Stability and Hopf Bifurcation of a Computer Virus Model with Infection Delay and Recovery Delay

A computer virus model with infection delay and recovery delay is considered. The sufficient conditions for the global stability of the virus infection equilibrium are established. We show that the time delay can destabilize the virus infection equilibrium and give rise to Hopf bifurcations and stable periodic orbits. By the normal form and center manifold theory, the direction of the Hopf bifurcation and stability of the bifurcating periodic orbits are determined. Numerical simulations are provided to support our theoretical conclusions.


Introduction
In recent years, the computer networks have become more and more popular, and people can find many useful things through computer networks.However, computer virus flows and spoils the correct operation of computer.As the computer networks become necessary tools in our daily life, computer virus becomes a major threat [1].
Cohen [2] found that there is high similarity between biological virus and computer virus.By the Kermack and McKendrick SIR epidemic model [3], the computer virus models were proposed in [4][5][6][7] which analyzed the spread of computer virus by epidemiological models.The model is as follows: d where (), (), and () denote susceptible, infected, and recovered computers, respectively.Here we assume that all the computers connect to the network. is the rate at which external computers are connected to the network,  is the recovery rate of infected computers because of the antivirus ability of the network,  is the infection rate, and  is the death rate of the classes (), (), and ().Dong et al. [8] considered the effect of immunization on susceptible state and exposed state by a delayed computer virus model.Zhang et al. [9] studied an impulse model for computer viruses and established the global dynamics of the model.Yang et al. [10] analyzed a computer virus model with graded cure rates and showed that the global dynamics are determined by the basic reproduction number.Then we can understand and control the computer virus propagation using the mathematical models.
Since there is a period of time from virus entering a host to active state [11], there exists an infection delay from infected to infectious computers.Similarly, there is a recovery delay from recovered to susceptible computers.Then the model is where 0 <  − 1 ≤ 1 is the survival probability of the infected computers,  1 is the infection delay, and ] is the rate at which one recovered computer reverts to the susceptible one.( −  2 ) denotes that a recovered computer moves into the susceptible class after time  2 .Ren et al. [12] obtained that the virus-free equilibrium is globally asymptotically stable when  0 < 1.When  0 > 1, we make a lot of improvement in the results in [12].Compared to [12], we show that the virus infection equilibrium is always locally asymptotically stable for  1 > 0 and  2 = 0, and Hopf bifurcation does not exist.For  1 > 0 and  2 > 0, Ren et al. [12] constructed a Lyapunov function by linearized equations of  * and obtained the sufficient conditions for global stability of the virus infection equilibrium.Obviously, this is inappropriate.Then we construct a suitable Lyapunov function and obtain the sufficient conditions of global stability for the virus infection equilibrium when  1 > 0 and  2 > 0. Furthermore, for  1 = 0 and  2 > 0, we study the direction of the Hopf bifurcation and stability of the bifurcating periodic orbits by the normal form and center manifold theory, and numerical simulations are given to support the theoretical conclusions.
The paper is organized as follows.In Section 2, the existence of equilibria is discussed and characteristic equation is given.In Section 3, we establish the local stability of the virus infection equilibrium  * for  1 > 0 and  2 = 0.In Section 4, for  1 = 0 and  2 > 0, we consider the local stability of  * and existence of local Hopf bifurcation, and the direction of Hopf bifurcation and stability of the bifurcating periodic solutions are considered.In Section 5, for  1 > 0 and  2 > 0, global stability of the virus infection equilibrium is obtained.Finally, we finish the paper with conclusions.
Remark 2. For  1 > 0 and  2 = 0, Ren et al. [12] showed that the time delay  1 can destabilize the infected equilibrium  * leading to Hopf bifurcations.However, we show that the virus infection equilibrium  * is always locally asymptotically stable for  1 > 0 and  2 = 0, and Hopf bifurcation does not exist.
We introduce a set of parameter values:  = 0.2,  = 0. Figure 1 shows that the infection equilibrium  * is locally asymptotically stable.
Let   ( = 1, 2) be the positive real roots such that () = 0 and   = √  ( = 1, 2).Denote where  ∈ [0, 2] is determined by Define Therefore,  0 is the first value of  when a pair of characteristic roots cross the imaginary axis at ± 0 .
From the above discussions, the sufficient conditions for the existence of Hopf bifurcation were given for  1 = 0 and  2 > 0. Thus, under the conditions of Theorem 3, we will study the direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits by normal form and center manifold theory (see, e.g., [46]).
Following the procedure in [46], we obtain the following coefficients: where where where ) .
Then  21 can be expressed definitely.From ( 27), (40), and (41), we have Then we have the following result.

Global Asymptotic Stability
The following result shows the global stability of the virus-free equilibrium.
Next we will prove the global stability of the virus infection equilibrium of system (2) From system (2) with initial conditions (3), we know that ()+ * = () ≤ /(∀ ≥ 0).By the method of constructing Lyapunov function [47], we obtain the following result.holds, then the virus infection equilibrium  * is globally asymptotically stable for  1 > 0 and  2 > 0.
Clearly, this is unreasonable.Thus, we define an appropriate Lyapunov function and obtain the sufficient conditions of global stability for the virus infection equilibrium  * .

Conclusions
We consider a computer virus model with infection delay and recovery delay.When  0 < 1, the virus-free equilibrium of system (2) is globally asymptotically stable and the virus fades out from the network.When  0 > 1, we obtain that the virus infection equilibrium is always locally asymptotically stable for  1 > 0 and  2 = 0; for  1 = 0 and  2 > 0, we establish that the time delay can destabilize the virus infection equilibrium and give rise to Hopf bifurcations and stable periodic orbits.For  1 > 0 and  2 > 0, we construct an appropriate Lyapunov function and get that the infection equilibrium is globally asymptotically stable under the sufficient conditions, and the virus finally persists at a constant endemic equilibrium level.
Our results show that we can take measures to make the basic reproduction number  0 to be less than one leading to the extinction of computer virus.We must take some effective measures (such as the installation of antivirus software) to decrease the infection rate  and increase recovery rate  of the infected computers.
However, our results should be viewed carefully because the model used in this paper is simplified and possibly does not explain all relevant dynamics of computer virus.Then we need more realistic computer virus models to study the real network.