Global Stability and SensitivityAnalysis of SEIQR ...The system (2.2) always has a unique worm endemic equilibrium, if R0 > 1. grows, while if R0 < 1, then the number of worm

Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 3833-3850© Research India Publicationshttp://www.ripublication.com/gjpam.htm

Global Stability and Sensitivity Analysis of SEIQRWorm Virus Propagation Model with Quarantined

State in Mobile Internet

Sanoe Koonprasert1 and Nichapa Channgam

Department of Mathematics,Faculty of Applied Sciences,

King Monkut’s University Technology North Bangkok,Bangkok 11000, Thailand.

Abstract

The mobile Internet plays a significant role in everyday life. In recent years, it hasdeveloped very quickly and continues to offers us more and more functionalitiesand facilities. At present, Wi-Fi is widely used for mobile devices to connect to theInternet, but there has been a rapid increase in computer worms and viruses in theWi-Fi base station. These worms and viruses expose the devices to the dangerousenvironment. In this paper, a worm propagation model with a quarantined stateis proposed for studying how to control worm propagation in Wi-Fi stations. Thequarantined state (Q) is introduced and used to protect the infected nodes fromtransmitting the worm sources to the Internet or Wi-Fi base station. We determinethe basic reproduction number R0 and sensitivity index that can show which pa-rameter has the most effect in reducing R0 < 1 rapidly. We then prove the localand global asymptotic stabilities at worm-free equilibrium P0 when R0 < 1. More-over, at the worm endemic equilibrium P ∗, we also prove the locally stable and theglobal asymptotic stability by applying Bennison criteria, when R0 > 1. Graphs ofnumerical results show that increasing the quarantined rate can reduce the wormspropagated on Wi-Fi network. These results indicate that our model can predict themechanism for combating worm propagating in Wi-Fi stations.

AMS subject classification:Keywords: SEIQR model, Wi-Fi station, Global stability analysis, Sensitivity anal-ysis.

1Corresponding author: [email protected]

2 Sanoe Koonprasert and Nichapa Channgam

1. Introduction

Malware epidemic models describing the dynamic behaviors and characteristics of com-puter worms and viruses in the Internet are of great issue in computer networks or Wi-Fibase stations because mathematics can help to devise effective mechanisms for control-ling the spread of computer worms and viruses, [1, 2, 3, 4], [5, 7, 8, 9]. A computerworm is a kind of computer program that can replicate itself. They can spread through-out the Internet network or Wi-Fi base station and send sources to other nodes withoutany userâŁ™s intervention. Moreover, computer worms can cause large scale networkcongestions [10]-[11],[12], [13], [14] and can rapidly infect millions of computers andbring about huge economic and financial losses. [15],[16]

According to a report by China Internet Network Information Center in 2015 [17],mobile devices have become increasingly pervasive as there are about 5.57 billion smart-phone users in China, which is far greater than the number of computer users. This largenumber attracts attackers to spread worm programs among mobile devices. However,most smartphones do not have any effective methods to prevent worm attacks, leavingthem vulnerable to such attacks. Smartphones affected by worms can cause great lossesto users, including the leakage of data, system damange, and even financial loss.

In the past decades, computer networks have been the main focus of researchers whohave proposed many different mathematical models to explore the dynamic behaviorsand characteristics of computer worms present in the Internet, which the Wi-Fi basestation can help control by either continuing or disrupting the connection. Specifically,Wi-Fi base stations can quarantine infected nodes by cutting off the connection betweenthe infected nodes and other nodes.

In this paper, we first define the Quarantined state(Q), to represent the infectednodes quarantined by the Wi-Fi base station, and then build a new propagation model,the SEIQR (Susceptible-Exposed-Infectious-Quarantined-Recovered) model, for mobiledevice worms. This model can describe the dynamic behaviors of the worms spreadingin the Wi-Fi environment as the following state transition.

Figure 1: State transition graph of the SEIQR model.

Global Stability and Sensitivity Analysis 3

2. SEIQR worm propagation model

In this model, a new state, the Quarantined state (Q) is introduced. If the mobile devicesare quarantined from the infectious state, then they cannot spread the worm. Therefore,the quarantine rate(ξ), has a significant impact on the propagation of the worm. Wecan take measures to remove the worm sources from quarantined mobile nodes so thatthese nodes will be in the recovered state forever. All the nodes (N) are divided into fiveclasses: the Susceptible nodes (S), the Exposed nodes (E), the Infected nodes (I ), theQuarantined nodes(Q) and the Recovered nodes (R). Hence,

S(t) + E(t) + I (t) + Q(t) + R(t) = N (2.1)

Based on these transition relationships, the SEIQR worm propagation model can beformulated by the following differential equations:

dS

dt= µN − βSI − µS,

dE

dt= βSI − ηE − εE − µE,

dI

dt= ηE − µI − ξI − γ I, (2.2)

dQ

dt= ξI − ϕQ − µQ,

dR

dt= εE + γ I + ϕQ − µR,

with an initial condition (S(0), E(0), I (0), Q(0), R(0)) ∈ R5+.The parameter µ is the Wi-Fi connection rate or disconnection rate, β is the rate at

which susceptible state which is attacked by the infected state, η is the state transitionrate from E to I , ε is the state transition rate of E to R, γ is the state transition ratefrom I to R, ξ is the quarantine rate that affects the worm propagation, and φ is the statetransition rate from Q to R . The feasible region of the system (2.2) can be described as�:

� = {(S, E, I, Q, R)|S, E, I, Q, R ≥ 0, S + E + I + Q + R = N}, (2.3)

which is a positively invariant for the system (2.2).

3. Model analysis

This section is devoted to study the system (2.2) theoretically. The analysis of this modelcomprises the existence of the worm-free equilibrium, the worm endemic equilibrium,and the basic reproduction number. Secondly, we prove the local and global stabilities ofthe worm-free equilibrium. Finally, the global stability of the worm endemic equilibriumis proved.


3.1. Equilibriums

Theorem 3.1. The worm propagation system (2.2) has the unique worm-free equilib-rium,

P0 = (N, 0, 0, 0, 0), (3.1)

and the worm endemic equilibrium: P1 = (S∗, E∗, I ∗, Q∗, R∗), where

S∗ = (η + ε + µ)(ξ + γ + µ)

ηβ, E∗ = µ(µ + ξ + γ )

βη(R0 − 1),

I ∗ = µ

β(R0 − 1), Q∗ = µ + ξ

β(ϕ + µ)(R0 − 1), (3.2)

R∗ =(

εµ + ξ + γ

η+ γ + ϕ

ξ

(ϕ + µ)

)(R0 − 1)

β,

where R0 = ηβN

(η + ε + µ)(µ + ξ + γ ).

Proof. Solving the following algebraic equations

−βS∗I ∗ + µN − µS∗ = 0,

βS∗I ∗ − ηE∗ − εE∗ − µE∗ = 0,

ηE∗ − µI ∗ − ξI ∗ − γ I ∗ = 0, (3.3)

ξI ∗ − ϕQ∗ − µQ∗ = 0,

εE∗ + γ I ∗ + ϕQ∗ − µR∗ = 0.

If I ∗ = 0, we obtain the unique worm-free equilibrium P0 = (N, 0, 0, 0, 0), otherwise,the unique worm endemic equilibrium, P1 = (S∗, E∗, I ∗, Q∗, R∗),

S∗ = (η + ε + µ)(ξ + γ + µ)

ηβ, E∗ = µ(µ + ξ + γ )

βη(R0 − 1), I ∗ = µ

β(R0 − 1),

Q∗ = µ + ξ

β(ϕ + µ)(R0 − 1), R∗ =

(εµ + ξ + γ

η+ γ + ϕ

ξ

(ϕ + µ)

)(R0 − 1)

β,

(3.4)

where R0 > 1. The proof is completed. �

Remark 3.2. The system (2.2) always has a unique worm endemic equilibrium, if R0 >

1. grows, while if R0 < 1, then the number of worm virus would decrease to zero.

3.2. The basic reproduction number (R0)

Now, we derive the basic reproduction number of system (2.2) which is the most funda-mental parameter used by epidemiologists. We apply the next generation method [21] to


derive the basic reproduction number. First, the model (2.2) can be written in the matrixform as

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

dE

dtdS

dtdI

dtdQ

dtdR

dt

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎝

βSI

0000

⎞⎟⎟⎟⎟⎠ −

⎛⎜⎜⎜⎜⎝

(η + ε + µ)E

βSI − µN − µS

(µ + ξ + γ )I − ηE

ϕQ + µQ − ξI

µR − εE − γ I − ϕQ

⎞⎟⎟⎟⎟⎠ = F − V. (3.5)

The 5 × 5 Jacobian matrices that are obtained

F =

⎡⎢⎢⎢⎢⎣

0 βI βS 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

⎤⎥⎥⎥⎥⎦ , and

V =

⎡⎢⎢⎢⎢⎣

(η + ε + µ) 0 0 0 00 βI − µ βS 0 0

−η 0 (µ + ξ + γ ) 0 00 0 −ξ ϕ + µ 0

−ε 0 −γ −ϕ µ

⎤⎥⎥⎥⎥⎦ . (3.6)

Substituting P0 = ( N, 0, 0, 0, 0 ) into both Jacobian matrices and then, finding the nextgeneration matrix is defined as FV−1,

FV−1 =

⎡⎢⎢⎢⎢⎢⎢⎣

ηβN

(η + ε + µ)(µ + ξ + γ )0

βN

(µ + ξ + γ )0 0

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎦

.

We determine its eigenvalues, (λ) from the characteristic equation

∣∣∣∣∣∣∣∣∣∣∣∣

ηβN

(η + ε + µ)(µ + ξ + γ )− λ 0

βN

(µ + ξ + γ )0 0

0 −λ 0 0 00 0 −λ 0 00 0 0 −λ 00 0 0 0 −λ

∣∣∣∣∣∣∣∣∣∣∣∣= 0, (3.7)


that gives λ1 = ηβN

(η + ε + µ)(µ + ξ + γ )and λ2 = λ3 = λ4 = 0 = λ5 = 0. Hence,

denote the basic reproduction number of system (2.2) which is determined by the spectralradius as follows:

R0 = ρ0(FV−1) = ηβN

(η + ε + µ)(µ + ξ + γ ). (3.8)

Remark 3.3. If R0 > 1, then the number of worm viruses grow, while if R0 < 1, thenthe number of worm viruses would decrease to zero.

3.3. Local stability of the worm-free equilibrium

Theorem 3.4. If R0 < 1 ,the worm-free equilibrium P0 of the system (2.2) is locallyasymptotically stable.

Proof. According to the system (2.2), the Jacobian matrix is given by

J =

⎡⎢⎢⎢⎢⎣

−βI0 − µ 0 −βS0 0 0βI0 −(η + ε + µ) βS0 0 00 η −(µ + ξ + γ ) 0 00 0 ξ −(ϕ + µ) 00 ε γ ϕ −µ

⎤⎥⎥⎥⎥⎦ . (3.9)

At the worm-free equilibrium P0 = (N, 0, 0, 0, 0) , we have the Jacobian matrix as:

J ∗0 =

⎡⎢⎢⎢⎢⎣

−µ 0 −βN 0 00 −(η + ε + µ) βN 0 00 η −(µ + ξ + γ ) 0 00 0 ξ −(ϕ + µ) 00 ε γ ϕ −µ

⎤⎥⎥⎥⎥⎦ , (3.10)

and the characteristic equation∣∣∣∣∣∣∣∣∣∣

λ + µ 0 βN 0 00 λ + (η + ε + µ) −βN 0 00 −η λ + (µ + ξ + γ ) 0 00 0 −ξ λ + (ϕ + µ) 00 −ε −γ −ϕ λ + µ

∣∣∣∣∣∣∣∣∣∣= 0, (3.11)

(λ + µ)2(λ + ϕ + µ)

∣∣∣∣ λ + (η + ε + µ) −βN

−η λ + (µ + ξ + γ )

∣∣∣∣ = 0.

Therefore, three of the eigenvalues of J ∗0 are λ1 = −µ,λ2 = −µ and λ3 = −(ϕ + µ)

which have negative real parts. The two remaining eigenvalues are solutions of thequadratic equation

λ2 + a1λ + a2 = 0, (3.12)


where

a1 = (ξ + γ + µ) + (η + ε + µ) > 0,

a2 = (η + ε + µ) + (ξ + γ + µ) − ηβN,

Using the Routh-Hurwitz conditions, we consider:

a1 = (ξ + γ + µ) + (η + ε + µ) > 0,

a1a2 = [(ξ + γ + µ) + (η + ε + µ)][(η + ε + µ) + (ξ + γ + µ) − ηβN ].= (ξ + γ + 2µ + η + ε)(η + ε + µ)(ξ + γ + µ)(1 − R0) > 0, R0 < 1.

So, a1 > 0 and a1a2 > 0, according to the Routh-Hurwitz Criterion, two roots ofEq. (3.12) have negative real parts, �(λ4) < 0, �(λ5) < 0, where R0 < 1, thus alleigenvalues of the characteristic equation (3.11) have negative real parts. Hence theworm-free equilibrium P0 of system (2.2) is locally asymptotically stable. �

3.4. Global stability of the worm-free equilibrium

In this section, We apply the Lyapunov theory to prove the globally asymptoticallystability of the system (2.2) at worm-free equilibrium P0.

Theorem 3.5. If R0 < 1, the worm-free equilibrium P0 of the system (2.2) is globallyasymptotically stable on the feasible region �, otherwise, unstable.

Proof. we construct the Lyapunov function

L(t) = ηE(t) + mI (t), m > 0.

Then, calculating the derivative of L along the solutions of the model (2.2), we obtain

dL

dt= η

dE

dt+ m

dI

dt= η[βSI − (η + ε + µ)E] + m[ηE − (µ + ξ + γ )I ]= ηβSI + η(m − η − ε − µ)E − m(µ + ξ + γ )I. (3.13)

The second term would be positive, we let m = η + ε + µ and S < N from the feasibleregion �.

dL

dt< [ηβN − (η + ε + µ)(µ + ξ + γ )]I

= (η + ε + µ)(µ + ξ + γ )[ ηβN

(η + ε + µ)(µ + ξ + γ )− 1]I

= (η + ε + µ)(µ + ξ + γ )(R0 − 1)I.

Hence, R0 < 0, it implies that L′(t) < 0 on �. According to LaSalle Invariance Principle[20], we obtain that the worm-free equilibrium P0 is globally asymptotically stable on�. �


Remark 3.6. Theorem 3.5 implies that computer worm viruses on Wi-Fi base stationswould tend to become extinct when the basic reproduction number is less than or equalto unity.

3.5. Local stability of the worm endemic equilibrium

Now we are ready to investigate the stability of the worm endemic equilibrium.

Theorem 3.7. If R0 > 1, the worm endemic equilibrium P1 in Eq. (3.4) of the system(2.2) is locally asymptotically stable.

Proof. The Jacobi matrix (3.9) of the system (2.2) at P1 is given by

JP ∗ =

⎡⎢⎢⎢⎢⎣

−βI ∗ − µ 0 −βS∗ 0 0βI ∗ −(η + ε + µ) βS∗ 0 0

0 η −(µ + ξ + γ ) 0 00 0 ξ −(ϕ + µ) 00 ε γ ϕ −µ

⎤⎥⎥⎥⎥⎦ . (3.14)

The characteristic equation is

∣∣∣∣∣∣∣∣∣∣

λ + βI ∗ + µ 0 βS∗ 0 0−βI ∗ λ + (η + ε + µ) −βS∗ 0 0

0 −η λ + (µ + ξ + γ ) 0 00 0 −ξ λ + (ϕ + µ) 00 −ε −γ −ϕ λ + µ

∣∣∣∣∣∣∣∣∣∣= 0.

Simplifying, we have

(λ + µ)[λ + (ϕ + µ)]∣∣∣∣∣∣

λ + βI ∗ + µ 0 βS∗−βI ∗ λ + (η + ε + µ) −βS∗

0 −η λ + (µ + ξ + γ )

∣∣∣∣∣∣ = 0.(3.15)

Two of the eigenvalues of J ∗0 are λ1 = −µ and λ2 = −(ϕ +µ) which have negative real

parts and other three remaining eigenvalues satisfy the following equation

λ3 + a1λ2 + a2λ + a3 = 0,

where

a1 = (η + ε + µ) + (βI ∗ + µ) + (µ + ξ + γ ) > 0,

a2 = (βI ∗ + µ)(η + ε + µ) + (η + ε + µ)(µ + ξ + γ )

+(βI ∗ + µ)(µ + ξ + γ ) − ηβS∗,a3 = (βI ∗ + µ)(η + ε + µ)(µ + ξ + γ ) − ηβS∗µ.


We consider

a1 = (η + ε + µ) + (βI ∗ + µ) + (µ + ξ + γ )

= η + ε + γ + µ(2 + R0) > 0,

a1a2 − a3 = [(η + ε + µ) + (βI ∗ + µ) + (µ + ξ + γ )][(η + ε + µ)(µ + ξ + γ ) + (µ + ξ + γ )(βI ∗ + µ)

+(βI ∗ + µ)(η + ε + µ) − ηβS∗]−[(η + ε + µ)(µ + ξ + γ )(βI ∗ + µ) − µηβS∗]

= [η + ε + ξ + γ ) + µ(2 + R0)](η + ε + µ)µR0

+[ε + γ + µ(1 + R0)](ε + γ + µ)µR0

+µ(η + ε + µ)(µ + ξ + γ ) > 0,

a3(a1a2 − a3) = (a1a2 − a3)[(η + ε + µ)(βI ∗ + µ)(µ + ξ + γ ) − µηβS∗]= (a1a2 − a3)(η + ε + µ)(µ + ξ + γ )(R0 − 1) > 0, R0 > 1.

So a1 > 0, a1a2 − a3 > 0, and a3(a1a2 − a3) > 0 satisfy the Routh-Hurwitz Criterion,which gives �(λ3) < 0, �(λ4) < 0, and �(λ5) < 0, where R0 > 1, thus all eigenvaluesof Eq. (3.15) have negative real parts. Hence the disease-free equilibrium P1 of system(2.2) is locally asymptotically stable. �

3.6. Globally asymptotically stability

We investigate the globally asymptotically stability of the worm endemic equilibriumP1 of the model (2.2) by using the BendixonâŁ™s criterion [18]. The existence of arobust Bendixon criterion for the system which is associated to the existence and unicityof an equilibrium determines the global dynamics of the system which requires the useof Lozinski Logarithmic norm. Since three populations S, E and I do not depend onthe populations Q and R, so the system (2.2) can be reduced to the model that has threeequations,

dS

dt= −βSI + µN − µS,

dE

dt= βSI − (η + ε + µ)E, (3.16)

dI

dt= ηE − (µ + ξ + γ )I.

Applying the method of Li and Muldowney[18], the Bendixson criterion can help toprove the global stability of an endemic equilibrium.

Theorem 3.8. If R0 > 1 , the worm endemic equilibrium P1 of the system (3.16) isglobally asymptotically stable on �, otherwise, unstable.


Proof. The Jacobain matrix of the system (3.16) is given by

J =⎡⎣ −βI − µ 0 −βS

βI −(η + ε + µ) βS

0 η −(µ + ξ + γ )

⎤⎦ , (3.17)

and its second additive compound Jacobian matrix J [2] is defined by

J [2] =⎡⎣ −βI − 2µ − η − ε βS βS

η −βI − 2µ − ξ − γ 00 βI −η − ε − 2µ − ξ − γ

⎤⎦ .(3.18)

We also define the matrix function P of E, I as:

P ==

⎡⎢⎢⎢⎣

1 0 0

0E

I0

0 0E

I

⎤⎥⎥⎥⎦ , P −1 =

⎡⎢⎢⎢⎣

1 0 0

0I

E0

0 0I

E

⎤⎥⎥⎥⎦ .

Calculation the derivative of matrix P with respect to t , we obtain

Pf = dP

dt=

⎡⎢⎢⎢⎢⎢⎣

d(1)

t0 0

0d

dt

(E

I

)0

0 0d

dt

(E

I

)

⎤⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎣

0 0 0

0E

I

(E

E− I

I

)0

0 0E

I

(E

E− I

I

)

⎤⎥⎥⎥⎥⎦ .

We have two products of the matrices as:

Pf P −1 =

⎡⎢⎢⎢⎣

0 0 0

0E

E− I

I0

0 0E

E− I

I

⎤⎥⎥⎥⎦ ,

and

PJ [2]P −1 =

⎡⎢⎢⎢⎣

−βI − 2µ − η − εβSI

E

βSI

EE

Iη (−βI − 2µ − ξ − γ ) 0

0 βI (−η − ε − 2µ − ξ − γ )

⎤⎥⎥⎥⎦ .

Then, the Bendixson matrix is defined by B = Pf P −1 + PJ [2]P −1 as:

B =

⎡⎢⎢⎢⎢⎢⎣

−βI − 2µ − η − εβSI

E

βSI

EE

Iη

E

E− I

I− βI − 2µ − ξ − γ 0

0 βIE

E− I

I− η − ε − 2µ − ξ − γ

⎤⎥⎥⎥⎥⎥⎦

,


which can be written in the partition matrix form as:

B =[

B11 B12

B21 B22

]. (3.19)

where

B11 = [−βI − 2µ − η − ε], B12 =[βSI

E,βSI

E

], B21 =

[E

Iη, 0

]T

,

B22 =⎡⎢⎣

E

E− I

I− βI − 2µ − ξ − γ 0

βIE

E− I

I− η − ε − 2µ − ξ − γ

⎤⎥⎦ .

Let (a1, a2, a3) be a vector in R3. We choose a vector norm in R3 as

|(a1, a2, a3)| = max {|a1|, |a2| + |a3|} (3.20)

and a matrix norm µ1 is defined by µ1(Bnn) = max(akk +

∑i,j �=k

|aik|)

as described in

[22].We consider the function g1(S, E, I) defined by

g1(S, E, I) = µ1(B11) + |B12|= max {−βI − 2µ − η − ε} + max

{∣∣∣∣βSI

E

∣∣∣∣,{∣∣∣∣βSI

E

∣∣∣∣}

= −βI − 2µ − η − ε + βSI

E.

From (3.3), we haveβSI

E= E

E+ (η + ε + µ), so it obtains

g1(S, E, I) = −βI − 2µ − η − ε +(

E

E+ (η + ε + µ)

)

= E

E− µ − βI ≤ E

E− µ,

and the function g2(S, E, I)

g2(S, E, I) = µ1(B22) + |B21|= max

{E

E− I

I− 2µ − ξ − γ ,

E

E− I

I− η − ε − 2µ − ξ − γ

}

+max

{E

Iη, 0

}

= E

E− I

I− µ − (µ − ξ + γ ) + E

Iη.


The relation −(µ + ξ + γ ) = I

I− ηE

Iin Eq.(3.3), we have

g2(S, E, I) = E

E− I

I− µ + I

I− ηE

I+ E

Iη = E

E− µ.

Let µ be the Lozinski measure with respect to this norm and the bound of Lozinskimeasure of Benison matrix is given by

µ(B) ≤ sup(g1, g2) = E

E− µ.

This holds along each solution (S(t), E(t), I (t)) of the system (3.16) with initial con-dition (S(0), E(0), I (0)) ∈ �, where � is the compact absorbing set. By Benisoncriterion, define the quantity q2 as:

q2 = limt→∞ sup sup

x∈K

1

t

∫ t

0µ(B(s, x0))ds

≤ limt→∞ sup sup

x∈K

1

t

∫ t

0

(E

E− µ

)ds.

= limt→∞ sup sup

x∈K

1

t

(lnE

∣∣∣t0− µs

∣∣∣t0

)


x∈K

1

t(lnE(t) − lnE(0) − µ(t − 0))


x∈K

1

t

(ln

E(t)

E(0)− µt

)= −µ ≤ −µ

2< 0,

which implies q2 ≤ −µ

2< 0, it implies that P1 is globally asymptotically stable. �

3.7. Sensitivity analysis of basic reproduction number R0

The basic reproduction numberR0 is a function that depends on five parameters ε, η, β, ξ, µ,

and γ . In order to reduce the computer worm viruses, it is necessary to control the pa-rameter values to make R0 < 1. We are therefore interested in finding the rate of changeof R0 as the parameter values are changed.

The rate of change of R0 for a change in value of parameter h can be estimated froma normalized sensitivity index, SI (h) defined as follows:

SI [h] = k

R0· ∂R0

∂k. (3.21)

The normalized sensitivity indices of the reproduction number with respect to ε, η, β, ξ, µ,


and γ are given by

SI [ε] = − ε

η + ε + µ, SI [η] = ε + µ

η + ε + µ,

SI [β] = 1, SI [ξ ] = − ξ

ξ + γ + µ, (3.22)

SI [µ] = − µ(µ2 + η + ε + ξ + γ )

(η + ε + µ)(ξ + γ + µ), SI [γ ] = − γ

ξ + γ + µ.

The sensitivity indices SI [η] and SI [β] are positive, i.e., the value of R0 decreasesas η and β values decrease. The remaining indices are negative, i.e., the value of R0decreases as ε, ξ , µ and γ values increase.

Table 1: Parameter values corresponding to the disease-free equilibrium point

Parameter Value Unitinfection rate of susceptible nodes by infected nodes (β) 0.000004 day−1

rate of losing immunity (ξ ) 0.05 day−1

rate for recovery (γ ) 0.01 day−1

transition rate (µ) 0.0000003 day−1

transition rate (η) 0.05 day−1

rate at which the exposed (ε) 0.0000003 day−1

As an example, we have computed the normalized sensitivity indices for the specialcase of parameter values given in Table 1. The sensitivity indices and corresponding% values in Table 2 represent the changes in parameter values required to give a 1%decrease in R0.

Table 2: Normalized sensitivity indices of R0 and change in parameter for 1% changein R0

Parameter Sensitivity indices of R0 corresponding % changesβ 1 1ξ −0.833 −1.200γ −0.167 −6µ −0.110 × 10−4 −90909.893η 1.200 × 10−5 83334.333ε −0.6 × 10−5 −1.667 × 10−5

In Table 2, the parameters that have the most effect on reducing the basic reproductionnumber (R0 < 1) are listed in descending order, from the rate (β) of susceptible state


that receives worm, the quarantine rate (ξ) , the recover rate (γ ) of state Q to state R,respectively. For example, in order to get a 1% decrease in the value of R0, it is necessaryto decrease the value of β by 1%. Also, in order to get a 1% decrease in the value of R0,it is necessary to increase the values of ξ and γ by 1.2% and 6.0%. Therefore, from thesensitivity indices, the most effective methods of reducing the value of R0 are to decreasethe rate of infected of susceptible node (β), decrease the growth rate of the population(ξ), and increase the rate of treatment rate (γ ), respectively.

4. Numerical simulation

In this section, some numerical simulations are given to show the geometric impressionof our results. With the set of the parameter values in Table 1, Fig 2 demonstrates theglobal stability of worm-free solution P0(100000, 0, 0, 0, 0) and R0 = 0.83 < 1 of thesystem (2.2) and various initial conditions.

(a) S(t) (b) E(t)

(c) I (t) (d) Q(t)

Figure 2: Graphs of numerical solutions for system (2.2) tends to P0(100000, 0, 0, 0, 0)


Table 3: Parameter values corresponding to the endemic equilibrium point

Parameter Value Unitinfection rate of susceptible nodes by infected nodes (β) 0.000004 day−1

rate of losing immunity (ξ ) 0.05 day−1

rate for recovery (γ ) 0.01 day−1

transition rate (µ) 0.0000003 day−1

transition rate (η) 0.05 day−1

rate at which the exposed (ε) 0.0000003 day−1

(a) S(t) (b) E(t)

(c) I (t) (d) Q(t)

Figure 3: Graphs of numerical solutions the for system (2.2) tends to P1

.

To demonstrate the permanence of system (2.2), we take the set of the parameter val-ues in Table 3, which runs for some choices of the initial conditions, In this case, we have


R0 = 6.67 > 1 and the endemic equilibriumP1(77501.005, 0.135, 0.022, 0.131, 22498.708)

that shows globally asymptotically stable in Fig 4.Consider system (2.2) with the parameter values in Table 3. For the transition rate

β = 3 × 10−6, 5 × 10−6, 10−5, 9 × 10−4, 10−3. Fig 4a shows how the infectiousindividuals I (t) evolves with time. It can be seen that a larger β favours computer wormvirus spreading.

For the quarantined rate ξ = 0, 0.01, 0.02, 0.05, 0.08, 1. Fig 4b shows how theinfectious group I (t) evolves with time. It can be seen that a smaller ξ favours computerworm virus spreading.

(a) (b)The transition rate β effects to I (t) The quarantined rate ξ effects to I (t)

(c) The rate from E(t) to R(t) (d)The rate from I (t) to R(t)

Figure 4: Graphs of numerical results show the impacts of β, ξ, ε, γ to the infectiousindividual I (t) of system (2.2)

Consider system (2.2) with the parameter values in Table 2. For the quarantined rateξ = 0, 0.01, 0.02, 0.05, 0.08, 1. Fig 4 shows how the infectious group I (t) evolves withtime. It can be seen that a smaller ξ favours computer worm virus spreading.


5. Discussion

In this paper, a worm propagation model with a quarantined state is proposed. Thequarantined state ( Q ) can protect the infected nodes from transmitting the worm sourcesto the Internet or Wi-Fi base station. We determine the basic reproduction number R0 andalso show sensitivity indices that show which parameter has the most effect in reducingR0 < 1 rapidly. We then prove the local and globally asymptotically stabilities at worm-free equilibrium P0 when R0 < 1. Moreover, at the worm endemic equilibrium P ∗,we also prove the locally stable and the globally asymptotically stability by applyingBennison criteria, when R0 > 1. Graphs of numerical results show that increasing thequarantined rate can reduce the worm propagated on Wi-Fi network.

Acknowledgment

The authors are grateful to the referees for their careful reading of the manuscript andtheir useful comments.

References

[1] J.R.C. Piqueira, A.A. de Vasconcelos, C.E.C.J. Gabriel, V.O. Araujo, Dynamicmodels for computer viruses, Comput. Secur. 27 (7–8)(2008) 355–359.

[2] L.X.Yang, X.Yang, The impact of nonlinear infection rate on the spread of computervirus, Nonlinear Dyn. 82 (1–2) (2015) 85–95.

[3] B.K. Mishra, D.K. Saini, SEIRS epidemic model with delay for transmission ofmalicious objects in computer network, Appl. Math. Comput. 188 (2) (2007) 1476–1482.

[4] L. Feng, X. Liao, H. Li, Q. Han, Hopf bifurcation analysis of a delayed viralinfection model in computer networks, Math. Comput. Model. 56 (7–8) (2012)167–179.

[5] Y. Yao, L. Guo, H. Guo, G. Yu, F. Gao, X. Tong, Pulse quarantine strategy ofinternet worm propagation: Modeling and analysis, Comput. Electr. Eng. 38 (9)(2012) 1047–1061.

[6] L.X. Yang, X. Yang, The pulse treatment of computer viruses: a modeling study,Nonlinear Dyn. 76 (2) (2014) 1379–1393.

[7] Y. Yao, X. Feng, W. Yang, W. Xiang, F. Gao, Analysis of a delayed internet wormpropagation model with impulsive quarantine strategy, Math. Probl. Eng. 2014(2014). Article ID 369360

[8] J. Amador, J.R. Artalejo, Stochastic modeling of computer virus spreading withwarning signals, J. Frankl. Inst. 350 (5) (2013) 1112–1138.

[9] J. Amador, The stochastic SIRA model for computer viruses, Appl. Math. Comput.232 (1) (2014) 1112–1124

[10] H. Berghel, The code red worm Commun. of the ACM. 12 (2001), 15–19.


[11] D. Moore, C. Shannon, K.Claffy, Code-red a case study on the spread and victimsof an Internet worm, In: Proc. of 2nd ACM SIGCOMM Workshop on Internetmeasurement. IEEE Secur Privacy (2002), 273–84.

[12] Z. Chen, L. Gao, K. Kwiat, Modeling the spread of active worms IEEE Com-put. Commun. IEEE Societies, INFOCOM, In: 22nd Annu. Joint Conf, 3; (2003)pp. 1890–900.

[13] C. Shannon, D. Moore, The spread of the witty worm. IEEE Secur Privacy. 3(2),(2004), 46–50.

[14] D. Moore, V. Paxson, S. Savage, C. Shannon, S. Staniford, N. Weaver, Inside theslammer worm, IEEE Secur Privacy (2003);1(4):33–9.

[15] HV. Poor, An introduction to signal detection and estimation, NewYork: Springer-Verlag, (1998).

[16] AL. Foster, Colleges brace for the next worm, Chronicle Higher Edu (2004), 50(28).

[17] The 35th statistical report on the development of the Internet in China. CNNIC,(2015). http://www.cnnic.cn/hlwfzyj/hlwxzbg/

[18] My. Li, JS. Muldowney, A geometric approach to global-stability problems, SIAMJ Math Anal (1996), 27(4):1070–83.

[19] JS RH. Martin, Logarithmic norms and projections applied to linear differentialsystems, J Math Anal Appl (1974);45:432–54.

[20] JP. LaSalle, The Stability of Dynamical Systems. Regional conference series inapplied mathematics. Phiadelphia, PA: SIAM; 1976.

[21] van den. Driessche, J. Watmough, The reproduction numbers and sub-threshold en-demic equilibria for compartmental models of disease transmission. MathematicalBiology Science. (2005): 1–21.

[22] R.H. Martin, Logarithmic norms and projections applied to linear differential sys-tems. Journal of mathematical Analysis and Applications. 45 (1974) 432–454.

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Global Stability and SensitivityAnalysis of SEIQR ...The system (2.2) always has a unique worm endemic equilibrium, if R0 > 1. grows, while if R0 < 1, then the number of worm