hopf bifurcation occurs and a branch of periodic solutions bifurcates from the positive equilibrium when the delay passes through the critical value 𝜏₀. Furthermore, the properties of the Hopf bifurcation such as direction and stability are determined by using the normal form theory and center manifold theorem. In order to verify the theoretical analysis, a numerical example is also included.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61273070), a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institution, Major Science Foundation Subject for the Education Department of Anhui Province under Project no. ZD200905, and a Project funded by the Ministry of Education (12YJA630136).