Hopf Bifurcation in a Delayed SEIQRS Model for the Transmission of Malicious Objects in Computer Network

A delayed SEIQRS model for the transmission of malicious objects in computer network is considered in this paper. Local stability of the positive equilibrium of the model and existence of local Hopf bifurcation are investigated by regarding the time delay due to the temporary immunity period after which a recovered computer may be infected again. Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem. Numerical simulations are also presented to support the theoretical results.


Introduction
The action of malicious objects throughout a network can be studied by using epidemic models due to the high similarity between malicious objects and biological viruses [1][2][3][4][5][6][7].In [2], Thommes and Coates proposed a modified version of the SEI model to predict virus propagation in a network.In [3], Mishra and Pandey proposed a SEIRS epidemic model for the transmission of worms with vertical transmission.Recently, antivirus counter measures such as virus immunization and quarantine strategy have been introduced into some epidemic models in order to study the prevalence of virus.In [7], Mishra and Jha proposed the following SEIQRS model for the transmission of malicious objects in computer network: where (), (), (), (), and () denote the sizes of nodes at time  in the states susceptible, exposed, infectious, quarantined, and recovered, respectively.The parameters , , and  are positive constants. is the recruitment rate of susceptible nodes to the computer network. is the crashing rate of nodes due to the reason other than the attack of malicious objects. is the crashing rate of nodes due to the attack of malicious objects. is the transmission rate., , , , and  are the state transition rates.Mishra and Jha [7] investigated the global stability of the unique endemic equilibrium for the system (1).It is well known that time delays can cause a stable equilibrium to become unstable and make a system bifurcate periodic solutions and dynamical systems with delay have been studied by many scholars [8][9][10][11][12][13][14][15].In [9], Feng et al. investigated the Hopf bifurcation of a delayed viral infection model in computer networks by using theories of stability and bifurcation.In [11] where  ≥ 0 is the time delay due to the temporary immunity period.
The main purpose is to investigate the effects of the delay on system (2) and this paper is organized as follows.Sufficient conditions for the local stability and existence of local Hopf bifurcation are obtained by regarding the delay due to the temporary immunity period after which a recovered computer may be infected again as a bifurcating parameter in Section 2. Direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are determined by the normal form method and center manifold theorem in Section 3. In Section 4, we give a numerical example to support the theoretical results in the paper.

Stability and Existence of Local Hopf Bifurcation
According to the analysis in [7], we can get that if the condition ( ) . ( Let () = () −  * , () = () −  * , () = () −  * , () = () −  * , and () = () −  * .Dropping the bars for the sake of simplicity, system (2) can be rewritten as the following system: where ( Then, we can get the linearized system of system (4) as follows: Thus, the characteristic equation of system (6) at the positive equilibrium  * is where For the existence of local Hopf bifurcation of system (2), we give the following result.
Proof.For  = 0, (7) becomes where Obviously,  1 =  4 > 0. Therefore, if condition ( 2 ): (11) holds, then the positive equilibrium  * is locally asymptotically stable without delay.Consider For  > 0, let  =  ( > 0) be a root of (7).Then, we can get from which we obtain with Let  2 = V, then (14) becomes In order to give the main results in this paper, we make the following assumption.
( 3 ) Equation ( 16) has at least one positive real root.Suppose that condition ( 3 ) holds.Without loss of generality, we suppose that ( 16) has five positive roots, which are denoted as V 1 , V 2 , . . ., V 5 , respectively.Then, ( 14) has five positive roots   = √ V  ,  = 1, 2, . . ., 5. For every fixed   , the corresponding critical value of time delay is where Let Taking the derivative of  with respect to  in (7), we obtain Then, we have where ) holds, then the transversality condition is satisfied.The proof of Theorem 1 is completed.

Properties of the Hopf Bifurcation
In the previous section, we have obtained the conditions under which system (2) undergoes Hopf bifurcation at the positive equilibrium  * ( * ,  * ,  * ,  * ,  * ) when the delay  crosses though the critical value  0 .In this section, we give the formula that determines the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (2). Define where  1 (0) is defined in the following analysis.Further, we give the following result with respect to the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (2).
Then, we can get the expression of  1 (0) as follows: Further we have where the sign  2 determines the direction of the Hopf bifurcation, the sign  2 determines the stability of the bifurcating periodic solutions, and the sign of  2 determines the period of the bifurcating periodic solutions.The proof of Theorem 2 is completed.

Numerical Simulation
In this section, we use a numerical example to support the theoretical analysis above in this paper.We take the following particular case of system (2) in which  = 0.

Conclusions
This paper is concerned with a delayed SEIQRS model for the transmission of malicious objects in computer network.The theoretical analysis for the delayed model is given and the main results are given in terms of local stability and local Hopf bifurcation.By regarding the delay due to the temporary  immunity period after which a recovered computer may be infected again, we have proven that when the delay passes through the critical value, the model undergoes a Hopf bifurcation.The occurrence of Hopf bifurcation means that the state of virus prevalence changes from a positive equilibrium to a limit cycle, which is not welcomed in networks.Hence, we should control the phenomenon by combining some bifurcation control strategies and other relative features of virus prevalence.Further, the properties of the Hopf bifurcation are studied by using the normal form method and center manifold theorem.Finally, some numerical simulations are presented to clarify our theoretical results.