Hopf Bifurcation Analysis for a Computer Virus Model with Two Delays
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 560804, 18 pages
http://dx.doi.org/10.1155/2013/560804
Research Article
Hopf Bifurcation Analysis for a Computer Virus
Model with Two Delays
Zizhen Zhang1,2 and Huizhong Yang1
1
2
Key Laboratory of Advanced Process Control for Light Industry, Jiangnan University, Wuxi 214122, China
School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
Correspondence should be addressed to Huizhong Yang;
Received 18 August 2013; Accepted 26 August 2013
Academic Editor: Luca Guerrini
Copyright © 2013 Z. Zhang and H. Yang. 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.
This paper is concerned with a computer virus model with two delays. Its dynamics are studied in terms of local stability and
Hopf bifurcation. Sufficient conditions for local stability of the positive equilibrium and existence of the local Hopf bifurcation are
obtained by regarding the possible combinations of the two delays as a bifurcation parameter. Furthermore, explicit formulae for
determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are obtained by using the normal
form method and center manifold theory. Finally, some numerical simulations are presented to support the theoretical results.
1. Introduction
Since the pioneering work of Kephart and White [1, 2],
many classical epidemic models such as SIR [3–5], SIRS [6–
8], SEIR [9, 10], and SEIRS [11, 12], SEIQV [13] have been
used to describe the spread of a computer virus in computer
network due to the high similarity between computer viruses
and biological viruses. In [9], Yuan and Chen proposed the
following SEIR model:
countermeasures by analyzing the equilibrium stability of
system (1).
It is well known that time delays of one or other reasons
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 [14–23].
Starting from this point and considering that the antivirus
software may use a period to clean the viruses in a computer,
Dong et al. [10] proposed the following model with delay:
𝑑𝑆 (𝑡)
= 𝜇𝑁 − V (𝑡) 𝑆 (𝑡) − 𝜌𝑆𝑅 𝑆 (𝑡) − 𝜇𝑆 (𝑡) ,
𝑑𝑡
𝑑𝑆 (𝑡)
= 𝜇𝑁 − 𝛽𝐼 (𝑡) 𝑆 (𝑡) − (𝜌𝑆𝑅 + 𝜇) 𝑆 (𝑡) ,
𝑑𝑡
𝑑𝐸 (𝑡)
= V (𝑡) 𝑆 (𝑡) − 𝑚𝐸 𝛼𝐸 (𝑡) ,
𝑑𝑡
𝑑𝐸 (𝑡)
= 𝛽𝐼 (𝑡) 𝑆 (𝑡) − (𝛼 + 𝜇) 𝐸 (𝑡) − 𝜌𝐸𝑅 𝐸 (𝑡 − 𝜏) ,
𝑑𝑡
𝑑𝐼 (𝑡)
= 𝛼𝐸 (𝑡) − (𝛾 + 𝜇) 𝐼 (𝑡) ,
𝑑𝑡
(1)
𝑑𝐼 (𝑡)
= 𝛼𝐸 (𝑡) − (𝛾 + 𝜇) 𝐼 (𝑡) ,
𝑑𝑡
(2)
𝑑𝑅 (𝑡)
= 𝜌𝑆𝑅 𝑆 (𝑡) + 𝜌𝐸𝑅 𝐸 (𝑡) + 𝛾𝐼 (𝑡) − 𝜇𝑅 (𝑡) ,
𝑑𝑡
𝑑𝑅 (𝑡)
= 𝜌𝑆𝑅 𝑆 (𝑡) + 𝜌𝐸𝑅 𝐸 (𝑡 − 𝜏) + 𝛾𝐼 (𝑡 − 𝜏) − 𝜇𝑅 (𝑡) ,
𝑑𝑡
where 𝑆(𝑡), 𝐸(𝑡), 𝐼(𝑡), and 𝑅(𝑡) denote the numbers of
nodes at time 𝑡 in states susceptible, exposed, infectious,
and recovered, respectively. Yuan and Chen [9] studied the
behaviors of virus propagation with the presence of antivirus
where 𝑁 is the total number of computers in a network.
𝜇 describes the impact of quarantine or replacement. 𝜌𝑆𝑅
describes the impact of implementing real-time immunization. 𝜌𝐸𝑅 describes the impact of cleaning the virus and
2
Abstract and Applied Analysis
immunizing the computers. 𝛽 is the transmission coefficient.
𝛼 and 𝛾 are the state transition rates. And 𝜏 is the period
which a computer uses antivirus software to clean viruses.
Dong et al. [10] discussed the local stability and existence of
local Hopf bifurcation of system (2). Properties of the Hopf
bifurcation were also investigated in [10].
However, Yuan and Chen [9] Dong et al. [10] supposed
that the recovered computers have a permanent immunization period and can no longer be infected. This is not consistent with real situation. In order to overcome limitation and
considering that the recovered computers may be infected
again after a temporary immunity period, we investigate the
following system with two delays in this paper:
𝑑𝑆 (𝑡)
= 𝜇𝑁 − 𝛽𝐼 (𝑡) 𝑆 (𝑡) − (𝜌𝑆𝑅 + 𝜇) 𝑆 (𝑡) + 𝜀𝑅 (𝑡 − 𝜏2 ) ,
𝑑𝑡
𝑑𝐸 (𝑡)
= 𝛽𝐼 (𝑡) 𝑆 (𝑡) − (𝛼 + 𝜇) 𝐸 (𝑡) − 𝜌𝐸𝑅 𝐸 (𝑡 − 𝜏1 ) ,
𝑑𝑡
𝑑𝐼 (𝑡)
= 𝛼𝐸 (𝑡) − (𝛾 + 𝜇) 𝐼 (𝑡) ,
𝑑𝑡
𝑑𝑅 (𝑡)
= 𝜌𝑆𝑅 𝑆 (𝑡) + 𝜌𝐸𝑅 𝐸 (𝑡 − 𝜏1 ) + 𝛾𝐼 (𝑡 − 𝜏1 )
𝑑𝑡
𝐸∗ =
𝑅∗ =
𝜇+𝛾
𝐼 ,
𝛼 ∗
𝜌𝑆𝑅 (𝜇 + 𝛾) (𝛼 + 𝜇 + 𝜌𝐸𝑅 ) + [𝛼𝛽𝛾 + 𝛽𝜌𝐸𝑅 (𝜇 + 𝛾)] 𝐼∗
.
𝛼𝛽 (𝜇 + 𝜀)
(5)
𝑅0 is called the basic reproduction number.
Linearizing system (3) at the positive equilibrium 𝐷∗
yields the following linear system:
𝑑𝑆 (𝑡)
= 𝑎11 𝑆 (𝑡) + 𝑎13 𝐼 (𝑡) + 𝑐14 𝑅 (𝑡 − 𝜏2 ) ,
𝑑𝑡
𝑑𝐸 (𝑡)
= 𝑎21 𝑆 (𝑡) + 𝑎22 𝐸 (𝑡) + 𝑎23 𝐼 (𝑡) + 𝑏22 𝐸 (𝑡 − 𝜏1 ) ,
𝑑𝑡
𝑑𝐼 (𝑡)
= 𝑎32 𝐸 (𝑡) + 𝑎33 𝐼 (𝑡) ,
𝑑𝑡
𝑑𝑅 (𝑡)
= 𝑎41 𝑆 (𝑡) + 𝑎44 𝑅 (𝑡) + 𝑏42 𝐸 (𝑡 − 𝜏1 ) + 𝑏43 𝐼 (𝑡 − 𝜏1 )
𝑑𝑡
+ 𝑐44 𝑅 (𝑡 − 𝜏2 ) ,
(6)
− 𝜇𝑅 (𝑡) − 𝜀𝑅 (𝑡 − 𝜏2 ) ,
(3)
where 𝜏1 is the period that a computer uses antivirus software
to clean viruses and 𝜏2 is the temporary immunity period after
which a recovered computer may be infected again. 𝜀 is the
transition rate from 𝑅 to 𝑆.
This paper is organized as follows. In Section 2, local
stability of the positive equilibrium and the existence of local
Hopf bifurcation are discussed. In Section 3, the direction
of the Hopf bifurcation and the stability of the bifurcating
periodic solutions are determined by using the normal form
theory and center manifold theorem. In order to illustrate the
validity of the theoretical analysis, some numerical simulations are presented in Section 4. Some main conclusions are
drawn in Section 5.
where
𝑎11 = − (𝜇 + 𝜌𝑆𝑅 + 𝛽𝐼∗ ) ,
𝑎21 = 𝛽𝐼∗ ,
𝑎22 = − (𝛼 + 𝜇) ,
𝑎32 = 𝛼,
𝑏22 = −𝜌𝐸𝑅 ,
𝑎23 = 𝛽𝑆∗ ,
𝑎33 = − (𝜇 + 𝛾) ,
𝑎41 = 𝜌𝑆𝑅 ,
𝑎44 = −𝜇,
𝑏42 = 𝜌𝐸𝑅 ,
𝑐14 = 𝜀,
(7)
𝑏43 = 𝛾,
𝑐44 = −𝜀.
Thus, the characteristic equation of system (3) at 𝐷∗ is
𝜆4 + 𝐴 3 𝜆3 + 𝐴 2 𝜆2 + 𝐴 1 𝜆 + 𝐴 0
2. Stability and Existence of Local
Hopf Bifurcation
+ (𝐵3 𝜆3 + 𝐵2 𝜆2 + 𝐵1 𝜆 + 𝐵0 ) 𝑒−𝜆𝜏1
+ (𝐶3 𝜆3 + 𝐶2 𝜆2 + 𝐶1 𝜆 + 𝐶0 ) 𝑒−𝜆𝜏2
In this section, we will study the stability of positive equilibrium and the existence of Hopf bifurcation. It is not difficult
to verify if
𝑅0 =
𝑎13 = −𝛽𝑆∗ ,
𝛼𝛽𝜇𝑁 (𝜇 + 𝜀) + 𝜀𝜌𝑆𝑅 (𝜇 + 𝛾) (𝛼 + 𝜇 + 𝜌𝐸𝑅 )
>1
(𝜇 + 𝜀) (𝜇 + 𝛾) (𝜇 + 𝜌𝑆𝑅 ) (𝛼 + 𝜇 + 𝜌𝐸𝑅 )
(4)
system (3) has a unique positive equilibrium 𝐷∗ (𝑆∗ , 𝐸∗ , 𝐼∗ ,
𝑅∗ ), where
(𝜇 + 𝛾) (𝛼 + 𝜇 + 𝜌𝐸𝑅 )
𝑆∗ =
,
𝛼𝛽
+ (𝐷2 𝜆2 + 𝐷1 𝜆 + 𝐷0 ) 𝑒−𝜆(𝜏1 +𝜏2 ) = 0,
where
𝐴 0 = 𝑎11 𝑎44 (𝑎22 𝑎33 − 𝑎23 𝑎32 ) + 𝑎13 𝑎21 𝑎32 𝑎44 ,
𝐴 1 = 𝑎23 𝑎32 (𝑎11 + 𝑎44 ) − 𝑎11 𝑎22 (𝑎33 + 𝑎44 ) − 𝑎13 𝑎21 𝑎32
− 𝑎33 𝑎44 (𝑎11 + 𝑎22 ) ,
𝐴 2 = (𝑎11 + 𝑎22 ) (𝑎33 + 𝑎44 ) + 𝑎11 𝑎22 + 𝑎33 𝑎44 − 𝑎23 𝑎32 ,
𝐼∗ = (𝛼𝛽𝜇𝑁 (𝜇 + 𝜀) + 𝜀𝜌𝑆𝑅 (𝜇 + 𝛾) (𝛼 + 𝜇 + 𝜌𝐸𝑅 )
𝐴 3 = − (𝑎11 + 𝑎22 + 𝑎33 + 𝑎44 ) ,
− (𝜇 + 𝜀) (𝜇 + 𝛾) (𝜇 + 𝜌𝑆𝑅 ) (𝛼 + 𝜇 + 𝜌𝐸𝑅 ) )
−1
× (𝛼𝛽𝜇 (𝜇 + 𝜀 + 𝛾) + 𝛽𝜇 (𝜇 + 𝛾) (𝜇 + 𝜀 + 𝜌𝐸𝑅 )) ,
𝐵0 = 𝑎11 𝑎33 𝑎44 𝑏22 ,
𝐵1 = −𝑎11 𝑎33 𝑏22 − 𝑎44 𝑏22 (𝑎11 + 𝑎33 ) ,
(8)
Abstract and Applied Analysis
3
𝐵2 = 𝑏22 (𝑎11 + 𝑎33 + 𝑎44 ) ,
𝐵3 = −𝑏22 ,
where
𝐶0 = (𝑎22 𝑎33 − 𝑎23 𝑎32 ) (𝑎11 𝑐44 − 𝑎41 𝑐14 ) + 𝑎13 𝑎21 𝑎32 𝑐44 ,
𝐴 23 = 𝐴 3 + 𝐶3 ,
𝐴 22 = 𝐴 2 + 𝐶2 ,
𝐶1 = 𝑎41 𝑐14 (𝑎22 + 𝑎33 ) (...truncated)
