^{1,2}

^{1}

^{1}

^{2}

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.

Since the pioneering work of Kephart and White [

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 [

Let

The main purpose of this paper is to investigate the effects of the time delay on the dynamics of system (

This paper is organized as follows. In Section

It is not difficult to verify that if

The Jacobian matrix of system (

Multiplying

When

By the Routh-Hurwitz criterion, the sufficient conditions for all roots of (

Thus, if the condition

For

It is well known that

Let

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

Suppose that the condition

Let

Taking the derivative of with respect to (

Obviously, if the condition

Suppose that the conditions (_{1})–(_{3}) hold. The positive equilibrium

In the previous section, we have obtained the conditions for the Hopf bifurcation to occur when

Let

By the Riesz representation theorem, there exists a

In fact, we choose

For

The adjoint operator

Let

From (

Then, we choose

Next, we can obtain the coefficients which will be used to determine the properties of the Hopf bifurcation by following the algorithms introduced in [

Therefore, we can calculate the following values:

For system (

In this section, we present a numerical example to verify the theoretical analysis in Sections

Then, we can get that

The track of the states

The phase plot of the states

The phase plot of the states

The track of the states

The phase plot of the states

The phase plot of the states

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 [

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).