research article modeling and stability analysis of worm...

Research ArticleModeling and Stability Analysis of Worm Propagation inWireless Sensor Network

Liping Feng1 Lipeng Song2 Qingshan Zhao1 and Hongbin Wang1

1Department of Computer Science of Xinzhou Normal University Xinzhou Shanxi 034000 China2School of Computer and Control Engineering North University of China Taiyuan Shanxi 030051 China

Correspondence should be addressed to Hongbin Wang whb-163163com

Received 28 April 2015 Revised 20 July 2015 Accepted 19 August 2015

Academic Editor Stefan Balint

Copyright copy 2015 Liping Feng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An improved SIRS model considering communication radius and distributed density of nodes is proposed The proposed modelcaptures both the spatial and temporal dynamics of worms spread process Using differential dynamical theories we investigatedynamics of worm propagation to time in wireless sensor networks (WSNs) Reproductive number which determines globaldynamics of worm propagation in WSNs is obtained Equilibriums and their stabilities are also found If reproductive numberis less than one the infected fraction of the sensor nodes disappears and if the reproduction number is greater than one theinfected fraction asymptotically stabilizes at the endemic equilibrium Based on the reproduction number we discuss the thresholdof worm propagation about communication radius and distributed density of nodes inWSNs Finally numerical simulations verifythe correctness of theoretical analysis

1 Introduction

A sensor network is composed of hundreds or even thou-sands of sensor nodes that are allowed randomdeployment ininaccessible terrains or disaster relief operations [1] Wirelesssensor networks (WSNs) as a kind of new information andcommunication network have gained worldwide attentionowing to their potential in civil and military applications forinstance intrusion detection perimeter monitoring infor-mation gathering and smart logistics support in an unknowndeployed area [2ndash4]

With widespread applications of WSNs research onWSNs has been a hot topic Some methods have beenproposed for prolonging the lifetime of WSNs focusing onenergy consumption [5ndash7] device placement [8] and topol-ogy management [9] Because sensor nodes are constrainedsources they have weak defenses and are attacking targetsfor worms Injecting malware into some nodes has become aserious threat [10] Recently malicious codes targeting wire-less devices have emerged which can spread directly fromdevice to device using wireless communication technologysuch as Wi-Fi and Bluetooth [11ndash14] For instance computerworm like Cabir uses the Bluetooth interface to spread among

cellphones which means that worms have committed thewireless domain and WSNs are also extremely vulnerable tomalware

Actions of malicious objects on the Internet have beenstudied by using epidemical models and have providedinsights for controlling worm prevalence in networks [15ndash20] In [15] the authors presented an E-mail virus modelthat accounts for behaviors of E-mail users and analyzedpropagation features of E-mail viruses in different networktopologies In [16ndash20] the authors proposed epidemic mod-els with time delay and analyzed dynamical features of wormprevalence To effectively defend against worm intrusionsit is necessary to deeply understand dynamical features ofworm propagation in WSNs Existing research prove thatepidemical models are valuable for portraying characteris-tics of worm propagation Since there is a basic similaritybetween worm propagation through wireless devices andtraditional worm spread on the Internet the epidemicalmodels extensively are applied to study worm spread inWSNs by some researchers in recent years [11 21ndash23] In [21]the authors proposed a SIRS malware propagation modelwith feedback controller and analyzed Hopf bifurcationdynamics ofmalware prevalence inmobile wireless networks

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 129598 8 pageshttpdxdoiorg1011552015129598

2 Mathematical Problems in Engineering

The authors in [22] presented an epidemic model withvaccination compartment which captures both the spatialand temporal dynamics of worm spread process and somemathematical analyses and numerical simulations were per-formed based on this model The common problem of theabove models is that the characteristics of wireless sensornetworks like energy consumption communication radiusand distributed density of nodes have not been considered inmodels In [11] the authors developed a mathematical modelfor the propagation that incorporates important parametersderived from the communication patterns of the protocolunder test Based on this model the authors analyzed thepropagation rate and the extent of spread of a malware overtypical broadcast protocols proposed in the paper Wang andLi derived an iSIR model describing the process of wormpropagation with energy consumption of nodes in WSNs[23] Numerical simulations are performed to observe theeffects of the network topology and energy consumption ofnodes on worm spread in WSNs However the authors havenot performedmathematical analyses based on thismodel Infact key parameters of affectingworm spread can be found byexplicit mathematical analyses

To better portrait the features of worm propagation inWSNs in this paper we study the attacking behavior ofpossible worms in WSNs by constructing an improved SIRSepidemicmodel In thismodel the following three factors areconsidered (i) energy consumptions of nodes (ii) communi-cation radius of nodes and (iii) distributed density of nodesin WSNs Based on this model we analyze the stability ofworm prevalence through finding the equilibriums of model

The rest of this paper is organized as follows in Section 2we analyze topology of WSNs and present the model for-mulation Section 3 derives the equilibriums of the modeland discusses the stability of worm propagation at theequilibriums In Sections 4 and 5 numerical simulations areperformed to verify the correctness of theoretical analysesand some conclusions are given respectively

2 The Proposed Model

21 System Description Wemodel a wireless sensor networkcomposed of119873nodesThenodes are uniformly distributed in119871 times 119871 area (nodes average density is 120588 = 1198731198712) and the wire-less communication range of every node is 119903 The topologicalstructure of a WSN is shown in Figure 1

Based on the existing 119878119868119877 epidemic model [24 25] thenodes in WSNs are classified into three states

(i) Susceptible state (119878) nodes in 119878have not been infectedbyworms and these nodes are vulnerable toworms inWSNs

(ii) Infected state (119868) nodes have been infected by wormsand have the ability to infect other nodes in WSNs

(iii) Recover state (119877) nodes have installed a detectiontool that can identify and remove worms or nodeshave installed a software patch to eliminate the nodevulnerability exploited by worms

L

L

r

Figure 1 The topology structure of a WSN

120583N

120583S

IS120574I

R

120583R120583I

120576R

120596S

120573S998400I

Figure 2 Transition relationship of states of nodes

We consider the following state transitions among thesethree states

(i) Users may immunize their nodes with countermea-sures in states 119878 and 119868 with probabilities 120596 and 120574respectively

(ii) As the energy of nodes is exhausted some nodesbecome dead nodes with probability 120583

(iii) Infected nodes 119868 infect susceptible 119878 with effectiveinfection rate 120573

(iv) Some recovered nodesmay become susceptible nodeswith probability 120576

Transition relationships among node states are described inFigure 2

22 Model Derivation The communication area of a node isdenoted by 119878

119903 and the density of susceptible nodes in a unit

area inWSNs is denoted by119901(119905)Then the following equationshold

119901 (119905) =119878 (119905)

1198712

119878119903= 1205871199032

1198781015840(119905) = 119878

119903119901 (119905)

(1)

Mathematical Problems in Engineering 3

From (1) we can get

1198781015840(119905) = 120587119903

2 119878 (119905)

1198712 (2)

According to state transition relationships in Figure 2 themathematical model of worm propagation in WSNs can bederived as follows

119889119878 (119905)

119889119905= 120583119873 minus

1205871199032

1198712120573119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877

119889119868 (119905)

119889119905=1205871199032

1198712120573119878 (119905) 119868 (119905) minus (120583 + 120574) 119868

119889119877 (119905)

119889119905= 120596119878 + 120574119868 minus (120583 + 120576) 119877

(3)

For convenience let

120585 =1205871199032

1198712120573 (4)

Then system (3) can be written as

119889119878

119889119905= 120583119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877

119889119868

119889119905= 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868

119889119877

119889119905= 120596119878 + 120574119868 minus (120583 + 120576) 119877

(5)

3 Stability Analysis of Equilibriums

In this section we will find the equilibriums of system (5) andinvestigate their stability The equilibriums of system (5) aregiven by solutions of

120583119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877 = 0

120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 = 0

120596119878 + 120574119868 minus (120583 + 120576) 119877 = 0

(6)

Let 120585119878(119905)119868(119905) minus (120583 + 120574)119868 = 0 (the second equation of (6)) wehave 119868lowast = 0 or 119868lowast gt 0 and 119878lowast = (120583 + 120574)120585 For the case of119868lowast= 0 we have worm-free equilibrium

1198760= (119878lowast

0 119868lowast

0 119877lowast

0) = (

(120583 + 120576)

120583 + 120576 + 120596119873 0

120596119878

120583 + 120576) (7)

For the case of 119868lowast gt 0 we have endemic equilibrium

119876lowast= (119878lowast

1 119868lowast

1 119877lowast

1) = (

120583 + 120574

120585

120585 (120583 + 120576)119873 minus (120583 + 120574) (120583 + 120576 + 120596)

120585 (120583 + 120576 + 120574)120596119878lowast

1+ 120574119868lowast

1

120583 + 120576)

(8)

Let

1198770=

120585 (120583 + 120576)

(120583 + 120574) (120583 + 120576 + 120596)119873 (9)

Notably the endemic equilibrium is meaningful only if 1198770gt

1

31 Worm-Free Equilibrium and Its Stability

Lemma 1 Theworm-free equilibrium is locally asymptoticallystable if 119877

0lt 1 and unstable if 119877

0gt 1

Proof According to 1198760= (((120583 + 120576)(120583 + 120576 + 120596))119873 0 120596119878(120583 +

120576)) the characteristic equation of system (5) at worm-freeequilibrium 119876

0is

det(minus(120583 + 120596) minus 120582 minus120585119878

lowast

0120576

0 120585119878 minus (120583 + 120574) minus 120582 0

120596 120574 (120583 + 120576) minus 120582

)

= 0

(10)

which is equivalent to

[120585119878lowast

0minus (120583 + 120574) minus 120582]

sdot [1205822minus (2120583 + 120576 + 120596) 120582 + 120583

2+ 120576120583 + 120583120596] = 0

(11)

Equation (11) has a characteristic root 1205821= 120585119878lowast

0minus (120583 + 120574) =

(120583 + 120574)(1198770minus 1) and the roots of equation

1205822+ (2120583 + 120576 + 120596) 120582 + 120583

2+ 120576120583 + 120583120596 = 0 (12)

Obviously in accordance with the relationship betweenroots and coefficients of quadratic equation there is nopositive real part characteristic root of (12) Hence when119877

0lt

1 (11) has no positive real root and worm-free equilibrium1198760is locally asymptotically stable When 119877

0gt 1 (11) has a

positive root thus worm-free equilibrium 1198760is an unstable

saddle-point

Furthermore the following theorem holds

Theorem 2 The worm-free equilibrium is globally asymptoti-cally stable if 119877

0le 1

Proof From the first equation of system (5)

119878 (119905) le (120583 + 120576)119873 minus (120583 + 120576 + 120596) 119878 (13)


Thus 119878(119905) le (120583 + 120576)119873(120583 + 120576 + 120596) + (119878(0) minus (120583 + 120576)(120583 + 120576 +

120596))exp[minus(120583 + 120576 + 120596)119905] When 119905 rarr infin we obtain

119878 (119905) le120583 + 120576

120583 + 120576 + 120596119873 (14)

Consider a Lyapunov function

119871 (119905) = 119864 (119905)

(119905) = 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

le [120585120583 + 120596

120583 + 120596 + 120576119873 minus (120583 + 120574)] 119868 (119905)

=1

120583 + 120574(1198770minus 1) 119868 (119905) le 0

(15)

So we prove that worm-free equilibrium 1198760is globally

asymptotically stable

32 Endemic Equilibrium and Its Stability Now we inves-tigate the local stability of endemic equilibrium 119876

lowast Thecharacteristic equation of system (5) at endemic equilibrium119876lowast is

det(minus120585119868lowast

1minus (120583 + 120596) minus 120582 minus120585119878

lowast

1120576

120585119868lowast

1120585119878lowast

1minus (120583 + 120574) minus 120582 0

120596 120574 (120583 + 120576) minus 120582

)

= 0

(16)


1205823+ 11990101205822+ 1199011120582 + 1199012= 0 (17)

where 1199010= 3120583 + 120576 + 120596 + ((120583 + 120576)(120583 + 120576 + 120574))119873 119901

1= (120583 +

120576)[2120583 + 120574 + ((120583 + 120576)(120583 + 120576 + 120574))119873] + 120583120596 + 1205852119878lowast

1119868lowast

1 and 119901

2=

120583120585(120583 + 120576)(119868lowast

1minus 119878lowast

1) + 120583(120583 + 120574)(120583 + 120576 + 120596) + 120583120585(120574119868

lowast

1minus 120596119878lowast

1)

Obviously 1199010

gt 0 1199011

gt 0 1199012

gt 0 and 11990101199011minus

1199012gt 0 According to the theorem of Routh-Hurwitz [26 27]

it follows that the roots of (17) have negative real partsTherefore the endemic equilibrium is locally asymptoticallystable

From the above discussion we can summarize the follow-ing conclusion

Lemma 3 If 1198770gt 1 then endemic equilibrium is locally


Note that the number of nodes in WSNs is relativelystable that is at time 119905 the number of nodes 119878(119905) 119868(119905) and119877(119905) in states 119878 119868 and 119877 respectively satisfies

119878 (119905) + 119868 (119905) + 119877 (119905) = 119873 (18)Hence the dynamics of system (5) is equivalent to thefollowing system

119889119878 (119905)

119889119905= (120583 + 120596)119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596 + 120576) 119878 (119905)

minus 120576119868 (119905)

119889119868 (119905)

119889119905= 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

(19)

Clearly when 1198770gt 1 system (19) has a unique positive

equilibrium 119876lowast(119878lowast

1 119868lowast

1) where

(119878lowast

1 119868lowast

1)

= (120583 + 120574

120585120585 (120583 + 120576)119873 minus (120583 + 120574) (120583 + 120576 + 120596)

120585 (120583 + 120576 + 120574))

(20)

Now we state and prove a result on the global stability of theendemic equilibrium (119878

lowast

1 119868lowast

1) of system (19)

Theorem 4 When 1198770gt 1 the endemic equilibrium (119878

lowast

1 119868lowast

1)

of system (19) is globally asymptotically stable

Proof Consider the following Lyapunov function [28]

119871 (119905) = int

119878

119878lowast

1

119909 minus 119878lowast

1

119909119889119909 + int

119868

119868lowast

1


1

119909119889119909 (21)

The time derivative of 119871(119905) along the solution of system (19)is given by

(119905) = (119878 minus 119878lowast

1

119878) 1198781015840+ (


1

119868) 1198681015840= (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873

minus 120585119878 (119905) 119868 (119905) minus (120583 + 120576 + 120596) 119878 (119905) minus 120576119868 (119905)] + (1

minus119868lowast

1

119868) [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)] le (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873

minus 120585119878 (119905) 119868 (119905) minus (120583 + 120576 + 120596) 119878 (119905)] + (1 minus119868lowast

1

119868)

sdot [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)]

= minus (120583 + 120576)119873119878

119878lowast

1

(119878lowast

1

119878minus 1)

2

le 0

(22)

The proof is completed

Remark 5 By Theorem 4 we obtain that 1198770

gt 1 thenendemic equilibrium 119876

lowast of system (5) is globally asymptoti-cally stable

4 Worm Propagation Threshold Analysis andNumerical Simulations

We have proved that the basic reproductive number 1198770

equaling zero is the threshold whether worms are eliminatedWhen 119877

0le 1 worms in WSNs can be eliminated and

system (5) will stabilize at worm-free equilibrium When1198770gt 1 worms in WSNs will exist consistently and system

(5) will stabilize at the endemic equilibrium For verifyingthe correctness of theoretical analysis we perform wormpropagation threshold analyses and numerical simulationsfrom the following two sides


1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)

Figure 3 Simulation results of different communication radius of nodes (a) 119903 = 01 (b) 119903 = 03 (c) 119903 = 08 and (d) 119903 = 1

41 Communication Radius of Nodes 119903 Let 1198770= (120585(120583 +

120576)(120583 + 120574)(120583 + 120576 + 120596))119873 = 1 we can get the thresholdof worm propagation about communication radius of nodes119903119888= 119871radic(120583 + 120574)(120583 + 120576 + 120596)120587120573(120583 + 120576)119873 that is when 119903 le 119903

119888

1198770le 1 according to Theorem 2 worms in WSNs can be

eliminated and system (5) will stabilize at the worm-freeequilibrium when 119903 gt 119903

119888 1198770gt 1 according to Remark 5

worms in WSNs will exist consistently and system (5) willstabilize at the endemic equilibrium

We choose a set of simulation parameters as follows119873 =

1000 119871 = 10 120583 = 0001 120573 = 00003 120596 = 0001 120576 = 00003and 120574 = 0002 By calculation we have 119903

119888= 07506 Initial

values of susceptible infected and recovered nodes in WSNs

are 119878(0) = 990 119868(0) = 10 and 119877(0) = 0 When 119903 takesdifferent values simulation results are depicted in Figures3(a)ndash3(d)

When 119903 = 01 lt 119903119888and 119903 = 03 lt 119903

119888 Figures 3(a) and 3(b)

show that system (5) stabilizes at worm-free equilibriumThesimulation results are consistent withTheorem 2

When 119903 = 08 gt 119903119888and 119903 = 1 gt 119903

119888 Figures 3(c) and 3(d)

show that system (5) stabilizes at the endemic equilibriumSimulation results are consistent with Remark 5

42 NodesDistributedDensity120588 Let1198770= (120585(120583+120576)(120583+120574)(120583+

120576 + 120596))119873 = (1205871199032120573(120583 + 120576)(120583 + 120574)(120583 + 120576 + 120596))120588 = 1 we can get

the threshold of worm propagation about node distributed


1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)

Figure 4 Simulation results of different nodes density (a) 120588 = 25 (b) 120588 = 16 (c) 120588 = 6944 and (d) 120588 = 10

density 120588119888= (120583 + 120574)(120583 + 120576 + 120596)120587119903

2120573(120583 + 120576) When 120588 lt 120588

119888

1198770le 1 system (5) has only a worm-free equilibrium and is

globally asymptotically stable when 120588 gt 120588119888 1198770gt 1 system

(5) has an endemic equilibrium besides the worm-free equi-librium and endemic equilibrium is globally asymptoticallystable


1000 119903 = 1120583 = 0001120573 = 00003120596 = 0001 120576 = 00003 and120574 = 0002 By calculation we get 120588

119888= 56345 Initial values

of system (5) are set as 119878(0) = 990 119868(0) = 10 and 119877(0) = 0When 119871 = 20 25 12 and 10 we can get 120588 = 119873119871

2= 25

16 6944 and 10 Simulation results are depicted in Figures4(a)ndash4(d)

When 120588 = 25 lt 120588119888and 120588 = 16 lt 120588

119888 Figures

4(a) and 4(b) indicate that system (5) stabilizes at the worm-free equilibrium and worm propagation is controlled finallySimulation results are consistent with theoretical analysis

When 120588 = 6944 gt 120588119888and 120588 = 10 gt 120588

119888 Figures 4(c)

and 4(d) show that the trajectories converge to the endemicequilibriumThe conclusions agree with theoretical analysis

5 Conclusions

In this paper we have proposed an improved SIRS modelfor analyzing dynamics of worm propagation in WSNs Thismodel can describe the process of worm propagation with


the energy consumption and different distributed densityof nodes Based on this model a control parameter 119877

0

that completely determines the global dynamics of wormpropagation has been obtained by the explicit mathematicalanalyses From Theorems 4 and 2 we learn out that wormwill be controlled in WSNs when 119877

0lt 1 and they will

be prevalent otherwise Finally based on 1198770 we discuss the

threshold of worm propagation about communication radiusand distributed densities of nodes inWSNs Numerical simu-lations verify the correctness of theoretical analysis Researchresults show that decreasing the value of communicationradius or reducing distributed density of nodes is an effectivemethod to prevent worms spread in WSNs Research ofthis paper provides the theoretical basis for predicting andcontrolling worm propagation in WSNs It is worth pointingout that we do not consider physical effects like ldquocollisionsrdquoand heterogeneous distribution of nodes on infection ratewhen modeling which is our focus in the future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61379125) the Natural Science Foun-dation of Shanxi Province (2015011053) Key ConstructionDisciplines of Xinzhou Normal University (ZDXK201204XK201307)

References

[1] S Tang and W Li ldquoQoS supporting and optimal energy allo-cation for a cluster based wireless sensor networkrdquo ComputerCommunications vol 29 no 13-14 pp 2569ndash2577 2006

[2] L H Zhu and H Y Zhao ldquoDynamical analysis and optimalcontrol for a malware propagation model in an informationnetworkrdquo Neurocomputing vol 149 pp 1370ndash1386 2015

[3] S S W Lee P-K Tseng and A Chen ldquoLink weight assignmentand loop-free routing table update for link state routing pro-tocols in energy-aware internetrdquo Future Generation ComputerSystems vol 28 no 2 pp 437ndash445 2012

[4] T Rault A Bouabdallah and Y Challal ldquoEnergy efficiencyin wireless sensor networks a top-down surveyrdquo ComputerNetworks vol 67 pp 104ndash122 2014

[5] M Chiang ldquoBalancing transport and physical layers in wirelessmultihop networks jointly optimal congestion control andpower controlrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 1 pp 104ndash116 2005

[6] H Shi W Wang and N Kwok ldquoEnergy dependent divisibleload theory for wireless sensor network workload allocationrdquoMathematical Problems in Engineering vol 2012 Article ID235289 16 pages 2012

[7] J Zhang and H-N Lee ldquoEnergy-efficient utility maximizationfor wireless networks withwithout multipath routingrdquo AEUmdashInternational Journal of Electronics and Communications vol64 no 2 pp 99ndash111 2010

[8] C-Y Chang J-P Sheu Y-C Chen and S-W Chang ldquoAnobstacle-free and power-efficient deployment algorithm forwireless sensor networksrdquo IEEE Transactions on Systems Manand Cybernetics Part A Systems and Humans vol 39 no 4 pp795ndash806 2009

[9] H Chen C K Tse and J Feng ldquoImpact of topology onperformance and energy efficiency in wireless sensor networksfor source extractionrdquo IEEE Transactions on Parallel and Dis-tributed Systems vol 20 no 6 pp 886ndash897 2009

[10] M H Khouzani and S Sarkar ldquoMaximum damage batterydepletion attack in mobile sensor networksrdquo IEEE Transactionson Automatic Control vol 56 no 10 pp 2358ndash2368 2011

[11] P De Y Liu and S K Das ldquoAn epidemic theoretic frameworkfor vulnerability analysis of broadcast protocols in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol8 no 3 pp 413ndash425 2009

[12] S Zanero ldquoWirelessmalware propagation a reality checkrdquo IEEESecurity amp Privacy vol 7 no 5 pp 70ndash74 2009

[13] S A Khayam and H Radha ldquoUsing signal processing tech-niques to model worm propagation over wireless sensor net-worksrdquo IEEE Signal Processing Magazine vol 23 no 2 pp 164ndash169 2006

[14] G Yan and S Eidenbenz ldquoModeling propagation dynamics ofbluetooth wormsrdquo IEEE Transactions onMobile Computing vol8 no 3 pp 353ndash367 2009

[15] C C Zou D Towsley and W B Gong ldquoModeling andsimulation study of the propagation and defense of internete-mail wormsrdquo IEEE Transactions on Dependable and SecureComputing vol 4 no 2 pp 105ndash118 2007

[16] B KMishra andD K Saini ldquoSEIRS epidemicmodel with delayfor transmission of malicious objects in computer networkrdquoAppliedMathematics and Computation vol 188 no 2 pp 1476ndash1482 2007

[17] S J Wang Q M Liu X F Yu and Y Ma ldquoBifurcation analysisof a model for network worm propagation with time delayrdquoMathematical and ComputerModelling vol 52 no 3-4 pp 435ndash447 2010

[18] Q Zhu X Yang L-X Yang and C Zhang ldquoOptimal control ofcomputer virus under a delayed modelrdquo Applied Mathematicsand Computation vol 218 no 23 pp 11613ndash11619 2012

[19] L Feng X Liao H Li and Q Han ldquoHopf bifurcation analysisof a delayed viral infection model in computer networksrdquoMathematical and Computer Modelling vol 56 no 7-8 pp 167ndash179 2012

[20] L Feng X Liao Q Han and H Li ldquoDynamical analysisand control strategies on malware propagation modelrdquo AppliedMathematical Modelling vol 37 no 16-17 pp 8225ndash8236 2013

[21] L H Zhu H Y Zhao and X M Wang ldquoBifurcation analysisof a delay reaction-diffusion malware propagation model withfeedback controlrdquo Communications in Nonlinear Science andNumerical Simulation vol 22 no 1ndash3 pp 747ndash768 2015

[22] B K Mishra and N Keshri ldquoMathematical model on thetransmission of worms in wireless sensor networkrdquo AppliedMathematical Modelling vol 37 no 6 pp 4103ndash4111 2013

[23] X M Wang and Y S Li ldquoAn improved SIR model foranalyzing the dynamics of worm propagation in wireless sensornetworksrdquo Chinese Journal of Electronics vol 18 no 1 pp 8ndash122009

[24] D J Daley and J Gani Epidemic Modeling An IntroductionCambridge University Press New York NY USA 1999


[25] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Security and Privacy pp 2ndash15 May 1993

[26] E A Barbashin Introduction to theTheory of Stability Walters-Noordhoff Groningen The Netherlands 1970

[27] J La Salle and S Lefschetz Stability by LiapunovsDirectMethodAcademic Press New York NY USA 1961

[28] H Yuan and G Q Chen ldquoNetwork virus-epidemic model withthe point-to-group information propagationrdquo Applied Mathe-matics and Computation vol 206 no 1 pp 357ndash367 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The authors in [22] presented an epidemic model withvaccination compartment which captures both the spatialand temporal dynamics of worm spread process and somemathematical analyses and numerical simulations were per-formed based on this model The common problem of theabove models is that the characteristics of wireless sensornetworks like energy consumption communication radiusand distributed density of nodes have not been considered inmodels In [11] the authors developed a mathematical modelfor the propagation that incorporates important parametersderived from the communication patterns of the protocolunder test Based on this model the authors analyzed thepropagation rate and the extent of spread of a malware overtypical broadcast protocols proposed in the paper Wang andLi derived an iSIR model describing the process of wormpropagation with energy consumption of nodes in WSNs[23] Numerical simulations are performed to observe theeffects of the network topology and energy consumption ofnodes on worm spread in WSNs However the authors havenot performedmathematical analyses based on thismodel Infact key parameters of affectingworm spread can be found byexplicit mathematical analyses

To better portrait the features of worm propagation inWSNs in this paper we study the attacking behavior ofpossible worms in WSNs by constructing an improved SIRSepidemicmodel In thismodel the following three factors areconsidered (i) energy consumptions of nodes (ii) communi-cation radius of nodes and (iii) distributed density of nodesin WSNs Based on this model we analyze the stability ofworm prevalence through finding the equilibriums of model

The rest of this paper is organized as follows in Section 2we analyze topology of WSNs and present the model for-mulation Section 3 derives the equilibriums of the modeland discusses the stability of worm propagation at theequilibriums In Sections 4 and 5 numerical simulations areperformed to verify the correctness of theoretical analysesand some conclusions are given respectively

2 The Proposed Model

21 System Description Wemodel a wireless sensor networkcomposed of119873nodesThenodes are uniformly distributed in119871 times 119871 area (nodes average density is 120588 = 1198731198712) and the wire-less communication range of every node is 119903 The topologicalstructure of a WSN is shown in Figure 1

Based on the existing 119878119868119877 epidemic model [24 25] thenodes in WSNs are classified into three states

(i) Susceptible state (119878) nodes in 119878have not been infectedbyworms and these nodes are vulnerable toworms inWSNs

(ii) Infected state (119868) nodes have been infected by wormsand have the ability to infect other nodes in WSNs

(iii) Recover state (119877) nodes have installed a detectiontool that can identify and remove worms or nodeshave installed a software patch to eliminate the nodevulnerability exploited by worms

L

L

r

Figure 1 The topology structure of a WSN

120583N

120583S

IS120574I

R

120583R120583I

120576R

120596S

120573S998400I

Figure 2 Transition relationship of states of nodes

We consider the following state transitions among thesethree states

(i) Users may immunize their nodes with countermea-sures in states 119878 and 119868 with probabilities 120596 and 120574respectively

(ii) As the energy of nodes is exhausted some nodesbecome dead nodes with probability 120583

(iii) Infected nodes 119868 infect susceptible 119878 with effectiveinfection rate 120573

(iv) Some recovered nodesmay become susceptible nodeswith probability 120576

Transition relationships among node states are described inFigure 2

22 Model Derivation The communication area of a node isdenoted by 119878

119903 and the density of susceptible nodes in a unit

area inWSNs is denoted by119901(119905)Then the following equationshold

119901 (119905) =119878 (119905)

1198712

119878119903= 1205871199032

1198781015840(119905) = 119878

119903119901 (119905)

(1)


From (1) we can get

1198781015840(119905) = 120587119903

2 119878 (119905)

1198712 (2)


119889119878 (119905)

119889119905= 120583119873 minus

1205871199032

1198712120573119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877

119889119868 (119905)

119889119905=1205871199032

1198712120573119878 (119905) 119868 (119905) minus (120583 + 120574) 119868

119889119877 (119905)

119889119905= 120596119878 + 120574119868 minus (120583 + 120576) 119877

(3)

For convenience let

120585 =1205871199032

1198712120573 (4)


119889119878

119889119905= 120583119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877

119889119868

119889119905= 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868

119889119877

119889119905= 120596119878 + 120574119868 minus (120583 + 120576) 119877

(5)



120583119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877 = 0

120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 = 0

120596119878 + 120574119868 minus (120583 + 120576) 119877 = 0

(6)


1198760= (119878lowast

0 119868lowast

0 119877lowast

0) = (

(120583 + 120576)

120583 + 120576 + 120596119873 0

120596119878

120583 + 120576) (7)



1 119868lowast

1 119877lowast

1) = (

120583 + 120574

120585

120585 (120583 + 120576)119873 minus (120583 + 120574) (120583 + 120576 + 120596)

120585 (120583 + 120576 + 120574)120596119878lowast

1+ 120574119868lowast

1

120583 + 120576)

(8)

Let

1198770=

120585 (120583 + 120576)

(120583 + 120574) (120583 + 120576 + 120596)119873 (9)


1




0gt 1



0is


lowast

0120576

0 120585119878 minus (120583 + 120574) minus 120582 0

120596 120574 (120583 + 120576) minus 120582

)

= 0

(10)


[120585119878lowast

0minus (120583 + 120574) minus 120582]

sdot [1205822minus (2120583 + 120576 + 120596) 120582 + 120583

2+ 120576120583 + 120583120596] = 0

(11)


0minus (120583 + 120574) =


1205822+ (2120583 + 120576 + 120596) 120582 + 120583

2+ 120576120583 + 120583120596 = 0 (12)


0lt


0gt 1 (11) has a


saddle-point



0le 1


119878 (119905) le (120583 + 120576)119873 minus (120583 + 120576 + 120596) 119878 (13)


Thus 119878(119905) le (120583 + 120576)119873(120583 + 120576 + 120596) + (119878(0) minus (120583 + 120576)(120583 + 120576 +


119878 (119905) le120583 + 120576

120583 + 120576 + 120596119873 (14)


119871 (119905) = 119864 (119905)

(119905) = 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

le [120585120583 + 120596

120583 + 120596 + 120576119873 minus (120583 + 120574)] 119868 (119905)

=1

120583 + 120574(1198770minus 1) 119868 (119905) le 0

(15)







lowast

1120576

120585119868lowast

1120585119878lowast

1minus (120583 + 120574) minus 120582 0

120596 120574 (120583 + 120576) minus 120582

)

= 0

(16)


1205823+ 11990101205822+ 1199011120582 + 1199012= 0 (17)

where 1199010= 3120583 + 120576 + 120596 + ((120583 + 120576)(120583 + 120576 + 120574))119873 119901

1= (120583 +

120576)[2120583 + 120574 + ((120583 + 120576)(120583 + 120576 + 120574))119873] + 120583120596 + 1205852119878lowast

1119868lowast

1 and 119901

2=

120583120585(120583 + 120576)(119868lowast

1minus 119878lowast

1) + 120583(120583 + 120574)(120583 + 120576 + 120596) + 120583120585(120574119868

lowast


1)

Obviously 1199010

gt 0 1199011

gt 0 1199012

gt 0 and 11990101199011minus








119889119878 (119905)

119889119905= (120583 + 120596)119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596 + 120576) 119878 (119905)

minus 120576119868 (119905)

119889119868 (119905)

119889119905= 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

(19)



1 119868lowast

1) where

(119878lowast

1 119868lowast

1)

= (120583 + 120574

120585120585 (120583 + 120576)119873 minus (120583 + 120574) (120583 + 120576 + 120596)

120585 (120583 + 120576 + 120574))

(20)


lowast

1 119868lowast

1) of system (19)


lowast

1 119868lowast

1)



119871 (119905) = int

119878

119878lowast

1


1

119909119889119909 + int

119868

119868lowast

1


1

119909119889119909 (21)


(119905) = (119878 minus 119878lowast

1

119878) 1198781015840+ (


1

119868) 1198681015840= (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873


minus119868lowast

1

119868) [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)] le (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873


1

119868)

sdot [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)]

= minus (120583 + 120576)119873119878

119878lowast

1

(119878lowast

1

119878minus 1)

2

le 0

(22)












1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)




119888







119888= 07506 Initial



When 119903 = 01 lt 119903119888and 119903 = 03 lt 119903



When 119903 = 08 gt 119903119888and 119903 = 1 gt 119903




120576 + 120596))119873 = (1205871199032120573(120583 + 120576)(120583 + 120574)(120583 + 120576 + 120596))120588 = 1 we can get



1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)


density 120588119888= (120583 + 120574)(120583 + 120576 + 120596)120587119903

2120573(120583 + 120576) When 120588 lt 120588

119888








2= 25


When 120588 = 25 lt 120588119888and 120588 = 16 lt 120588

119888 Figures


When 120588 = 6944 gt 120588119888and 120588 = 10 gt 120588

119888 Figures 4(c)


5 Conclusions




0


0lt 1 and they will





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










From (1) we can get

1198781015840(119905) = 120587119903

2 119878 (119905)

1198712 (2)


119889119878 (119905)

119889119905= 120583119873 minus

1205871199032

1198712120573119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877

119889119868 (119905)

119889119905=1205871199032

1198712120573119878 (119905) 119868 (119905) minus (120583 + 120574) 119868

119889119877 (119905)

119889119905= 120596119878 + 120574119868 minus (120583 + 120576) 119877

(3)

For convenience let

120585 =1205871199032

1198712120573 (4)


119889119878

119889119905= 120583119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877

119889119868

119889119905= 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868

119889119877

119889119905= 120596119878 + 120574119868 minus (120583 + 120576) 119877

(5)



120583119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596) 119878 + 120576119877 = 0

120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 = 0

120596119878 + 120574119868 minus (120583 + 120576) 119877 = 0

(6)


1198760= (119878lowast

0 119868lowast

0 119877lowast

0) = (

(120583 + 120576)

120583 + 120576 + 120596119873 0

120596119878

120583 + 120576) (7)



1 119868lowast

1 119877lowast

1) = (

120583 + 120574

120585

120585 (120583 + 120576)119873 minus (120583 + 120574) (120583 + 120576 + 120596)

120585 (120583 + 120576 + 120574)120596119878lowast

1+ 120574119868lowast

1

120583 + 120576)

(8)

Let

1198770=

120585 (120583 + 120576)

(120583 + 120574) (120583 + 120576 + 120596)119873 (9)


1




0gt 1



0is


lowast

0120576

0 120585119878 minus (120583 + 120574) minus 120582 0

120596 120574 (120583 + 120576) minus 120582

)

= 0

(10)


[120585119878lowast

0minus (120583 + 120574) minus 120582]

sdot [1205822minus (2120583 + 120576 + 120596) 120582 + 120583

2+ 120576120583 + 120583120596] = 0

(11)


0minus (120583 + 120574) =


1205822+ (2120583 + 120576 + 120596) 120582 + 120583

2+ 120576120583 + 120583120596 = 0 (12)


0lt


0gt 1 (11) has a


saddle-point



0le 1


119878 (119905) le (120583 + 120576)119873 minus (120583 + 120576 + 120596) 119878 (13)


Thus 119878(119905) le (120583 + 120576)119873(120583 + 120576 + 120596) + (119878(0) minus (120583 + 120576)(120583 + 120576 +


119878 (119905) le120583 + 120576

120583 + 120576 + 120596119873 (14)


119871 (119905) = 119864 (119905)

(119905) = 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

le [120585120583 + 120596

120583 + 120596 + 120576119873 minus (120583 + 120574)] 119868 (119905)

=1

120583 + 120574(1198770minus 1) 119868 (119905) le 0

(15)







lowast

1120576

120585119868lowast

1120585119878lowast

1minus (120583 + 120574) minus 120582 0

120596 120574 (120583 + 120576) minus 120582

)

= 0

(16)


1205823+ 11990101205822+ 1199011120582 + 1199012= 0 (17)

where 1199010= 3120583 + 120576 + 120596 + ((120583 + 120576)(120583 + 120576 + 120574))119873 119901

1= (120583 +

120576)[2120583 + 120574 + ((120583 + 120576)(120583 + 120576 + 120574))119873] + 120583120596 + 1205852119878lowast

1119868lowast

1 and 119901

2=

120583120585(120583 + 120576)(119868lowast

1minus 119878lowast

1) + 120583(120583 + 120574)(120583 + 120576 + 120596) + 120583120585(120574119868

lowast


1)

Obviously 1199010

gt 0 1199011

gt 0 1199012

gt 0 and 11990101199011minus








119889119878 (119905)

119889119905= (120583 + 120596)119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596 + 120576) 119878 (119905)

minus 120576119868 (119905)

119889119868 (119905)

119889119905= 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

(19)



1 119868lowast

1) where

(119878lowast

1 119868lowast

1)

= (120583 + 120574

120585120585 (120583 + 120576)119873 minus (120583 + 120574) (120583 + 120576 + 120596)

120585 (120583 + 120576 + 120574))

(20)


lowast

1 119868lowast

1) of system (19)


lowast

1 119868lowast

1)



119871 (119905) = int

119878

119878lowast

1


1

119909119889119909 + int

119868

119868lowast

1


1

119909119889119909 (21)


(119905) = (119878 minus 119878lowast

1

119878) 1198781015840+ (


1

119868) 1198681015840= (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873


minus119868lowast

1

119868) [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)] le (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873


1

119868)

sdot [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)]

= minus (120583 + 120576)119873119878

119878lowast

1

(119878lowast

1

119878minus 1)

2

le 0

(22)












1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)




119888







119888= 07506 Initial



When 119903 = 01 lt 119903119888and 119903 = 03 lt 119903



When 119903 = 08 gt 119903119888and 119903 = 1 gt 119903




120576 + 120596))119873 = (1205871199032120573(120583 + 120576)(120583 + 120574)(120583 + 120576 + 120596))120588 = 1 we can get



1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)


density 120588119888= (120583 + 120574)(120583 + 120576 + 120596)120587119903

2120573(120583 + 120576) When 120588 lt 120588

119888








2= 25


When 120588 = 25 lt 120588119888and 120588 = 16 lt 120588

119888 Figures


When 120588 = 6944 gt 120588119888and 120588 = 10 gt 120588

119888 Figures 4(c)


5 Conclusions




0


0lt 1 and they will





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thus 119878(119905) le (120583 + 120576)119873(120583 + 120576 + 120596) + (119878(0) minus (120583 + 120576)(120583 + 120576 +


119878 (119905) le120583 + 120576

120583 + 120576 + 120596119873 (14)


119871 (119905) = 119864 (119905)

(119905) = 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

le [120585120583 + 120596

120583 + 120596 + 120576119873 minus (120583 + 120574)] 119868 (119905)

=1

120583 + 120574(1198770minus 1) 119868 (119905) le 0

(15)







lowast

1120576

120585119868lowast

1120585119878lowast

1minus (120583 + 120574) minus 120582 0

120596 120574 (120583 + 120576) minus 120582

)

= 0

(16)


1205823+ 11990101205822+ 1199011120582 + 1199012= 0 (17)

where 1199010= 3120583 + 120576 + 120596 + ((120583 + 120576)(120583 + 120576 + 120574))119873 119901

1= (120583 +

120576)[2120583 + 120574 + ((120583 + 120576)(120583 + 120576 + 120574))119873] + 120583120596 + 1205852119878lowast

1119868lowast

1 and 119901

2=

120583120585(120583 + 120576)(119868lowast

1minus 119878lowast

1) + 120583(120583 + 120574)(120583 + 120576 + 120596) + 120583120585(120574119868

lowast


1)

Obviously 1199010

gt 0 1199011

gt 0 1199012

gt 0 and 11990101199011minus








119889119878 (119905)

119889119905= (120583 + 120596)119873 minus 120585119878 (119905) 119868 (119905) minus (120583 + 120596 + 120576) 119878 (119905)

minus 120576119868 (119905)

119889119868 (119905)

119889119905= 120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)

(19)



1 119868lowast

1) where

(119878lowast

1 119868lowast

1)

= (120583 + 120574

120585120585 (120583 + 120576)119873 minus (120583 + 120574) (120583 + 120576 + 120596)

120585 (120583 + 120576 + 120574))

(20)


lowast

1 119868lowast

1) of system (19)


lowast

1 119868lowast

1)



119871 (119905) = int

119878

119878lowast

1


1

119909119889119909 + int

119868

119868lowast

1


1

119909119889119909 (21)


(119905) = (119878 minus 119878lowast

1

119878) 1198781015840+ (


1

119868) 1198681015840= (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873


minus119868lowast

1

119868) [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)] le (1 minus

119878lowast

1

119878)

sdot [(120583 + 120576)119873


1

119868)

sdot [120585119878 (119905) 119868 (119905) minus (120583 + 120574) 119868 (119905)]

= minus (120583 + 120576)119873119878

119878lowast

1

(119878lowast

1

119878minus 1)

2

le 0

(22)












1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)




119888







119888= 07506 Initial



When 119903 = 01 lt 119903119888and 119903 = 03 lt 119903



When 119903 = 08 gt 119903119888and 119903 = 1 gt 119903




120576 + 120596))119873 = (1205871199032120573(120583 + 120576)(120583 + 120574)(120583 + 120576 + 120596))120588 = 1 we can get



1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)


density 120588119888= (120583 + 120574)(120583 + 120576 + 120596)120587119903

2120573(120583 + 120576) When 120588 lt 120588

119888








2= 25


When 120588 = 25 lt 120588119888and 120588 = 16 lt 120588

119888 Figures


When 120588 = 6944 gt 120588119888and 120588 = 10 gt 120588

119888 Figures 4(c)


5 Conclusions




0


0lt 1 and they will





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)




119888







119888= 07506 Initial



When 119903 = 01 lt 119903119888and 119903 = 03 lt 119903



When 119903 = 08 gt 119903119888and 119903 = 1 gt 119903




120576 + 120596))119873 = (1205871199032120573(120583 + 120576)(120583 + 120574)(120583 + 120576 + 120596))120588 = 1 we can get



1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)


density 120588119888= (120583 + 120574)(120583 + 120576 + 120596)120587119903

2120573(120583 + 120576) When 120588 lt 120588

119888








2= 25


When 120588 = 25 lt 120588119888and 120588 = 16 lt 120588

119888 Figures


When 120588 = 6944 gt 120588119888and 120588 = 10 gt 120588

119888 Figures 4(c)


5 Conclusions




0


0lt 1 and they will





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(a)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(b)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000

Num

ber o

f nod

es

S(t)

I(t)

R(t)

(c)

1000 2000 3000 4000 50000t

0

100

200

300

400

500

600

700

800

900

1000N

umbe

r of n

odes

S(t)

I(t)

R(t)

(d)


density 120588119888= (120583 + 120574)(120583 + 120576 + 120596)120587119903

2120573(120583 + 120576) When 120588 lt 120588

119888








2= 25


When 120588 = 25 lt 120588119888and 120588 = 16 lt 120588

119888 Figures


When 120588 = 6944 gt 120588119888and 120588 = 10 gt 120588

119888 Figures 4(c)


5 Conclusions




0


0lt 1 and they will





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0


0lt 1 and they will





Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article modeling and stability analysis of worm...

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