Hopf Bifurcation of an Improved SLBS Model under the Influence of Latent Period

Amodel applicable to describe the propagation of computer virus is developed and studied, along with the latent time incorporated. We regard time delay as a bifurcating parameter to study the dynamical behaviors including local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when the time delay passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal formmethod and centermanifold theorem. Finally, illustrative examples are given to support the theoretical results.


Introduction
With the rapid development of computer technologies and network applications, the Internet has become a powerful mechanism for propagating computer virus.Because of this, computers connected to the Internet become much vulnerable to digital threats.
It is quite urgent to understand how computer viruses spread over the Internet and to propose effective measures to cope with this issue.To achieve this goal, and in view of the fact that the spread of virus among computers resembles that of biological virus among a population, it is suitable to establish dynamical models describing the propagation of computer viruses across the Internet by appropriately modifying epidemic models [1].
(H1) All computers connected to the Internet are partitioned into three compartments: susceptible computers (S-computers), infected computers that are latent (L-computers), and infected computers that are breaking out (B-computers).
(H2) All newly connected computers are virus-free.
(H3) External computers are connected to the Internet at constant rate .Meanwhile, internal computers are disconnected from the Internet at rate .
(H4) Each virus-free computer is infected by a virulent computer at constant rate , and the ratio of previously virus-free computers that are infected exactly at time  is  ( + ) [16].
(H5) Breaking-out computers are cured at constant rate .
(H6) Latent computers break out at constant rate .
According to the above assumptions, the authors of [14,15] proposed the proposed the following SLBS model, which is formulated as It is well known that some computer viruses would delay a period to break out after the computer is infected.However, the above model fails to consider the concrete time of the delay.Thus, this paper aims to establish a model to incorporate the unconsidered factor, by adding a delay item to the above model.First, we give another assumption as (H7): L-computers turn out to be B-computers with constant time delay .
According to the above assumptions (H1)-(H7), the new model with time delay is formulated as Here, we let (), (), and () represent the percentage of S-, L-, and B-computers in all internal computers at time , respectively.Then we get () + () + () ≡ 1 and consider the following equivalent two-dimensional subsystem: The initial conditions of (3) are given by () =  1 () > 0, () =  2 () > 0, and  ∈ [−, 0], where ( The remainder of this paper is organized as follows.In Section 2, the stability of trivial solutions and the existence of Hopf bifurcation are discussed.In Section 3, a formula for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions will be given by using the normal form and center manifold theorem introduced by Hassard et al. in [17].In Section 4, numerical simulations aimed at justifying the theoretical analysis will be reported.

Stability of the Equilibria and Existence of Hopf Bifurcation
This section is intended to study model (3) theoretically, by analyzing the stability of its solutions and the existence of Hopf bifurcation.For the convenience of the following description, we define the basic reproduction number of system (3) as We have the following result with respect to the stable state of system (3).Theorem 1.Consider system (3) with  = 0. Then the unique virus-free equilibrium  0 = (0, 0) is globally asymptotically stable if  0 < 1, whereas  0 becomes unstable and the unique positive equilibrium  * = (( + )(1 − 1/ 0 )/( +  + ), (1 − 1/ 0 )/( +  + )) is locally asymptotically stable if  0 > 1.
The proof is omitted here (see [14] for details).
Proof.Suppose that  =  ( > 0) is a root of (7); then separating the real and imaginary parts of (7), we have Adding up the squares of (8) yields Since  +  ≥ , we derive the following equations: Therefore, (9) exists as a unique positive solution  0 , and the characteristic equation ( 7) has a pair of pure imaginary roots ± 0 .By (8), we have By Theorem 1, for  = 0, the positive equilibrium  * is locally asymptotically stable, and hence by Butler's Lemma [18],  * remains stable for  <  0 .Now, we need to show This will signify that there exists at least one eigenvalue with positive real part for  >  0 .Moreover, the conditions for Hopf bifurcation [19] are then satisfied yielding the required periodic solution.Now, by differentiating ( 9) with respect to , we get This gives Thus, Since As Therefore, the transversality condition holds and thus Hopf bifurcation occurs at  =  0 .The proof is complete.

Direction of the Hopf Bifurcation
In this section, we derive explicit formulae for computing the direction of the Hopf bifurcation and the stability of bifurcation periodic solution at critical value  0 by using the normal form theory and center manifold reduction.

Then we have
Comparing the coefficients of the above equation with those in (41), we have In what follows, we focus on the computation of  20 () and  11 ().For the expression of  21 , we have We can easily obtain the solutions of (58) as (59) Substituting ( 59) and ( 61) into (62) and noticing that we can obtain which leads to ) . (66) Hence, we know  20 and then we can obtain  21 .The following parameters can be calculated: Theorem 3.Under the condition of Theorem 1, one has the following.
(2) The direction of Hopf bifurcation is determined by the sign of  2 : if  2 > 0, the Hopf bifurcation is supercritical; if  2 < 0, the Hopf bifurcation is subcritical.
(3) The stability of bifurcating periodic solutions is determined by  2 : if  2 < 0, the periodic solutions are stable; if  2 > 0, they are unstable.

Numerical Examples
In this section, some numerical examples of system (3) are presented to justify the previous theorem above.It follows by (11) that  0 = 3.5705.First, we choose  = 3 <  0 .For a set of initial conditions satisfying (0) = 0.1 and (0) = 0.1, Figure 1 demonstrates the evolutions from which it can be seen that the equilibrium is asymptotically stable.Second, we choose  = 3.7 >  0 .For a set of initial conditions satisfying, the corresponding wave form and phase plots are shown in Figure 2, from which it is easy to see that a Hopf bifurcation occurs.

Conclusions
In this paper, we have constructed a computer virus model with time delay depending on the SLBS model.The theoretical analyses for the computer virus models are given.Furthermore, it is proved that there exists a Hopf bifurcation when time crosses through the critical value.Finally, the numerical simulations illustrate our results.

Figure 4 :
Figure 4: The bifurcation periodic solution is stable.