# Dynamical Analysis of a Viral Infection Model with Delays in Computer Networks

Authors: Zizhen Zhang; Huizhong Yang
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper is devoted to the study of an SIRS computer virus propagation model with two
delays and multistate antivirus measures. We demonstrate that the system loses its stability
and a Hopf bifurcation occurs when the delay passes through the corresponding critical value
by choosing the possible combination of the two delays as the bifurcation parameter. Moreover,
the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are
determined by means of the center manifold theorem and the normal form theory. Finally,
some numerical simulations are performed to illustrate the obtained results.


---

## Body

## 1. Introduction

With the rapid development of computer technologies and network applications, the threat of computer viruses to the world would become increasingly serious. It is of vital importance to understand how computer viruses spread over computer network and to control the computer viruses’ propagation in computer networks. To this end, many mathematical models have been studied to illustrate the dynamical behavior of computer viruses spreading since Murray [1] suggested that computer viruses share some traits of biological viruses. In [2, 3], Kephart and White used the SIS model to describe the propagation of computer viruses. In [4], Zou et al. investigated how the spread of red worms is affected by the worm characteristics based on the SIR model. In [5, 6], Yuan et al. proposed the SEIR computer virus model and studied the dynamics of the model, respectively. In [7], Mishra and Pandey formulated an SEIRS model for the transmission of worms in computer network through vertical transmission. In addition, there are also some researchers who proposed the computer virus models with vaccination and quarantine strategy [8–10].In fact, many computer viruses have different kinds of delays when the viruses spread, such as latent period delay [11, 12], immunity period delay [12–15], and the delay due to the period that the anti-virus software needs to clean the viruses [6]. In [12], Feng et al. proposed the following computer virus propagation model with dual delays and multistate antivirus measures based on the classical SIR epidemic model in [16]:
(1)
d
S
t
d
t
=
p
A
-
β
S
t
-
τ
1
I
t
-
τ
1
-
μ
+
γ
S
t
+
δ
R
(
t
-
τ
2
)
,
d
I
(
t
)
d
t
=
β
S
(
t
-
τ
1
)
I
(
t
-
τ
1
)
-
(
μ
+
α
)
I
(
t
)
,
d
R
(
t
)
d
t
=
(
1
-
p
)
A
+
γ
S
(
t
)
+
α
I
(
t
)
-
δ
R
(
t
-
τ
2
)
-
μ
R
(
t
)
,
where S
(
t
), I
(
t
), and R
(
t
) represent the numbers of susceptible, infected, and recovered hosts in computer networks at time t, respectively. A is the number of the hosts which are attached to the computer networks and p is the proportion of the new hosts which are susceptible. μ is the death rate of the hosts. α, β, γ, and δ are the state transition rates between the classes S, I, and R. τ
1
≥
0 is the latent period of the computer viruses and τ
2
≥
0 is the temporary immune period of the recovered hosts. For the convenience of analysis, Feng et al. [12] let τ
1
=
τ
2; then, system (1) becomes the following form:
(2)
d
S
t
d
t
=
p
A
-
β
S
t
-
τ
I
t
-
τ
-
μ
+
γ
S
t
+
δ
R
(
t
-
τ
)
,
d
I
(
t
)
d
t
=
β
S
(
t
-
τ
)
I
(
t
-
τ
)
-
(
μ
+
α
)
I
(
t
)
,
d
R
(
t
)
d
t
=
(
1
-
p
)
A
+
γ
S
(
t
)
+
α
I
(
t
)
-
δ
R
(
t
-
τ
)
-
μ
R
(
t
)
.By regarding the time delayτ as the bifurcation parameter, Feng et al. [12] studied the existence and properties of Hopf bifurcation of system (2). As is known, it needs some time to clean the viruses in the infected hosts for the antivirus software. Therefore, it is reasonable to take into account the time delay due to the period that the antivirus software uses to clean the viruses in the infected hosts in system (2). To this end, we consider the following system with two delays:
(3)
d
S
t
d
t
=
p
A
-
β
S
t
-
τ
1
I
t
-
τ
1
-
μ
+
γ
S
t
+
δ
R
(
t
-
τ
1
)
,
d
I
(
t
)
d
t
=
β
S
(
t
-
τ
1
)
I
(
t
-
τ
1
)
-
μ
I
(
t
)
-
α
I
(
t
-
τ
2
)
,
d
R
t
d
t
=
1
-
p
A
+
γ
S
t
+
α
I
t
-
τ
2
-
δ
R
t
-
τ
1
-
μ
R
(
t
)
,
where τ
1
≥
0 is the time delay due to the latent period of the computer viruses and the temporary immune period of the recovered hosts. τ
2
≥
0 is the time delay due to the period that the antivirus software uses to clean the viruses in the infected hosts.The remaining materials of this paper are organized in this fashion: local stability and existence of local Hopf bifurcation are discussed in Section2. Properties of the Hopf bifurcation such as the direction and stability are investigated in Section 3. Some numerical simulations are carried out to verify the theoretical results in Section 4 and, finally, this work is summarized in Section 5.


## 2. Local Stability and Existence of Local Hopf Bifurcation

By direct computation, it can be concluded that ifR
0
=
(
(
μ
+
δ
)
p
A
β
+
(
1
-
p
)
β
δ
A
+
(
μ
+
α
)
δ
γ
)
/
(
μ
+
α
)
(
μ
+
γ
)
(
μ
+
δ
)
>
1, then system (3) has a unique positive equilibrium E
*
(
S
*
,
I
*
,
R
*
), where
(4)
S
*
=
μ
+
α
β
,
I
*
=
μ
+
δ
p
A
β
+
1
-
p
β
δ
A
+
μ
+
α
δ
γ
-
μ
+
α
μ
+
γ
μ
+
δ
×
β
(
μ
+
α
)
(
μ
+
δ
)
-
α
β
δ
-
1
,
R
*
=
(
1
-
p
)
A
+
α
I
*
+
γ
S
*
μ
+
δ
.
The characteristic equation of system (3) at E
* is
(5)
λ
-
a
11
-
b
11
e
-
λ
τ
1
-
b
12
e
-
λ
τ
1
-
b
13
e
-
λ
τ
1
-
b
21
e
-
λ
τ
1
λ
-
a
22
-
b
22
e
-
λ
τ
1
-
c
22
e
-
λ
τ
2
0
-
a
31
-
c
32
e
-
λ
τ
2
λ
-
a
33
-
b
33
e
-
λ
τ
1
=
0
,
from which one can obtain
(6)
λ
3
+
A
2
λ
2
+
A
1
λ
+
A
0
+
B
2
λ
2
+
B
1
λ
+
B
0
e
-
λ
τ
1
+
C
2
λ
2
+
C
1
λ
+
C
0
e
-
λ
τ
2
+
D
1
λ
+
D
0
e
-
λ
τ
1
+
τ
2
+
(
E
1
λ
+
E
0
)
e
-
2
λ
τ
1
+
F
0
e
-
λ
(
2
τ
1
+
τ
2
)
=
0
,
where
(7)
A
0
=
-
a
11
a
22
a
33
,
A
1
=
a
11
a
22
+
a
11
a
33
+
a
22
a
33
,
A
2
=
-
(
a
11
+
a
22
+
a
33
)
,
B
0
=
a
22
a
31
b
13
-
a
11
a
22
b
33
-
a
11
a
33
b
22
-
a
22
a
33
b
11
,
B
1
=
b
11
a
22
+
a
33
+
b
22
a
11
+
a
33
+
b
33
(
a
11
+
a
22
)
-
a
31
b
13
,
B
2
=
-
(
b
11
+
b
22
+
b
33
)
,
C
0
=
-
a
11
a
33
c
22
,
C
1
=
c
22
(
a
11
+
a
33
)
,
C
2
=
-
c
22
,
D
0
=
c
22
(
a
31
b
13
-
a
11
b
33
-
a
33
b
11
)
,
D
1
=
c
22
(
b
11
+
b
33
)
,
E
0
=
a
31
b
13
b
22
-
b
33
(
a
11
b
22
+
a
22
b
11
)
,
E
1
=
b
33
(
b
11
+
b
22
)
,
F
0
=
-
(
b
11
b
33
c
22
+
b
13
b
21
c
32
)
,
with
(8)
a
11
=
-
μ
+
γ
,
a
22
=
-
μ
,
a
31
=
γ
,
a
33
-
μ
,
b
11
=
-
β
I
*
,
b
12
=
-
β
S
*
,
b
13
=
δ
,
b
21
=
β
I
*
,
b
22
=
β
S
*
,
b
33
=
-
δ
,
c
22
=
-
α
,
c
32
=
α
.From the expressions ofb
11, b
13, b
21, b
33, c
22, and c
32, one can obtain F
0
=
0. Therefore, (6) can be transformed into the following form:
(9)
λ
3
+
A
2
λ
2
+
A
1
λ
+
A
0
+
B
2
λ
2
+
B
1
λ
+
B
0
e
-
λ
τ
1
+
C
2
λ
2
+
C
1
λ
+
C
0
e
-
λ
τ
2
+
D
1
λ
+
D
0
e
-
λ
τ
1
+
τ
2
+
(
E
1
λ
+
E
0
)
e
-
2
λ
τ
1
=
0
.Case 1 (τ
1
=
τ
2
=
0).
Whenτ
1
=
τ
2
=
0, (9) is equivalent to
(10)
λ
3
+
A
12
λ
2
+
A
11
λ
+
A
10
=
0
,
where
(11)
A
10
=
A
0
+
B
0
+
C
0
+
D
0
+
E
0
,
A
11
=
A
1
+
B
1
+
C
1
+
D
1
+
E
1
,
A
12
=
A
2
+
B
2
+
C
2
.
It is easy to get thatA
12
=
2
μ
+
γ
+
δ
+
β
I
*
>
0. Therefore, according to the Routh-Hurwitz criterion, we can conclude that if A
12
A
11
>
A
10
>
0, then the positive equilibrium E
* of system (3) is locally asymptotically stable when τ
1
=
τ
2
=
0.Case 2 (τ
1
>
0, τ
2
=
0).
Whenτ
1
>
0 and τ
2
=
0, (9) becomes the following:
(12)
λ
3
+
A
22
λ
2
+
A
21
λ
+
A
20
+
B
22
λ
2
+
B
21
λ
+
B
20
e
-
λ
τ
1
+
(
E
21
λ
+
E
20
)
e
-
2
λ
τ
1
=
0
,
where
(13)
A
20
=
A
0
+
C
0
,
A
21
=
A
1
+
C
1
,
A
22
=
A
2
+
C
2
,
B
20
=
B
0
+
D
0
,
B
21
=
B
1
+
D
1
,
B
22
=
B
2
,
E
20
=
E
0
,
E
21
=
E
1
.
Multiplying e
λ
τ
1 on both sides of (12), it is easy to get
(14)
B
22
λ
2
+
B
21
λ
+
B
20
+
λ
3
+
A
22
λ
2
+
A
21
λ
+
A
20
e
λ
τ
1
+
(
E
21
λ
+
E
20
)
e
-
λ
τ
1
=
0
.
Let λ
=
i
ω
1
(
ω
1
>
0
) be the root of (14). Then,
(15)
A
20
+
E
20
+
A
22
ω
1
2
cos
⁡
τ
1
ω
1
-
(
A
21
ω
1
-
E
21
ω
1
-
ω
1
3
)
sin
τ
1
ω
1
=
B
22
ω
1
2
-
B
20
,
A
20
-
E
20
+
A
22
ω
1
2
sin
τ
1
ω
1
+
(
A
21
ω
1
+
E
21
ω
1
-
ω
1
3
)
cos
⁡
τ
1
ω
1
=
-
B
21
ω
1
.
Then, one can obtain
(16)
cos
⁡
τ
1
ω
1
=
p
24
ω
1
4
+
p
22
ω
1
2
+
p
20
ω
1
6
+
q
24
ω
1
4
+
q
22
ω
1
2
+
q
20
,
sin
τ
1
ω
1
=
p
25
ω
1
5
+
p
23
ω
1
3
+
p
21
ω
1
ω
1
6
+
q
24
ω
1
4
+
q
22
ω
1
2
+
q
20
,
where
(17)
p
20
=
B
20
(
E
20
-
A
20
)
,
p
21
=
B
20
(
A
21
+
E
21
)
-
B
21
(
A
20
+
E
20
)
,
P
22
=
B
22
(
A
20
-
E
20
)
-
B
21
(
A
21
-
E
21
)
+
A
22
B
00
,
p
23
=
A
22
B
21
-
B
20
-
B
22
(
A
21
+
E
21
)
,
p
24
=
B
21
-
A
22
B
22
,
p
25
=
B
22
,
q
20
=
A
20
2
-
E
20
2
,
q
22
=
A
21
2
-
E
21
2
-
2
A
20
A
22
,
q
24
=
A
22
2
-
2
A
21
.
Since cos
⁡
2
τ
1
ω
1
+
sin
2
τ
1
ω
1
=
1, we have
(18)
ω
1
12
+
e
25
ω
1
10
+
e
24
ω
1
8
+
e
23
ω
1
6
+
e
22
ω
1
4
+
e
21
ω
1
2
+
e
20
=
0
,
where
(19)
e
20
=
q
20
2
-
p
20
2
,
e
21
=
2
q
20
q
22
-
2
p
20
p
22
-
p
21
2
,
e
22
=
q
22
2
-
p
22
2
+
2
q
20
q
24
-
2
p
20
p
24
-
2
p
21
p
23
,
e
23
=
2
q
20
+
2
q
22
q
24
-
2
p
22
p
24
-
2
p
21
p
25
-
p
23
2
,
e
24
=
q
24
2
-
p
24
2
+
2
q
22
-
2
p
23
p
25
,
e
25
=
2
q
24
-
p
25
2
.
Let ω
1
2
=
v
1; then, (18) becomes
(20)
v
1
6
+
e
25
v
1
5
+
e
24
v
1
4
+
e
23
v
1
3
+
e
22
v
1
2
+
e
21
v
1
+
e
20
=
0
.
In order to give the main results in the present paper, we make the following assumption.(
H
21
)
Equation (20) has at least one positive real root.
If the condition(
H
21
) holds, then there exists a positive root v
10 of (20) which can make (14) have a pair of purely imaginary roots ±
i
ω
10
=
±
i
v
10. For ω
10, the corresponding critical value of delay is
(21)
τ
10
=
1
ω
10
arccos
p
24
ω
10
4
+
p
22
ω
10
2
+
p
20
ω
10
6
+
q
24
ω
10
4
+
q
22
ω
10
2
+
q
20
.
Differentiating (14) with respect to τ
1, we get
(22)
d
λ
d
τ
1
-
1
=
-
2
B
22
λ
+
B
21
+
3
λ
2
+
2
A
22
λ
+
A
21
e
λ
τ
1
d
λ
d
τ
1
-
1
+
E
21
e
-
λ
τ
1
d
λ
d
τ
1
-
1
×
E
21
λ
2
+
E
20
λ
e
-
λ
τ
1
d
λ
d
τ
1
-
1
-
λ
4
+
A
22
λ
3
+
A
21
λ
2
+
A
20
λ
e
λ
τ
1
-
1
d
λ
d
τ
1
-
1
-
τ
1
λ
.
Thus,
(23)
Re
d
λ
d
τ
1
τ
1
=
τ
10
-
1
=
P
2
R
Q
2
R
+
P
2
I
Q
2
I
Q
2
R
2
+
Q
2
I
2
,
where
(24)
P
2
R
=
A
21
+
E
21
-
3
ω
10
2
cos
⁡
τ
10
ω
10
-
2
A
22
ω
10
sin
τ
10
ω
10
+
B
21
,
P
2
I
=
A
21
-
E
21
-
3
ω
10
2
sin
τ
10
ω
10
+
2
A
22
ω
10
cos
⁡
τ
10
ω
10
+
2
B
22
ω
10
,
Q
2
R
=
A
21
ω
10
2
-
E
21
ω
10
2
-
ω
10
4
cos
⁡
τ
10
ω
10
-
(
A
22
ω
10
3
-
A
20
ω
10
-
E
20
ω
10
)
sin
τ
10
ω
10
,
Q
2
I
=
A
21
ω
10
2
+
E
21
ω
10
2
-
ω
10
4
sin
τ
10
ω
10
+
(
A
22
ω
10
3
-
A
20
ω
10
+
E
20
ω
10
)
cos
⁡
τ
10
ω
10
.
It is obvious that if the condition(
H
21
)
P
2
R
Q
2
R
+
P
2
I
Q
2
I
≠
0 holds, then Re
[
d
λ
/
d
τ
1
]
τ
=
τ
10
-
1
≠
0. According to the Hopf bifurcation theorem in [17], the following results hold.Theorem 1.
If the conditions(
H
21
)-(
H
22
) hold, the positive equilibrium E
*
(
S
*
,
I
*
,
R
*
) of system (3) is locally asymptotically stable for τ
1
∈
[
0
,
τ
10
) and system (3) undergoes a Hopf bifurcation at the positive equilibrium E
*
(
S
*
,
I
*
,
R
*
) when τ
1
=
τ
10.Case 3 (τ
1
=
0, τ
2
>
0).
Whenτ
1
=
0 and τ
2
>
0, (9) becomes
(25)
λ
3
+
A
32
λ
2
+
A
31
λ
+
A
30
+
(
C
32
λ
2
+
C
31
λ
+
C
30
)
e
-
λ
τ
2
=
0
,
where
(26)
A
30
=
A
0
+
B
0
+
E
0
,
A
31
=
A
1
+
B
1
+
E
1
,
A
32
=
A
2
+
B
2
,
C
30
=
C
0
+
D
0
,
C
31
=
C
1
+
D
1
,
C
32
=
C
2
.
Let λ
=
i
ω
2
(
ω
2
>
0
) be the root of (25). Then,
(27)
C
31
ω
2
sin
τ
2
ω
2
+
(
C
30
-
C
32
ω
2
2
)
cos
⁡
τ
2
ω
2
=
A
32
ω
2
2
-
A
30
,
C
31
ω
2
cos
⁡
τ
2
ω
2
-
(
C
30
-
C
32
ω
2
2
)
sin
τ
2
ω
2
=
ω
2
3
-
A
31
ω
2
,
which follows that
(28)
ω
2
6
+
e
32
ω
2
4
+
e
31
ω
2
2
+
e
30
=
0
,
with
(29)
e
30
=
A
30
2
-
C
30
2
,
e
31
=
A
31
2
-
C
31
2
-
2
A
30
A
32
+
2
C
30
C
32
,
e
32
=
A
32
2
-
C
32
2
-
2
A
31
.
Let ω
2
2
=
v
2; then, (28) becomes
(30)
v
2
3
+
e
32
v
2
2
+
e
31
v
2
+
e
30
=
0
.
Let
(31)
f
2
(
v
2
)
=
v
2
3
+
e
32
v
2
2
+
e
31
v
2
+
e
30
.
Discussion about the roots of (30) is similar to that in [18].Lemma 2.
(i) Ife
30
<
0, then (30) has at least one positive root.
(ii) Ife
30
≥
0 and e
32
2
-
3
e
31
≤
0, then (30) has no positive root.
(iii) Ife
30
≥
0 and e
32
2
-
3
e
31
>
0, then (30) has positive root if and only if v
2
*
=
(
-
e
32
+
e
32
2
-
3
e
31
)
/
3
>
0 and f
2
(
v
2
*
)
<
0.In what follows, we suppose that the coefficients in (30) satisfy the following condition:(
H
31
)
:
(a)e
30
<
0 or (b) e
30
≥
0, e
32
2
-
3
e
31
>
0, v
2
*
=
(
-
e
32
+
e
32
2
-
3
e
31
)
/
3
>
0, and f
2
(
v
2
*
)
<
0.If the condition(
H
31
) holds, we know that there exists a positive root v
20 of (30) such that (25) has a pair of purely imaginary roots ±
i
ω
20
=
±
i
v
20. For ω
20, the corresponding critical value of time delay is
(32)
τ
20
=
1
ω
20
arccos
⁡
(
C
31
-
A
32
C
32
)
ω
20
4
C
32
ω
20
4
+
C
31
2
-
2
C
30
C
32
ω
20
2
+
C
30
2
-
1
+
A
30
C
32
-
A
31
C
31
+
A
32
C
30
ω
20
2
-
A
30
C
30
ω
20
4
C
31
-
A
32
C
32
ω
20
4
×
C
32
ω
20
4
+
(
C
31
2
-
2
C
30
C
32
)
ω
20
2
+
C
30
2
-
1
.
Differentiating two sides of (25) with respect to τ
2, we have
(33)
d
λ
d
τ
2
-
1
=
-
3
λ
2
+
2
A
32
λ
+
A
31
λ
λ
3
+
A
32
λ
2
+
A
31
λ
+
A
30
+
2
C
32
λ
+
C
31
λ
C
32
λ
2
+
C
31
λ
+
C
30
-
τ
2
λ
.
Thus,
(34)
Re
d
λ
d
τ
2
τ
2
=
τ
20
-
1
=
f
2
′
(
v
2
*
)
B
21
ω
10
-
B
23
ω
10
3
2
+
B
24
ω
20
4
-
B
22
ω
20
2
+
B
20
2
,
where f
2
(
v
2
)
=
v
2
3
+
e
32
v
2
2
+
e
31
v
2
+
e
30 and v
2
*
=
ω
20
2.Obviously, if the condition(
H
32
)
f
2
′
(
v
2
*
)
≠
0 holds, then Re
[
d
λ
/
d
τ
2
]
τ
2
=
τ
20
-
1
≠
0. According to the Hopf bifurcation theorem in [17], the following results hold.Theorem 3.
If the conditions(
H
31
)-(
H
32
) hold, the positive equilibrium E
*
(
S
*
,
I
*
,
R
*
) of system (3) is locally asymptotically stable for τ
2
∈
[
0
,
τ
20
) and system (3) undergoes a Hopf bifurcation at E
*
(
S
*
,
I
*
,
R
*
) when τ
2
=
τ
20.Case 4 (τ
1
>
0, τ
2
>
0, τ
2
∈
(
0
,
τ
20
)).
We consider (9) with τ
2 in its stable interval and choose τ
1 as a bifurcation parameter. Multiplying by e
λ
τ
1, (9) becomes
(35)
B
2
λ
2
+
B
1
λ
+
B
0
+
λ
3
+
A
2
λ
2
+
A
1
λ
+
A
0
e
λ
τ
1
+
E
1
λ
+
E
0
e
-
λ
τ
1
+
D
1
λ
+
D
0
e
-
λ
τ
2
+
(
C
2
λ
2
+
C
1
λ
+
C
0
)
e
λ
(
τ
1
-
τ
2
)
=
0
.
Let λ
=
i
ω
1
*
(
ω
1
*
>
0
) be the root of (35). Then,
(36)
M
41
cos
⁡
τ
1
ω
1
*
-
M
42
sin
τ
1
ω
1
*
=
M
43
,
M
44
sin
τ
1
ω
1
*
+
M
45
cos
⁡
τ
1
ω
1
*
=
M
46
,
where
(37)
M
41
=
A
0
+
E
0
-
A
2
ω
1
*
2
+
C
0
-
C
2
ω
1
*
2
cos
⁡
τ
2
ω
1
*
+
C
1
ω
1
*
sin
τ
2
ω
1
*
,
M
42
=
A
1
ω
1
*
-
E
1
ω
1
*
-
ω
1
*
3
-
C
0
-
C
2
ω
1
*
2
sin
τ
2
ω
1
*
+
C
1
ω
1
*
cos
⁡
τ
2
ω
1
*
,
M
43
=
B
2
ω
1
*
2
-
B
0
-
D
1
ω
1
*
sin
τ
2
ω
1
*
-
D
0
cos
⁡
τ
2
ω
1
*
,
M
44
=
A
0
-
E
0
-
A
2
ω
1
*
2
+
C
0
-
C
2
ω
1
*
2
cos
⁡
τ
2
ω
1
*
+
C
1
ω
1
*
sin
τ
2
ω
1
*
,
M
45
=
A
1
ω
1
*
+
E
1
ω
1
*
-
ω
1
*
3
-
C
0
-
C
2
ω
1
*
2
sin
τ
2
ω
1
*
+
C
1
ω
1
*
cos
⁡
τ
2
ω
1
*
,
M
46
=
-
B
1
ω
1
*
+
D
0
sin
τ
2
ω
1
*
-
D
1
ω
1
*
cos
⁡
τ
2
ω
1
*
.
Then, we can obtain
(38)
cos
⁡
τ
1
ω
1
*
=
h
40
(
ω
1
*
)
+
h
41
(
ω
1
*
)
cos
⁡
τ
2
ω
1
*
+
h
42
(
ω
1
*
)
sin
τ
2
ω
1
*
g
40
(
ω
1
*
)
+
g
41
(
ω
1
*
)
cos
⁡
τ
2
ω
1
*
+
g
42
(
ω
1
*
)
sin
τ
2
ω
1
*
,
sin
τ
1
ω
1
*
=
h
40
′
(
ω
1
*
)
+
h
41
′
(
ω
1
*
)
cos
⁡
τ
2
ω
1
*
+
h
42
′
(
ω
1
*
)
sin
τ
2
ω
1
*
g
40
(
ω
1
*
)
+
g
41
(
ω
1
*
)
cos
⁡
τ
2
ω
1
*
+
g
42
(
ω
1
*
)
sin
τ
2
ω
1
*
,
where
(39)
g
40
ω
1
*
=
ω
1
*
6
+
A
2
2
+
C
2
2
-
2
A
1
ω
1
*
4
+
A
1
2
+
C
1
2
-
E
1
2
-
2
A
0
A
2
-
2
C
0
C
2
ω
1
*
2
+
A
0
2
+
C
0
2
-
C
0
2
,
g
41
ω
1
*
=
2
A
2
C
2
-
C
1
ω
1
*
4
+
2
A
1
C
1
-
A
0
C
2
-
A
2
C
0
ω
1
*
2
+
2
A
0
C
0
,
g
42
ω
1
*
=
-
2
C
2
ω
1
*
5
+
2
A
1
C
2
-
A
2
C
1
-
C
0
ω
1
*
3
+
2
(
A
0
C
1
-
A
1
C
0
)
ω
1
*
,
h
40
ω
1
*
=
B
1
+
A
2
B
2
ω
1
*
4
+
A
2
B
0
-
C
1
D
1
+
C
2
D
0
-
B
1
A
1
-
E
1
+
B
2
A
0
-
E
0
ω
1
*
2
+
B
0
(
E
0
-
A
0
)
-
C
0
D
0
,
h
41
ω
1
*
=
D
1
-
B
2
C
2
ω
1
*
4
+
A
2
D
0
+
B
0
C
2
-
B
1
C
1
+
B
2
C
0
+
D
1
E
1
-
A
1
ω
1
*
2
+
D
0
(
E
0
-
A
0
)
-
B
0
C
0
,
h
42
ω
1
*
=
A
2
D
1
-
B
1
C
2
+
B
2
C
1
-
D
0
ω
1
*
3
+
D
0
(
A
1
-
E
1
)
-
D
1
(
A
0
-
E
0
)
+
B
1
C
0
-
B
0
C
1
ω
1
*
,
h
40
′
ω
1
*
=
B
2
ω
1
*
5
+
A
2
B
1
+
C
2
D
1
-
B
0
-
B
2
A
1
+
E
1
ω
1
*
3
+
B
0
(
A
1
+
E
1
)
-
B
1
(
A
0
+
E
0
)
-
C
0
D
1
+
C
1
D
0
ω
1
*
,
h
41
′
ω
1
*
=
A
2
D
1
+
B
1
C
2
-
B
2
C
1
-
D
0
ω
1
*
3
+
D
0
(
A
1
+
E
1
)
-
D
1
(
A
0
+
E
0
)
+
B
0
C
1
-
B
1
C
0
ω
1
*
,
h
42
′
ω
1
*
=
-
B
2
C
2
+
D
1
ω
1
*
4
+
B
0
C
2
-
B
1
C
1
-
A
2
D
0
+
B
2
C
0
+
D
1
A
1
+
E
1
ω
1
*
2
+
D
0
(
A
0
+
E
0
)
-
B
0
C
0
.
Then, we can get a function with respect to ω
1
*:
(40)
cos
⁡
2
τ
1
ω
1
*
+
sin
2
τ
1
ω
1
*
=
1
.
Next, we suppose that (
H
41
):   (40) has at least one positive real root.
If the condition(
H
41
) holds, then there exists a ω
10
* such that (35) has a pair of purely imaginary roots ±
i
ω
10
*. For ω
10
*, the corresponding critical value of time delay is
(41)
τ
10
*
=
1
ω
10
*
arccos
h
40
ω
10
*
+
h
41
ω
10
*
cos
⁡
τ
2
ω
10
*
+
g
42
ω
10
*
sin
τ
2
ω
10
*
-
1
+
h
42
(
ω
10
*
)
sin
τ
2
ω
10
*
×
g
40
(
ω
10
*
)
+
g
41
(
ω
10
*
)
cos
⁡
τ
2
ω
10
*
+
g
42
(
ω
10
*
)
sin
τ
2
ω
10
*
-
1
.
Taking the derivative with respect to τ
1 in (35), we get
(42)
d
λ
d
τ
1
-
1
=
g
41
(
λ
)
g
42
(
λ
)
-
τ
1
λ
,
where
(43)
g
41
λ
=
2
B
2
λ
+
B
1
+
E
1
e
-
λ
τ
1
+
3
λ
2
+
2
A
2
λ
+
A
1
e
λ
τ
1
-
τ
2
D
1
λ
-
D
1
+
τ
2
D
0
e
-
λ
τ
2
-
[
τ
2
C
2
λ
2
-
(
2
C
2
-
τ
2
C
1
)
λ
+
C
1
-
τ
2
C
0
]
e
λ
(
τ
1
-
τ
2
)
,
g
42
λ
=
E
1
λ
2
+
E
0
λ
e
-
λ
τ
1
-
λ
4
+
A
2
λ
3
+
A
1
λ
2
+
A
0
λ
e
λ
τ
1
-
(
C
2
λ
3
+
C
1
λ
2
+
C
0
λ
)
e
λ
(
τ
1
-
τ
2
)
.
Thus,
(44)
Re
d
λ
d
τ
1
τ
1
=
τ
10
*
-
1
=
P
4
R
Q
4
R
+
P
4
I
Q
4
I
Q
4
R
2
+
Q
4
I
2
,
where
(45)
P
4
R
=
B
1
+
D
1
-
τ
2
D
0
cos
⁡
τ
2
ω
10
*
-
τ
2
D
1
ω
10
*
sin
τ
2
ω
10
*
+
A
1
+
E
1
-
3
ω
10
*
2
+
τ
2
C
2
ω
10
*
2
+
τ
2
C
0
-
C
1
cos
⁡
τ
2
ω
10
*
+
2
C
2
-
τ
2
C
1
sin
τ
2
ω
10
*
ω
10
*
2
cos
⁡
τ
10
*
ω
10
*
+
τ
2
C
2
ω
10
*
2
+
τ
2
C
0
-
C
1
sin
τ
2
ω
10
*
-
2
C
2
-
τ
2
C
1
-
2
A
2
ω
10
*
ω
10
*
2
sin
τ
10
*
ω
10
*
,
P
4
I
=
2
B
2
ω
10
*
-
D
1
-
τ
2
D
0
sin
τ
2
ω
10
*
-
τ
1
D
1
ω
10
*
cos
⁡
τ
2
ω
10
*
+
A
1
-
E
1
-
3
ω
10
*
2
+
τ
2
C
2
ω
10
*
2
+
τ
2
C
0
-
C
1
cos
⁡
τ
2
ω
10
*
+
2
C
2
-
τ
2
C
1
sin
τ
2
ω
10
*
ω
10
*
2
sin
τ
10
*
ω
10
*
+
2
A
2
ω
10
*
-
τ
2
C
2
ω
10
*
2
+
τ
2
C
0
-
C
1
cos
⁡
τ
2
ω
10
*
+
2
C
2
-
τ
2
C
1
sin
τ
2
ω
10
*
ω
10
*
2
cos
⁡
τ
10
*
ω
10
*
,
Q
4
R
=
A
0
ω
10
*
+
E
0
ω
10
*
-
A
2
ω
10
*
3
+
C
1
ω
10
*
2
sin
τ
2
ω
10
*
-
C
2
ω
10
*
3
-
C
0
ω
10
*
cos
⁡
τ
2
ω
10
*
sin
τ
10
*
ω
10
*
+
A
1
ω
10
*
2
-
E
1
ω
10
*
2
-
ω
10
*
4
+
C
1
ω
10
*
2
cos
⁡
τ
2
ω
10
*
+
C
2
ω
10
*
3
-
C
0
ω
10
*
sin
τ
2
ω
10
*
cos
⁡
τ
10
*
ω
10
*
,
Q
4
I
=
A
2
ω
10
*
3
-
A
0
ω
10
*
+
E
0
ω
10
*
-
C
1
ω
10
*
2
sin
τ
2
ω
10
*
+
C
2
ω
10
*
3
-
C
0
ω
10
*
cos
⁡
τ
2
ω
10
*
cos
⁡
τ
10
*
ω
10
*
+
A
1
ω
10
*
2
+
E
1
ω
10
*
2
-
ω
10
*
4
+
C
1
ω
10
*
2
cos
⁡
τ
2
ω
10
*
+
C
2
ω
10
*
3
-
C
0
ω
10
*
sin
τ
2
ω
10
*
sin
τ
10
*
ω
10
*
.
Thus, if the condition(
H
42
)
P
4
R
Q
4
R
+
P
4
I
Q
4
I
≠
0 holds, then Re
[
d
λ
/
d
τ
]
τ
1
=
τ
10
*
-
1
≠
0, which implies that the transversality condition is satisfied. According to the Hopf bifurcation theorem in [17], we can conclude the discussions above as follows.Theorem 4.
If the conditions(
H
41
)-(
H
42
) hold and τ
2
∈
(
0
,
τ
20
), the positive equilibrium E
*
(
S
*
,
I
*
,
R
*
) of system (3) is locally asymptotically stable for τ
1
∈
[
0
,
τ
10
*
) and system (3) undergoes a Hopf bifurcation at E
*
(
S
*
,
I
*
,
R
*
) when τ
1
=
τ
10
*.Case 5 (τ
1
>
0, τ
2
>
0 and τ
1
∈
(
0
,
τ
10
)).
We consider (9) with τ
1 in its stable interval and τ
2 is considered as a bifurcation parameter.
Letλ
=
i
ω
2
*
(
ω
2
*
>
0
) be the root of (9). Then,
(46)
M
51
sin
τ
2
ω
2
*
+
M
52
cos
⁡
τ
2
ω
2
*
=
M
53
,
M
51
cos
⁡
τ
2
ω
2
*
-
M
52
sin
τ
2
ω
2
*
=
M
54
,
where
(47)
M
51
=
C
1
ω
2
*
-
D
0
sin
τ
1
ω
2
*
+
D
1
ω
2
*
cos
⁡
τ
1
ω
2
*
,
M
52
=
C
0
-
C
2
ω
2
*
2
+
D
0
cos
⁡
τ
1
ω
2
*
+
D
1
ω
2
*
sin
τ
1
ω
2
*
,
M
53
=
A
2
ω
2
*
2
-
A
0
-
B
1
ω
2
*
sin
τ
1
ω
2
*
-
B
0
-
B
2
ω
2
*
2
cos
⁡
τ
1
ω
2
*
-
E
1
ω
2
*
sin
2
τ
1
ω
2
*
-
E
0
cos
⁡
2
τ
1
ω
2
*
,
M
54
=
ω
2
*
3
-
A
1
ω
2
*
-
B
1
ω
2
*
cos
⁡
τ
1
ω
2
*
+
B
0
-
B
2
ω
2
*
2
sin
τ
1
ω
2
*
-
E
1
ω
2
*
cos
⁡
2
τ
1
ω
2
*
+
E
0
sin
2
τ
1
ω
2
*
.
Then, we have
(48)
g
50
ω
2
*
+
g
51
ω
2
*
cos
⁡
τ
1
ω
2
*
+
g
52
ω
2
*
sin
τ
1
ω
2
*
+
g
53
(
ω
2
*
)
cos
⁡
2
τ
1
ω
2
*
+
g
54
(
ω
2
*
)
sin
2
τ
1
ω
2
*
=
0
,
where
(49)
g
50
ω
2
*
=
ω
2
*
6
+
A
2
2
+
B
2
2
-
C
2
2
-
2
A
1
ω
2
*
4
+
A
1
2
+
B
1
2
-
C
1
2
-
D
1
2
-
E
1
2
-
2
A
0
A
2
-
2
B
0
B
2
+
2
C
0
C
2
E
1
2
ω
2
*
2
+
A
0
2
-
C
0
2
-
D
0
2
+
E
0
2
,
g
51
ω
2
*
=
2
A
2
B
2
-
B
1
ω
2
*
4
+
2
A
1
B
1
-
A
0
B
2
-
A
2
B
0
-
C
1
D
1
+
C
2
D
0
ω
2
*
2
+
2
(
A
0
B
0
-
C
0
D
0
)
,
g
52
ω
2
*
=
-
2
B
2
ω
2
*
5
+
2
A
1
B
2
-
A
2
B
1
+
C
2
D
1
+
B
0
ω
2
*
3
+
2
(
A
0
B
1
-
A
1
B
0
-
C
0
D
1
+
C
1
D
0
)
ω
2
*
,
g
53
(
ω
2
*
)
=
-
2
E
1
ω
2
*
4
+
2
(
A
1
E
1
-
A
2
E
0
)
ω
2
*
2
+
2
A
0
E
0
,
g
54
ω
2
*
=
2
E
0
-
A
2
E
1
ω
2
*
3
+
2
A
0
E
1
-
A
1
E
0
ω
2
*
.
Similar to Case4, we suppose that (
H
51
):   (48) has at least one positive real root. If the condition (
H
51
) holds, then there exists a ω
20
* such that (9) has a pair of purely imaginary roots ±
i
ω
20
*. For ω
20
*, the corresponding critical value of time delay is
(50)
τ
20
*
=
1
ω
20
*
arccos
M
51
×
M
54
+
M
52
×
M
53
M
51
2
+
M
52
2
τ
2
=
τ
20
*
.
Differentiating (9) with respect to τ
2, we have
(51)
d
λ
d
τ
2
-
1
=
g
51
(
λ
)
g
52
(
λ
)
-
τ
2
λ
,
where
(52)
g
51
λ
=
3
λ
2
+
2
A
2
λ
+
A
1
+
2
B
2
-
τ
1
B
1
λ
-
τ
1
B
2
λ
2
+
B
1
-
τ
1
B
0
e
-
λ
τ
1
+
2
C
2
λ
+
C
1
e
-
λ
τ
2
+
D
1
-
τ
1
D
0
-
τ
1
D
1
λ
e
-
λ
τ
1
+
τ
2
+
(
E
1
-
2
τ
1
E
0
-
2
τ
1
E
1
λ
)
e
-
2
λ
τ
1
,
g
52
λ
=
C
2
λ
3
+
C
1
λ
2
+
C
0
λ
e
-
λ
τ
2
+
D
1
λ
2
+
D
0
λ
e
-
λ
(
τ
1
+
τ
2
)
.
Define
(53)
Re
d
λ
d
τ
2
τ
2
=
τ
20
*
-
1
=
P
5
R
Q
5
R
+
P
5
I
Q
5
I
Q
5
R
2
+
Q
5
I
2
.
If the condition(
H
52
)
P
5
R
Q
5
R
+
P
5
I
Q
5
I
≠
0 holds, then Re
[
d
λ
/
d
τ
1
]
τ
2
=
τ
20
*
-
1
≠
0. Therefore, according to the Hopf bifurcation theorem in [17], we can conclude the discussions above as follows.Theorem 5.
If the conditions(
H
51
)-(
H
52
) hold and τ
1
∈
(
0
,
τ
10
), the positive equilibrium E
*
(
S
*
,
I
*
,
R
*
) of system (3) is locally asymptotically stable for τ
2
∈
[
0
,
τ
20
*
) and system (3) undergoes a Hopf bifurcation at E
*
(
S
*
,
I
*
,
R
*
) when τ
2
=
τ
20
*.


## 3. Direction and Stability of the Hopf Bifurcation

In this section, we determine the properties of the Hopf bifurcation of system (3) with respect to τ
2 for τ
1
∈
(
0
,
τ
10
). Throughout this section, we assume that τ
1
*
<
τ
20
*, where τ
1
∈
(
0
,
τ
10
).Letτ
2
=
τ
20
*
+
μ, μ
∈
R; then, μ
=
0 is the Hopf bifurcation value of system (3). Rescale the time delay t
→
(
t
/
τ
2
). Let u
1
(
t
)
=
S
(
t
)
-
S
*, let u
2
(
t
)
=
I
(
t
)
-
I
*, and let u
3
(
t
)
=
R
(
t
)
-
R
*; then, system (3) can be transformed into an FDE in C
=
C
(
[
-
1,0
]
,
R
3
):
(54)
u
˙
t
=
L
μ
u
t
+
F
μ
,
u
t
,
where u
(
t
)
=
(
u
1
(
t
)
,
u
2
(
t
)
,
u
3
(
t
)
)
T and L
μ
:
C
→
R
3 and F
:
R
×
C
→
R
3 are given, respectively, by
(55)
L
μ
ϕ
=
τ
20
*
+
μ
A
′
ϕ
0
+
B
′
ϕ
-
τ
1
*
τ
20
*
+
C
′
ϕ
-
1
,
F
(
μ
,
ϕ
)
=
(
τ
20
*
+
μ
)
-
β
ϕ
1
(
-
τ
1
*
τ
20
*
)
ϕ
2
(
-
τ
1
*
τ
20
*
)
β
ϕ
1
(
-
τ
1
*
τ
20
*
)
ϕ
2
(
-
τ
1
*
τ
20
*
)
0
,
with
(56)
A
′
=
a
11
0
0
0
a
22
0
a
31
0
a
33
,
B
′
=
b
11
b
12
b
13
b
21
b
22
0
0
0
b
33
,
C
′
=
0
0
0
0
c
22
0
0
c
32
0
.By the Riesz representation theorem, there exists a functionη
(
θ
,
μ
) of bounded variation for θ
∈
[
-
1,0
] such that
(57)
L
μ
ϕ
=
∫
-
1
0
‍
d
η
(
θ
,
μ
)
ϕ
(
θ
)
,
ϕ
∈
C
(
[
-
1,0
]
,
R
3
)
.
In fact, we can choose
(58)
η
(
θ
,
μ
)
=
(
τ
20
*
+
μ
)
(
A
′
+
B
′
+
C
′
)
,
θ
=
0
,
(
τ
20
*
+
μ
)
(
B
′
+
C
′
)
,
θ
∈
-
τ
1
*
τ
20
*
,
0
,
(
τ
20
*
+
μ
)
C
′
,
θ
∈
-
1
,
-
τ
1
*
τ
20
*
,
0
,
θ
=
-
1
.
For ϕ
∈
C
(
[
-
1,0
]
,
R
3
), we define
(59)
A
(
μ
)
ϕ
=
d
ϕ
(
θ
)
d
θ
,
-
1
≤
θ
<
0
,
∫
-
1
0
‍
d
η
(
θ
,
μ
)
ϕ
(
θ
)
,
θ
=
0
,
R
(
μ
)
ϕ
=
0
,
-
1
≤
θ
<
0
,
F
(
μ
,
ϕ
)
,
θ
=
0
.
Then, system (54) is equivalent to
(60)
u
˙
t
=
A
μ
u
t
+
R
μ
u
t
,
where u
t
=
u
(
t
+
θ
) for θ
∈
[
-
1,0
].Forφ
∈
C
(
[
-
1,0
]
,
(
R
3
)
*
), define
(61)
A
*
(
φ
)
=
-
d
φ
(
s
)
d
s
,
0
<
s
≤
1
,
∫
-
1
0
‍
d
η
T
(
s
,
0
)
φ
(
-
s
)
,
s
=
0
,
and the bilinear form
(62)
φ
s
,
ϕ
θ
=
φ
¯
0
ϕ
0
-
∫
θ
=
-
1
0
‍
∫
ξ
=
0
θ
‍
φ
¯
ξ
-
θ
d
η
θ
ϕ
ξ
d
ξ
,
where η
(
θ
)
=
η
(
θ
,
0
).Letq
(
θ
)
=
(
1
,
q
2
,
q
3
)
T
e
i
ω
20
*
τ
20
*
θ and q
*
(
s
)
=
V
(
1
,
q
2
*
,
q
3
*
)
e
i
ω
20
*
τ
20
*
s be the eigenvectors of A and A
* corresponding to +
i
ω
20
*
τ
20
* and -
i
ω
20
*
τ
20
*, respectively. By a direction computation, we get
(63)
q
2
=
b
21
e
-
i
ω
20
*
τ
1
*
i
ω
20
*
-
a
22
-
b
22
e
-
i
ω
20
*
τ
1
*
-
c
22
e
-
i
ω
20
*
τ
20
*
,
q
3
=
i
ω
20
*
-
a
11
-
b
11
e
-
i
ω
20
*
τ
1
*
-
b
12
e
-
i
ω
20
*
τ
1
*
q
2
b
13
e
-
i
ω
20
*
τ
1
*
,
q
2
*
=
-
b
12
e
-
i
ω
20
*
τ
1
*
+
c
32
e
-
i
ω
20
*
τ
20
*
q
3
*
i
ω
20
*
+
a
22
+
b
22
e
-
i
ω
20
*
τ
1
*
+
c
22
e
-
i
ω
20
*
τ
20
*
,
q
3
*
=
-
b
13
e
-
i
ω
20
*
τ
1
*
i
ω
20
*
+
a
33
+
b
33
e
-
i
ω
20
*
τ
1
*
.
From (62), we obtain
(64)
V
¯
=
1
+
q
2
q
¯
2
*
+
q
3
q
¯
3
*
+
τ
20
*
e
-
i
ω
20
*
τ
20
*
q
2
(
c
22
q
¯
2
*
+
c
32
q
¯
3
*
)
+
τ
1
*
e
-
i
ω
20
*
τ
1
*
b
11
+
b
12
q
2
+
b
13
q
3
+
q
¯
2
*
(
b
21
+
b
22
q
2
)
+
b
33
q
3
q
¯
3
*
e
-
i
ω
20
*
τ
20
*
-
1
.
Then, one can see that 〈
q
*
,
q
〉
=
1 and 〈
q
*
,
q
¯
〉
=
0.Next, we can obtain the coefficients determining the properties of the Hopf bifurcation by the algorithms introduced in [17] and using a computation process similar to that in [19, 20]:
(65)
g
20
=
2
β
τ
20
*
V
¯
q
¯
2
*
-
1
q
1
-
τ
1
*
τ
20
*
q
2
-
τ
1
*
τ
20
*
,
g
11
=
β
τ
20
*
V
¯
q
¯
2
*
-
1
q
1
-
τ
1
*
τ
20
*
q
¯
2
-
τ
1
*
τ
20
*
+
q
¯
1
-
τ
1
*
τ
20
*
q
2
-
τ
1
*
τ
20
*
,
g
02
=
2
β
τ
20
*
V
¯
q
¯
2
*
-
1
q
¯
1
-
τ
1
*
τ
20
*
q
¯
2
-
τ
1
*
τ
20
*
,
g
21
=
2
β
τ
20
*
V
¯
(
q
¯
2
*
-
1
)
W
11
1
(
-
τ
1
*
τ
20
*
)
q
2
(
-
τ
1
*
τ
20
*
)
+
1
2
W
20
1
-
τ
1
*
τ
20
*
q
¯
2
-
τ
1
*
τ
20
*
+
W
11
2
-
τ
1
*
τ
20
*
q
1
-
τ
1
*
τ
20
*
+
1
2
W
20
2
-
τ
1
*
τ
20
*
q
¯
1
-
τ
1
*
τ
20
*
,
with
(66)
W
20
θ
=
i
g
20
q
0
ω
20
*
τ
20
*
e
i
ω
20
*
τ
20
*
θ
+
i
g
¯
02
q
¯
0
3
ω
20
*
τ
20
*
e
-
i
ω
20
*
τ
20
*
θ
+
E
20
e
2
i
ω
20
*
τ
20
*
θ
,
W
11
(
θ
)
=
-
i
g
11
q
(
0
)
ω
20
*
τ
20
*
e
i
ω
20
*
τ
20
*
θ
+
i
g
¯
11
q
¯
(
0
)
ω
20
*
τ
20
*
e
-
i
ω
20
*
τ
20
*
θ
+
E
11
,
where E
20 and E
11 can be calculated by the following two equations:
(67)
a
11
′
-
b
12
e
-
2
i
ω
20
*
τ
1
*
-
b
13
e
-
2
i
ω
20
*
τ
1
*
-
b
21
e
-
2
i
ω
20
*
τ
1
*
a
22
′
0
-
a
31
-
c
32
e
-
2
i
ω
20
*
τ
20
*
a
33
′
E
20
=
2
E
20
1
E
20
2
0
,
a
11
+
b
11
b
12
b
13
b
21
a
22
+
b
22
+
c
22
0
a
31
c
32
a
33
+
b
33
E
11
=
-
E
11
(
1
)
E
11
(
2
)
0
with
(68)
a
11
′
=
2
i
ω
20
*
-
a
11
-
b
11
e
-
2
i
ω
20
*
τ
1
*
,
a
22
′
=
2
i
ω
20
*
-
a
22
-
b
22
e
-
2
i
ω
20
*
τ
1
*
-
c
22
e
-
2
i
ω
20
*
τ
20
*
,
a
33
′
=
2
i
ω
20
*
-
a
33
-
b
33
e
-
2
i
ω
20
*
τ
1
*
,
E
20
(
1
)
=
-
β
q
1
-
τ
1
*
τ
20
*
q
2
-
τ
1
*
τ
20
*
,
E
20
(
2
)
=
β
q
1
-
τ
1
*
τ
20
*
q
2
-
τ
1
*
τ
20
*
,
E
11
1
=
-
β
q
1
-
τ
1
*
τ
20
*
q
¯
2
-
τ
1
*
τ
20
*
+
q
¯
1
-
τ
1
*
τ
20
*
q
2
-
τ
1
*
τ
20
*
,
E
11
2
=
β
q
1
-
τ
1
*
τ
20
*
q
¯
2
-
τ
1
*
τ
20
*
+
q
¯
1
-
τ
1
*
τ
20
*
q
2
-
τ
1
*
τ
20
*
.
Then, we can get the following coefficients:
(69)
C
1
(
0
)
=
i
2
ω
20
*
τ
20
*
g
11
g
20
-
2
g
11
2
-
g
02
2
3
+
g
21
2
,
μ
2
=
-
Re
{
C
1
(
0
)
}
Re
{
λ
′
(
τ
20
*
)
}
,
β
2
=
2
Re
C
1
0
,
T
2
=
-
Im
⁡
C
1
0
+
μ
2
Im
⁡
λ
′
τ
20
*
ω
20
*
τ
20
*
.By the discussion above, we have the following results about the properties of the Hopf bifurcation.Theorem 6.
For system (3), the direction of the Hopf bifurcation is determined by the sign of μ
2: if μ
2
>
0
(
μ
2
<
0
), the Hopf bifurcation is supercritical (subcritical); the stability of bifurcating periodic solutions is determined by the sign of β
2: if β
2
<
0
(
β
2
>
0
), the bifurcating periodic solutions are stable (unstable); the period of the bifurcating periodic solutions is determined by the sign of T
2: if T
2
>
0
(
T
2
<
0
), the period of the bifurcating periodic solutions increases (decreases).


## 4. Numerical Simulations and Discussion

In this section, in order to support our theoretical results, we will show the interesting dynamical behaviors of system (3) by a special case of system (3). Let p
=
0.9, A
=
1, let α
=
0.3, β
=
0.6, γ
=
0.2, δ
=
0.7, and μ
=
0.3 and we consider the following system:
(70)
d
S
t
d
t
=
0.9
-
0.6
S
t
-
τ
1
I
t
-
τ
1
-
0.5
S
t
+
0.7
R
(
t
-
τ
1
)
,
d
I
(
t
)
d
t
=
0.6
S
(
t
-
τ
1
)
I
(
t
-
τ
1
)
-
0.3
I
(
t
)
-
0.3
I
(
t
-
τ
2
)
,
d
R
t
d
t
=
0.1
+
0.2
S
t
+
0.3
I
t
-
τ
2
-
0.7
R
t
-
τ
1
-
0.3
R
(
t
)
,
from which one can get R
0
=
2.2200 and the unique positive equilibrium E
*
(
1.0000,1.5641,0.7692
). By computing, we obtain A
10
=
0.3661, A
11
=
1.8616, and A
12
=
2.4385. Obviously, A
12
A
11
>
A
10
>
0.Forτ
1
>
0, τ
2
=
0. Equation (18) has a unique positive root ω
10
=
0.5833 and one can obtain τ
10
=
2.8957 from (21). Further, the characteristic equation (14) has a pair of purely imaginary roots ±
i
ω
10. The computer simulations in Figures 1 and 2 show that E
*
(
1.0000,1.5641,0.7692
) is asymptotically stable when τ
1
<
τ
10
=
2.8957 and when τ
1 passes through the critical value τ
10
=
2.8957, E
*
(
1.0000,1.5641,0.7692
) loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from E
*
(
1.0000,1.5641,0.7692
). Similarly, we obtain ω
20
=
3.2514 and τ
20
=
8.4669. The corresponding trajectories graphs and phase graphs are shown in Figures 3 and 4.Figure 1
E
* is locally asymptotically stable for τ
1
=
2.3750
<
τ
10
=
2.8957.Figure 2
E
* is unstable for τ
1
=
3.2950
>
τ
10
=
2.8957.Figure 3
E
* is locally asymptotically stable for τ
2
=
7.2500
<
τ
20
=
8.4669.Figure 4
E
* is unstable for τ
2
=
10.7500
>
τ
20
=
8.4669.Letτ
2
=
1.05
∈
(
0
,
τ
20
) and choose τ
1 as a bifurcation parameter. Then, we have ω
10
*
=
1.0514 and τ
10
*
=
2.5386. The computer simulations in Figures 5 and 6 show that E
*
(
1.0000,1.5641,0.7692
) is asymptotically stable when τ
1
<
τ
10
*
=
2.5386 and E
*
(
1.0000,1.5641,0.7692
) loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from E
*
(
1.0000,1.5641, 0.7692
). Similarly, by some complex computations, we have ω
20
*
=
0.8545 and τ
20
*
=
2.9170 when τ
1
=
1.65
∈
(
0
,
τ
10
) and choose τ
2 as a bifurcation parameter. The corresponding trajectories graphs and phase graphs are shown in Figures 7 and 8. Furthermore, we can compute and obtain λ
′
(
τ
20
*
)
=
2.3606
+
1.7054
i and C
1
(
0
)
=
-
17.9318
+
26.0921
i. It follows from (69) that μ
2
=
7.5963
>
0, β
2
=
-
35.8636
<
0, and T
2
=
-
15.3981
<
0. According to Theorem 6, we can conclude that the Hopf bifurcation of system (70) is supercritical, the bifurcating periodic solutions are stable, and the period of the periodic solutions decreases.Figure 5
E
* is locally asymptotically stable for τ
1
=
2.0300
<
τ
10
*
=
2.5386 and τ
2
=
1.05.Figure 6
E
* is unstable for τ
1
=
2.9732
>
τ
10
*
=
2.5386 and τ
2
=
1.05.Figure 7
E
* is locally asymptotically stable for τ
2
=
2.3700
<
τ
20
*
=
2.9170 and τ
1
=
1.65.Figure 8
E
* is unstable for τ
2
=
4.3500
>
τ
20
*
=
2.9170 and τ
1
=
1.65.In addition, it can be seen from the expression of the positive equilibrium of system (3) that the more the hosts are attached to the computer networks, the more the hosts in networks will be infected. Therefore, the managers of the real networks should properly control the number of the new hosts attached to networks. According to the numerical simulations, we also find that the onset of the Hopf bifurcation can be delayed by the values of the parameters A and p in system (3), which can be controlled by the managers of the real networks. Therefore, the managers of the real networks should properly control the number of the hosts attached to the networks and properly strengthen the immunization of the new hosts in order to control the onset of the Hopf bifurcation, so as to make the propagation of computer viruses predicted and controlled easily.


## 5. Conclusion

In this paper, an SIRS computer virus propagation model with two delays and multistate antivirus measures is investigated. By choosing the possible combination of the two delays as the bifurcation parameter and analyzing the distribution of the roots of the associated characteristic equation, sufficient conditions for the local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained. Furthermore, the properties of the Hopf bifurcation are determined by using the method in [17].Compared to the model considered in [12], we consider not only the delay due to the latent period of computer virus and the delay due to the temporary immune period of the recovered hosts, but also the delay due to the period that the antivirus software uses to clean the viruses in the infected hosts. All the possible delays are incorporated into the model and the model considered in this paper is more general. Our analysis shows that the new delay we incorporate into the model can also change the stability of the positive equilibrium of the model and numerical simulations show that our results obtained in the present paper improve some of the existing results on this system that are obtained in [12].



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*Source: 280856-2015-01-22.xml*