research article modeling computer virus and its...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 842614 5 pageshttpdxdoiorg1011552013842614

Research ArticleModeling Computer Virus and Its Dynamics

Mei Peng1 Xing He2 Junjian Huang3 and Tao Dong4

1 College of Mathematical and Computer Science Yangtze Normal University Chongqing 400084 China2 College of Computer Science Chongqing University Chongqing 400030 China3 School of Computer Science Chongqing University of Education Chongqing 400067 China4College of Software and Engineering Chongqing University of Posts and Telecommunications Chongqing 400065 China

Correspondence should be addressed to Mei Peng pmgsqqcom

Received 27 March 2013 Accepted 9 June 2013

Academic Editor Tingwen Huang

Copyright copy 2013 Mei Peng et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on that the computer will be infected by infected computer and exposed computer and some of the computers which arein suscepitible status and exposed status can get immunity by antivirus ability a novel coumputer virus model is established Thedynamic behaviors of this model are investigated First the basic reproduction number 119877

0 which is a threshold of the computer

virus spreading in internet is determined Second this model has a virus-free equilibrium 1198750 which means that the infected part

of the computer disappears and the virus dies out and 1198750 is a globally asymptotically stable equilibrium if 119877

0lt 1 Third if 119877

0gt 1

then this model has only one viral equilibrium 119875lowast which means that the computer persists at a constant endemic level and 119875

lowast isalso globally asymptotically stable Finally some numerical examples are given to demonstrate the analytical results

1 Introduction

Computer virus is a malicious mobile code which includingvirus Trojan horses worm and logic bomb It is a programthat can copy itself and attack other computers And theyare residing by erasing data damaging files or modifyingthe normal operation Due to the high similarity betweencomputer virus and biological virus [1] various computervirus propagationmodels are proposed [2ndash4]This dynamicalmodeling of the spread process of computer virus is aneffective approach to the understanding of the behaviorof computer viruses because on this basis some effectivemeasures can be posed to prevent infection

The computer virus has a latent period during whichindividuals are exposed to a computer virus but are notyet infectious An infected computer which is in latencycalled exposed computer will not infect other computersimmediately however it still can be infected Based on thesecharacteristics delay is used in some models of computervirus to describe that although the exposed computer doesnot infect other computers it still has infectivity [5 6] Yanget al [7 8] proposed an SLB and SLBS models in thesemodels the authors considered that the computer virus has

latency and the computer also has infectivity in the periodof latency However they do not show the length of latencyand take into account the impact of artificial immunizationways such as installing antivirus software And the newlyentered in the internet from the susceptible status to exposedstatus the contact rate is the same as that of susceptible statusentering into infected status In this paper a novel modelof computer virus known as SEIR model is put forwardto describe the susceptible computer which can be infectedby the other infected or exposed computer and come intothe exposed status In the SEIR model based on artificialimmunity we consider the bilinear incidence rate for thelatent and infection status Assume that the computers whichnewly entered the internet are susceptible the computerscorrespond with exposed computers and their adequatecontact rate is denoted by 120573

1 and computers also correspond

with infected computers and their adequate contact rateis denoted by 120573

2 So the fraction of the computer which

newly entered the internet will enter the class 119877 by anti-virussoftware the fraction of computers contact with exposed andinfected computer will stay latent before becoming infectiousand enter the class119864 It is shown that the dynamic behavior ofthe proposedmodel is determined by a threshold 119877

0 and this

2 Mathematical Problems in Engineering

model has a virus-free equilibrium 1198750 and 119875

0 is a globallyasymptotically stable equilibrium if 119877

0lt 1 if 119877

0gt 1 this

model has only one viral equilibrium 119875lowast and it is globally

asymptotically stableThis paper is organized as follows Section 2 formulates

a novel computer virus mode Section 3 proves the globalstability of the virus-free equilibrium Section 4 discussesthe stability of the viral equilibrium In Section 5 numericalsimulations are given to present the effectiveness of thetheoretic results Finally Section 6 summarizes this work

2 Model Formulation

At any time a computer is classified as internal and externaldepending on weather it is connected to internet or notAt that time all of the internet computers are further cate-gorized into four classes (1) susceptible computers that isuninfected computers and new computers which connectedto network (2) exposed computers that is infected butnot yet broken-out (3) infectious computers (4) recoveredcomputers that is virus-free computer having immunity Let119878(119905) 119864(119905) 119868(119905) 119877(119905) denote their corresponding numbers attime 119905 without ambiguity 119878(119905) 119864(119905) 119868(119905) 119877(119905)will be abbre-viated as 119878 119864 119868 119877 respectively The model is formulated asthe following system of differential equations

1198781015840= (1 minus 119901)119873 minus 120573

1119878119868 minus 120573

2119878119864 minus 119901119878 minus 120583119878

1198641015840= 1205731119878119868 + 120573

2119878119864 minus 119896119864 minus 120572119864 minus 120583119864

1198681015840= 120572119864 minus 119903119868 minus 120583119868

1198771015840= 119901119878 + 119896119864 + 119903119868

(1)

119873(119905) = 119878 (119905) + 119864 (119905) + 119868 (119905) + 119877 (119905) (2)

We may see that the first three equations in (1) areindependent of the fourth equation and therefore the fourthequation can be omitted without loss of generality Hencesystem (1) can be rewritten as

1198781015840= 119860 minus 120573

1119878119868 minus 120573

2119878119864 minus 119886119878

1198641015840= 1205731119878119868 + 120573

2119878119864 minus 119887119864

1198681015840= 120572119864 minus 119888119868

(3)

Therefore

119886 = 119901 + 120583 119887 = 119896 + 120572 + 120583

119888 = 119903 + 120583 (1 minus 119901)119873 = 119860

(4)

where 119873 denotes the rate at which external computersare connected to the network 119901 denotes the recovery rateof susceptible computer due to the anti-virus ability ofnetwork 119896 denotes the recovery rate of exposed computerdue to the anti-virus ability of network 120573

1denotes the

rate at which when having a connection to one infectedcomputer one susceptible computer can become exposedbut has not broken-out 120573

2denotes the rate of which when

having connection to one exposed computer one susceptiblecomputer can become exposed 120572 denotes the rate of theexposed computers cannot be cured by anti-virus softwareand broken-out 119903 denotes the recovery rate of infectedcomputers that are cured 120583 denotes the rate at which onecomputer is removed from the network All the parametersare nonnegative

Moreover all feasible solutions of the system (3) arebounded and enter the region 119863 where

119863 = (119878 119864 119868) isin 1198773

+| 119878 ge 0 119864 ge 0 119868 ge 0 119878 + 119864 + 119868 le

119860

119886

(5)

Referring to [9] we define the basic reproduction numberof the infection as

1198770=

119860 (1205731120572 + 1205732119888)

119886119887119888 (6)

For system (3) there always exists the virus-free equilib-rium which is 119875

0(119860119886 0 0) if 119877

0gt 1 then there also exists

a viral equilibrium 119875lowast(119878lowast 119864lowast 119868lowast)

Therefore

119878lowast=

119860

1198861198770

119864lowast=

119860 (1198770minus 1)

1198871198770

119868lowast=

119860120572 (1198770minus 1)

1198871198881198770

(7)

3 The Virus-Free Equilibrium and Its Stability

Theorem 1 1198750 is locally asymptotically stable if 119877

0lt 1

Whereas 1198750 is unstable if 1198770gt 1

Proof The characteristic equation of (3) at 1198750 is given by

det(

120582 + 119886 minus1205732119878 minus120573

1119878

0 120582 minus (1205732119878 minus 119887) 120573

1119878

0 120572 120582 + 119888

) = 0 (8)

which equals to

(120582 + 119886) [1205822minus (12057321198780minus 119887 minus 119888) 120582 minus 119887119888 (119877

0minus 1)] = 0 (9)

Then (9) has negative real part characteristic roots

1205821= minus119886

12058223

=

(1205732119878 minus 119887 minus 119888) plusmn radic(120573

2119878 minus 119887 minus 119888)

2+ 4119886119887119888 (119877

0minus 1)

2

(10)

where

1205732119878 minus 119887 minus 119888 lt 0 (11)

Mathematical Problems in Engineering 3

When 1198770lt 1 there are no positive real roots of (9) and

thus 1198750 is a local asymptotically stable equilibrium While

1198770gt 1 (9) has positive real roots whichmeans1198750 is unstableThe proof is completed

Theorem 2 1198750 is globally asymptotically stable with respect to

119863 if 1198770lt 1

Proof Let 119871 = ((1205731119888 + 1205732120572)119887119888)119864 + 120573

2119868119888

Obviously

119871 gt 0(12)

thus

1198711015840=

(1205731119888 + 1205732120572)

1198871198881198641015840+

1205732

1198881198681015840

=(1205731119888 + 1205732120572)

119887119888(1205731119878119868 + 120573

2119878119864 minus 119887119864) +

1205732

119888(120572119864 minus 119888119868)

=(1205731119888 + 1205732120572)

119887119888(1205731119868 + 1205732119864) 119878 minus

(1205731119888 + 1205732120572)

119887119888119887119864

+1205732120572119864

119888minus 1205732119868

=(1205731119888 + 1205732120572)

119887119888(1205731119868 + 1205732119864) 119878 minus 120573

1119864 minus 1205732119868

= (1205731119864 + 1205732119868) [

(1205731119888 + 1205732120572)

119887119888119878 minus 1]

= (1205731119864 + 1205732119868) [

1

119887119888sdot119860

119886sdot1198770119886119887119888

119860minus 1]

= (1205731119864 + 1205732119868) (1198770minus 1) lt 0

(13)

The proof is completed

4 The Viral Equilibrium and Its Stability

Theorem 3 119875lowast is locally asymptotically stable if 119877

0gt 1

Proof The Jacobin matrix of system (3) about 119875lowast is given by

119869lowast= (

minus1198861198770

minus1205732119878lowast


119886 (1198770minus 1) 120573

2119878lowastminus 119887 120573

1119878lowast

0 120572 minus119888

) (14)

which equals to

119891 (120582) = 11988601205823+ 11988611205822+ 1198862120582 + 1198863= 0 (15)

where

1198860= 1

1198861= 1198861198770minus (1205732119878lowastminus 119887 minus 119888) gt 0

1198862= 119886119887119877

0+ 119886119888119877

0minus 1198861205732119878lowast

gt 1198861198871198770+ 119886119888119877

0minus 119886 (119887 + 119888)

= 119886 (119887 + 119888) (1198770minus 1) gt 0

(16)

where

1205732119878lowastlt 119887 + 119888

1198863= 119886119887119888 (119877

0minus 1) gt 0

(17)

Thus

Δ1= 1198861gt 0

Δ2=

10038161003816100381610038161003816100381610038161003816

1198861

1

1198863

1198862

10038161003816100381610038161003816100381610038161003816

= 11988611198862minus 1198863gt 0

Δ3=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198861

1 0

1198863

1198862

1198861

0 0 1198863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 1198863(11988611198862minus 1198863) gt 0

(18)

According to the Hurwitz criterion all roots of (15) havenegative real pats Thus the claimed result followsThe proofis completed

The following result can be proved in the same way (see[9])

Theorem 4 119875lowast is uniquely globally asymptotically stable if

1198770gt 1


119869lowast= (

minus1198861198770



119886 (1198770minus 1) 120573

2119878lowastminus 119887 120573

1119878lowast

0 120572 minus119888

) (19)

The second compound matrix 119869[2] of the Jacobin matrix

can be calculated as follows (see [10 11])

119869[2]

= (

minus1198861198770+ 1205732119878lowastminus 119887 120573

1119878lowast

1205731119878lowast

120572 minus1198861198770minus 119888 minus120573

2119878lowast

0 119886 (1198770minus 1) 120573

2119878 minus 119887 minus 119888

)

(20)

Set 119875 as the following diagonal matrix

119875 (119909) = (1119864

119868119864

119868) (21)

Denote that

119875119891119875minus1

= diag(01198641015840

119864minus

1198681015840

1198681198641015840

119864minus

1198681015840

119868) (22)


Therefore thematrix119861 = 119875119891119875minus1

+119875119869[2]

119875minus1 can be written

in the following block form

119861 = (11986111

11986112

11986121

11986122

) (23)

with11986111

= minus1198861198770+ 1205732119878lowastminus 119887

11986112

=119868

1198641205731119878lowast(1 1)

11986121

=119864

119868(120572 0)119879

11986122

= (

1198641015840

119864minus

1198681015840

119868minus (119886119877

0+ 119888) minus120573

2119878lowast

119886 (1198770minus 1)

1198641015840

119864minus

1198681015840

119868+ (1205732119878 lowast minus119887 minus 119888)

)

(24)

thus1205831(11986111) = 1205732119878lowastminus 1198861198770minus 119887

1205831(11986122) = max119864

1015840

119864minus

1198681015840

119868minus 1198861198770minus 119888 + 119886119877

0minus 119886

1198641015840

119864minus

1198681015840

119868+ 1205732119878 lowast minus119887 minus 119888 minus 120573

2119878lowast

=1198641015840

119864minus

1198681015840

119868minus 119888 minus 119886

(25)

The vector norm sdot in 1198773cong 119877(3

2) is choosen as

(119906 V 119908) = max |119906| |V + 119908| (26)

The Lozinskii measure 120583(119861) with respect to sdot is as follows(see [12])

120583 (119861) le sup 1198921 1198922 (27)

where

1198921= 1205831(11986111) +

1003816100381610038161003816119861121003816100381610038161003816 = 1205732119878lowastminus 1198861198770minus 119887 +

119868

1198641205731119878lowast

1198922= 1205831(11986122) +

1003816100381610038161003816119861211003816100381610038161003816 =

1198641015840

119864minus

1198681015840

119868minus 119886 minus 119888 +

119864

119868120572

(28)

From (3) we find that

119868

1198641205731119878lowast=

1198641015840

119864minus 1205732119878lowast+ 119887

1198681015840

119868=

119864

119868120572 minus 119888

(29)

thus

1198921=

1198641015840

119864minus 1198861198770

1198922=

1198641015840

119864minus 119886

(30)

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)

Figure 1 Dynamical behavior of system (3) Time series of suscep-tible exposed and infectious computers 119878(119905) 119864(119905) 119868(119905)with119877

0gt 1

Relations (28)ndash(30) imply that

120583 (119861) le1198641015840

119864minus 119886

(31)

Thus

1

119905int

119905

0

120583 (119861) d120591 le1

119905int

119905

0

(1198641015840

119864minus 119886) d120591 =

1

119905ln 119864 (119905)

119864 (0)minus 119886 (32)

If 1198770

gt 1 then the virus-free equilibrium is unstableby Theorem 1 Moreover the behavior of the local dynamicnear 119863

0as described in Theorem 1 implies that the system

(3) is uniformly persistent in119863 that is there exists a constant1198881gt 0 and 119879 gt 0 such that 119905 gt 119879 implies that

lim119905rarrinfin

inf 119878 (119905) gt 1198881

lim119905rarrinfin

inf 119864 (119905) gt 1198881

lim119905rarrinfin

inf 119868 (119905) gt 1198881

lim119905rarrinfin

inf [1 minus 119878 (119905) minus 119864 (119905) minus 119868 (119905)] gt 1198881

(33)

For all (119878(0) 119864(0) 119868(0) isin 119863) (see [13 14])

119902 = lim119905rarrinfin

sup sup119909isin119870

1

119905int

119905

0

120583 (119861) 119889120591 le minus119886

2lt 0 (34)

The proof is complete

5 Numerical Examples

For the system (3) Theorem 2 implies that the virus diesout if 119877

0lt 1 and Theorem 4 implies that the virus persists

if 1198770gt 1 Now we present two numerical examples

Let 119901 = 05 120583 = 002 119896 = 04 120572 = 06 119903 = 06 119873 =

100 1205731= 07 120573

2= 08 then 119877

0= 138 gt 1 and 120573119878

lowastlt 119887 + 119888

Figure 1 shows the solution of system (3) when 1198770gt 1 We


0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)

Figure 2 Dynamical behavior of system (3) Time series ofsusceptible exposed and infectious computers 119878(119905) 119864(119905) 119868(119905) with1198770lt 1

can see that the viral equilibrium 119875lowast of system (3) is globally

asymptotically stableLet 119901 = 07 120583 = 0001 119896 = 002 120572 = 009 119903 =

004 119873 = 10 1205731= 0002 120573

2= 0003 then 119877

0= 01808 lt 1

and 12057321198780lt 119887 + 119888 Figure 2 shows the solution of system (3)

when 1198770lt 1 We can see that the virus-free equilibrium 119875

0

of the system (3) is globally asymptotically stable

6 Conclusion

We assume that the virus process has a latent period andin these times the infected computers have infectivity alsoA compartmental SEIR model for transmission of virus incomputer network is formulated In this paper the dynamicsof this model have been fully studied

The results show that we should try our best to make1198770less than 1 The most effective way is to increase the

parameters 119901 119896 119903 and decrease 1205731 1205732 120572 and so on Maybe

in such way the computer virus can be well predicted andthus controlled

Acknowledgment

The work described in this paper was supported by theScience and Technology Project of Chongqing EducationCommittee under Grant KJ130519

References

[1] C Sun and Y-H Hsieh ldquoGlobal analysis of an SEIRmodel withvarying population size and vaccinationrdquoAppliedMathematicalModelling vol 34 no 10 pp 2685ndash2697 2010

[2] L-P Song Z Jin and G-Q Sun ldquoModeling and analyzing ofbotnet interactionsrdquo Physica A vol 390 no 2 pp 347ndash358 2011

[3] J Ren X Yang L-X Yang Y Xu and F Yang ldquoA delayedcomputer virus propagation model and its dynamicsrdquo ChaosSolitons amp Fractals vol 45 no 1 pp 74ndash79 2012

[4] B K Mishra and S K Pandey ldquoDynamic model of worms withvertical transmission in computer networkrdquoAppliedMathemat-ics and Computation vol 217 no 21 pp 8438ndash8446 2011

[5] X Han and Q Tan ldquoDynamical behavior of computer virus onInternetrdquoAppliedMathematics and Computation vol 217 no 6pp 2520ndash2526 2010

[6] Q Zhu X Yang and J Ren ldquoModeling and analysis of thespread of computer virusrdquo Communications in Nonlinear Sci-ence and Numerical Simulation vol 17 no 12 pp 5117ndash51242012

[7] L X Yang X Yang Q Zhu and L Wen ldquoA computer virusmodel with graded cure ratesrdquo Nonlinear Analysis Real WorldApplications vol 14 no 1 pp 414ndash422 2013

[8] L X Yang X Yang L Wen and J Liu ldquoA novel computer viruspropagation model and its dynamicsrdquo International Journal ofComputer Mathematics vol 89 no 17 pp 2307ndash2314 2012

[9] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[10] M Fiedler ldquoAdditive compound matrices and an inequality foreigenvalues of symmetric stochastic matricesrdquo CzechoslovakMathematical Journal vol 24(99) pp 392ndash402 1974

[11] J SMuldowney ldquoCompoundmatrices and ordinary differentialequationsrdquoTheRockyMountain Journal ofMathematics vol 20no 4 pp 857ndash872 1990

[12] G Butler H I Freedman and P Waltman ldquoUniformly persis-tent systemsrdquoProceedings of the AmericanMathematical Societyvol 96 no 3 pp 425ndash430 1986

[13] H I Freedman S G Ruan and M X Tang ldquoUniform per-sistence and flows near a closed positively invariant setrdquo Journalof Dynamics and Differential Equations vol 6 no 4 pp 583ndash600 1994

[14] P Waltman ldquoA brief survey of persistence in dynamical sys-temsrdquo in Delay Differential Equations and Dynamical Systems(Claremont CA 1990) S Busenberg and M Martelli Eds vol1475 pp 31ndash40 Springer Berlin Germany 1991

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model has a virus-free equilibrium 1198750 and 119875

0 is a globallyasymptotically stable equilibrium if 119877

0lt 1 if 119877

0gt 1 this

model has only one viral equilibrium 119875lowast and it is globally

asymptotically stableThis paper is organized as follows Section 2 formulates

a novel computer virus mode Section 3 proves the globalstability of the virus-free equilibrium Section 4 discussesthe stability of the viral equilibrium In Section 5 numericalsimulations are given to present the effectiveness of thetheoretic results Finally Section 6 summarizes this work

2 Model Formulation

At any time a computer is classified as internal and externaldepending on weather it is connected to internet or notAt that time all of the internet computers are further cate-gorized into four classes (1) susceptible computers that isuninfected computers and new computers which connectedto network (2) exposed computers that is infected butnot yet broken-out (3) infectious computers (4) recoveredcomputers that is virus-free computer having immunity Let119878(119905) 119864(119905) 119868(119905) 119877(119905) denote their corresponding numbers attime 119905 without ambiguity 119878(119905) 119864(119905) 119868(119905) 119877(119905)will be abbre-viated as 119878 119864 119868 119877 respectively The model is formulated asthe following system of differential equations

1198781015840= (1 minus 119901)119873 minus 120573

1119878119868 minus 120573

2119878119864 minus 119901119878 minus 120583119878

1198641015840= 1205731119878119868 + 120573

2119878119864 minus 119896119864 minus 120572119864 minus 120583119864

1198681015840= 120572119864 minus 119903119868 minus 120583119868

1198771015840= 119901119878 + 119896119864 + 119903119868

(1)

119873(119905) = 119878 (119905) + 119864 (119905) + 119868 (119905) + 119877 (119905) (2)

We may see that the first three equations in (1) areindependent of the fourth equation and therefore the fourthequation can be omitted without loss of generality Hencesystem (1) can be rewritten as

1198781015840= 119860 minus 120573

1119878119868 minus 120573

2119878119864 minus 119886119878

1198641015840= 1205731119878119868 + 120573

2119878119864 minus 119887119864

1198681015840= 120572119864 minus 119888119868

(3)

Therefore

119886 = 119901 + 120583 119887 = 119896 + 120572 + 120583

119888 = 119903 + 120583 (1 minus 119901)119873 = 119860

(4)

where 119873 denotes the rate at which external computersare connected to the network 119901 denotes the recovery rateof susceptible computer due to the anti-virus ability ofnetwork 119896 denotes the recovery rate of exposed computerdue to the anti-virus ability of network 120573

1denotes the

rate at which when having a connection to one infectedcomputer one susceptible computer can become exposedbut has not broken-out 120573

2denotes the rate of which when

having connection to one exposed computer one susceptiblecomputer can become exposed 120572 denotes the rate of theexposed computers cannot be cured by anti-virus softwareand broken-out 119903 denotes the recovery rate of infectedcomputers that are cured 120583 denotes the rate at which onecomputer is removed from the network All the parametersare nonnegative

Moreover all feasible solutions of the system (3) arebounded and enter the region 119863 where

119863 = (119878 119864 119868) isin 1198773

+| 119878 ge 0 119864 ge 0 119868 ge 0 119878 + 119864 + 119868 le

119860

119886

(5)

Referring to [9] we define the basic reproduction numberof the infection as

1198770=

119860 (1205731120572 + 1205732119888)

119886119887119888 (6)

For system (3) there always exists the virus-free equilib-rium which is 119875

0(119860119886 0 0) if 119877

0gt 1 then there also exists

a viral equilibrium 119875lowast(119878lowast 119864lowast 119868lowast)

Therefore

119878lowast=

119860

1198861198770

119864lowast=

119860 (1198770minus 1)

1198871198770

119868lowast=

119860120572 (1198770minus 1)

1198871198881198770

(7)

3 The Virus-Free Equilibrium and Its Stability

Theorem 1 1198750 is locally asymptotically stable if 119877

0lt 1

Whereas 1198750 is unstable if 1198770gt 1

Proof The characteristic equation of (3) at 1198750 is given by

det(

120582 + 119886 minus1205732119878 minus120573

1119878

0 120582 minus (1205732119878 minus 119887) 120573

1119878

0 120572 120582 + 119888

) = 0 (8)

which equals to

(120582 + 119886) [1205822minus (12057321198780minus 119887 minus 119888) 120582 minus 119887119888 (119877

0minus 1)] = 0 (9)

Then (9) has negative real part characteristic roots

1205821= minus119886

12058223

=

(1205732119878 minus 119887 minus 119888) plusmn radic(120573

2119878 minus 119887 minus 119888)

2+ 4119886119887119888 (119877

0minus 1)

2

(10)

where

1205732119878 minus 119887 minus 119888 lt 0 (11)






119863 if 1198770lt 1

Proof Let 119871 = ((1205731119888 + 1205732120572)119887119888)119864 + 120573

2119868119888

Obviously

119871 gt 0(12)

thus

1198711015840=

(1205731119888 + 1205732120572)

1198871198881198641015840+

1205732

1198881198681015840

=(1205731119888 + 1205732120572)

119887119888(1205731119878119868 + 120573

2119878119864 minus 119887119864) +

1205732

119888(120572119864 minus 119888119868)

=(1205731119888 + 1205732120572)

119887119888(1205731119868 + 1205732119864) 119878 minus

(1205731119888 + 1205732120572)

119887119888119887119864

+1205732120572119864

119888minus 1205732119868

=(1205731119888 + 1205732120572)

119887119888(1205731119868 + 1205732119864) 119878 minus 120573

1119864 minus 1205732119868

= (1205731119864 + 1205732119868) [

(1205731119888 + 1205732120572)

119887119888119878 minus 1]

= (1205731119864 + 1205732119868) [

1

119887119888sdot119860

119886sdot1198770119886119887119888

119860minus 1]

= (1205731119864 + 1205732119868) (1198770minus 1) lt 0

(13)




0gt 1


119869lowast= (

minus1198861198770



119886 (1198770minus 1) 120573

2119878lowastminus 119887 120573

1119878lowast

0 120572 minus119888

) (14)

which equals to

119891 (120582) = 11988601205823+ 11988611205822+ 1198862120582 + 1198863= 0 (15)

where

1198860= 1


1198862= 119886119887119877

0+ 119886119888119877

0minus 1198861205732119878lowast

gt 1198861198871198770+ 119886119888119877

0minus 119886 (119887 + 119888)

= 119886 (119887 + 119888) (1198770minus 1) gt 0

(16)

where

1205732119878lowastlt 119887 + 119888

1198863= 119886119887119888 (119877

0minus 1) gt 0

(17)

Thus

Δ1= 1198861gt 0

Δ2=

10038161003816100381610038161003816100381610038161003816

1198861

1

1198863

1198862

10038161003816100381610038161003816100381610038161003816

= 11988611198862minus 1198863gt 0

Δ3=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198861

1 0

1198863

1198862

1198861

0 0 1198863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 1198863(11988611198862minus 1198863) gt 0

(18)




1198770gt 1


119869lowast= (

minus1198861198770



119886 (1198770minus 1) 120573

2119878lowastminus 119887 120573

1119878lowast

0 120572 minus119888

) (19)



119869[2]

= (


1119878lowast

1205731119878lowast


2119878lowast

0 119886 (1198770minus 1) 120573

2119878 minus 119887 minus 119888

)

(20)


119875 (119909) = (1119864

119868119864

119868) (21)

Denote that

119875119891119875minus1

= diag(01198641015840

119864minus

1198681015840

1198681198641015840

119864minus

1198681015840

119868) (22)



+119875119869[2]



119861 = (11986111

11986112

11986121

11986122

) (23)

with11986111


11986112

=119868

1198641205731119878lowast(1 1)

11986121

=119864

119868(120572 0)119879

11986122

= (

1198641015840

119864minus

1198681015840

119868minus (119886119877

0+ 119888) minus120573

2119878lowast

119886 (1198770minus 1)

1198641015840

119864minus

1198681015840


)

(24)


1205831(11986122) = max119864

1015840

119864minus

1198681015840

119868minus 1198861198770minus 119888 + 119886119877

0minus 119886

1198641015840

119864minus

1198681015840


2119878lowast

=1198641015840

119864minus

1198681015840

119868minus 119888 minus 119886

(25)


2) is choosen as

(119906 V 119908) = max |119906| |V + 119908| (26)


120583 (119861) le sup 1198921 1198922 (27)

where

1198921= 1205831(11986111) +


119868

1198641205731119878lowast

1198922= 1205831(11986122) +

1003816100381610038161003816119861211003816100381610038161003816 =

1198641015840

119864minus

1198681015840

119868minus 119886 minus 119888 +

119864

119868120572

(28)


119868

1198641205731119878lowast=

1198641015840

119864minus 1205732119878lowast+ 119887

1198681015840

119868=

119864

119868120572 minus 119888

(29)

thus

1198921=

1198641015840

119864minus 1198861198770

1198922=

1198641015840

119864minus 119886

(30)

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)


0gt 1


120583 (119861) le1198641015840

119864minus 119886

(31)

Thus

1

119905int

119905

0

120583 (119861) d120591 le1

119905int

119905

0

(1198641015840

119864minus 119886) d120591 =

1

119905ln 119864 (119905)

119864 (0)minus 119886 (32)

If 1198770




lim119905rarrinfin

inf 119878 (119905) gt 1198881

lim119905rarrinfin

inf 119864 (119905) gt 1198881

lim119905rarrinfin

inf 119868 (119905) gt 1198881

lim119905rarrinfin


(33)

For all (119878(0) 119864(0) 119868(0) isin 119863) (see [13 14])



1

119905int

119905

0

120583 (119861) 119889120591 le minus119886

2lt 0 (34)






Let 119901 = 05 120583 = 002 119896 = 04 120572 = 06 119903 = 06 119873 =

100 1205731= 07 120573

2= 08 then 119877

0= 138 gt 1 and 120573119878

lowastlt 119887 + 119888



0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)




004 119873 = 10 1205731= 0002 120573

2= 0003 then 119877

0= 01808 lt 1



0


6 Conclusion





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119863 if 1198770lt 1

Proof Let 119871 = ((1205731119888 + 1205732120572)119887119888)119864 + 120573

2119868119888

Obviously

119871 gt 0(12)

thus

1198711015840=

(1205731119888 + 1205732120572)

1198871198881198641015840+

1205732

1198881198681015840

=(1205731119888 + 1205732120572)

119887119888(1205731119878119868 + 120573

2119878119864 minus 119887119864) +

1205732

119888(120572119864 minus 119888119868)

=(1205731119888 + 1205732120572)

119887119888(1205731119868 + 1205732119864) 119878 minus

(1205731119888 + 1205732120572)

119887119888119887119864

+1205732120572119864

119888minus 1205732119868

=(1205731119888 + 1205732120572)

119887119888(1205731119868 + 1205732119864) 119878 minus 120573

1119864 minus 1205732119868

= (1205731119864 + 1205732119868) [

(1205731119888 + 1205732120572)

119887119888119878 minus 1]

= (1205731119864 + 1205732119868) [

1

119887119888sdot119860

119886sdot1198770119886119887119888

119860minus 1]

= (1205731119864 + 1205732119868) (1198770minus 1) lt 0

(13)




0gt 1


119869lowast= (

minus1198861198770



119886 (1198770minus 1) 120573

2119878lowastminus 119887 120573

1119878lowast

0 120572 minus119888

) (14)

which equals to

119891 (120582) = 11988601205823+ 11988611205822+ 1198862120582 + 1198863= 0 (15)

where

1198860= 1


1198862= 119886119887119877

0+ 119886119888119877

0minus 1198861205732119878lowast

gt 1198861198871198770+ 119886119888119877

0minus 119886 (119887 + 119888)

= 119886 (119887 + 119888) (1198770minus 1) gt 0

(16)

where

1205732119878lowastlt 119887 + 119888

1198863= 119886119887119888 (119877

0minus 1) gt 0

(17)

Thus

Δ1= 1198861gt 0

Δ2=

10038161003816100381610038161003816100381610038161003816

1198861

1

1198863

1198862

10038161003816100381610038161003816100381610038161003816

= 11988611198862minus 1198863gt 0

Δ3=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198861

1 0

1198863

1198862

1198861

0 0 1198863

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 1198863(11988611198862minus 1198863) gt 0

(18)




1198770gt 1


119869lowast= (

minus1198861198770



119886 (1198770minus 1) 120573

2119878lowastminus 119887 120573

1119878lowast

0 120572 minus119888

) (19)



119869[2]

= (


1119878lowast

1205731119878lowast


2119878lowast

0 119886 (1198770minus 1) 120573

2119878 minus 119887 minus 119888

)

(20)


119875 (119909) = (1119864

119868119864

119868) (21)

Denote that

119875119891119875minus1

= diag(01198641015840

119864minus

1198681015840

1198681198641015840

119864minus

1198681015840

119868) (22)



+119875119869[2]



119861 = (11986111

11986112

11986121

11986122

) (23)

with11986111


11986112

=119868

1198641205731119878lowast(1 1)

11986121

=119864

119868(120572 0)119879

11986122

= (

1198641015840

119864minus

1198681015840

119868minus (119886119877

0+ 119888) minus120573

2119878lowast

119886 (1198770minus 1)

1198641015840

119864minus

1198681015840


)

(24)


1205831(11986122) = max119864

1015840

119864minus

1198681015840

119868minus 1198861198770minus 119888 + 119886119877

0minus 119886

1198641015840

119864minus

1198681015840


2119878lowast

=1198641015840

119864minus

1198681015840

119868minus 119888 minus 119886

(25)


2) is choosen as

(119906 V 119908) = max |119906| |V + 119908| (26)


120583 (119861) le sup 1198921 1198922 (27)

where

1198921= 1205831(11986111) +


119868

1198641205731119878lowast

1198922= 1205831(11986122) +

1003816100381610038161003816119861211003816100381610038161003816 =

1198641015840

119864minus

1198681015840

119868minus 119886 minus 119888 +

119864

119868120572

(28)


119868

1198641205731119878lowast=

1198641015840

119864minus 1205732119878lowast+ 119887

1198681015840

119868=

119864

119868120572 minus 119888

(29)

thus

1198921=

1198641015840

119864minus 1198861198770

1198922=

1198641015840

119864minus 119886

(30)

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)


0gt 1


120583 (119861) le1198641015840

119864minus 119886

(31)

Thus

1

119905int

119905

0

120583 (119861) d120591 le1

119905int

119905

0

(1198641015840

119864minus 119886) d120591 =

1

119905ln 119864 (119905)

119864 (0)minus 119886 (32)

If 1198770




lim119905rarrinfin

inf 119878 (119905) gt 1198881

lim119905rarrinfin

inf 119864 (119905) gt 1198881

lim119905rarrinfin

inf 119868 (119905) gt 1198881

lim119905rarrinfin


(33)

For all (119878(0) 119864(0) 119868(0) isin 119863) (see [13 14])



1

119905int

119905

0

120583 (119861) 119889120591 le minus119886

2lt 0 (34)






Let 119901 = 05 120583 = 002 119896 = 04 120572 = 06 119903 = 06 119873 =

100 1205731= 07 120573

2= 08 then 119877

0= 138 gt 1 and 120573119878

lowastlt 119887 + 119888



0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)




004 119873 = 10 1205731= 0002 120573

2= 0003 then 119877

0= 01808 lt 1



0


6 Conclusion





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











+119875119869[2]



119861 = (11986111

11986112

11986121

11986122

) (23)

with11986111


11986112

=119868

1198641205731119878lowast(1 1)

11986121

=119864

119868(120572 0)119879

11986122

= (

1198641015840

119864minus

1198681015840

119868minus (119886119877

0+ 119888) minus120573

2119878lowast

119886 (1198770minus 1)

1198641015840

119864minus

1198681015840


)

(24)


1205831(11986122) = max119864

1015840

119864minus

1198681015840

119868minus 1198861198770minus 119888 + 119886119877

0minus 119886

1198641015840

119864minus

1198681015840


2119878lowast

=1198641015840

119864minus

1198681015840

119868minus 119888 minus 119886

(25)


2) is choosen as

(119906 V 119908) = max |119906| |V + 119908| (26)


120583 (119861) le sup 1198921 1198922 (27)

where

1198921= 1205831(11986111) +


119868

1198641205731119878lowast

1198922= 1205831(11986122) +

1003816100381610038161003816119861211003816100381610038161003816 =

1198641015840

119864minus

1198681015840

119868minus 119886 minus 119888 +

119864

119868120572

(28)


119868

1198641205731119878lowast=

1198641015840

119864minus 1205732119878lowast+ 119887

1198681015840

119868=

119864

119868120572 minus 119888

(29)

thus

1198921=

1198641015840

119864minus 1198861198770

1198922=

1198641015840

119864minus 119886

(30)

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)


0gt 1


120583 (119861) le1198641015840

119864minus 119886

(31)

Thus

1

119905int

119905

0

120583 (119861) d120591 le1

119905int

119905

0

(1198641015840

119864minus 119886) d120591 =

1

119905ln 119864 (119905)

119864 (0)minus 119886 (32)

If 1198770




lim119905rarrinfin

inf 119878 (119905) gt 1198881

lim119905rarrinfin

inf 119864 (119905) gt 1198881

lim119905rarrinfin

inf 119868 (119905) gt 1198881

lim119905rarrinfin


(33)

For all (119878(0) 119864(0) 119868(0) isin 119863) (see [13 14])



1

119905int

119905

0

120583 (119861) 119889120591 le minus119886

2lt 0 (34)






Let 119901 = 05 120583 = 002 119896 = 04 120572 = 06 119903 = 06 119873 =

100 1205731= 07 120573

2= 08 then 119877

0= 138 gt 1 and 120573119878

lowastlt 119887 + 119888



0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)




004 119873 = 10 1205731= 0002 120573

2= 0003 then 119877

0= 01808 lt 1



0


6 Conclusion





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

Time (t)

S(t)

E(t)

I(t)

S(t)E(t)I(t)




004 119873 = 10 1205731= 0002 120573

2= 0003 then 119877

0= 01808 lt 1



0


6 Conclusion





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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