Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 196214,10 pages http://dx.doi.org/10.1155/2013/196214

Research Article

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

Chunming Zhang,1 Wanping Liu,2 Jing Xiao,1 and Yun Zhao1

1 School of Information Engineering, Guangdong Medical College, Dongguan 523808, China

2 College of Computer Science, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Chunming Zhang; chunfei2002@163.com Received 12 June 2013; Accepted 15 August 2013 Academic Editor: Fazal M. Mahomed

Copyright © 2013 Chunming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A model 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 form method and center manifold theorem. Finally, illustrative examples are given to support the theoretical results.

1. 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].

Based on the fact that infectivity is one of the common features shared by computer viruses and their biological counterparts [2], some classic epidemic models were established for computer virus propagation, such as the SIRS model [3-7], the SEIR model [8, 9], the SEIRS model [10], the SEIQV model [11], and the SEIQRS model [12]. In [13-15] the authors made the following assumptions.

(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 S. Meanwhile, internal computers are disconnected from the Internet at rate S.

(H4) Each virus-free computer is infected by a virulent computer at constant rate p, and the ratio of previously virus-free computers that are infected exactly at time t is ¡3S(L + B) [16].

(H5) Breaking-out computers are cured at constant rate y.

(H6) Latent computers break out at constant rate a.

According to the above assumptions, the authors of [14, 15] proposed the proposed the following SLBS model, which is formulated as

S = S- ¡3S(L + B) + yB-8S, L = ¡3S(L + B)-aL- SL, (1)

B = aL-yB- SB.

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 ¿-computers with constant time delay t.

According to the above assumptions (H1)-(H7), the new model with time delay is formulated as

S = S- ¡3S(L + B)+yB-8S,

L = ¡3S(L + B)-aL(t-T)- SL, (2)

B = aL(t-r)-yB- SB.

Here, we let S(t), L(t), and B(t) represent the percentage of S-, L-, and ¿-computers in all internal computers at time t, respectively. Then we get S(t) + L(t) + B(t) = 1 and consider the following equivalent two-dimensional subsystem:

L = ß(1-L-B)(L + B)-aL(t-r)-8L, È = aL(t-T)-yB- SB.

The initial conditions of (3)aregivenbyL(d) = ^1(d) > 0, B(0) = $2(G) > 0, and d e [-r,0], where (^1(d),^2(d)) e C([-t, 0], R+), the Banach space of the continuous functions mapping the interval [-T, 0] into R+ = {(x1, x2) : xi > 0, i = 1,2}.

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.

2. 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

ß(a + y + 8) (a + S) (y + S)

We have the following result with respect to the stable state of system (3).

Theorem 1. Consider system (3) with t = 0. Then the unique virus-free equilibrium E0 = (0,0) is globally asymptotically stable if R0 < 1, whereas E0 becomes unstable and the unique positive equilibrium Et = ((y + S)(1-1/R0)l(a + S + y), a(1-1/R0)/(a + S + y)) is locally asymptotically stable if R0 > 1.

The proof is omitted here (see [14] for details).

The linearized equations of (3) are as follows:

„ 2ß

B-aL(t-r),

È = aL(t-r)-yB- SB.

The determinant of the Jacobian matrix of the system (5) at Et is given by \ yE- A1 - B1e-yT\ = 0, where

-ß+2l-s-ß+2l R0 RQ

0 -V-S

-a 0 a 0

Let d = - (2(a + S)(y + S))/(a + y + S), andwe can obtain the following characteristic equation:

y2 + m1y + m0 + (n1y + n0) e yr = 0,

where m1 = y + 28 - d, m0 = (y + 8)(8 - d), n1 = a, \ a(y + 8 - d).

Theorem 2. Suppose that R0 > 1, y + 8 > a, and (y + 8) (8 - d) -a2(y + 8-d) <0 hold; then the positive equilibrium Et is asymptotically stable for t e [0, r0) and system (3) undergoes a Hopf bifurcation at Et when r = t0 .

Proof. Suppose that y = iw (w > 0) is a root of (7); then separating the real and imaginary parts of (7), we have

m1œ = hq sin œr - n1w cos <mt,

w - m0 = n0 cos wt + n1w sin wr.

Adding up the squares of (8) yields

4(2- 2\ 2 2 2 n

w + (m1 - 2m0 -n1)w +m0 -n0 = 0. Since y + 8 > a, we derive the following equations:

22 m1 - 2m0 - n1

= (y + 28-d)2 -2(y + 8) (S-d)- a2 = (8 - d)2 + (y + 8 + a) (y + 8 - a) > 0, m2 - % = (Y + à)2(8 - d)2 - oi(y + 8 - d)2 < 0.

Therefore, (9) exists as a unique positive solution w0, and the characteristic equation (7) has a pair of pure imaginary roots ±iw0. By (8), we have

1 n0 arccos-

(w^ - m0 ) - m1 n^

Tn = — arccos- 2 2 2

w0 n2 + n^2

+-, n = 0,1,2,

By Theorem 1, for t = 0, the positive equilibrium Et is locally asymptotically stable, and hence by Butler's Lemma [18], Et remains stable for t < t0. Now, we need to show

d(Re y)

This will signify that there exists at least one eigenvalue with positive real part for t > t0. Moreover, the conditions for Hopf bifurcation [19] are then satisfied yielding the required periodic solution. Now, by differentiating (9) with respect to r, we get

(2y + m1 + n1e yr - r(n1y + n0) e yr)

= y (n1y + n0) e yr.

This gives

2y + m1 + n1e yr -t (n1y + n0) e yr

y (n1y + n0) e yT

2y + m1

y{*1 y + no)e yr y{n.1y + no) y

-y2 (y2 +m1y + m0) y2 (n1y + n0) y

, d(Re y) 1

Sign 1"dHt

= Sign

= Sign -

-y2 (y2 + m1y + m0) -nn

72 (n1y + nQ)

= sign

(w20 + mQ) (w20 - mQ)

u>2 [(m0 -u2)2 + (m1<M0)2

w2 (n2 + n2w2)

Since (mQ - œ20)2 + (mlwQ)2 =n20 + n^œ'^, we have that . [ d(Re y) 1

Sign t" L

= sign -

= Sign -

[u>l + m0) (up - m0) nQ

\ [(m0 - u2)2 + (m^)2] u2 (n2 + n2w2)

"ft (n0 + n1œ0,)

As m0 -n0 < 0, thus

d(Re y)

Therefore, the transversality condition holds and thus Hopf bifurcation occurs at t = rQ. The proof is complete. □

3. 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 tq by using the normal form theory and center manifold reduction.

Letting t = st, ul = L - L„, u2 = B - Bt, ui(t) = ui(rt), and t = tq + p, system (3) is transformed to an FDE as

ù(t) = L v (ut ) + f(p,ut),

where u(t) = (u1 (t),u2(t))T e R2, ut(9) = u(t + 9), 9 e [0,1], L v :C^R, }:RxC^R,

Lv (ut)

= (^0 + V)

^2(a + 8)(y + 8) s -ß+2(a + 8)(y + 8)

91 (0)

92 (0)

a + y + 8 0

+ (^0 + V)

a + y + S -y - S

-a 0 a 0

91 (-1)

92 (-1)

= (*0 + y)A 1

91 (0)

92 (0)

+ fo + y) B1

91 (-1)

92 (-1)

f (v ut) = (T0 + y)

-ßtf (0)-2ß<p1 (0)V2 (0) - ß<p2 (0) 0

Using the Riesz representation theorem, there exists a function q(9, p) of bounded variation for 9 e [0,1], such that

L v<p = J dq(6,n)<p(6), <peC. (20)

In fact, we can choose

q (9, p) = (t0 + p) A2S (9) + (t0 + y)B28(9+1), (21)

where 8(9) is Dirac delta function. In the following, for f e [0,1], we define

A(y)9 =

J dv(s,^)9(s), 9 = 0, (22)

r(h)9 =

9 e [-1,0),

f(^,9), d = 0.

% +n0 - m0

Then system (18) can be rewritten as

ù(t) = A (p) ut + R (p) ut,

dut (9) du(t + d) du(t + d) dut (9)

For 0 eC* = C([0,1],(R2)*), the adjoint operator A* of A is defined by

A* (0)$(s) =

d$(s) ds '

s e (0,1]

dr)T (t, 0) $ (-t), s = 0, where rjT is the transpose of the matrix We define

($,<p) = $(0)<p(0)

Î0 ¡-e

ï(Ç-e) dn(d)f(Ç) d£

-1 J$=0

where q(9) = q(9,0). We know that iw0T0 is an eigenvalue of A(0), so -iw0T0 is also an eigenvalue of A* (0). We can get q(9) = (1,q1)TeiaoT'e.

From the above discussion, it is easy to know that

Aq (9) = i^oToq (9),

tqa 1q(0)+ToB1q(-1) = i^oToq(0).

Hence, we obtain

(iw0 + y + S) eta°r° Suppose that the eigenvector

* / \ 1 /1 iWntfiS

q (s) = ~(1,q1 ) e 00 .

Then the following relationship is obtained: A*q* (0) = -iw0?0i (0).

Hence, we obtain

* -, 1 J2(a+s)(r:s)-ß)- (3D

^ (y + S - iw0)\ a + y + S 11 V '

(q*,q) = h

{q,q) = q* (0)?(0)-f f qT (^-6) d^(9)9(^) d^

= 1(1 + ml)

(0 f 1

To -[1,q; ][A 2Ô(9)+B2Ô(9+1)]

J0=-1 J£=0 p

e,r°WotidÇ d9

= 1(1 + q1q1 ) + To 1a(1-q**)e-= 1.

Hence, we obtain

p=(1+qi q*) + T0«(1-q*)e-""". (33)

In the remainder of this section, by using the same notations as in the work by Hassard et al. [17], we first compute the coordinates f or describing the center manifold C0 at ^ = 0. Letting ut be the solution of (18) with ^ = 0,we define z(t) = (q*,ut),and

W(t,9) = ut -2 Re [z(t)q(9)}.

On the center manifold C0 we have

W(t,9) = W(z,z,t),

(27) where

2 — —2

W(z,z,t) = W20 (9) 2 + Wu (9) — + W02 (9) — + ••• .

In fact, z and z are local coordinates for C0 in the direction of q and q*. Note that if ut is, we will deal with real solutions only. Since ^ = 0,

z(t) = (q*,ut) = {q*,A(^)ut + R(p)ut) = (q*>Aut) + (q*,Rut)

= ir0w0z + q* (0) •f(0,W (t, 0) + 2 Re [z (t) q (0)]).

Rewrite (37) as

z (t) = h0w0z + g (z, z),

— ^ — ^ ^ ^ A\

g(z,z) = g20 Y + 011ZZ + 0O2 2 + 021—+••• . (39)

t t (a) (b)

L(t) (c)

Figure 1: The positive equilibrium E, of system (3) is asymptotically stable.

From (18) and (38), we have W = ùt - zq- zq

AW-2 Re [q* (0) f (z,z) q(9)], -1 < 9 < 0, AW-2 Re [q* (0) f (z,z)q(9)] + f, 9 = 0.

W = AW + H(z,z,9),

H (z,z, 9) = H20 (9) J + Hn (9) zz + HQ 2 (0)y+ •

Taking the derivative of W with respect to t in (36), we have

W = Wz z + Wjz.

Substituting (36) and (38) into (43), we obtain

W = (W2QZ + Wx{z + ■■■) (îtquQz + g)

+ (Wn z + wQ2~z + ■■■) {-îtQwQz + g).

Then substituting (36) and (41) into (42), we have the following results:

W = (AW20 + H20)- + (AWU + Hu) zz

+ (AW02 +H02)j +

Comparing the coefficients of (44) with (45), the following equations hold:

(A-HTQ WQ)W2Q (9) = -H2Q (9), (46)

AWn (9) = -Hn (9). (47)

0.04 -

50 100 150 200 250

0 50 100 150 200 250

0 0.02 0.04 0.06 0.08 L(t)

Figure 2: The bifurcation periodic solution is stable.

Since ut = u(t + d) = W(z, z,d) + zq + z ■ q, we have

= (W(1) (z,z,d) Ut =(W(2) (z,z,9)

Thus, we can obtain

1 \ iu0e - ( 1 e 0 + z\

2 -2 <Pi (0) = z + z + W^ (0) y + W1(;) (0) zz + W^ (0) j,

<h (0) = zqi + zq! + W™ (0) 2 + W® (0) zz + w02 (0) — .

So, we have

Vi (0) <P2 (0) = (hz1 + q{z2 + (q1 + qj zz

+ (< + \< +W^q1 + \W% q^Z,

q>2 (0) = z2 + z2 + 2zz + (wH + 2W1(I))z2z, (p2 (0) = q(z2 + q^z2 + 2q1q(zz + (q-jW^ + 2q1w('^)) z2z.

It follows from (38) and (39) that

f (z,z) = (K(Z2 + K2zI + K3z2 + K4z2z

K( =-ßT0 (1 + q2( +2q(), K2 = -ß^0(2 + 2qih + 2qi + 2ql), K = -ßt0 (1 + q2i +2qi), K = - ß%0 (2WÏ? + W^ + 2Wl1)qi + W^qi

+ W21 + 2Wl( + q(W^ + 2qiW22)).

0.115 0.11 0.105 0.1 0.095 0.09 0.085 0.08 0.075

0.1 0.11 0.12 0.13 0.14 0.15 0.16

Figure 3: The positive equilibrium E* of system is asymptotically stable.

Then we have

g(z,z)=l-(1,q1 1 P

K, z2 + K2zz + K3z2 + K4z2z

Comparing the coefficients of the above equation with those in (41), we have

020 = ~K1, P

011 = ~K2, P

021 = ~KA. P

In what follows, we focus on the computation of W20(d) and W11 (d). For the expression of g21, we have

H (z, z,9)= -2 Re (q* (0) f (z, z) q(9)) + Rut

1 2 - 1 -2

^02OZ +011ZZ+^0O2Z +•••

1- -2 - - 1- 2

^02OZ +011ZZ+^0O2Z +•••

q(d) q(9)

+ Rut.

Comparing the coefficients of the above equation, we can obtain that

H20 (0) =-0201(d)-002Ï(d)> de [-1,0), (56)

Hu (d) = -guq(d)-guq(d), de [-1,0). (57)

Substituting (56) into (46) and (57) into (47), respectively, we get

W20 (0) = 2iT0^0W20 (0) + g20q (V) + 002q (d), wn (d) = gnq(d) + gnq(d). We can easily obtain the solutions of (58) as

W20 (d) = ^q (0) e'r°q(0) e-,r°a°e T0U0 3lT0^0

+ Eïe2ir°a°e

^ 1 (9) = ^ q(0)e' ITo^o

09 011 -

I To^o

-hoWo 8

0.25 |-.-.-.- 0.2

Figure 4: The bifurcation periodic solution is stable.

We will determine E1 and E2. Form the definition of A in (23), we have

W20 (0) = 2ir0u0W20 (0)-H20 (0),

iw0I - I e

-iw0I - I e

| eii0° 9d^(d))q(0) = 0,

| ° e~iw°dd^(d))q(0) = 0,

J d^(d)wn (o) = -Hu (0).

From (59), (56) and (57), we have

Ho (d) = -0201 (d) - M (V) + (K,0)T, (61)

Hn (d) = -gnq(d)-gnq(d) + (K2,0)T. (62) Substituting (59) and (61) into (62) and noticing that

we can obtain

(2iw0I - | e2iT°a°9dn (d)) E1 = K 0)T, (64)

which leads to

2iw0 ° ß°

2(g + S)(Y + 8) i5ixc-^ro ß 2(a + 8)(y + 8)

a + y + 8

a + y + 8 ) Et 2iw0 + y + S j

' ß 2(a + 8)(y + 8) ß 2(a + 8)(y + 8)\ ^

a + y + 8

a + y + 8 )e2 =

y + 8 J

It follows that

2iw0 - ß -

2(g + S)(y + S) ^ | . 2(g + S)(y + S)\-I --- + ö + ae 00 ß---- \

a + y + S

a + y + S 2iw0 + y + S

L(0) = 0.1 and B(0) = 0.1, Figure 3 demonstrates the evolutions from which it can be seen that the equilibrium is asymptotically stable. Second, we choose t = 4.2 > t0. For a set of initial conditions satisfying L(0) = 0.1 and B(0) = 0.1, the corresponding wave form and phase plots are shown in Figure 4, from which it is easy to see that a Hopf bifurcation

2(a + S)(y + 8) ^ 2(a + 8)(y + 8) \ /K -p-----h« + a p---- \ / a2

ß2 = ( a + y + 8 a + y + 8

Hence, we know W20 and then we can obtain g21. The following parameters can be calculated:

C1 (0) = 2mt (020011 - 2\0n\2 - 3^002 ^ + ~2'

Re [C! (0)}

T2 = -

Re [A' (To)}'

Im C (0)}+k Im {A' (to)}

ß2 = 2 Re [C (0)}, (67)

Theorem 3. Under the condition of Theorem 1, one has the following.

(1) p = 0 is Hopf bifurcation value of system (18).

(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 p2:if ¡i2 < 0, theperiodicsolutionsarestable; if P2 > 0, they are unstable.

4. Numerical Examples

In this section, some numerical examples of system (3) are presented to justify the previous theorem above.

Example 1. Consider system (3) with parameters a = 0.8, p = 0.8, y = 0.8, and S = 0.4. Then R0 = 1.1111, E„ = (0.06, 0.04), and (7) has one positive real root w = 0.3470. It follows by (11) that t0 = 3.5705. First, we choose t = 3 < t0. For a set of initial conditions satisfying L(0) = 0.1 and B(0) = 0.1, Figure 1 demonstrates the evolutions from which it can be seen that the equilibrium is asymptotically stable. Second, we choose t =3.7 > t0. 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.

Example 2. Consider system (3) with parameters a = 0.7, p = 0.8, y = 0.5, and S = 0.4. Then R0 = 1.2929, E, = (0.1274,0.0991), and (7) has one positive real root w = 0.4463. It follows by (11) that t0 = 4.0204. First, we choose t = 3.3 < t0. For a set of initial conditions satisfying

5. 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.

Acknowledgments

The authors were greatly indebted to the anonymous reviewers for their valuable suggestions. This paper was supported by the National Natural Science Foundation of China (no. 61170320), the Natural Science Foundation of Guangdong Province (no. S2011040002981), and Nature Science Foundation of Guangdong Medical College (B2012053).

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