the spread of computer viruses under the influence of removable storage devices

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The spread of computer viruses under the influence of removable storage devices Lu-Xing Yang , Xiaofan Yang School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China College of Computer Science, Chongqing University, Chongqing 400044, China article info Keywords: Computer virus Epidemiological model Virose equilibrium Global asymptotical stability Control policy abstract Removable storage devices provide a way other than the Internet for the spread of com- puter viruses. However, nearly all previous epidemiological models of viruses considered only the Internet route of spread of viruses, neglecting the removable device route at all. In this paper, a new spread model of viruses, which incorporates the effect of removable devices, is suggested. Different from previous models, the epidemic threshold for this model vanishes. Moreover, the model admits a unique virose equilibrium, which is shown to be globally asymptotically stable. This result implies that any effort in eradicating viruses cannot succeed. By analyzing the respective influences of system parameters, a number of policies are recommended so as to restrict the density of infected computers to below an acceptable threshold. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Inspired by the compelling analogies between computer viruses and their biological counterparts, Cohen [1] and Murray [2] suggested to exploit the mathematical techniques developed in the epidemiology of infectious diseases to study the spread of computer viruses. Like its biological counterpart, computer virus epidemiology is intended to understand how computer viruses spread on networks and, thereby, to work out policies of inhibiting their prevalence. Kephart and White [3] were the first to investigate computer viruses spreading on the Internet with an epidemic model (i.e., the SIS model). In the past decade, the work on computer virus epidemiology was mainly focused on the following two topics: Viruses spreading on fully-connected networks (i.e., networks in which each computer is equally likely to be accessed by any other computer). Most work toward this direction has been done by adapting biological epidemic models to the sce- nario of computer viruses [4–21]. Viruses spreading on complex networks, which was stimulated by the discovery that the Internet follows a power-law degree distribution [22–24]. Nearly all previous work toward this direction was based on the SI model [25,26], or the SIS model [27–36], or the SIR model [35–38]. The most surprising finding is that, in this case, the epidemic threshold van- ishes [27], implying that traditional methods for eradicating viruses cannot succeed. Recently, some defects of previous models were reported [39]: some models overlook the fact that all infected computers possess infectivity, while some other models have all infected computers in a single compartment, neglecting the marked 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.027 Corresponding author at: School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China. E-mail addresses: [email protected] (L.-X. Yang), [email protected] (X. Yang). Applied Mathematics and Computation 219 (2012) 3914–3922 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

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Page 1: The spread of computer viruses under the influence of removable storage devices

Applied Mathematics and Computation 219 (2012) 3914–3922

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

The spread of computer viruses under the influence of removablestorage devices

Lu-Xing Yang ⇑, Xiaofan YangSchool of Electronic and Information Engineering, Southwest University, Chongqing 400715, ChinaCollege of Computer Science, Chongqing University, Chongqing 400044, China

a r t i c l e i n f o a b s t r a c t

Keywords:Computer virusEpidemiological modelVirose equilibriumGlobal asymptotical stabilityControl policy

0096-3003/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.amc.2012.10.027

⇑ Corresponding author at: School of Electronic aE-mail addresses: [email protected] (L.-X. Y

Removable storage devices provide a way other than the Internet for the spread of com-puter viruses. However, nearly all previous epidemiological models of viruses consideredonly the Internet route of spread of viruses, neglecting the removable device route at all.In this paper, a new spread model of viruses, which incorporates the effect of removabledevices, is suggested. Different from previous models, the epidemic threshold for thismodel vanishes. Moreover, the model admits a unique virose equilibrium, which is shownto be globally asymptotically stable. This result implies that any effort in eradicatingviruses cannot succeed. By analyzing the respective influences of system parameters, anumber of policies are recommended so as to restrict the density of infected computersto below an acceptable threshold.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Inspired by the compelling analogies between computer viruses and their biological counterparts, Cohen [1] and Murray[2] suggested to exploit the mathematical techniques developed in the epidemiology of infectious diseases to study thespread of computer viruses. Like its biological counterpart, computer virus epidemiology is intended to understand howcomputer viruses spread on networks and, thereby, to work out policies of inhibiting their prevalence. Kephart and White[3] were the first to investigate computer viruses spreading on the Internet with an epidemic model (i.e., the SIS model).In the past decade, the work on computer virus epidemiology was mainly focused on the following two topics:

� Viruses spreading on fully-connected networks (i.e., networks in which each computer is equally likely to be accessed byany other computer). Most work toward this direction has been done by adapting biological epidemic models to the sce-nario of computer viruses [4–21].� Viruses spreading on complex networks, which was stimulated by the discovery that the Internet follows a power-law

degree distribution [22–24]. Nearly all previous work toward this direction was based on the SI model [25,26], or theSIS model [27–36], or the SIR model [35–38]. The most surprising finding is that, in this case, the epidemic threshold van-ishes [27], implying that traditional methods for eradicating viruses cannot succeed.

Recently, some defects of previous models were reported [39]: some models overlook the fact that all infected computerspossess infectivity, while some other models have all infected computers in a single compartment, neglecting the marked

. All rights reserved.

nd Information Engineering, Southwest University, Chongqing 400715, China.ang), [email protected] (X. Yang).

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L.-X. Yang, X. Yang / Applied Mathematics and Computation 219 (2012) 3914–3922 3915

difference between latent computers and breaking-out computers. Very recently, a number of epidemiological models ofviruses were proposed to overcome these defects [40–42].

As we all known, removable storage devices provide a way other than the Internet for the spread of viruses. However,nearly all previous models ignore the effect of removable devices on the spread of viruses.

This paper addresses the influence of removable storage devices on the spread of viruses. By taking this effect into ac-count, a new epidemiological model of computer viruses is established. As with the model presented in [42], the new modeladmits no virus-free equilibrium and is proved to admit a globally asymptotically stable virose equilibrium. By analyzing therespective influences of system parameters, a collection of policies are worked out to keep the density of infected computersbelow an acceptable threshold.

The subsequent materials of this paper are organized in this fashion: Section 2 describes the new model. Section 3 is de-voted to the proof of the global asymptotic stability of the virose equilibrium. In Section 4, the influences of system param-eters on the prevalence of viruses are analyzed. Finally, Section 5 summarizes this work.

2. Model formulation

A computer is referred to as internal or external depending on whether it is connected to the Internet or not. A computer isreferred to as infected or uninfected depending on whether there is a virus staying in it or not. An infected computer is re-ferred to as latent or breaking-out depending on whether all viruses in it are in their respective latent periods or at leastone virus in it is in its breaking-out period. All internal computers worldwide are classified as the following three categories:

� S-computers, i.e., uninfected internal computers.� L-computers, i.e., latent internal computers.� A-computers, i.e., breaking-out internal computers.

At time t, let SðtÞ; LðtÞ, and AðtÞ denote the densities of S-, L-, and A-computers in all internal computers, respectively. Withoutambiguity, they will be abbreviated as S; L, and A, respectively.

The new model is based on the following assumptions:

(A1) The total number of internal computers is constant.(A2) All external computers are virus-free when connected to the Internet.(A3) Due to that uninfected external computers are constantly connected to the Internet, the percentage of S-computers

increases at constant rate d.(A4) Each internal computer is disconnected from the Internet with probability d.(A5) Due to the influence of removable media, each S-computer is infected with constant probability h.(A6) Due to the influence of L- or A-computers, at time t each S-computer is infected with probability b1LðtÞ þ b2AðtÞ, where

b1 and b2 are positive constants.(A7) Due to the outbreak of viruses, each L-computer breaks out and therefore becomes an A-computer with constant prob-

ability a.(A8) Each A-computer is cured with constant probability c.

Based on this collection of hypotheses, the new model is formulated as

_S ¼ d� b1SL� b2SAþ cA� dS� hS;_L ¼ b1SLþ b2SAþ hS� aL� dL;_A ¼ aL� cA� dA;

8><>: ð1Þ

with initial conditions ðSð0Þ; Lð0Þ;Að0ÞÞ 2 R3þ.

Because SðtÞ þ LðtÞ þ AðtÞ � 1, system (1) can be reduced to the following two-dimensional system:

_L ¼ b1ð1� L� AÞLþ b2ð1� L� AÞAþ hð1� L� AÞ � aL� dL;_A ¼ aL� cA� dA;

(ð2Þ

with initial conditions ðLð0Þ;Að0ÞÞ 2 R2þ. It is easy to verify that the simply connected compact set

X ¼ fðL;AÞ 2 R2þ : Lþ A 6 1g ð3Þ

is positively invariant for this system.

3. Model analysis

This section examines the dynamic properties of system (2).

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3916 L.-X. Yang, X. Yang / Applied Mathematics and Computation 219 (2012) 3914–3922

3.1. Equilibrium

Theorem 3.1. System (2) has a unique equilibrium E� ¼ ðL�;A�Þ within X, where

L� ¼�a1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1 � 4a0a2

q2a0

; ð4Þ

A� ¼a

cþ dL�; ð5Þ

a0 ¼a

cþ dþ 1

� ��b1 � b2

acþ d

� �; ð6Þ

a1 ¼ b1 þ b2a

cþ d� h

acþ d

þ 1� �

� a� d; ð7Þ

a2 ¼ h: ð8Þ

Moreover, this equilibrium is virose, that is, L� þ A� > 0.

Proof. Solving the linear system

b1ð1� L� AÞLþ b2ð1� L� AÞAþ hð1� L� AÞ � aL� dL ¼ 0;aL� cA� dA ¼ 0;

we get E� ¼ ðL�;A�Þ as the unique solution within X. It is trivial to verify that L� þ A� > 0. h

Remark 1. From this theorem, it can be concluded that the system has no virus-free equilibrium.

3.2. Local analysis

Now, let us investigate the local stability of equilibrium E�.

Lemma 3.1. E� is locally asymptotically stable.

Proof. The Jacobian of the linearized system of system (2) evaluated at E� is

b1ð1� 2L� þ A�Þ � b2A� � a� d� h b2ð1� L� � 2A�Þ � b1L� � h

a �c� d

� �:

So, the corresponding characteristic equation is

k2 þ b1kþ b2 ¼ 0; ð9Þ

where

b1 ¼ b1ð2L� þ A�Þ þ b2A� þ aþ cþ hþ 2d� b1;

b2 ¼ ½b1ð2L� þ A�Þ þ b2A� þ aþ dþ h� b1�ðcþ dÞ þ a½b2ðL� þ 2A�Þ þ b1L� þ h� b2�:

Let S� ¼ 1� L� � A�. Since

b1S�L� þb2acþd

S�L� þ hS� � ðaþ dÞL� ¼ 0;

we have

b1S� þb2acþ d

S� < aþ d: ð10Þ

That is, S� <ðcþdÞðaþdÞ

b1ðcþdÞþb2a. Hence,

b1 ¼ b1L� þ b2A� � b1S� þ aþ cþ hþ 2d > b1L� þ b2A� þ cþ hþ d > 0:

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L.-X. Yang, X. Yang / Applied Mathematics and Computation 219 (2012) 3914–3922 3917

On the other hand,

b2 ¼ 2b1ðaþ cþ dÞ þ 2b2ðaþ cþ dÞ acþ d

� �L�

� b1ðcþ dÞ þ b2a� hðaþ cþ dÞ � ðaþ dÞðcþ dÞ½ �

¼ 2b1ðaþ cþ dÞ þ 2b2ðaþ cþ dÞ acþ d

� ��a1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1 � 4a0a2

q2a0

� b1ðcþ dÞ þ b2a� hðaþ cþ dÞ � ðaþ dÞðcþ dÞ½ �

¼ ðcþ dÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1 � 4a0a2

q> 0;

By the Hurwitz criterion [43], the two roots of Eq. (9) both have negative real parts. Hence, the claimed result follows by theLyapunov theorem [43]. h

3.3. Global analysis

Now, it is the turn to examine the global behavior of equilibrium E�. To begin with, two lemmas are presented.

Lemma 3.2. System (2) admits no periodic orbit in the interior of X.

Proof. Let

f1ðL;AÞ ¼ b1ð1� L� AÞLþ b2ð1� L� AÞAþ hð1� L� AÞ � aL� dL;

f2ðL;AÞ ¼ aL� cA� dA:

Define DðL;AÞ ¼ 1L. Then,

@ðDf1Þ@L

þ @ðDf2Þ@A

¼ �b1 �b2A

L2 ð1� AÞ � h

L2 ð1� AÞ � cþ dL

< 0:

The claimed result follows by applying the Bendixson–Dulac criterion [43]. h

Lemma 3.3. System (2) admits no periodic orbit that passes through a point on @X, the boundary of X.

Proof. By the smoothness of all orbits of system (2), it can be concluded that

(a) there is no periodic orbit that passes through a corner of X, i.e., either ð0;0Þ or ð0;1Þ or ð1;0Þ, and(b) if there is a periodic orbit that passes through a non-corner point on @X, then this orbit must be tangent to @X at this

point.

On the contrary, suppose there is a periodic orbit C that passes through a non-corner point ðL;AÞ on @X, then there arethree possibilities.

Case 1: 0 < L < 1; A ¼ 0. Then _AjðL;AÞ ¼ aL > 0, implying that C is not tangent to @X at this point, which leads to acontradiction.

Case 2: L ¼ 0; 0 < A < 1. Then _LjðL;AÞ ¼ b2ð1� AÞAþ hð1� AÞ > 0, implying that C is not tangent to @X at this point, again acontradiction.

Case 3: Lþ A ¼ 1; L – 0; A – 0. Then dðLþAÞdt jðL;AÞ ¼ �cA < 0, implying that C is not tangent to @X at this point, also a

contradiction.Combining the above discussions, we conclude that, for system (2), there is no periodic orbit that passes through a point

on @X. h

We are ready to present the main result of this paper.

Theorem 3.2. E� is globally asymptotically stable.

Proof. The claimed result follows by combining Lemmas 3.1–3.3 with the generalized Poincaré–Bendixson theorem[43]. h

Example 1. Consider system (2) with b1 ¼ 0:05; b2 ¼ 0:1; a ¼ 0:3; c ¼ 0:2; d ¼ 0:2, and h ¼ 0:1. ThenðL�;A�Þ ¼ ð0:1703;0:1277Þ. See Fig. 1 for the phase portrait of this system.

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3918 L.-X. Yang, X. Yang / Applied Mathematics and Computation 219 (2012) 3914–3922

Example 2. Consider system (2) with b1 ¼ 0:05; b2 ¼ 0:1; a ¼ 0:1; c ¼ 0:1; d ¼ 0:1, and h ¼ 0:3. ThenðL�;A�Þ ¼ ð0:2410;0:7231Þ. See Fig. 2 for the phase portrait of this system.

Example 3. Consider system (2) with b1 ¼ 0:04; b2 ¼ 0:05; a ¼ 0:1; c ¼ 0:05; d ¼ 0:05, and h ¼ 0:08. ThenðL�;A�Þ ¼ ð0:2933;0:2933Þ. See Fig. 3 for the phase portrait of this system.

Example 4. Consider system (2) with b1 ¼ 0:1; b2 ¼ 0:15; a ¼ 0:1; c ¼ 0:15; d ¼ 0:05, and h ¼ 0:08. ThenðL�;A�Þ ¼ ð0:4;0:2Þ. See Fig. 4 for the phase portrait of this system.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L

A

Fig. 1. Phase portrait for the system given in Example 1.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L

A

Fig. 2. Phase portrait for the system given in Example 2.

Page 6: The spread of computer viruses under the influence of removable storage devices

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L

A

Fig. 3. Phase portrait for the system given in Example 3.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

L

A

Fig. 4. Phase portrait for the system given in Example 4.

L.-X. Yang, X. Yang / Applied Mathematics and Computation 219 (2012) 3914–3922 3919

4. Further discussions

Due to the fact that the proposed model has no virus-free equilibrium, it can be concluded that any effort in eradicatingvirus is doomed to failure. In actual situations, the optimal achievable target is to make the infection prevalence, U ¼ L� þ A�,below an acceptable threshold. For that purpose, let us check the influence of some system parameters on that percentage.

First, check how the infection prevalence varies with the infection probabilities of infected computers and removablestorage media, respectively. We have.

Theorem 4.1. @U@b1

> 0; @U@b2

> 0; @U@h > 0.

Proof. Clearly, we have

c0U2 þ c1U þ c2 ¼ 0;

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3920 L.-X. Yang, X. Yang / Applied Mathematics and Computation 219 (2012) 3914–3922

where

c0 ¼ b1ðcþ dÞ þ b2a;c1 ¼ ðaþ dÞðcþ dÞ þ hðaþ cþ dÞ � b2a� b1ðcþ dÞ;c2 ¼ �hðaþ cþ dÞ:

Straightforward calculations yield

@U@b1¼ cþ d

2c20

c0 þ c1 �c1 þ 2c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

1 � 4c0c2

q c0 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

1 � 4c0c2

q264

375 > 0;

@U@b2¼ a

2c20

c0 þ c1 �c1 þ 2c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

1 � 4c0c2

q c0 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

1 � 4c0c2

q264

375 > 0;

@U@h¼ aþ cþ d

2c0

c1 þ 2c0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

1 � 4c0c2

q � 1

0B@

1CA > 0: �

Remark 2. From Theorem 4.1 one can see that U is increasing with b1; b2, and h, respectively. Therefore, for the purpose ofinhibiting the infection prevalence we need to take effective steps so that these three system parameters diminish. In par-ticular, it should be emphasized that the use of removable storage media must be preempted by the check of their safety.

Second, let us have a closer look at the influence of the infection probability of removable media on the virus infectionprevalence. Indeed, we have.

Theorem 4.2. U 6 T if and only if

a0ðcþ dÞ2T2 þ a1ðcþ dÞðaþ cþ dÞT þ a2ðaþ cþ dÞ2 6 0; ð11Þ

where a0; a1 and a2 are given by (6)–(8).

Proof. This result follows by substituting Eqs. (4) and (5) into L� þ A� 6 T and simplifying the resulting inequality. h

As a consequence of this theorem, we have.

Corollary 1. U 6 T if and only if h 6 hmax where

hmax ¼ðaþ dÞðcþ dÞ � ð1� TÞ½b1ðcþ dÞ þ b2a�

ðaþ cþ dÞð1� TÞ T: ð12Þ

Proof. The claimed result holds by combining Theorem 4.1 and inequality (10). h

0 0.05 0.1 0.15 0.20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

T

θ max

Fig. 5. hmax versus T for the class of systems given in Example 5.

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L.-X. Yang, X. Yang / Applied Mathematics and Computation 219 (2012) 3914–3922 3921

Example 5. Consider a class of systems of the form (2) and with a ¼ 0:1; c ¼ 0:1; d ¼ 0:2; b1 ¼ 0:08, and b2 ¼ 0:1. To keepthe infection prevalence less than or equal to T, it is sufficient, by Corollary 1, that the inequality h 6 hmax ¼ 9

40T

1�T � 17200 T is

satisfied. See Fig. 5 for how hmax varies with T.

Remark 3. Corollary 1 shows that, in order for the infection prevalence of computer viruses to be bounded by a given level,effectual measures must be taken so as to keep the infection probability of removable media not to exceed the level given byhmax. In practical situations, it is crucial to adjust the system parameters present in formula (12) so that hmax can beminimized.

5. Summary

By taking into account the influence of removable storage devices, a new epidemiological model of computer viruses hasbeen proposed. This model has been shown to have a globally asymptotically stable virose equilibrium. By analyzing the ef-fect of system parameters, some policies have been suggested to suppress the percentage of infected computers. In our opin-ion, this work provides a new train of thoughts in modeling the diffusion of computer viruses, which can be modified orgeneralized to produce a series of valuable models.

Our next work is to study the proposed model on scale-free networks.

Acknowledgments

The authors are grateful to the two anonymous reviewers for their valuable suggestions. This work is supported by Doc-torate Foundation of Educational Ministry of China (Grant No. 20110191110022).

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