research article modeling computer virus and its dynamics
TRANSCRIPT
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
minus1205731119878lowast
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
Proof The Jacobin matrix of system (3) about 119875lowast is given by
119869lowast= (
minus1198861198770
minus1205732119878lowast
minus1205731119878lowast
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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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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
minus1205731119878lowast
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
Proof The Jacobin matrix of system (3) about 119875lowast is given by
119869lowast= (
minus1198861198770
minus1205732119878lowast
minus1205731119878lowast
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
minus1205731119878lowast
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
Proof The Jacobin matrix of system (3) about 119875lowast is given by
119869lowast= (
minus1198861198770
minus1205732119878lowast
minus1205731119878lowast
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of