dynamics of a computer virus propagation model with delays...
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Research ArticleDynamics of a Computer Virus Propagation Modelwith Delays and Graded Infection Rate
Zizhen Zhang1 and Limin Song2
1School of Management Science and Engineering Anhui University of Finance and Economics Bengbu 233030 China2Department of Computer Liaocheng College of Education Liaocheng 252004 China
Correspondence should be addressed to Zizhen Zhang zzzhaida163com
Received 14 July 2016 Accepted 1 September 2016 Published 4 January 2017
Academic Editor Xiao-Jun Yang
Copyright copy 2017 Z Zhang and L Song This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A four-compartment computer virus propagationmodel with two delays and graded infection rate is investigated in this paperThecritical values where a Hopf bifurcation occurs are obtained by analyzing the distribution of eigenvalues of the correspondingcharacteristic equation In succession direction and stability of the Hopf bifurcation when the two delays are not equal aredetermined by using normal form theory and center manifold theorem Finally some numerical simulations are also carried outto justify the obtained theoretical results
1 Introduction
In recent years with the fast development and populariza-tion of computer technologies and network Internet hasoffered numerous functionalities and facilities to the worldMeanwhile Internet has also become a powerful mechanismfor propagating computer viruses Computer viruses arecomputer programs which have serious effects on individualand corporate computer systems in the network such asmodifying data and formatting disks [1 2]
In order to analyze the propagation laws of computerviruses in the network many epidemiological models havebeen borrowed to depict the spread of computer virusesbecause of the high similarity between the computer virusesand the biological viruses [3ndash5] In [6ndash11] Mishra et al pro-posed SIRS computer virus models in different forms Yuanand Chen presented the SEIR computer virus propagationmodel in [12] and they studied the stability of the modelBased on the the work in [12] Dong et al proposed theSEIR computer virus model with time delay in [13] and theyinvestigated the Hopf bifurcation of the modelThere are alsosome other different computer virus models which have beenproposed by other scholars in recent years and one can referto [14ndash18] However all the computer virus models abovewhich incorporate the latent status of the viruses assumethat the latent computers have no infection ability This is
not consistent with the reality because an infected computerwhich is in latency can also infect other computers throughfile copying or file downloading Based on this fact Yang et alestablished a computer virus propagation model with gradedinfection rate in [19]119889119878 (119905)119889119905 = 120583 minus 1205731119878 (119905) 119871 (119905) minus 1205732119878 (119905) 119860 (119905) + 120572119877 (119905)minus 120583119878 (119905) 119889119871 (119905)119889119905 = 1205731119878 (119905) 119871 (119905) + 1205732119878 (119905) 119860 (119905) minus 120576119871 (119905) minus 120583119871 (119905) 119889119860 (119905)119889119905 = 120576119871 (119905) minus 120574119860 (119905) minus 120583119860 (119905) 119889119877 (119905)119889119905 = 120574119860 (119905) minus 120572119877 (119905) minus 120583119877 (119905)
(1)
where 119878(119905) 119871(119905) 119860(119905) and 119877(119905) are the percentages of sus-ceptible computers latent computers active computers andrecovered computers on the Internet at time 119905 respectively 120583is the rate at which external computers are connected to theInternet and it is also the rate at which internal computersare disconnected from the Internet 1205731 is the infected rateof the susceptible computers by the latent computers 1205732 isthe infected rate of the susceptible computers by the active
HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 4514935 13 pageshttpsdoiorg10115520174514935
2 Advances in Mathematical Physics
computers 120572 is the rate at which the recovered computersbecome susceptibly virus-free again 120576 is the rate at whichthe latent computers break out and 120574 is the rate at which theactive computers are cured by the antivirus software
As pointed out in [9] one of the typical features ofcomputer viruses is their latent characteristic Therefore theyneed a period to become active computers for the latent onesLikewise the antivirus software needs a period to clean theviruses in the active computers Based on this and motivatedby the work about the dynamical system with delay in [20ndash24] we incorporate two delays into system (1) and obtain thefollowing delayed computer virus model119889119878 (119905)119889119905 = 120583 minus 1205731119878 (119905) 119871 (119905) minus 1205732119878 (119905) 119860 (119905) + 120572119877 (119905)minus 120583119878 (119905) 119889119871 (119905)119889119905 = 1205731119878 (119905) 119871 (119905) + 1205732119878 (119905) 119860 (119905) minus 120576119871 (119905 minus 1205911)minus 120583119871 (119905) 119889119860 (119905)119889119905 = 120576119871 (119905 minus 1205911) minus 120574119860 (119905 minus 1205912) minus 120583119860 (119905) 119889119877 (119905)119889119905 = 120574119860 (119905 minus 1205912) minus 120572119877 (119905) minus 120583119877 (119905)
(2)
where 1205911 is the latent period of the computer viruses and 1205912is the period that the antivirus software needs to clean theviruses in the active computers
The rest of this paper is organized as follows In Section 2we present the existence of the viral equilibrium and con-ditions for the local stability of the viral equilibrium andexistence of the Hopf bifurcation are derived Direction andstability of the Hopf bifurcation are studied in Section 3and some numerical simulations are performed in Section 4to justify the obtained theoretical findings by taking somerelevant values of the parameters in system (2) and usingthe Matlab software package Finally we end this paper withconcluding remarks in Section 5
2 Existence of Local Hopf Bifurcation
By a simple computation we know that if (120576+120583)(120574+120583)(1205731(120574+120583) + 1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) then system(2) has a unique viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) where119878lowast = (120574 + 120583) (120576 + 120583)1205731 (120574 + 120583) + 1205732120576 119871lowast = 120574 + 120583120576 119860lowast119877lowast = 120574120572 + 120583119860lowast119860lowast = 1198601lowast1198602lowast 1198601lowast = 120583 minus 120583 (120576 + 120583) (120574 + 120583)1205731 (120574 + 120583) + 1205732120576 1198602lowast = (120576 + 120583) (120574 + 120583)120576 minus 120572120574120572 + 120583
(3)
Let 119878(119905) = 119878(119905) minus 119878lowast 119871(119905) = 119871(119905) minus 119871lowast 119860(119905) = 119860(119905) minus 119860lowast119877(119905) = 119877(119905) minus 119877lowast Dropping the bars system (2) becomes
119889119878 (119905)119889119905 = 1198861119878 (119905) + 1198862119871 (119905) + 1198863119860 (119905) + 1198864119877 (119905)minus 1205731119878 (119905) 119871 (119905) minus 1205732119878 (119905) 119860 (119905) 119889119871 (119905)119889119905 = 1198865119878 (119905) + 1198866119871 (119905) + 1198867119860 (119905) + 1198871119871 (119905 minus 1205911)+ 1205731119878 (119905) 119871 (119905) + 1205732119878 (119905) 119860 (119905) 119889119860 (119905)119889119905 = 1198868119860 (119905) + 1198872119871 (119905 minus 1205911) + 1198881119860 (119905 minus 1205912) 119889119877 (119905)119889119905 = 1198869119877 (119905) + 1198882119860 (119905 minus 1205912) (4)
where 1198861 = minus (1205731119871lowast + 1205732119860lowast + 120583) 1198862 = minus1205731119878lowast1198863 = minus1205732119878lowast1198864 = 1205721198865 = 1205731119871lowast + 1205732119860lowast1198871 = minus1205761198866 = 1205731119878lowast minus 1205831198867 = 1205732119878lowast1198868 = minus1205831198872 = 1205761198881 = minus1205741198869 = minus (120572 + 120583)119877lowast1198882 = 120574
(5)
The linear system of system (4) is
119889119878 (119905)119889119905 = 1198861119878 (119905) + 1198862119871 (119905) + 1198863119860 (119905) + 1198864119877 (119905) 119889119871 (119905)119889119905 = 1198865119878 (119905) + 1198866119871 (119905) + 1198867119860 (119905) + 1198871119871 (119905 minus 1205911) 119889119860 (119905)119889119905 = 1198868119860 (119905) + 1198872119871 (119905 minus 1205911) + 1198881119860 (119905 minus 1205912) 119889119877 (119905)119889119905 = 1198869119877 (119905) + 1198882119860 (119905 minus 1205912) (6)
Advances in Mathematical Physics 3
The corresponding characteristic equation is1205824 + 119886031205823 + 119886021205822 + 11988601120582 + 11988600+ (119887031205823 + 119887021205822 + 11988701120582 + 11988700) 119890minus1205821205911+ (119888031205823 + 119888021205822 + 11988801120582 + 11988800) 119890minus1205821205912+ (119889021205822 + 11988901120582 + 11988900) 119890minus120582(1205911+1205912) = 0(7)
where 11988600 = (11988611198866 minus 11988621198865) 1198868119886911988601 = (11988621198865 minus 11988611198866) (1198868 + 1198869) 11988602 = 11988611198866 minus 11988621198865 + 11988681198869 + (1198861 + 1198866) (1198868 + 1198869) 11988603 = minus (1198861 + 1198866 + 1198868 + 1198869) 11988700 = 1198861119886811988691198871 + (11988631198865 minus 11988611198867) 1198869119887211988701 = 11988671198872 (1198861 + 1198869) minus 1198871 (11988611198868 + 11988611198869 + 11988681198869) 11988702 = 1198871 (1198861 + 1198868 + 1198869) minus 1198867119887211988703 = minus119887111988800 = (11988611198866 minus 11988621198865) 1198869119888111988801 = 1198881 (11988621198865 minus 11988611198866 minus 11988611198869 minus 11988661198869) 11988802 = 1198881 (1198861 + 1198866 + 1198869) 11988803 = minus119888111988900 = 1198861119886911988711198881 + 119886411988651198872119888211988901 = minus11988711198881 (1198861 + 1198869) 11988902 = 11988711198881
(8)
Case 1 (1205911 = 1205912 = 0) For 1205911 = 1205912 = 0 (7) becomes1205824 + 119886131205823 + 119886121205822 + 11988611120582 + 11988610 = 0 (9)
where 11988610 = 11988600 + 11988700 + 11988800 + 1198890011988611 = 11988601 + 11988701 + 11988801 + 1198890111988612 = 11988602 + 11988702 + 11988802 + 1198890211988613 = 11988603 + 11988703 + 11988803 (10)
Thus according to the Routh-Hurwithz theorem weknow that if conditions (11986711) 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt11988611 hold then viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system(2) without delay is locally asymptotically stable
Case 2 (1205911 gt 0 1205912 = 0) For 1205911 gt 0 and 1205912 = 0 we can get thefollowing from (7)1205824 + 119886231205823 + 119886221205822 + 11988621120582 + 11988620+ (119887231205823 + 119887221205822 + 11988721120582 + 11988720) 119890minus1205821205911 = 0 (11)
where 11988620 = 11988600 + 1198880011988621 = 11988601 + 1198880111988622 = 11988602 + 1198880211988623 = 11988603 + 1198880311988720 = 11988700 + 1198890011988721 = 11988701 + 1198890111988722 = 11988702 + 1198890211988723 = 11988703(12)
We assume that 120582 = 1198941205961 (1205961 gt 0) is a root of (11) Then(119887211205961 minus 1198872312059631) sin 12059111205961 + (11988720 minus 1198872212059621) cos 12059111205961= 1198862212059621 minus 12059641 minus 11988620(119887211205961 minus 1198872312059631) cos 12059111205961 minus (11988720 minus 1198872212059621) sin 12059111205961= 1198862312059631 minus 119886211205961(13)
which implies that12059681 + 1198922312059661 + 1198922212059641 + 1198922112059621 + 11989220 = 0 (14)
with 11989220 = 119886220 minus 11988722011989221 = 119886221 minus 119887221 minus 21198862011988622 + 2119887201198872211989222 = 119886222 minus 119887222 + 21198872111988723 minus 21198862111988623 + 21198862011989223 = 119886223 minus 119887223 minus 211988622(15)
Let 12059621 = V1 then (14) becomes
V41 + 11989223V31 + 11989222V21 + 11989221V1 + 11989220 = 0 (16)
Discussion about distribution of roots for (16) is similarto that in [25] Therefore we directly assume that (11986721) (16)has at least one positive equilibrium V10
If (11986721) holds we know that (14) has at least one positiveroot 12059610 = radicV10 such that (11) has a pair of purely imaginaryroots plusmn11989412059610 For 12059610
12059110 = 112059610 arccos ℎ21 (12059610)ℎ22 (12059610) (17)
4 Advances in Mathematical Physics
whereℎ21 (12059610) = (11988722 minus 1198862311988723) 120596610+ (1198862111988723 + 1198862311988721 minus 1198862211988722 minus 11988720) 120596410+ (1198862011988722 + 1198862211988720 minus 1198862111988721) 120596210minus 1198862011988720ℎ22 (12059610) = 119887223120596610 + (119887222 minus 21198872111988723)120596410+ (119887221 minus 21198872011988722) 120596210 + 119887220(18)
Differentiating both sides of (11) with respect to 1205911 one canobtain[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886231205822 + 211988622120582 + 11988621120582 (1205824 + 119886231205823 + 119886221205822 + 11988621120582 + 11988620)+ 3119887231205822 + 211988722120582 + 11988721120582 (119887231205823 + 119887221205822 + 11988721120582 + 11988720) minus 1205911120582 (19)
Thus
Re [ 1198891205821198891205911 ]minus11205911=12059110 = 11989110158401 (Vlowast1 )ℎ22 (12059610) (20)
where Vlowast1 = 120596210 and 1198911(V1) = V41 + 11989223V31 + 11989222V21 + 11989221V1 +11989220 Therefore if condition (11986722) 11989110158401(Vlowast1 ) = 0 holds thenRe[1198891205821198891205911]minus11205911=12059110 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 1 If conditions (11986721)-(11986722) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205911 isin [0 12059110)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 12059110 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 3 (1205911 = 0 1205912 gt 0) For 1205911 = 0 and 1205912 gt 0 (7) becomes1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630+ (119888331205823 + 119888321205822 + 11988831120582 + 11988830) 119890minus1205821205912 = 0 (21)
where 11988630 = 11988600 + 1198870011988631 = 11988601 + 1198870111988632 = 11988602 + 1198870211988633 = 11988603 + 1198870311988830 = 11988800 + 1198890011988831 = 11988801 + 1198890111988832 = 11988802 + 1198890211988833 = 11988803(22)
Let 120582 = 1198941205962 (1205962 gt 0) be a root of (21) Then(119888311205962 minus 1198883312059632) sin 12059121205962 + (11988830 minus 1198883212059622) cos 12059121205962= 1198863212059622 minus 12059642 minus 11988630(119888311205962 minus 1198883312059632) cos 12059121205962 minus (11988830 minus 1198883212059622) sin 12059121205962= 1198863312059632 minus 119886311205962(23)
It follows that12059682 + 1198923312059662 + 1198923212059642 + 1198923112059622 + 11989230 = 0 (24)
with 11989230 = 119886230 minus 11988823011989231 = 119886231 minus 119888231 minus 21198863011988632 + 2119888301198873211989232 = 119886232 minus 119888232 + 21198883111988833 minus 21198863111988633 + 21198863011989233 = 119886233 minus 119887233 minus 211988632(25)
Let 12059622 = V2 then we have
V42 + 11989233V32 + 11989232V22 + 11989231V2 + 11989230 = 0 (26)
Similar to Case 2 we make the following assumption(11986731) (26) has at least one positive root V20 If condition (11986731)holds then there exists 12059620 = radicV20 such that (21) has a pairof purely imaginary roots plusmn11989412059620 For 1205962012059120 = 112059620 arccos ℎ31 (12059620)ℎ32 (12059620) (27)
whereℎ31 (12059620) = (11988832 minus 1198863311988833) 120596620+ (1198863111988833 minus 1198863211988832 + 1198863311988831 minus 11988830) 120596420+ (1198863011988832 minus 1198863111988831 + 1198863211988830) 120596220minus 1198863011988830ℎ32 (12059620) = 119888233120596620 + (119887232 minus 21198883111988833) 120596420+ (119888231 minus 21198883011988832)120596220 + 119888230(28)
In addition we have[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886331205822 + 211988632120582 + 11988631120582 (1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630)+ 3119888331205822 + 211988832120582 + 11988831120582 (119888331205823 + 119888321205822 + 11988831120582 + 11988830) minus 1205912120582 (29)
Further
Re [ 1198891205821198891205912 ]minus11205912=12059120 = 11989110158402 (Vlowast2 )ℎ32 (12059620) (30)
Advances in Mathematical Physics 5
where Vlowast2 = 120596220 and 1198912(V2) = V42 + 11989233V32 + 11989232V22 + 11989231V2 +11989230 Therefore if condition (11986732) 11989110158402(Vlowast2 ) = 0 holds thenRe[1198891205821198891205912]minus11205912=12059120 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 2 If conditions (11986731)-(11986732) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205912 isin [0 12059120)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205912 = 12059120 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 4 (1205911 = 1205912 = 120591 gt 0) For 1205911 = 1205912 = 120591 gt 0 we have1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640+ (119887431205823 + 119887421205822 + 11988741120582 + 11988740) 119890minus120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus2120582120591 = 0 (31)
with 11988640 = 1198860011988641 = 1198860111988642 = 1198860211988643 = 1198860311988740 = 11988700 + 1198880011988741 = 11988701 + 1198880111988742 = 11988702 + 1198880211988743 = 11988703 + 1198880311988940 = 1198890011988941 = 1198890111988942 = 11988902
(32)
Multiplying by 119890120582120591 (31) becomes the following119887431205823 + 119887421205822 + 11988741120582 + 11988740+ (1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640) 119890120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus120582120591 = 0 (33)
Let 120582 = 119894120596 (120596 gt 0) be a root of (37) it is easy to get11989241 (120596) cos 120591120596 minus 11989242 (120596) sin 120591120596 = 11989243 (120596) 11989244 (120596) sin 120591120596 + 11989245 (120596) cos 120591120596 = 11989246 (120596) (34)
where 11989241 (120596) = 1205964 minus (11988642 + 11988942) 1205962 + 11988640 + 1198894011989242 (120596) = (11988641 minus 11988941) 120596 minus 11988643120596311989243 (120596) = 119887421205962 minus 1198874011989244 (120596) = 1205964 minus (11988642 minus 11988942) 1205962 + 11988640 minus 1198894011989245 (120596) = (11988641 + 11988941) 120596 minus 11988643120596311989246 (120596) = 119887431205963 minus 11988741120596(35)
It leads to
cos 120591120596 = 11989242 (120596) times 11989246 (120596) + 11989243 (120596) times 11989244 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) sin 120591120596 = 11989241 (120596) times 11989246 (120596) minus 11989243 (120596) times 11989245 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) (36)
Thus we can get the following equation with respect to 120596cos2120591120596 + sin2120591120596 = 1 (37)
Next we make the following assumption (11986741) (37) hasat least one positive root 1205960 Then for 1205960 we have1205910 = 11205960sdot arccos 11989242 (1205960) times 11989246 (1205960) + 11989243 (1205960) times 11989244 (1205960)11989241 (1205960) times 11989244 (1205960) + 11989242 (1205960) times 11989245 (1205960) (38)
Taking the derivative of 120582 with respect to 120591 we obtain[119889120582119889120591]minus1 = minus11989247 (120582)11989248 (120582) minus 120591120582 (39)
where11989247 (120582) = 3119887431205822 + 211988742120582 + 11988741+ (41205823 + 3119886431205822 + 211988642120582 + 11988641) 119890120582120591+ (211988942120582 + 11988941) 119890minus12058212059111989248 (120582) = (1205825 + 119886431205824 + 119886421205823 + 119886411205822 + 11988640120582) 119890120582120591minus (119889421205823 + 119889411205822 + 11988940120582) 119890minus120582120591(40)
Then we can get that
Re[119889120582119889120591]minus1120591=1205910= minusℎ41 (1205960) times ℎ43 (1205960) + ℎ42 (1205960) times ℎ44 (1205960)ℎ243 (1205960) + ℎ244 (1205960) (41)
6 Advances in Mathematical Physics
whereℎ41 (1205960)= (11988641 + 11988941 minus 31198864312059620) cos 12059101205960minus 2 ((11988642 minus 11988942) 1205960 minus 212059630) sin 12059101205960 + 11988741minus 31198874312059620 ℎ42 (1205960)= (11988641 minus 11988941 minus 31198864312059620) sin 12059101205960+ 2 ((11988642 + 11988942) 1205960 minus 212059630) cos 12059101205960 + 2119887421205960ℎ43 (1205960)= (1198864312059640 minus (11988641 + 11988941) 12059620) cos 12059101205960minus (12059650 minus (11988642 + 11988942) 12059630 + (11988640 minus 11988940) 1205960) sin 12059101205960ℎ44 (1205960)= (1198864312059640 minus (11988641 minus 11988941) 12059620) sin 12059101205960+ (12059650 minus (11988642 + 11988942) 12059630 + (11988640 + 11988940) 1205960) cos 12059101205960
(42)
We assume that (11986742) ℎ41(1205960) times ℎ43(1205960) + ℎ42(1205960) timesℎ44(1205960) = 0 Clearly if condition (11986742) holds then we canconclude that Re[119889120582119889120591]minus1120591=1205910 = 0 Therefore according to theHopf bifurcation theorem in [26] we obtain the following
Theorem 3 If conditions (11986741)-(11986742) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 120591 isin [0 1205910)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) when 120591 = 1205910 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 5 (1205911 gt 0 1205912 isin (0 12059120)) Let 120582 = 11989412059610158401 be the root of (7)Then11989251 (12059610158401) sin 120591112059610158401 + 11989252 (12059610158401) cos 120591112059610158401 = 11989253 (12059610158401) 11989251 (12059610158401) cos 120591112059610158401 minus 11989252 (12059610158401) sin 120591112059610158401 = 11989254 (12059610158401) (43)
where11989251 (12059610158401) = 1198870112059610158401 minus 11988703 (12059610158401)3 + 1198890112059610158401 cos 120591212059610158401minus (11988900 minus 11988902 (12059610158401)2) sin 12059121205961015840111989252 (12059610158401) = 11988700 minus 11988702 (12059610158401)2 + 1198890112059610158401 sin 120591212059610158401+ (11988900 minus 11988902 (12059610158401)2) cos 12059121205961015840111989253 (12059610158401) = 11988602 (12059610158401)2 minus (12059610158401)4 minus 11988600minus (1198880112059610158401 minus 11988803 (12059610158401)3) sin 120591212059610158401minus (11988800 minus 11988802 (12059610158401)2) cos 12059121205961015840111989254 (12059610158401) = 11988603 (12059610158401)3 minus 1198860112059610158401minus (1198880112059610158401 minus 11988803 (12059610158401)3) cos 120591212059610158401+ (11988800 minus 11988802 (12059610158401)2) sin 120591212059610158401
(44)
Thus one can get the following equation with respect to 12059610158401119892251 (12059610158401) + 119892252 (12059610158401) = 119892253 (12059610158401) + 119892254 (12059610158401) (45)
Similar to Case 4 we assume that (11986751) (45) has at leastone positive root 120596101584010 Then for 120596101584010 we have120591101584010 = 1120596101584010sdot arccos 11989251 (120596101584010) times 11989254 (120596101584010) + 11989252 (120596101584010) times 11989253 (120596101584010)119892251 (120596101584010) + 119892252 (120596101584010) (46)
Differentiating (7) with respect to 1205911 we get[119889120582119889120591]minus1 = 11989255 (120582)11989256 (120582) minus 1205911120582 (47)
with
11989255 (120582) = 41205823 + 3119886031205822 + 211988602120582 + 11988601 + (3119887031205822+ 211988702120582 + 11988701) 119890minus1205821205911 + [(311988803 minus 120591211988802) 1205822 minus 1205912119888031205823+ (211988802 minus 120591211988801) 120582 + 11988801 minus 120591211988800] 119890minus1205821205912+ [(211988902 minus 120591211988901) 120582 minus 1205912119889021205822 + 11988901 minus 120591211988900]sdot 119890minus120582(1205911+1205912)11989256 (120582) = (119888031205824 + 119888021205823 + 119888011205822 + 11988800120582) 119890minus1205821205912+ (119889021205823 + 119889011205822 + 11988900120582) 119890minus120582(1205911+1205912)
(48)
Then we obtain
Re [ 1198891205821198891205911 ]minus11205911=120591101584010= ℎ51 (120596101584010) times ℎ53 (120596101584010) + ℎ52 (120596101584010) times ℎ54 (120596101584010)ℎ253 (120596101584010) + ℎ254 (120596101584010) (49)
Advances in Mathematical Physics 7
whereℎ51 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] sin 120591101584010120596101584010+ [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] cos 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] sin 1205912120596101584010+ [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] cos 1205912120596101584010+ 11988601 minus 311988603 (120596101584010)2 ℎ52 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] cos 120591101584010120596101584010minus [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] sin 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] cos 1205912120596101584010minus [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] sin 1205912120596101584010+ 11988602120596101584010 minus 4 (120596101584010)3 ℎ53 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] sin 1205912120596101584010 + [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] cos 1205912120596101584010ℎ54 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] cos 1205912120596101584010 minus [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] sin 12059110158402120596101584010
(50)
We assume that (11986752) ℎ51(120596101584010) times ℎ53(120596101584010) + ℎ52(120596101584010) timesℎ54(120596101584010) = 0 Thus we know that Re[1198891205821198891205911]minus11205911=120591101584010 = 0
if condition (11986752) holds Therefore according to the Hopfbifurcation theorem in [26] we obtain the following
Theorem 4 If conditions (11986751)-(11986752) hold and 1205912 isin (0 12059120)then
(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) islocally asymptotically stable for 1205911 isin [0 120591101584010)
(ii) system (2) undergoes a Hopf bifurcation at viral equi-librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 120591101584010 and a familyof periodic solutions bifurcate from119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
3 Properties of the Hopf Bifurcation
In this section we shall investigate direction of the Hopfbifurcation and stability of the bifurcating periodic solutionof system (2) when 1205911 = 120591101584010 and 1205912 = 120591101584020 isin (0 12059120) by usingthe center manifold theorem and the normal form theorywhich has been developed by Hassard et al [26]
Let 1205911 = 120591101584010 + 120583 1199061 = 119878(1205911119905) 1199062 = 119871(1205911119905) 1199063 = 119860(1205911119905)1199064 = 119877(1205911119905) 120583 isin 119877 Then 120583 = 0 is the Hopf bifurcation valueof system (2) and system (2) can be rewritten as (119905) = 119871120583 (119906119905) + 119865 (120583 119906119905) (51)
where 119906(119905) = (1199061 1199062 1199063 1199064)119879 isin 119862 = 119862([minus1 0] 1198774) and 119871120583119862 rarr 1198774 and 119865 119877 times 119862 rarr 1198774 are given respectively by119871120583120601= (120591101584010 + 120583)(119860lowast120601 (0) + 119862lowast120601(minus120591101584020120591101584010) + 119861lowast120601 (minus1)) (52)
with
119860lowast = [[[[[[1198861 1198862 1198863 11988641198865 1198866 1198867 00 0 1198868 00 0 0 1198869
]]]]]] 119861lowast = [[[[[[
0 0 0 00 1198871 0 00 1198872 0 00 0 0 0]]]]]]
119862lowast = [[[[[[0 0 0 00 0 0 00 0 1198881 00 0 1198882 0
]]]]]] 1198651 = minus12057311206011 (0) 1206012 (0) minus 12057321206011 (0) 1206013 (0) 1198652 = 12057311206011 (0) 1206012 (0) + 12057321206011 (0) 1206013 (0) 120601 = (1206011 1206012 1206013 1206014)119879 isin 119862 ([minus1 0] 1198774)
(53)
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
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Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
computers 120572 is the rate at which the recovered computersbecome susceptibly virus-free again 120576 is the rate at whichthe latent computers break out and 120574 is the rate at which theactive computers are cured by the antivirus software
As pointed out in [9] one of the typical features ofcomputer viruses is their latent characteristic Therefore theyneed a period to become active computers for the latent onesLikewise the antivirus software needs a period to clean theviruses in the active computers Based on this and motivatedby the work about the dynamical system with delay in [20ndash24] we incorporate two delays into system (1) and obtain thefollowing delayed computer virus model119889119878 (119905)119889119905 = 120583 minus 1205731119878 (119905) 119871 (119905) minus 1205732119878 (119905) 119860 (119905) + 120572119877 (119905)minus 120583119878 (119905) 119889119871 (119905)119889119905 = 1205731119878 (119905) 119871 (119905) + 1205732119878 (119905) 119860 (119905) minus 120576119871 (119905 minus 1205911)minus 120583119871 (119905) 119889119860 (119905)119889119905 = 120576119871 (119905 minus 1205911) minus 120574119860 (119905 minus 1205912) minus 120583119860 (119905) 119889119877 (119905)119889119905 = 120574119860 (119905 minus 1205912) minus 120572119877 (119905) minus 120583119877 (119905)
(2)
where 1205911 is the latent period of the computer viruses and 1205912is the period that the antivirus software needs to clean theviruses in the active computers
The rest of this paper is organized as follows In Section 2we present the existence of the viral equilibrium and con-ditions for the local stability of the viral equilibrium andexistence of the Hopf bifurcation are derived Direction andstability of the Hopf bifurcation are studied in Section 3and some numerical simulations are performed in Section 4to justify the obtained theoretical findings by taking somerelevant values of the parameters in system (2) and usingthe Matlab software package Finally we end this paper withconcluding remarks in Section 5
2 Existence of Local Hopf Bifurcation
By a simple computation we know that if (120576+120583)(120574+120583)(1205731(120574+120583) + 1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) then system(2) has a unique viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) where119878lowast = (120574 + 120583) (120576 + 120583)1205731 (120574 + 120583) + 1205732120576 119871lowast = 120574 + 120583120576 119860lowast119877lowast = 120574120572 + 120583119860lowast119860lowast = 1198601lowast1198602lowast 1198601lowast = 120583 minus 120583 (120576 + 120583) (120574 + 120583)1205731 (120574 + 120583) + 1205732120576 1198602lowast = (120576 + 120583) (120574 + 120583)120576 minus 120572120574120572 + 120583
(3)
Let 119878(119905) = 119878(119905) minus 119878lowast 119871(119905) = 119871(119905) minus 119871lowast 119860(119905) = 119860(119905) minus 119860lowast119877(119905) = 119877(119905) minus 119877lowast Dropping the bars system (2) becomes
119889119878 (119905)119889119905 = 1198861119878 (119905) + 1198862119871 (119905) + 1198863119860 (119905) + 1198864119877 (119905)minus 1205731119878 (119905) 119871 (119905) minus 1205732119878 (119905) 119860 (119905) 119889119871 (119905)119889119905 = 1198865119878 (119905) + 1198866119871 (119905) + 1198867119860 (119905) + 1198871119871 (119905 minus 1205911)+ 1205731119878 (119905) 119871 (119905) + 1205732119878 (119905) 119860 (119905) 119889119860 (119905)119889119905 = 1198868119860 (119905) + 1198872119871 (119905 minus 1205911) + 1198881119860 (119905 minus 1205912) 119889119877 (119905)119889119905 = 1198869119877 (119905) + 1198882119860 (119905 minus 1205912) (4)
where 1198861 = minus (1205731119871lowast + 1205732119860lowast + 120583) 1198862 = minus1205731119878lowast1198863 = minus1205732119878lowast1198864 = 1205721198865 = 1205731119871lowast + 1205732119860lowast1198871 = minus1205761198866 = 1205731119878lowast minus 1205831198867 = 1205732119878lowast1198868 = minus1205831198872 = 1205761198881 = minus1205741198869 = minus (120572 + 120583)119877lowast1198882 = 120574
(5)
The linear system of system (4) is
119889119878 (119905)119889119905 = 1198861119878 (119905) + 1198862119871 (119905) + 1198863119860 (119905) + 1198864119877 (119905) 119889119871 (119905)119889119905 = 1198865119878 (119905) + 1198866119871 (119905) + 1198867119860 (119905) + 1198871119871 (119905 minus 1205911) 119889119860 (119905)119889119905 = 1198868119860 (119905) + 1198872119871 (119905 minus 1205911) + 1198881119860 (119905 minus 1205912) 119889119877 (119905)119889119905 = 1198869119877 (119905) + 1198882119860 (119905 minus 1205912) (6)
Advances in Mathematical Physics 3
The corresponding characteristic equation is1205824 + 119886031205823 + 119886021205822 + 11988601120582 + 11988600+ (119887031205823 + 119887021205822 + 11988701120582 + 11988700) 119890minus1205821205911+ (119888031205823 + 119888021205822 + 11988801120582 + 11988800) 119890minus1205821205912+ (119889021205822 + 11988901120582 + 11988900) 119890minus120582(1205911+1205912) = 0(7)
where 11988600 = (11988611198866 minus 11988621198865) 1198868119886911988601 = (11988621198865 minus 11988611198866) (1198868 + 1198869) 11988602 = 11988611198866 minus 11988621198865 + 11988681198869 + (1198861 + 1198866) (1198868 + 1198869) 11988603 = minus (1198861 + 1198866 + 1198868 + 1198869) 11988700 = 1198861119886811988691198871 + (11988631198865 minus 11988611198867) 1198869119887211988701 = 11988671198872 (1198861 + 1198869) minus 1198871 (11988611198868 + 11988611198869 + 11988681198869) 11988702 = 1198871 (1198861 + 1198868 + 1198869) minus 1198867119887211988703 = minus119887111988800 = (11988611198866 minus 11988621198865) 1198869119888111988801 = 1198881 (11988621198865 minus 11988611198866 minus 11988611198869 minus 11988661198869) 11988802 = 1198881 (1198861 + 1198866 + 1198869) 11988803 = minus119888111988900 = 1198861119886911988711198881 + 119886411988651198872119888211988901 = minus11988711198881 (1198861 + 1198869) 11988902 = 11988711198881
(8)
Case 1 (1205911 = 1205912 = 0) For 1205911 = 1205912 = 0 (7) becomes1205824 + 119886131205823 + 119886121205822 + 11988611120582 + 11988610 = 0 (9)
where 11988610 = 11988600 + 11988700 + 11988800 + 1198890011988611 = 11988601 + 11988701 + 11988801 + 1198890111988612 = 11988602 + 11988702 + 11988802 + 1198890211988613 = 11988603 + 11988703 + 11988803 (10)
Thus according to the Routh-Hurwithz theorem weknow that if conditions (11986711) 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt11988611 hold then viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system(2) without delay is locally asymptotically stable
Case 2 (1205911 gt 0 1205912 = 0) For 1205911 gt 0 and 1205912 = 0 we can get thefollowing from (7)1205824 + 119886231205823 + 119886221205822 + 11988621120582 + 11988620+ (119887231205823 + 119887221205822 + 11988721120582 + 11988720) 119890minus1205821205911 = 0 (11)
where 11988620 = 11988600 + 1198880011988621 = 11988601 + 1198880111988622 = 11988602 + 1198880211988623 = 11988603 + 1198880311988720 = 11988700 + 1198890011988721 = 11988701 + 1198890111988722 = 11988702 + 1198890211988723 = 11988703(12)
We assume that 120582 = 1198941205961 (1205961 gt 0) is a root of (11) Then(119887211205961 minus 1198872312059631) sin 12059111205961 + (11988720 minus 1198872212059621) cos 12059111205961= 1198862212059621 minus 12059641 minus 11988620(119887211205961 minus 1198872312059631) cos 12059111205961 minus (11988720 minus 1198872212059621) sin 12059111205961= 1198862312059631 minus 119886211205961(13)
which implies that12059681 + 1198922312059661 + 1198922212059641 + 1198922112059621 + 11989220 = 0 (14)
with 11989220 = 119886220 minus 11988722011989221 = 119886221 minus 119887221 minus 21198862011988622 + 2119887201198872211989222 = 119886222 minus 119887222 + 21198872111988723 minus 21198862111988623 + 21198862011989223 = 119886223 minus 119887223 minus 211988622(15)
Let 12059621 = V1 then (14) becomes
V41 + 11989223V31 + 11989222V21 + 11989221V1 + 11989220 = 0 (16)
Discussion about distribution of roots for (16) is similarto that in [25] Therefore we directly assume that (11986721) (16)has at least one positive equilibrium V10
If (11986721) holds we know that (14) has at least one positiveroot 12059610 = radicV10 such that (11) has a pair of purely imaginaryroots plusmn11989412059610 For 12059610
12059110 = 112059610 arccos ℎ21 (12059610)ℎ22 (12059610) (17)
4 Advances in Mathematical Physics
whereℎ21 (12059610) = (11988722 minus 1198862311988723) 120596610+ (1198862111988723 + 1198862311988721 minus 1198862211988722 minus 11988720) 120596410+ (1198862011988722 + 1198862211988720 minus 1198862111988721) 120596210minus 1198862011988720ℎ22 (12059610) = 119887223120596610 + (119887222 minus 21198872111988723)120596410+ (119887221 minus 21198872011988722) 120596210 + 119887220(18)
Differentiating both sides of (11) with respect to 1205911 one canobtain[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886231205822 + 211988622120582 + 11988621120582 (1205824 + 119886231205823 + 119886221205822 + 11988621120582 + 11988620)+ 3119887231205822 + 211988722120582 + 11988721120582 (119887231205823 + 119887221205822 + 11988721120582 + 11988720) minus 1205911120582 (19)
Thus
Re [ 1198891205821198891205911 ]minus11205911=12059110 = 11989110158401 (Vlowast1 )ℎ22 (12059610) (20)
where Vlowast1 = 120596210 and 1198911(V1) = V41 + 11989223V31 + 11989222V21 + 11989221V1 +11989220 Therefore if condition (11986722) 11989110158401(Vlowast1 ) = 0 holds thenRe[1198891205821198891205911]minus11205911=12059110 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 1 If conditions (11986721)-(11986722) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205911 isin [0 12059110)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 12059110 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 3 (1205911 = 0 1205912 gt 0) For 1205911 = 0 and 1205912 gt 0 (7) becomes1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630+ (119888331205823 + 119888321205822 + 11988831120582 + 11988830) 119890minus1205821205912 = 0 (21)
where 11988630 = 11988600 + 1198870011988631 = 11988601 + 1198870111988632 = 11988602 + 1198870211988633 = 11988603 + 1198870311988830 = 11988800 + 1198890011988831 = 11988801 + 1198890111988832 = 11988802 + 1198890211988833 = 11988803(22)
Let 120582 = 1198941205962 (1205962 gt 0) be a root of (21) Then(119888311205962 minus 1198883312059632) sin 12059121205962 + (11988830 minus 1198883212059622) cos 12059121205962= 1198863212059622 minus 12059642 minus 11988630(119888311205962 minus 1198883312059632) cos 12059121205962 minus (11988830 minus 1198883212059622) sin 12059121205962= 1198863312059632 minus 119886311205962(23)
It follows that12059682 + 1198923312059662 + 1198923212059642 + 1198923112059622 + 11989230 = 0 (24)
with 11989230 = 119886230 minus 11988823011989231 = 119886231 minus 119888231 minus 21198863011988632 + 2119888301198873211989232 = 119886232 minus 119888232 + 21198883111988833 minus 21198863111988633 + 21198863011989233 = 119886233 minus 119887233 minus 211988632(25)
Let 12059622 = V2 then we have
V42 + 11989233V32 + 11989232V22 + 11989231V2 + 11989230 = 0 (26)
Similar to Case 2 we make the following assumption(11986731) (26) has at least one positive root V20 If condition (11986731)holds then there exists 12059620 = radicV20 such that (21) has a pairof purely imaginary roots plusmn11989412059620 For 1205962012059120 = 112059620 arccos ℎ31 (12059620)ℎ32 (12059620) (27)
whereℎ31 (12059620) = (11988832 minus 1198863311988833) 120596620+ (1198863111988833 minus 1198863211988832 + 1198863311988831 minus 11988830) 120596420+ (1198863011988832 minus 1198863111988831 + 1198863211988830) 120596220minus 1198863011988830ℎ32 (12059620) = 119888233120596620 + (119887232 minus 21198883111988833) 120596420+ (119888231 minus 21198883011988832)120596220 + 119888230(28)
In addition we have[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886331205822 + 211988632120582 + 11988631120582 (1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630)+ 3119888331205822 + 211988832120582 + 11988831120582 (119888331205823 + 119888321205822 + 11988831120582 + 11988830) minus 1205912120582 (29)
Further
Re [ 1198891205821198891205912 ]minus11205912=12059120 = 11989110158402 (Vlowast2 )ℎ32 (12059620) (30)
Advances in Mathematical Physics 5
where Vlowast2 = 120596220 and 1198912(V2) = V42 + 11989233V32 + 11989232V22 + 11989231V2 +11989230 Therefore if condition (11986732) 11989110158402(Vlowast2 ) = 0 holds thenRe[1198891205821198891205912]minus11205912=12059120 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 2 If conditions (11986731)-(11986732) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205912 isin [0 12059120)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205912 = 12059120 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 4 (1205911 = 1205912 = 120591 gt 0) For 1205911 = 1205912 = 120591 gt 0 we have1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640+ (119887431205823 + 119887421205822 + 11988741120582 + 11988740) 119890minus120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus2120582120591 = 0 (31)
with 11988640 = 1198860011988641 = 1198860111988642 = 1198860211988643 = 1198860311988740 = 11988700 + 1198880011988741 = 11988701 + 1198880111988742 = 11988702 + 1198880211988743 = 11988703 + 1198880311988940 = 1198890011988941 = 1198890111988942 = 11988902
(32)
Multiplying by 119890120582120591 (31) becomes the following119887431205823 + 119887421205822 + 11988741120582 + 11988740+ (1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640) 119890120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus120582120591 = 0 (33)
Let 120582 = 119894120596 (120596 gt 0) be a root of (37) it is easy to get11989241 (120596) cos 120591120596 minus 11989242 (120596) sin 120591120596 = 11989243 (120596) 11989244 (120596) sin 120591120596 + 11989245 (120596) cos 120591120596 = 11989246 (120596) (34)
where 11989241 (120596) = 1205964 minus (11988642 + 11988942) 1205962 + 11988640 + 1198894011989242 (120596) = (11988641 minus 11988941) 120596 minus 11988643120596311989243 (120596) = 119887421205962 minus 1198874011989244 (120596) = 1205964 minus (11988642 minus 11988942) 1205962 + 11988640 minus 1198894011989245 (120596) = (11988641 + 11988941) 120596 minus 11988643120596311989246 (120596) = 119887431205963 minus 11988741120596(35)
It leads to
cos 120591120596 = 11989242 (120596) times 11989246 (120596) + 11989243 (120596) times 11989244 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) sin 120591120596 = 11989241 (120596) times 11989246 (120596) minus 11989243 (120596) times 11989245 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) (36)
Thus we can get the following equation with respect to 120596cos2120591120596 + sin2120591120596 = 1 (37)
Next we make the following assumption (11986741) (37) hasat least one positive root 1205960 Then for 1205960 we have1205910 = 11205960sdot arccos 11989242 (1205960) times 11989246 (1205960) + 11989243 (1205960) times 11989244 (1205960)11989241 (1205960) times 11989244 (1205960) + 11989242 (1205960) times 11989245 (1205960) (38)
Taking the derivative of 120582 with respect to 120591 we obtain[119889120582119889120591]minus1 = minus11989247 (120582)11989248 (120582) minus 120591120582 (39)
where11989247 (120582) = 3119887431205822 + 211988742120582 + 11988741+ (41205823 + 3119886431205822 + 211988642120582 + 11988641) 119890120582120591+ (211988942120582 + 11988941) 119890minus12058212059111989248 (120582) = (1205825 + 119886431205824 + 119886421205823 + 119886411205822 + 11988640120582) 119890120582120591minus (119889421205823 + 119889411205822 + 11988940120582) 119890minus120582120591(40)
Then we can get that
Re[119889120582119889120591]minus1120591=1205910= minusℎ41 (1205960) times ℎ43 (1205960) + ℎ42 (1205960) times ℎ44 (1205960)ℎ243 (1205960) + ℎ244 (1205960) (41)
6 Advances in Mathematical Physics
whereℎ41 (1205960)= (11988641 + 11988941 minus 31198864312059620) cos 12059101205960minus 2 ((11988642 minus 11988942) 1205960 minus 212059630) sin 12059101205960 + 11988741minus 31198874312059620 ℎ42 (1205960)= (11988641 minus 11988941 minus 31198864312059620) sin 12059101205960+ 2 ((11988642 + 11988942) 1205960 minus 212059630) cos 12059101205960 + 2119887421205960ℎ43 (1205960)= (1198864312059640 minus (11988641 + 11988941) 12059620) cos 12059101205960minus (12059650 minus (11988642 + 11988942) 12059630 + (11988640 minus 11988940) 1205960) sin 12059101205960ℎ44 (1205960)= (1198864312059640 minus (11988641 minus 11988941) 12059620) sin 12059101205960+ (12059650 minus (11988642 + 11988942) 12059630 + (11988640 + 11988940) 1205960) cos 12059101205960
(42)
We assume that (11986742) ℎ41(1205960) times ℎ43(1205960) + ℎ42(1205960) timesℎ44(1205960) = 0 Clearly if condition (11986742) holds then we canconclude that Re[119889120582119889120591]minus1120591=1205910 = 0 Therefore according to theHopf bifurcation theorem in [26] we obtain the following
Theorem 3 If conditions (11986741)-(11986742) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 120591 isin [0 1205910)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) when 120591 = 1205910 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 5 (1205911 gt 0 1205912 isin (0 12059120)) Let 120582 = 11989412059610158401 be the root of (7)Then11989251 (12059610158401) sin 120591112059610158401 + 11989252 (12059610158401) cos 120591112059610158401 = 11989253 (12059610158401) 11989251 (12059610158401) cos 120591112059610158401 minus 11989252 (12059610158401) sin 120591112059610158401 = 11989254 (12059610158401) (43)
where11989251 (12059610158401) = 1198870112059610158401 minus 11988703 (12059610158401)3 + 1198890112059610158401 cos 120591212059610158401minus (11988900 minus 11988902 (12059610158401)2) sin 12059121205961015840111989252 (12059610158401) = 11988700 minus 11988702 (12059610158401)2 + 1198890112059610158401 sin 120591212059610158401+ (11988900 minus 11988902 (12059610158401)2) cos 12059121205961015840111989253 (12059610158401) = 11988602 (12059610158401)2 minus (12059610158401)4 minus 11988600minus (1198880112059610158401 minus 11988803 (12059610158401)3) sin 120591212059610158401minus (11988800 minus 11988802 (12059610158401)2) cos 12059121205961015840111989254 (12059610158401) = 11988603 (12059610158401)3 minus 1198860112059610158401minus (1198880112059610158401 minus 11988803 (12059610158401)3) cos 120591212059610158401+ (11988800 minus 11988802 (12059610158401)2) sin 120591212059610158401
(44)
Thus one can get the following equation with respect to 12059610158401119892251 (12059610158401) + 119892252 (12059610158401) = 119892253 (12059610158401) + 119892254 (12059610158401) (45)
Similar to Case 4 we assume that (11986751) (45) has at leastone positive root 120596101584010 Then for 120596101584010 we have120591101584010 = 1120596101584010sdot arccos 11989251 (120596101584010) times 11989254 (120596101584010) + 11989252 (120596101584010) times 11989253 (120596101584010)119892251 (120596101584010) + 119892252 (120596101584010) (46)
Differentiating (7) with respect to 1205911 we get[119889120582119889120591]minus1 = 11989255 (120582)11989256 (120582) minus 1205911120582 (47)
with
11989255 (120582) = 41205823 + 3119886031205822 + 211988602120582 + 11988601 + (3119887031205822+ 211988702120582 + 11988701) 119890minus1205821205911 + [(311988803 minus 120591211988802) 1205822 minus 1205912119888031205823+ (211988802 minus 120591211988801) 120582 + 11988801 minus 120591211988800] 119890minus1205821205912+ [(211988902 minus 120591211988901) 120582 minus 1205912119889021205822 + 11988901 minus 120591211988900]sdot 119890minus120582(1205911+1205912)11989256 (120582) = (119888031205824 + 119888021205823 + 119888011205822 + 11988800120582) 119890minus1205821205912+ (119889021205823 + 119889011205822 + 11988900120582) 119890minus120582(1205911+1205912)
(48)
Then we obtain
Re [ 1198891205821198891205911 ]minus11205911=120591101584010= ℎ51 (120596101584010) times ℎ53 (120596101584010) + ℎ52 (120596101584010) times ℎ54 (120596101584010)ℎ253 (120596101584010) + ℎ254 (120596101584010) (49)
Advances in Mathematical Physics 7
whereℎ51 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] sin 120591101584010120596101584010+ [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] cos 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] sin 1205912120596101584010+ [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] cos 1205912120596101584010+ 11988601 minus 311988603 (120596101584010)2 ℎ52 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] cos 120591101584010120596101584010minus [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] sin 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] cos 1205912120596101584010minus [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] sin 1205912120596101584010+ 11988602120596101584010 minus 4 (120596101584010)3 ℎ53 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] sin 1205912120596101584010 + [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] cos 1205912120596101584010ℎ54 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] cos 1205912120596101584010 minus [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] sin 12059110158402120596101584010
(50)
We assume that (11986752) ℎ51(120596101584010) times ℎ53(120596101584010) + ℎ52(120596101584010) timesℎ54(120596101584010) = 0 Thus we know that Re[1198891205821198891205911]minus11205911=120591101584010 = 0
if condition (11986752) holds Therefore according to the Hopfbifurcation theorem in [26] we obtain the following
Theorem 4 If conditions (11986751)-(11986752) hold and 1205912 isin (0 12059120)then
(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) islocally asymptotically stable for 1205911 isin [0 120591101584010)
(ii) system (2) undergoes a Hopf bifurcation at viral equi-librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 120591101584010 and a familyof periodic solutions bifurcate from119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
3 Properties of the Hopf Bifurcation
In this section we shall investigate direction of the Hopfbifurcation and stability of the bifurcating periodic solutionof system (2) when 1205911 = 120591101584010 and 1205912 = 120591101584020 isin (0 12059120) by usingthe center manifold theorem and the normal form theorywhich has been developed by Hassard et al [26]
Let 1205911 = 120591101584010 + 120583 1199061 = 119878(1205911119905) 1199062 = 119871(1205911119905) 1199063 = 119860(1205911119905)1199064 = 119877(1205911119905) 120583 isin 119877 Then 120583 = 0 is the Hopf bifurcation valueof system (2) and system (2) can be rewritten as (119905) = 119871120583 (119906119905) + 119865 (120583 119906119905) (51)
where 119906(119905) = (1199061 1199062 1199063 1199064)119879 isin 119862 = 119862([minus1 0] 1198774) and 119871120583119862 rarr 1198774 and 119865 119877 times 119862 rarr 1198774 are given respectively by119871120583120601= (120591101584010 + 120583)(119860lowast120601 (0) + 119862lowast120601(minus120591101584020120591101584010) + 119861lowast120601 (minus1)) (52)
with
119860lowast = [[[[[[1198861 1198862 1198863 11988641198865 1198866 1198867 00 0 1198868 00 0 0 1198869
]]]]]] 119861lowast = [[[[[[
0 0 0 00 1198871 0 00 1198872 0 00 0 0 0]]]]]]
119862lowast = [[[[[[0 0 0 00 0 0 00 0 1198881 00 0 1198882 0
]]]]]] 1198651 = minus12057311206011 (0) 1206012 (0) minus 12057321206011 (0) 1206013 (0) 1198652 = 12057311206011 (0) 1206012 (0) + 12057321206011 (0) 1206013 (0) 120601 = (1206011 1206012 1206013 1206014)119879 isin 119862 ([minus1 0] 1198774)
(53)
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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
Advances in Mathematical Physics 3
The corresponding characteristic equation is1205824 + 119886031205823 + 119886021205822 + 11988601120582 + 11988600+ (119887031205823 + 119887021205822 + 11988701120582 + 11988700) 119890minus1205821205911+ (119888031205823 + 119888021205822 + 11988801120582 + 11988800) 119890minus1205821205912+ (119889021205822 + 11988901120582 + 11988900) 119890minus120582(1205911+1205912) = 0(7)
where 11988600 = (11988611198866 minus 11988621198865) 1198868119886911988601 = (11988621198865 minus 11988611198866) (1198868 + 1198869) 11988602 = 11988611198866 minus 11988621198865 + 11988681198869 + (1198861 + 1198866) (1198868 + 1198869) 11988603 = minus (1198861 + 1198866 + 1198868 + 1198869) 11988700 = 1198861119886811988691198871 + (11988631198865 minus 11988611198867) 1198869119887211988701 = 11988671198872 (1198861 + 1198869) minus 1198871 (11988611198868 + 11988611198869 + 11988681198869) 11988702 = 1198871 (1198861 + 1198868 + 1198869) minus 1198867119887211988703 = minus119887111988800 = (11988611198866 minus 11988621198865) 1198869119888111988801 = 1198881 (11988621198865 minus 11988611198866 minus 11988611198869 minus 11988661198869) 11988802 = 1198881 (1198861 + 1198866 + 1198869) 11988803 = minus119888111988900 = 1198861119886911988711198881 + 119886411988651198872119888211988901 = minus11988711198881 (1198861 + 1198869) 11988902 = 11988711198881
(8)
Case 1 (1205911 = 1205912 = 0) For 1205911 = 1205912 = 0 (7) becomes1205824 + 119886131205823 + 119886121205822 + 11988611120582 + 11988610 = 0 (9)
where 11988610 = 11988600 + 11988700 + 11988800 + 1198890011988611 = 11988601 + 11988701 + 11988801 + 1198890111988612 = 11988602 + 11988702 + 11988802 + 1198890211988613 = 11988603 + 11988703 + 11988803 (10)
Thus according to the Routh-Hurwithz theorem weknow that if conditions (11986711) 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt11988611 hold then viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system(2) without delay is locally asymptotically stable
Case 2 (1205911 gt 0 1205912 = 0) For 1205911 gt 0 and 1205912 = 0 we can get thefollowing from (7)1205824 + 119886231205823 + 119886221205822 + 11988621120582 + 11988620+ (119887231205823 + 119887221205822 + 11988721120582 + 11988720) 119890minus1205821205911 = 0 (11)
where 11988620 = 11988600 + 1198880011988621 = 11988601 + 1198880111988622 = 11988602 + 1198880211988623 = 11988603 + 1198880311988720 = 11988700 + 1198890011988721 = 11988701 + 1198890111988722 = 11988702 + 1198890211988723 = 11988703(12)
We assume that 120582 = 1198941205961 (1205961 gt 0) is a root of (11) Then(119887211205961 minus 1198872312059631) sin 12059111205961 + (11988720 minus 1198872212059621) cos 12059111205961= 1198862212059621 minus 12059641 minus 11988620(119887211205961 minus 1198872312059631) cos 12059111205961 minus (11988720 minus 1198872212059621) sin 12059111205961= 1198862312059631 minus 119886211205961(13)
which implies that12059681 + 1198922312059661 + 1198922212059641 + 1198922112059621 + 11989220 = 0 (14)
with 11989220 = 119886220 minus 11988722011989221 = 119886221 minus 119887221 minus 21198862011988622 + 2119887201198872211989222 = 119886222 minus 119887222 + 21198872111988723 minus 21198862111988623 + 21198862011989223 = 119886223 minus 119887223 minus 211988622(15)
Let 12059621 = V1 then (14) becomes
V41 + 11989223V31 + 11989222V21 + 11989221V1 + 11989220 = 0 (16)
Discussion about distribution of roots for (16) is similarto that in [25] Therefore we directly assume that (11986721) (16)has at least one positive equilibrium V10
If (11986721) holds we know that (14) has at least one positiveroot 12059610 = radicV10 such that (11) has a pair of purely imaginaryroots plusmn11989412059610 For 12059610
12059110 = 112059610 arccos ℎ21 (12059610)ℎ22 (12059610) (17)
4 Advances in Mathematical Physics
whereℎ21 (12059610) = (11988722 minus 1198862311988723) 120596610+ (1198862111988723 + 1198862311988721 minus 1198862211988722 minus 11988720) 120596410+ (1198862011988722 + 1198862211988720 minus 1198862111988721) 120596210minus 1198862011988720ℎ22 (12059610) = 119887223120596610 + (119887222 minus 21198872111988723)120596410+ (119887221 minus 21198872011988722) 120596210 + 119887220(18)
Differentiating both sides of (11) with respect to 1205911 one canobtain[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886231205822 + 211988622120582 + 11988621120582 (1205824 + 119886231205823 + 119886221205822 + 11988621120582 + 11988620)+ 3119887231205822 + 211988722120582 + 11988721120582 (119887231205823 + 119887221205822 + 11988721120582 + 11988720) minus 1205911120582 (19)
Thus
Re [ 1198891205821198891205911 ]minus11205911=12059110 = 11989110158401 (Vlowast1 )ℎ22 (12059610) (20)
where Vlowast1 = 120596210 and 1198911(V1) = V41 + 11989223V31 + 11989222V21 + 11989221V1 +11989220 Therefore if condition (11986722) 11989110158401(Vlowast1 ) = 0 holds thenRe[1198891205821198891205911]minus11205911=12059110 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 1 If conditions (11986721)-(11986722) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205911 isin [0 12059110)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 12059110 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 3 (1205911 = 0 1205912 gt 0) For 1205911 = 0 and 1205912 gt 0 (7) becomes1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630+ (119888331205823 + 119888321205822 + 11988831120582 + 11988830) 119890minus1205821205912 = 0 (21)
where 11988630 = 11988600 + 1198870011988631 = 11988601 + 1198870111988632 = 11988602 + 1198870211988633 = 11988603 + 1198870311988830 = 11988800 + 1198890011988831 = 11988801 + 1198890111988832 = 11988802 + 1198890211988833 = 11988803(22)
Let 120582 = 1198941205962 (1205962 gt 0) be a root of (21) Then(119888311205962 minus 1198883312059632) sin 12059121205962 + (11988830 minus 1198883212059622) cos 12059121205962= 1198863212059622 minus 12059642 minus 11988630(119888311205962 minus 1198883312059632) cos 12059121205962 minus (11988830 minus 1198883212059622) sin 12059121205962= 1198863312059632 minus 119886311205962(23)
It follows that12059682 + 1198923312059662 + 1198923212059642 + 1198923112059622 + 11989230 = 0 (24)
with 11989230 = 119886230 minus 11988823011989231 = 119886231 minus 119888231 minus 21198863011988632 + 2119888301198873211989232 = 119886232 minus 119888232 + 21198883111988833 minus 21198863111988633 + 21198863011989233 = 119886233 minus 119887233 minus 211988632(25)
Let 12059622 = V2 then we have
V42 + 11989233V32 + 11989232V22 + 11989231V2 + 11989230 = 0 (26)
Similar to Case 2 we make the following assumption(11986731) (26) has at least one positive root V20 If condition (11986731)holds then there exists 12059620 = radicV20 such that (21) has a pairof purely imaginary roots plusmn11989412059620 For 1205962012059120 = 112059620 arccos ℎ31 (12059620)ℎ32 (12059620) (27)
whereℎ31 (12059620) = (11988832 minus 1198863311988833) 120596620+ (1198863111988833 minus 1198863211988832 + 1198863311988831 minus 11988830) 120596420+ (1198863011988832 minus 1198863111988831 + 1198863211988830) 120596220minus 1198863011988830ℎ32 (12059620) = 119888233120596620 + (119887232 minus 21198883111988833) 120596420+ (119888231 minus 21198883011988832)120596220 + 119888230(28)
In addition we have[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886331205822 + 211988632120582 + 11988631120582 (1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630)+ 3119888331205822 + 211988832120582 + 11988831120582 (119888331205823 + 119888321205822 + 11988831120582 + 11988830) minus 1205912120582 (29)
Further
Re [ 1198891205821198891205912 ]minus11205912=12059120 = 11989110158402 (Vlowast2 )ℎ32 (12059620) (30)
Advances in Mathematical Physics 5
where Vlowast2 = 120596220 and 1198912(V2) = V42 + 11989233V32 + 11989232V22 + 11989231V2 +11989230 Therefore if condition (11986732) 11989110158402(Vlowast2 ) = 0 holds thenRe[1198891205821198891205912]minus11205912=12059120 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 2 If conditions (11986731)-(11986732) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205912 isin [0 12059120)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205912 = 12059120 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 4 (1205911 = 1205912 = 120591 gt 0) For 1205911 = 1205912 = 120591 gt 0 we have1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640+ (119887431205823 + 119887421205822 + 11988741120582 + 11988740) 119890minus120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus2120582120591 = 0 (31)
with 11988640 = 1198860011988641 = 1198860111988642 = 1198860211988643 = 1198860311988740 = 11988700 + 1198880011988741 = 11988701 + 1198880111988742 = 11988702 + 1198880211988743 = 11988703 + 1198880311988940 = 1198890011988941 = 1198890111988942 = 11988902
(32)
Multiplying by 119890120582120591 (31) becomes the following119887431205823 + 119887421205822 + 11988741120582 + 11988740+ (1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640) 119890120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus120582120591 = 0 (33)
Let 120582 = 119894120596 (120596 gt 0) be a root of (37) it is easy to get11989241 (120596) cos 120591120596 minus 11989242 (120596) sin 120591120596 = 11989243 (120596) 11989244 (120596) sin 120591120596 + 11989245 (120596) cos 120591120596 = 11989246 (120596) (34)
where 11989241 (120596) = 1205964 minus (11988642 + 11988942) 1205962 + 11988640 + 1198894011989242 (120596) = (11988641 minus 11988941) 120596 minus 11988643120596311989243 (120596) = 119887421205962 minus 1198874011989244 (120596) = 1205964 minus (11988642 minus 11988942) 1205962 + 11988640 minus 1198894011989245 (120596) = (11988641 + 11988941) 120596 minus 11988643120596311989246 (120596) = 119887431205963 minus 11988741120596(35)
It leads to
cos 120591120596 = 11989242 (120596) times 11989246 (120596) + 11989243 (120596) times 11989244 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) sin 120591120596 = 11989241 (120596) times 11989246 (120596) minus 11989243 (120596) times 11989245 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) (36)
Thus we can get the following equation with respect to 120596cos2120591120596 + sin2120591120596 = 1 (37)
Next we make the following assumption (11986741) (37) hasat least one positive root 1205960 Then for 1205960 we have1205910 = 11205960sdot arccos 11989242 (1205960) times 11989246 (1205960) + 11989243 (1205960) times 11989244 (1205960)11989241 (1205960) times 11989244 (1205960) + 11989242 (1205960) times 11989245 (1205960) (38)
Taking the derivative of 120582 with respect to 120591 we obtain[119889120582119889120591]minus1 = minus11989247 (120582)11989248 (120582) minus 120591120582 (39)
where11989247 (120582) = 3119887431205822 + 211988742120582 + 11988741+ (41205823 + 3119886431205822 + 211988642120582 + 11988641) 119890120582120591+ (211988942120582 + 11988941) 119890minus12058212059111989248 (120582) = (1205825 + 119886431205824 + 119886421205823 + 119886411205822 + 11988640120582) 119890120582120591minus (119889421205823 + 119889411205822 + 11988940120582) 119890minus120582120591(40)
Then we can get that
Re[119889120582119889120591]minus1120591=1205910= minusℎ41 (1205960) times ℎ43 (1205960) + ℎ42 (1205960) times ℎ44 (1205960)ℎ243 (1205960) + ℎ244 (1205960) (41)
6 Advances in Mathematical Physics
whereℎ41 (1205960)= (11988641 + 11988941 minus 31198864312059620) cos 12059101205960minus 2 ((11988642 minus 11988942) 1205960 minus 212059630) sin 12059101205960 + 11988741minus 31198874312059620 ℎ42 (1205960)= (11988641 minus 11988941 minus 31198864312059620) sin 12059101205960+ 2 ((11988642 + 11988942) 1205960 minus 212059630) cos 12059101205960 + 2119887421205960ℎ43 (1205960)= (1198864312059640 minus (11988641 + 11988941) 12059620) cos 12059101205960minus (12059650 minus (11988642 + 11988942) 12059630 + (11988640 minus 11988940) 1205960) sin 12059101205960ℎ44 (1205960)= (1198864312059640 minus (11988641 minus 11988941) 12059620) sin 12059101205960+ (12059650 minus (11988642 + 11988942) 12059630 + (11988640 + 11988940) 1205960) cos 12059101205960
(42)
We assume that (11986742) ℎ41(1205960) times ℎ43(1205960) + ℎ42(1205960) timesℎ44(1205960) = 0 Clearly if condition (11986742) holds then we canconclude that Re[119889120582119889120591]minus1120591=1205910 = 0 Therefore according to theHopf bifurcation theorem in [26] we obtain the following
Theorem 3 If conditions (11986741)-(11986742) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 120591 isin [0 1205910)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) when 120591 = 1205910 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 5 (1205911 gt 0 1205912 isin (0 12059120)) Let 120582 = 11989412059610158401 be the root of (7)Then11989251 (12059610158401) sin 120591112059610158401 + 11989252 (12059610158401) cos 120591112059610158401 = 11989253 (12059610158401) 11989251 (12059610158401) cos 120591112059610158401 minus 11989252 (12059610158401) sin 120591112059610158401 = 11989254 (12059610158401) (43)
where11989251 (12059610158401) = 1198870112059610158401 minus 11988703 (12059610158401)3 + 1198890112059610158401 cos 120591212059610158401minus (11988900 minus 11988902 (12059610158401)2) sin 12059121205961015840111989252 (12059610158401) = 11988700 minus 11988702 (12059610158401)2 + 1198890112059610158401 sin 120591212059610158401+ (11988900 minus 11988902 (12059610158401)2) cos 12059121205961015840111989253 (12059610158401) = 11988602 (12059610158401)2 minus (12059610158401)4 minus 11988600minus (1198880112059610158401 minus 11988803 (12059610158401)3) sin 120591212059610158401minus (11988800 minus 11988802 (12059610158401)2) cos 12059121205961015840111989254 (12059610158401) = 11988603 (12059610158401)3 minus 1198860112059610158401minus (1198880112059610158401 minus 11988803 (12059610158401)3) cos 120591212059610158401+ (11988800 minus 11988802 (12059610158401)2) sin 120591212059610158401
(44)
Thus one can get the following equation with respect to 12059610158401119892251 (12059610158401) + 119892252 (12059610158401) = 119892253 (12059610158401) + 119892254 (12059610158401) (45)
Similar to Case 4 we assume that (11986751) (45) has at leastone positive root 120596101584010 Then for 120596101584010 we have120591101584010 = 1120596101584010sdot arccos 11989251 (120596101584010) times 11989254 (120596101584010) + 11989252 (120596101584010) times 11989253 (120596101584010)119892251 (120596101584010) + 119892252 (120596101584010) (46)
Differentiating (7) with respect to 1205911 we get[119889120582119889120591]minus1 = 11989255 (120582)11989256 (120582) minus 1205911120582 (47)
with
11989255 (120582) = 41205823 + 3119886031205822 + 211988602120582 + 11988601 + (3119887031205822+ 211988702120582 + 11988701) 119890minus1205821205911 + [(311988803 minus 120591211988802) 1205822 minus 1205912119888031205823+ (211988802 minus 120591211988801) 120582 + 11988801 minus 120591211988800] 119890minus1205821205912+ [(211988902 minus 120591211988901) 120582 minus 1205912119889021205822 + 11988901 minus 120591211988900]sdot 119890minus120582(1205911+1205912)11989256 (120582) = (119888031205824 + 119888021205823 + 119888011205822 + 11988800120582) 119890minus1205821205912+ (119889021205823 + 119889011205822 + 11988900120582) 119890minus120582(1205911+1205912)
(48)
Then we obtain
Re [ 1198891205821198891205911 ]minus11205911=120591101584010= ℎ51 (120596101584010) times ℎ53 (120596101584010) + ℎ52 (120596101584010) times ℎ54 (120596101584010)ℎ253 (120596101584010) + ℎ254 (120596101584010) (49)
Advances in Mathematical Physics 7
whereℎ51 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] sin 120591101584010120596101584010+ [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] cos 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] sin 1205912120596101584010+ [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] cos 1205912120596101584010+ 11988601 minus 311988603 (120596101584010)2 ℎ52 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] cos 120591101584010120596101584010minus [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] sin 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] cos 1205912120596101584010minus [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] sin 1205912120596101584010+ 11988602120596101584010 minus 4 (120596101584010)3 ℎ53 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] sin 1205912120596101584010 + [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] cos 1205912120596101584010ℎ54 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] cos 1205912120596101584010 minus [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] sin 12059110158402120596101584010
(50)
We assume that (11986752) ℎ51(120596101584010) times ℎ53(120596101584010) + ℎ52(120596101584010) timesℎ54(120596101584010) = 0 Thus we know that Re[1198891205821198891205911]minus11205911=120591101584010 = 0
if condition (11986752) holds Therefore according to the Hopfbifurcation theorem in [26] we obtain the following
Theorem 4 If conditions (11986751)-(11986752) hold and 1205912 isin (0 12059120)then
(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) islocally asymptotically stable for 1205911 isin [0 120591101584010)
(ii) system (2) undergoes a Hopf bifurcation at viral equi-librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 120591101584010 and a familyof periodic solutions bifurcate from119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
3 Properties of the Hopf Bifurcation
In this section we shall investigate direction of the Hopfbifurcation and stability of the bifurcating periodic solutionof system (2) when 1205911 = 120591101584010 and 1205912 = 120591101584020 isin (0 12059120) by usingthe center manifold theorem and the normal form theorywhich has been developed by Hassard et al [26]
Let 1205911 = 120591101584010 + 120583 1199061 = 119878(1205911119905) 1199062 = 119871(1205911119905) 1199063 = 119860(1205911119905)1199064 = 119877(1205911119905) 120583 isin 119877 Then 120583 = 0 is the Hopf bifurcation valueof system (2) and system (2) can be rewritten as (119905) = 119871120583 (119906119905) + 119865 (120583 119906119905) (51)
where 119906(119905) = (1199061 1199062 1199063 1199064)119879 isin 119862 = 119862([minus1 0] 1198774) and 119871120583119862 rarr 1198774 and 119865 119877 times 119862 rarr 1198774 are given respectively by119871120583120601= (120591101584010 + 120583)(119860lowast120601 (0) + 119862lowast120601(minus120591101584020120591101584010) + 119861lowast120601 (minus1)) (52)
with
119860lowast = [[[[[[1198861 1198862 1198863 11988641198865 1198866 1198867 00 0 1198868 00 0 0 1198869
]]]]]] 119861lowast = [[[[[[
0 0 0 00 1198871 0 00 1198872 0 00 0 0 0]]]]]]
119862lowast = [[[[[[0 0 0 00 0 0 00 0 1198881 00 0 1198882 0
]]]]]] 1198651 = minus12057311206011 (0) 1206012 (0) minus 12057321206011 (0) 1206013 (0) 1198652 = 12057311206011 (0) 1206012 (0) + 12057321206011 (0) 1206013 (0) 120601 = (1206011 1206012 1206013 1206014)119879 isin 119862 ([minus1 0] 1198774)
(53)
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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OptimizationJournal 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
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
whereℎ21 (12059610) = (11988722 minus 1198862311988723) 120596610+ (1198862111988723 + 1198862311988721 minus 1198862211988722 minus 11988720) 120596410+ (1198862011988722 + 1198862211988720 minus 1198862111988721) 120596210minus 1198862011988720ℎ22 (12059610) = 119887223120596610 + (119887222 minus 21198872111988723)120596410+ (119887221 minus 21198872011988722) 120596210 + 119887220(18)
Differentiating both sides of (11) with respect to 1205911 one canobtain[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886231205822 + 211988622120582 + 11988621120582 (1205824 + 119886231205823 + 119886221205822 + 11988621120582 + 11988620)+ 3119887231205822 + 211988722120582 + 11988721120582 (119887231205823 + 119887221205822 + 11988721120582 + 11988720) minus 1205911120582 (19)
Thus
Re [ 1198891205821198891205911 ]minus11205911=12059110 = 11989110158401 (Vlowast1 )ℎ22 (12059610) (20)
where Vlowast1 = 120596210 and 1198911(V1) = V41 + 11989223V31 + 11989222V21 + 11989221V1 +11989220 Therefore if condition (11986722) 11989110158401(Vlowast1 ) = 0 holds thenRe[1198891205821198891205911]minus11205911=12059110 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 1 If conditions (11986721)-(11986722) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205911 isin [0 12059110)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 12059110 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 3 (1205911 = 0 1205912 gt 0) For 1205911 = 0 and 1205912 gt 0 (7) becomes1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630+ (119888331205823 + 119888321205822 + 11988831120582 + 11988830) 119890minus1205821205912 = 0 (21)
where 11988630 = 11988600 + 1198870011988631 = 11988601 + 1198870111988632 = 11988602 + 1198870211988633 = 11988603 + 1198870311988830 = 11988800 + 1198890011988831 = 11988801 + 1198890111988832 = 11988802 + 1198890211988833 = 11988803(22)
Let 120582 = 1198941205962 (1205962 gt 0) be a root of (21) Then(119888311205962 minus 1198883312059632) sin 12059121205962 + (11988830 minus 1198883212059622) cos 12059121205962= 1198863212059622 minus 12059642 minus 11988630(119888311205962 minus 1198883312059632) cos 12059121205962 minus (11988830 minus 1198883212059622) sin 12059121205962= 1198863312059632 minus 119886311205962(23)
It follows that12059682 + 1198923312059662 + 1198923212059642 + 1198923112059622 + 11989230 = 0 (24)
with 11989230 = 119886230 minus 11988823011989231 = 119886231 minus 119888231 minus 21198863011988632 + 2119888301198873211989232 = 119886232 minus 119888232 + 21198883111988833 minus 21198863111988633 + 21198863011989233 = 119886233 minus 119887233 minus 211988632(25)
Let 12059622 = V2 then we have
V42 + 11989233V32 + 11989232V22 + 11989231V2 + 11989230 = 0 (26)
Similar to Case 2 we make the following assumption(11986731) (26) has at least one positive root V20 If condition (11986731)holds then there exists 12059620 = radicV20 such that (21) has a pairof purely imaginary roots plusmn11989412059620 For 1205962012059120 = 112059620 arccos ℎ31 (12059620)ℎ32 (12059620) (27)
whereℎ31 (12059620) = (11988832 minus 1198863311988833) 120596620+ (1198863111988833 minus 1198863211988832 + 1198863311988831 minus 11988830) 120596420+ (1198863011988832 minus 1198863111988831 + 1198863211988830) 120596220minus 1198863011988830ℎ32 (12059620) = 119888233120596620 + (119887232 minus 21198883111988833) 120596420+ (119888231 minus 21198883011988832)120596220 + 119888230(28)
In addition we have[ 1198891205821198891205911 ]minus1 = minus 41205823 + 3119886331205822 + 211988632120582 + 11988631120582 (1205824 + 119886331205823 + 119886321205822 + 11988631120582 + 11988630)+ 3119888331205822 + 211988832120582 + 11988831120582 (119888331205823 + 119888321205822 + 11988831120582 + 11988830) minus 1205912120582 (29)
Further
Re [ 1198891205821198891205912 ]minus11205912=12059120 = 11989110158402 (Vlowast2 )ℎ32 (12059620) (30)
Advances in Mathematical Physics 5
where Vlowast2 = 120596220 and 1198912(V2) = V42 + 11989233V32 + 11989232V22 + 11989231V2 +11989230 Therefore if condition (11986732) 11989110158402(Vlowast2 ) = 0 holds thenRe[1198891205821198891205912]minus11205912=12059120 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 2 If conditions (11986731)-(11986732) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205912 isin [0 12059120)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205912 = 12059120 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 4 (1205911 = 1205912 = 120591 gt 0) For 1205911 = 1205912 = 120591 gt 0 we have1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640+ (119887431205823 + 119887421205822 + 11988741120582 + 11988740) 119890minus120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus2120582120591 = 0 (31)
with 11988640 = 1198860011988641 = 1198860111988642 = 1198860211988643 = 1198860311988740 = 11988700 + 1198880011988741 = 11988701 + 1198880111988742 = 11988702 + 1198880211988743 = 11988703 + 1198880311988940 = 1198890011988941 = 1198890111988942 = 11988902
(32)
Multiplying by 119890120582120591 (31) becomes the following119887431205823 + 119887421205822 + 11988741120582 + 11988740+ (1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640) 119890120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus120582120591 = 0 (33)
Let 120582 = 119894120596 (120596 gt 0) be a root of (37) it is easy to get11989241 (120596) cos 120591120596 minus 11989242 (120596) sin 120591120596 = 11989243 (120596) 11989244 (120596) sin 120591120596 + 11989245 (120596) cos 120591120596 = 11989246 (120596) (34)
where 11989241 (120596) = 1205964 minus (11988642 + 11988942) 1205962 + 11988640 + 1198894011989242 (120596) = (11988641 minus 11988941) 120596 minus 11988643120596311989243 (120596) = 119887421205962 minus 1198874011989244 (120596) = 1205964 minus (11988642 minus 11988942) 1205962 + 11988640 minus 1198894011989245 (120596) = (11988641 + 11988941) 120596 minus 11988643120596311989246 (120596) = 119887431205963 minus 11988741120596(35)
It leads to
cos 120591120596 = 11989242 (120596) times 11989246 (120596) + 11989243 (120596) times 11989244 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) sin 120591120596 = 11989241 (120596) times 11989246 (120596) minus 11989243 (120596) times 11989245 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) (36)
Thus we can get the following equation with respect to 120596cos2120591120596 + sin2120591120596 = 1 (37)
Next we make the following assumption (11986741) (37) hasat least one positive root 1205960 Then for 1205960 we have1205910 = 11205960sdot arccos 11989242 (1205960) times 11989246 (1205960) + 11989243 (1205960) times 11989244 (1205960)11989241 (1205960) times 11989244 (1205960) + 11989242 (1205960) times 11989245 (1205960) (38)
Taking the derivative of 120582 with respect to 120591 we obtain[119889120582119889120591]minus1 = minus11989247 (120582)11989248 (120582) minus 120591120582 (39)
where11989247 (120582) = 3119887431205822 + 211988742120582 + 11988741+ (41205823 + 3119886431205822 + 211988642120582 + 11988641) 119890120582120591+ (211988942120582 + 11988941) 119890minus12058212059111989248 (120582) = (1205825 + 119886431205824 + 119886421205823 + 119886411205822 + 11988640120582) 119890120582120591minus (119889421205823 + 119889411205822 + 11988940120582) 119890minus120582120591(40)
Then we can get that
Re[119889120582119889120591]minus1120591=1205910= minusℎ41 (1205960) times ℎ43 (1205960) + ℎ42 (1205960) times ℎ44 (1205960)ℎ243 (1205960) + ℎ244 (1205960) (41)
6 Advances in Mathematical Physics
whereℎ41 (1205960)= (11988641 + 11988941 minus 31198864312059620) cos 12059101205960minus 2 ((11988642 minus 11988942) 1205960 minus 212059630) sin 12059101205960 + 11988741minus 31198874312059620 ℎ42 (1205960)= (11988641 minus 11988941 minus 31198864312059620) sin 12059101205960+ 2 ((11988642 + 11988942) 1205960 minus 212059630) cos 12059101205960 + 2119887421205960ℎ43 (1205960)= (1198864312059640 minus (11988641 + 11988941) 12059620) cos 12059101205960minus (12059650 minus (11988642 + 11988942) 12059630 + (11988640 minus 11988940) 1205960) sin 12059101205960ℎ44 (1205960)= (1198864312059640 minus (11988641 minus 11988941) 12059620) sin 12059101205960+ (12059650 minus (11988642 + 11988942) 12059630 + (11988640 + 11988940) 1205960) cos 12059101205960
(42)
We assume that (11986742) ℎ41(1205960) times ℎ43(1205960) + ℎ42(1205960) timesℎ44(1205960) = 0 Clearly if condition (11986742) holds then we canconclude that Re[119889120582119889120591]minus1120591=1205910 = 0 Therefore according to theHopf bifurcation theorem in [26] we obtain the following
Theorem 3 If conditions (11986741)-(11986742) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 120591 isin [0 1205910)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) when 120591 = 1205910 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 5 (1205911 gt 0 1205912 isin (0 12059120)) Let 120582 = 11989412059610158401 be the root of (7)Then11989251 (12059610158401) sin 120591112059610158401 + 11989252 (12059610158401) cos 120591112059610158401 = 11989253 (12059610158401) 11989251 (12059610158401) cos 120591112059610158401 minus 11989252 (12059610158401) sin 120591112059610158401 = 11989254 (12059610158401) (43)
where11989251 (12059610158401) = 1198870112059610158401 minus 11988703 (12059610158401)3 + 1198890112059610158401 cos 120591212059610158401minus (11988900 minus 11988902 (12059610158401)2) sin 12059121205961015840111989252 (12059610158401) = 11988700 minus 11988702 (12059610158401)2 + 1198890112059610158401 sin 120591212059610158401+ (11988900 minus 11988902 (12059610158401)2) cos 12059121205961015840111989253 (12059610158401) = 11988602 (12059610158401)2 minus (12059610158401)4 minus 11988600minus (1198880112059610158401 minus 11988803 (12059610158401)3) sin 120591212059610158401minus (11988800 minus 11988802 (12059610158401)2) cos 12059121205961015840111989254 (12059610158401) = 11988603 (12059610158401)3 minus 1198860112059610158401minus (1198880112059610158401 minus 11988803 (12059610158401)3) cos 120591212059610158401+ (11988800 minus 11988802 (12059610158401)2) sin 120591212059610158401
(44)
Thus one can get the following equation with respect to 12059610158401119892251 (12059610158401) + 119892252 (12059610158401) = 119892253 (12059610158401) + 119892254 (12059610158401) (45)
Similar to Case 4 we assume that (11986751) (45) has at leastone positive root 120596101584010 Then for 120596101584010 we have120591101584010 = 1120596101584010sdot arccos 11989251 (120596101584010) times 11989254 (120596101584010) + 11989252 (120596101584010) times 11989253 (120596101584010)119892251 (120596101584010) + 119892252 (120596101584010) (46)
Differentiating (7) with respect to 1205911 we get[119889120582119889120591]minus1 = 11989255 (120582)11989256 (120582) minus 1205911120582 (47)
with
11989255 (120582) = 41205823 + 3119886031205822 + 211988602120582 + 11988601 + (3119887031205822+ 211988702120582 + 11988701) 119890minus1205821205911 + [(311988803 minus 120591211988802) 1205822 minus 1205912119888031205823+ (211988802 minus 120591211988801) 120582 + 11988801 minus 120591211988800] 119890minus1205821205912+ [(211988902 minus 120591211988901) 120582 minus 1205912119889021205822 + 11988901 minus 120591211988900]sdot 119890minus120582(1205911+1205912)11989256 (120582) = (119888031205824 + 119888021205823 + 119888011205822 + 11988800120582) 119890minus1205821205912+ (119889021205823 + 119889011205822 + 11988900120582) 119890minus120582(1205911+1205912)
(48)
Then we obtain
Re [ 1198891205821198891205911 ]minus11205911=120591101584010= ℎ51 (120596101584010) times ℎ53 (120596101584010) + ℎ52 (120596101584010) times ℎ54 (120596101584010)ℎ253 (120596101584010) + ℎ254 (120596101584010) (49)
Advances in Mathematical Physics 7
whereℎ51 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] sin 120591101584010120596101584010+ [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] cos 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] sin 1205912120596101584010+ [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] cos 1205912120596101584010+ 11988601 minus 311988603 (120596101584010)2 ℎ52 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] cos 120591101584010120596101584010minus [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] sin 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] cos 1205912120596101584010minus [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] sin 1205912120596101584010+ 11988602120596101584010 minus 4 (120596101584010)3 ℎ53 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] sin 1205912120596101584010 + [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] cos 1205912120596101584010ℎ54 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] cos 1205912120596101584010 minus [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] sin 12059110158402120596101584010
(50)
We assume that (11986752) ℎ51(120596101584010) times ℎ53(120596101584010) + ℎ52(120596101584010) timesℎ54(120596101584010) = 0 Thus we know that Re[1198891205821198891205911]minus11205911=120591101584010 = 0
if condition (11986752) holds Therefore according to the Hopfbifurcation theorem in [26] we obtain the following
Theorem 4 If conditions (11986751)-(11986752) hold and 1205912 isin (0 12059120)then
(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) islocally asymptotically stable for 1205911 isin [0 120591101584010)
(ii) system (2) undergoes a Hopf bifurcation at viral equi-librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 120591101584010 and a familyof periodic solutions bifurcate from119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
3 Properties of the Hopf Bifurcation
In this section we shall investigate direction of the Hopfbifurcation and stability of the bifurcating periodic solutionof system (2) when 1205911 = 120591101584010 and 1205912 = 120591101584020 isin (0 12059120) by usingthe center manifold theorem and the normal form theorywhich has been developed by Hassard et al [26]
Let 1205911 = 120591101584010 + 120583 1199061 = 119878(1205911119905) 1199062 = 119871(1205911119905) 1199063 = 119860(1205911119905)1199064 = 119877(1205911119905) 120583 isin 119877 Then 120583 = 0 is the Hopf bifurcation valueof system (2) and system (2) can be rewritten as (119905) = 119871120583 (119906119905) + 119865 (120583 119906119905) (51)
where 119906(119905) = (1199061 1199062 1199063 1199064)119879 isin 119862 = 119862([minus1 0] 1198774) and 119871120583119862 rarr 1198774 and 119865 119877 times 119862 rarr 1198774 are given respectively by119871120583120601= (120591101584010 + 120583)(119860lowast120601 (0) + 119862lowast120601(minus120591101584020120591101584010) + 119861lowast120601 (minus1)) (52)
with
119860lowast = [[[[[[1198861 1198862 1198863 11988641198865 1198866 1198867 00 0 1198868 00 0 0 1198869
]]]]]] 119861lowast = [[[[[[
0 0 0 00 1198871 0 00 1198872 0 00 0 0 0]]]]]]
119862lowast = [[[[[[0 0 0 00 0 0 00 0 1198881 00 0 1198882 0
]]]]]] 1198651 = minus12057311206011 (0) 1206012 (0) minus 12057321206011 (0) 1206013 (0) 1198652 = 12057311206011 (0) 1206012 (0) + 12057321206011 (0) 1206013 (0) 120601 = (1206011 1206012 1206013 1206014)119879 isin 119862 ([minus1 0] 1198774)
(53)
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
Advances in Mathematical Physics 5
where Vlowast2 = 120596220 and 1198912(V2) = V42 + 11989233V32 + 11989232V22 + 11989231V2 +11989230 Therefore if condition (11986732) 11989110158402(Vlowast2 ) = 0 holds thenRe[1198891205821198891205912]minus11205912=12059120 = 0 Based on the discussion above andaccording to the Hopf bifurcation theorem in [26] we obtainthe following
Theorem 2 If conditions (11986731)-(11986732) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 1205912 isin [0 12059120)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205912 = 12059120 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 4 (1205911 = 1205912 = 120591 gt 0) For 1205911 = 1205912 = 120591 gt 0 we have1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640+ (119887431205823 + 119887421205822 + 11988741120582 + 11988740) 119890minus120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus2120582120591 = 0 (31)
with 11988640 = 1198860011988641 = 1198860111988642 = 1198860211988643 = 1198860311988740 = 11988700 + 1198880011988741 = 11988701 + 1198880111988742 = 11988702 + 1198880211988743 = 11988703 + 1198880311988940 = 1198890011988941 = 1198890111988942 = 11988902
(32)
Multiplying by 119890120582120591 (31) becomes the following119887431205823 + 119887421205822 + 11988741120582 + 11988740+ (1205824 + 119886431205823 + 119886421205822 + 11988641120582 + 11988640) 119890120582120591+ (119889421205822 + 11988941120582 + 11988940) 119890minus120582120591 = 0 (33)
Let 120582 = 119894120596 (120596 gt 0) be a root of (37) it is easy to get11989241 (120596) cos 120591120596 minus 11989242 (120596) sin 120591120596 = 11989243 (120596) 11989244 (120596) sin 120591120596 + 11989245 (120596) cos 120591120596 = 11989246 (120596) (34)
where 11989241 (120596) = 1205964 minus (11988642 + 11988942) 1205962 + 11988640 + 1198894011989242 (120596) = (11988641 minus 11988941) 120596 minus 11988643120596311989243 (120596) = 119887421205962 minus 1198874011989244 (120596) = 1205964 minus (11988642 minus 11988942) 1205962 + 11988640 minus 1198894011989245 (120596) = (11988641 + 11988941) 120596 minus 11988643120596311989246 (120596) = 119887431205963 minus 11988741120596(35)
It leads to
cos 120591120596 = 11989242 (120596) times 11989246 (120596) + 11989243 (120596) times 11989244 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) sin 120591120596 = 11989241 (120596) times 11989246 (120596) minus 11989243 (120596) times 11989245 (120596)11989241 (120596) times 11989244 (120596) + 11989242 (120596) times 11989245 (120596) (36)
Thus we can get the following equation with respect to 120596cos2120591120596 + sin2120591120596 = 1 (37)
Next we make the following assumption (11986741) (37) hasat least one positive root 1205960 Then for 1205960 we have1205910 = 11205960sdot arccos 11989242 (1205960) times 11989246 (1205960) + 11989243 (1205960) times 11989244 (1205960)11989241 (1205960) times 11989244 (1205960) + 11989242 (1205960) times 11989245 (1205960) (38)
Taking the derivative of 120582 with respect to 120591 we obtain[119889120582119889120591]minus1 = minus11989247 (120582)11989248 (120582) minus 120591120582 (39)
where11989247 (120582) = 3119887431205822 + 211988742120582 + 11988741+ (41205823 + 3119886431205822 + 211988642120582 + 11988641) 119890120582120591+ (211988942120582 + 11988941) 119890minus12058212059111989248 (120582) = (1205825 + 119886431205824 + 119886421205823 + 119886411205822 + 11988640120582) 119890120582120591minus (119889421205823 + 119889411205822 + 11988940120582) 119890minus120582120591(40)
Then we can get that
Re[119889120582119889120591]minus1120591=1205910= minusℎ41 (1205960) times ℎ43 (1205960) + ℎ42 (1205960) times ℎ44 (1205960)ℎ243 (1205960) + ℎ244 (1205960) (41)
6 Advances in Mathematical Physics
whereℎ41 (1205960)= (11988641 + 11988941 minus 31198864312059620) cos 12059101205960minus 2 ((11988642 minus 11988942) 1205960 minus 212059630) sin 12059101205960 + 11988741minus 31198874312059620 ℎ42 (1205960)= (11988641 minus 11988941 minus 31198864312059620) sin 12059101205960+ 2 ((11988642 + 11988942) 1205960 minus 212059630) cos 12059101205960 + 2119887421205960ℎ43 (1205960)= (1198864312059640 minus (11988641 + 11988941) 12059620) cos 12059101205960minus (12059650 minus (11988642 + 11988942) 12059630 + (11988640 minus 11988940) 1205960) sin 12059101205960ℎ44 (1205960)= (1198864312059640 minus (11988641 minus 11988941) 12059620) sin 12059101205960+ (12059650 minus (11988642 + 11988942) 12059630 + (11988640 + 11988940) 1205960) cos 12059101205960
(42)
We assume that (11986742) ℎ41(1205960) times ℎ43(1205960) + ℎ42(1205960) timesℎ44(1205960) = 0 Clearly if condition (11986742) holds then we canconclude that Re[119889120582119889120591]minus1120591=1205910 = 0 Therefore according to theHopf bifurcation theorem in [26] we obtain the following
Theorem 3 If conditions (11986741)-(11986742) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 120591 isin [0 1205910)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) when 120591 = 1205910 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 5 (1205911 gt 0 1205912 isin (0 12059120)) Let 120582 = 11989412059610158401 be the root of (7)Then11989251 (12059610158401) sin 120591112059610158401 + 11989252 (12059610158401) cos 120591112059610158401 = 11989253 (12059610158401) 11989251 (12059610158401) cos 120591112059610158401 minus 11989252 (12059610158401) sin 120591112059610158401 = 11989254 (12059610158401) (43)
where11989251 (12059610158401) = 1198870112059610158401 minus 11988703 (12059610158401)3 + 1198890112059610158401 cos 120591212059610158401minus (11988900 minus 11988902 (12059610158401)2) sin 12059121205961015840111989252 (12059610158401) = 11988700 minus 11988702 (12059610158401)2 + 1198890112059610158401 sin 120591212059610158401+ (11988900 minus 11988902 (12059610158401)2) cos 12059121205961015840111989253 (12059610158401) = 11988602 (12059610158401)2 minus (12059610158401)4 minus 11988600minus (1198880112059610158401 minus 11988803 (12059610158401)3) sin 120591212059610158401minus (11988800 minus 11988802 (12059610158401)2) cos 12059121205961015840111989254 (12059610158401) = 11988603 (12059610158401)3 minus 1198860112059610158401minus (1198880112059610158401 minus 11988803 (12059610158401)3) cos 120591212059610158401+ (11988800 minus 11988802 (12059610158401)2) sin 120591212059610158401
(44)
Thus one can get the following equation with respect to 12059610158401119892251 (12059610158401) + 119892252 (12059610158401) = 119892253 (12059610158401) + 119892254 (12059610158401) (45)
Similar to Case 4 we assume that (11986751) (45) has at leastone positive root 120596101584010 Then for 120596101584010 we have120591101584010 = 1120596101584010sdot arccos 11989251 (120596101584010) times 11989254 (120596101584010) + 11989252 (120596101584010) times 11989253 (120596101584010)119892251 (120596101584010) + 119892252 (120596101584010) (46)
Differentiating (7) with respect to 1205911 we get[119889120582119889120591]minus1 = 11989255 (120582)11989256 (120582) minus 1205911120582 (47)
with
11989255 (120582) = 41205823 + 3119886031205822 + 211988602120582 + 11988601 + (3119887031205822+ 211988702120582 + 11988701) 119890minus1205821205911 + [(311988803 minus 120591211988802) 1205822 minus 1205912119888031205823+ (211988802 minus 120591211988801) 120582 + 11988801 minus 120591211988800] 119890minus1205821205912+ [(211988902 minus 120591211988901) 120582 minus 1205912119889021205822 + 11988901 minus 120591211988900]sdot 119890minus120582(1205911+1205912)11989256 (120582) = (119888031205824 + 119888021205823 + 119888011205822 + 11988800120582) 119890minus1205821205912+ (119889021205823 + 119889011205822 + 11988900120582) 119890minus120582(1205911+1205912)
(48)
Then we obtain
Re [ 1198891205821198891205911 ]minus11205911=120591101584010= ℎ51 (120596101584010) times ℎ53 (120596101584010) + ℎ52 (120596101584010) times ℎ54 (120596101584010)ℎ253 (120596101584010) + ℎ254 (120596101584010) (49)
Advances in Mathematical Physics 7
whereℎ51 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] sin 120591101584010120596101584010+ [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] cos 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] sin 1205912120596101584010+ [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] cos 1205912120596101584010+ 11988601 minus 311988603 (120596101584010)2 ℎ52 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] cos 120591101584010120596101584010minus [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] sin 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] cos 1205912120596101584010minus [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] sin 1205912120596101584010+ 11988602120596101584010 minus 4 (120596101584010)3 ℎ53 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] sin 1205912120596101584010 + [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] cos 1205912120596101584010ℎ54 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] cos 1205912120596101584010 minus [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] sin 12059110158402120596101584010
(50)
We assume that (11986752) ℎ51(120596101584010) times ℎ53(120596101584010) + ℎ52(120596101584010) timesℎ54(120596101584010) = 0 Thus we know that Re[1198891205821198891205911]minus11205911=120591101584010 = 0
if condition (11986752) holds Therefore according to the Hopfbifurcation theorem in [26] we obtain the following
Theorem 4 If conditions (11986751)-(11986752) hold and 1205912 isin (0 12059120)then
(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) islocally asymptotically stable for 1205911 isin [0 120591101584010)
(ii) system (2) undergoes a Hopf bifurcation at viral equi-librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 120591101584010 and a familyof periodic solutions bifurcate from119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
3 Properties of the Hopf Bifurcation
In this section we shall investigate direction of the Hopfbifurcation and stability of the bifurcating periodic solutionof system (2) when 1205911 = 120591101584010 and 1205912 = 120591101584020 isin (0 12059120) by usingthe center manifold theorem and the normal form theorywhich has been developed by Hassard et al [26]
Let 1205911 = 120591101584010 + 120583 1199061 = 119878(1205911119905) 1199062 = 119871(1205911119905) 1199063 = 119860(1205911119905)1199064 = 119877(1205911119905) 120583 isin 119877 Then 120583 = 0 is the Hopf bifurcation valueof system (2) and system (2) can be rewritten as (119905) = 119871120583 (119906119905) + 119865 (120583 119906119905) (51)
where 119906(119905) = (1199061 1199062 1199063 1199064)119879 isin 119862 = 119862([minus1 0] 1198774) and 119871120583119862 rarr 1198774 and 119865 119877 times 119862 rarr 1198774 are given respectively by119871120583120601= (120591101584010 + 120583)(119860lowast120601 (0) + 119862lowast120601(minus120591101584020120591101584010) + 119861lowast120601 (minus1)) (52)
with
119860lowast = [[[[[[1198861 1198862 1198863 11988641198865 1198866 1198867 00 0 1198868 00 0 0 1198869
]]]]]] 119861lowast = [[[[[[
0 0 0 00 1198871 0 00 1198872 0 00 0 0 0]]]]]]
119862lowast = [[[[[[0 0 0 00 0 0 00 0 1198881 00 0 1198882 0
]]]]]] 1198651 = minus12057311206011 (0) 1206012 (0) minus 12057321206011 (0) 1206013 (0) 1198652 = 12057311206011 (0) 1206012 (0) + 12057321206011 (0) 1206013 (0) 120601 = (1206011 1206012 1206013 1206014)119879 isin 119862 ([minus1 0] 1198774)
(53)
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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
6 Advances in Mathematical Physics
whereℎ41 (1205960)= (11988641 + 11988941 minus 31198864312059620) cos 12059101205960minus 2 ((11988642 minus 11988942) 1205960 minus 212059630) sin 12059101205960 + 11988741minus 31198874312059620 ℎ42 (1205960)= (11988641 minus 11988941 minus 31198864312059620) sin 12059101205960+ 2 ((11988642 + 11988942) 1205960 minus 212059630) cos 12059101205960 + 2119887421205960ℎ43 (1205960)= (1198864312059640 minus (11988641 + 11988941) 12059620) cos 12059101205960minus (12059650 minus (11988642 + 11988942) 12059630 + (11988640 minus 11988940) 1205960) sin 12059101205960ℎ44 (1205960)= (1198864312059640 minus (11988641 minus 11988941) 12059620) sin 12059101205960+ (12059650 minus (11988642 + 11988942) 12059630 + (11988640 + 11988940) 1205960) cos 12059101205960
(42)
We assume that (11986742) ℎ41(1205960) times ℎ43(1205960) + ℎ42(1205960) timesℎ44(1205960) = 0 Clearly if condition (11986742) holds then we canconclude that Re[119889120582119889120591]minus1120591=1205910 = 0 Therefore according to theHopf bifurcation theorem in [26] we obtain the following
Theorem 3 If conditions (11986741)-(11986742) hold then(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) is
locally asymptotically stable for 120591 isin [0 1205910)(ii) system (2) undergoes a Hopf bifurcation at viral equi-
librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) when 120591 = 1205910 and a familyof periodic solutions bifurcate from 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
Case 5 (1205911 gt 0 1205912 isin (0 12059120)) Let 120582 = 11989412059610158401 be the root of (7)Then11989251 (12059610158401) sin 120591112059610158401 + 11989252 (12059610158401) cos 120591112059610158401 = 11989253 (12059610158401) 11989251 (12059610158401) cos 120591112059610158401 minus 11989252 (12059610158401) sin 120591112059610158401 = 11989254 (12059610158401) (43)
where11989251 (12059610158401) = 1198870112059610158401 minus 11988703 (12059610158401)3 + 1198890112059610158401 cos 120591212059610158401minus (11988900 minus 11988902 (12059610158401)2) sin 12059121205961015840111989252 (12059610158401) = 11988700 minus 11988702 (12059610158401)2 + 1198890112059610158401 sin 120591212059610158401+ (11988900 minus 11988902 (12059610158401)2) cos 12059121205961015840111989253 (12059610158401) = 11988602 (12059610158401)2 minus (12059610158401)4 minus 11988600minus (1198880112059610158401 minus 11988803 (12059610158401)3) sin 120591212059610158401minus (11988800 minus 11988802 (12059610158401)2) cos 12059121205961015840111989254 (12059610158401) = 11988603 (12059610158401)3 minus 1198860112059610158401minus (1198880112059610158401 minus 11988803 (12059610158401)3) cos 120591212059610158401+ (11988800 minus 11988802 (12059610158401)2) sin 120591212059610158401
(44)
Thus one can get the following equation with respect to 12059610158401119892251 (12059610158401) + 119892252 (12059610158401) = 119892253 (12059610158401) + 119892254 (12059610158401) (45)
Similar to Case 4 we assume that (11986751) (45) has at leastone positive root 120596101584010 Then for 120596101584010 we have120591101584010 = 1120596101584010sdot arccos 11989251 (120596101584010) times 11989254 (120596101584010) + 11989252 (120596101584010) times 11989253 (120596101584010)119892251 (120596101584010) + 119892252 (120596101584010) (46)
Differentiating (7) with respect to 1205911 we get[119889120582119889120591]minus1 = 11989255 (120582)11989256 (120582) minus 1205911120582 (47)
with
11989255 (120582) = 41205823 + 3119886031205822 + 211988602120582 + 11988601 + (3119887031205822+ 211988702120582 + 11988701) 119890minus1205821205911 + [(311988803 minus 120591211988802) 1205822 minus 1205912119888031205823+ (211988802 minus 120591211988801) 120582 + 11988801 minus 120591211988800] 119890minus1205821205912+ [(211988902 minus 120591211988901) 120582 minus 1205912119889021205822 + 11988901 minus 120591211988900]sdot 119890minus120582(1205911+1205912)11989256 (120582) = (119888031205824 + 119888021205823 + 119888011205822 + 11988800120582) 119890minus1205821205912+ (119889021205823 + 119889011205822 + 11988900120582) 119890minus120582(1205911+1205912)
(48)
Then we obtain
Re [ 1198891205821198891205911 ]minus11205911=120591101584010= ℎ51 (120596101584010) times ℎ53 (120596101584010) + ℎ52 (120596101584010) times ℎ54 (120596101584010)ℎ253 (120596101584010) + ℎ254 (120596101584010) (49)
Advances in Mathematical Physics 7
whereℎ51 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] sin 120591101584010120596101584010+ [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] cos 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] sin 1205912120596101584010+ [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] cos 1205912120596101584010+ 11988601 minus 311988603 (120596101584010)2 ℎ52 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] cos 120591101584010120596101584010minus [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] sin 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] cos 1205912120596101584010minus [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] sin 1205912120596101584010+ 11988602120596101584010 minus 4 (120596101584010)3 ℎ53 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] sin 1205912120596101584010 + [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] cos 1205912120596101584010ℎ54 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] cos 1205912120596101584010 minus [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] sin 12059110158402120596101584010
(50)
We assume that (11986752) ℎ51(120596101584010) times ℎ53(120596101584010) + ℎ52(120596101584010) timesℎ54(120596101584010) = 0 Thus we know that Re[1198891205821198891205911]minus11205911=120591101584010 = 0
if condition (11986752) holds Therefore according to the Hopfbifurcation theorem in [26] we obtain the following
Theorem 4 If conditions (11986751)-(11986752) hold and 1205912 isin (0 12059120)then
(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) islocally asymptotically stable for 1205911 isin [0 120591101584010)
(ii) system (2) undergoes a Hopf bifurcation at viral equi-librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 120591101584010 and a familyof periodic solutions bifurcate from119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
3 Properties of the Hopf Bifurcation
In this section we shall investigate direction of the Hopfbifurcation and stability of the bifurcating periodic solutionof system (2) when 1205911 = 120591101584010 and 1205912 = 120591101584020 isin (0 12059120) by usingthe center manifold theorem and the normal form theorywhich has been developed by Hassard et al [26]
Let 1205911 = 120591101584010 + 120583 1199061 = 119878(1205911119905) 1199062 = 119871(1205911119905) 1199063 = 119860(1205911119905)1199064 = 119877(1205911119905) 120583 isin 119877 Then 120583 = 0 is the Hopf bifurcation valueof system (2) and system (2) can be rewritten as (119905) = 119871120583 (119906119905) + 119865 (120583 119906119905) (51)
where 119906(119905) = (1199061 1199062 1199063 1199064)119879 isin 119862 = 119862([minus1 0] 1198774) and 119871120583119862 rarr 1198774 and 119865 119877 times 119862 rarr 1198774 are given respectively by119871120583120601= (120591101584010 + 120583)(119860lowast120601 (0) + 119862lowast120601(minus120591101584020120591101584010) + 119861lowast120601 (minus1)) (52)
with
119860lowast = [[[[[[1198861 1198862 1198863 11988641198865 1198866 1198867 00 0 1198868 00 0 0 1198869
]]]]]] 119861lowast = [[[[[[
0 0 0 00 1198871 0 00 1198872 0 00 0 0 0]]]]]]
119862lowast = [[[[[[0 0 0 00 0 0 00 0 1198881 00 0 1198882 0
]]]]]] 1198651 = minus12057311206011 (0) 1206012 (0) minus 12057321206011 (0) 1206013 (0) 1198652 = 12057311206011 (0) 1206012 (0) + 12057321206011 (0) 1206013 (0) 120601 = (1206011 1206012 1206013 1206014)119879 isin 119862 ([minus1 0] 1198774)
(53)
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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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
Advances in Mathematical Physics 7
whereℎ51 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] sin 120591101584010120596101584010+ [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] cos 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] sin 1205912120596101584010+ [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] cos 1205912120596101584010+ 11988601 minus 311988603 (120596101584010)2 ℎ52 (120596101584010) = [211988702120596101584010 + (211988902 minus 120591211988901) cos 1205912120596101584010minus (1205912 (120596101584010)2 + 11988901 minus 120591211988900) sin 1205912120596101584010] cos 120591101584010120596101584010minus [11988701 minus 311988703 (120596101584010)2 + (211988902 minus 120591211988901) sin 1205912120596101584010+ (1205912 (120596101584010)2 + 11988901 minus 120591211988900) cos 1205912120596101584010] sin 120591101584010120596101584010+ [(211988802 minus 120591211988801) 120596101584010 + 120591211988803 (120596101584010)3] cos 1205912120596101584010minus [11988801 minus 120591211988800 minus (311988803 minus 120591211988802) (120596101584010)2] sin 1205912120596101584010+ 11988602120596101584010 minus 4 (120596101584010)3 ℎ53 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] sin 1205912120596101584010 + [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] cos 1205912120596101584010ℎ54 (120596101584010) = [11988800120596101584010 minus 11988802 (120596101584010)3+ (11988900120596101584010 minus 11988902 (120596101584010)3) cos 120591101584010120596101584010+ 11988901 (120596101584010)2 sin 120591101584010120596101584010] cos 1205912120596101584010 minus [11988803 (120596101584010)4minus 11988801 (120596101584010)2 + (11988900120596101584010 minus 11988902 (120596101584010)3) sin 120591101584010120596101584010minus 11988901 (120596101584010)2 cos 120591101584010120596101584010] sin 12059110158402120596101584010
(50)
We assume that (11986752) ℎ51(120596101584010) times ℎ53(120596101584010) + ℎ52(120596101584010) timesℎ54(120596101584010) = 0 Thus we know that Re[1198891205821198891205911]minus11205911=120591101584010 = 0
if condition (11986752) holds Therefore according to the Hopfbifurcation theorem in [26] we obtain the following
Theorem 4 If conditions (11986751)-(11986752) hold and 1205912 isin (0 12059120)then
(i) viral equilibrium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast) of system (2) islocally asymptotically stable for 1205911 isin [0 120591101584010)
(ii) system (2) undergoes a Hopf bifurcation at viral equi-librium 119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)when 1205911 = 120591101584010 and a familyof periodic solutions bifurcate from119864lowast(119878lowast 119871lowast 119860lowast 119877lowast)
3 Properties of the Hopf Bifurcation
In this section we shall investigate direction of the Hopfbifurcation and stability of the bifurcating periodic solutionof system (2) when 1205911 = 120591101584010 and 1205912 = 120591101584020 isin (0 12059120) by usingthe center manifold theorem and the normal form theorywhich has been developed by Hassard et al [26]
Let 1205911 = 120591101584010 + 120583 1199061 = 119878(1205911119905) 1199062 = 119871(1205911119905) 1199063 = 119860(1205911119905)1199064 = 119877(1205911119905) 120583 isin 119877 Then 120583 = 0 is the Hopf bifurcation valueof system (2) and system (2) can be rewritten as (119905) = 119871120583 (119906119905) + 119865 (120583 119906119905) (51)
where 119906(119905) = (1199061 1199062 1199063 1199064)119879 isin 119862 = 119862([minus1 0] 1198774) and 119871120583119862 rarr 1198774 and 119865 119877 times 119862 rarr 1198774 are given respectively by119871120583120601= (120591101584010 + 120583)(119860lowast120601 (0) + 119862lowast120601(minus120591101584020120591101584010) + 119861lowast120601 (minus1)) (52)
with
119860lowast = [[[[[[1198861 1198862 1198863 11988641198865 1198866 1198867 00 0 1198868 00 0 0 1198869
]]]]]] 119861lowast = [[[[[[
0 0 0 00 1198871 0 00 1198872 0 00 0 0 0]]]]]]
119862lowast = [[[[[[0 0 0 00 0 0 00 0 1198881 00 0 1198882 0
]]]]]] 1198651 = minus12057311206011 (0) 1206012 (0) minus 12057321206011 (0) 1206013 (0) 1198652 = 12057311206011 (0) 1206012 (0) + 12057321206011 (0) 1206013 (0) 120601 = (1206011 1206012 1206013 1206014)119879 isin 119862 ([minus1 0] 1198774)
(53)
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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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
8 Advances in Mathematical Physics
Using Riesz representation theorem there exists matrix120578(120579 120583) [minus1 0] rarr 1198774 such that119871120583120601 = int0minus1119889120578 (120579 120583) 120601 (120579) 120601 isin 119862 ([minus1 0] 1198774) (54)
In fact choose120578 (120579 120583)=
(120591101584010 + 120583) (119860lowast + 119861lowast + 119862lowast) 120579 = 0(120591101584010 + 120583) (1198611015840 + 119862lowast) 120579 isin [minus120591101584020120591101584010 0) (120591101584010 + 120583) 119861lowast 120579 isin (minus1 minus120591101584020120591101584010) 0 120579 = minus1(55)
For 120601 isin 119862([minus1 0] 1198774) define119860 (120583) 120601 =
119889120601 (120579)119889120579 minus1 le 120579 lt 0int0minus1119889120578 (120579 120583) 120601 (120579) 120579 = 0
119877 (120583) 120601 = 0 minus1 le 120579 lt 0119865 (120583 120601) 120579 = 0(56)
Then system (51) can be rewritten in the following form (119905) = 119860 (120583) 119906119905 + 119877 (120583) 119906119905 (57)
For 120601 isin 119862([minus1 0] 1198774) 120593 isin 1198621([minus1 0] (1198774)lowast)119860lowast (120593) =
minus119889120593 (119904)119889119904 0 lt 119904 le 1int0minus1119889120578119879 (119904 0) 120593 (minus119904) 119904 = 0 (58)
and bilinear form⟨120593 120601⟩ = 120593 (0) 120601 (0)minus int0120579=minus1
int120579120585=0
120593 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585 (59)
are defined where 120578(120579) = 120578(120579 0) 119860 = 119860(0) and 119860lowast areadjoint operators
Based on the discussion above one can conclude thatplusmn119894120596101584010120591101584010 are common eigenvalues of 119860(0) and 119860lowast Theeigenvectors of 119860(0) and 119860lowast can be calculated correspond-ing to +119894120596101584010120591101584010 and minus119894120596101584010120591101584010 respectively Let 119902(120579) =(1 1199022 1199023 1199024)119879119890119894120596101584010120591101584010120579 be the eigenvector of 119860(0) correspond-ing to +119894120596101584010120591101584010 and 119902lowast(120579) = 119870(1 119902lowast2 119902lowast3 119902lowast4 )119890119894120596101584010120591101584010119904 be
the eigenvector of 119860lowast corresponding to minus119894120596101584010120591101584010 By somecomplex computations we obtain
1199022 = 119886511990220 1199023 = 1198872119890minus1198941205911015840101205961015840101199022119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1199024 = 1198882119890minus1198941205911015840201205961015840101199023119894120596101584010 minus 1198869 119902lowast2 = minus119894120596101584010 + 11988611198865 119902lowast3 = 1198862 + (119894120596101584010 + 1198866 + 1198871119890119894120591101584010120596101584010) 119902lowast21198872119890119894120591101584010120596101584010 119902lowast4 = minus1198863 + 1198867119902lowast2 + (119894120596101584010 + 1198868 + 1198881119890119894120591101584020120596101584010) 119902lowast31198882119890119894120591101584020120596101584010
(60)
with
11990220 = 119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 minus 11988671198872119890minus119894120591101584010120596101584010119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 (61)
From (59) we get
119870 = [1 + 1199022119902lowast2 + 1199023119902lowast3 + 1199024119902lowast4+ 120591101584010119890minus1198941205961015840101205911015840101199022 (1198871119902lowast2 + 1198872119902lowast3 )+ 120591101584020119890minus1198941205961015840101205911015840101199023 (1198881119902lowast3 + 1198882119902lowast4 )]minus1 (62)
such that ⟨119902lowast 119902⟩ = 1 In what follows we can obtain thecoefficients by using the method introduced in [26]
11989220 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989211 = 120591101584010119870 (119902lowast2 minus 1) (1205731 Re 1199022 + 1205732 Re 1199023) 11989202 = 2120591101584010119870 (119902lowast2 minus 1) (12057311199022 + 12057321199023) 11989221 = 2120591101584010119870 (119902lowast2 minus 1) (1205731 (119882(1)11 (0) 1199022 + 12119882(1)20 (0) 1199022+119882(2)11 (0) + 12119882(2)20 (0)) + 1205732 (119882(1)11 (0) 1199023+ 12119882(1)20 (0) 1199023 +119882(3)11 (0) + 12119882(3)20 (0))) (63)
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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
Advances in Mathematical Physics 9
with
11988220 (120579) = 11989411989220119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989202119902 (0)3120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198641119890211989412059110158401012059610158401012057911988211 (120579) = minus 11989411989211119902 (0)120591101584010120596101584010 119890119894120591101584010120596101584010120579 + 11989411989211119902 (0)120591101584010120596101584010 119890minus119894120591101584010120596101584010120579+ 1198642(64)
where
1198641 = 2((1198861lowast minus1198862 minus1198863 minus1198864minus1198865 1198866lowast 1198867 00 minus1198872119890minus119894120591101584010120596101584010 1198868lowast 00 0 minus1198882119890minus119894120591101584020120596101584010 1198869lowast
))minus1
sdot(minus12057311199022 minus 1205732119902312057311199022 + 1205732119902300 )
1198642 = minus(1198861 1198862 1198863 11988641198865 1198866 + 1198871 1198867 00 1198872 1198868 + 1198881 00 0 1198882 1198869)minus1
sdot(minus1205731 Re 1199022 minus 1205732 Re 11990231205731 Re 1199022 + 1205732 Re 119902300 )
(65)
with
1198861lowast = 2119894120596101584010 minus 11988611198866lowast = 2119894120596101584010 minus 1198866 minus 1198871119890minus119894120591101584010120596101584010 1198868lowast = 2119894120596101584010 minus 1198868 minus 1198881119890minus119894120591101584020120596101584010 1198869lowast = 2119894120596101584010 minus 1198869(66)
Thus we can compute the following coefficients1198621 (0) = 1198942120596101584010120591101584010 (1198921111989220 minus 2 10038161003816100381610038161198921110038161003816100381610038162 minus 100381610038161003816100381611989202100381610038161003816100381623 )+ 119892212 1205832 = minus Re 1198621 (0)Re 1205821015840 (120591101584010) 1205732 = 2Re 1198621 (0) 1198792 = minus Im 1198621 (0) + 1205832 Im 1205821015840 (120591101584010)120596101584010120591101584010
(67)
In conclusion we have the following results in this section
Theorem 5 For system (2)
(i) the direction of the Hopf bifurcation is determined by1205832 if 1205832 gt 0 the Hopf bifurcation is supercritical if1205832 lt 0 the Hopf bifurcation is subcritical(ii) the stability of the bifurcating periodic solutions is
determined by 1205732 if 1205732 lt 0 the bifurcating periodicsolutions are stable if 1205732 gt 0 the bifurcating periodicsolutions are unstable
(iii) the period of the bifurcating periodic solution is deter-mined by 1198792 if 1198792 gt 0 the period of the bifurcatingperiodic solutions increases if 1198792 lt 0 the period of thebifurcating periodic solutions decreases
4 Numerical Simulations
In this section we present some numerical simulation resultsof system (2) to illustrate our theoretical results We choose aset of parameters as follows 120583 = 0001 1205731 = 01 1205732 = 015120572 = 005 120576 = 005 and 120574 = 002 Then we get the followingsystem119889119878 (119905)119889119905 = 0001 minus 01119878 (119905) 119871 (119905) minus 015119878 (119905) 119860 (119905)+ 005119877 (119905) minus 0001119878 (119905) 119889119871 (119905)119889119905 = 01119878 (119905) 119871 (119905) + 015119878 (119905) 119860 (119905)minus 005119871 (119905 minus 1205911) minus 0001119871 (119905) 119889119860 (119905)119889119905 = 005119871 (119905 minus 1205911) minus 002119860 (119905 minus 1205912)minus 0001119860 (119905) 119889119877 (119905)119889119905 = 002119860 (119905 minus 1205912) minus 005119877 (119905) minus 0001119877 (119905)
(68)
It is easy to verify that (120576 + 120583)(120574 + 120583)(1205731(120574 + 120583) +1205732120576) lt 1 and (120576 + 120583)(120574 + 120583)120576 gt 120572120574(120572 + 120583) which ensuresthe fact that system (68) has a unique viral equilibrium119864lowast(01116 02073 04936 01936)
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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
10 Advances in Mathematical Physics
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 1 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2236 lt284522 = 12059110
005 01 015 02 025 03 035
0204
0608
S(t)
A(t)
00102030405
L(t)
Figure 2 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2236 lt284522 = 12059110For 1205911 = 1205912 = 0 by direct computation by Matlab 70 we
can get 11988610 gt 0 11988613 gt 0 and 1198861311988612 gt 11988611 which means thatviral equilibrium 119864lowast(01116 02073 04936 01936) is locallyasymptotically stable
For 1205911 gt 0 1205912 = 0 we obtain that (16) has a uniquepositive root V10 = 00013 and (14) has a unique positiveroot 12059610 = 00363 Further we get the critical value of delay12059110 = 284522 and 11989110158401(Vlowast1 ) = 00612 gt 0 Thus we cansee that the conditions in Theorem 1 hold true It followsthat viral equilibrium 119864lowast(01116 02073 04936 01936) islocally asymptotically stable for 1205911 isin [0 284522) andsystem (68) undergoes aHopf bifurcation at viral equilibrium119864lowast(01116 02073 04936 01936) when 1205911 = 284522 Thisproperty can be depicted in Figures 1ndash4 Similarly we can alsoobtain 12059620 = 02756 12059120 = 959606 for 1205911 = 0 1205912 gt 0 and1205960 = 07114 1205910 = 242508 for 1205911 = 1205912 = 120591 gt 0 respectivelyThe corresponding phase plots are depicted in Figures 5ndash8and Figures 9ndash12 respectively
For 1205911 gt 0 and 1205912 = 6525 isin (0 12059120) we obtain120596101584010 = 00089 120591101584010 = 186255 The simulation results canbe seen in Figures 13ndash16 In addition we obtain 1198621(0) =minus02107 + 00062119894 1205821015840(120591101584010) = 00031 minus 01086119894 by somecomplex computations Further we have 1205832 = 679677 gt 01205732 = minus04214 lt 0 1198792 = 444907 gt 0 Therefore wecan conclude that the Hopf bifurcation is supercritical thebifurcating periodic solutions are stable and the period of thebifurcating periodic solutions increases
005 01 015 02 025 03 035
0204
0608
S(t)A(t)
01
02
03
04
R(t)
Figure 3 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 2949 gt284522 = 12059110
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
0
02
04
06
L(t)
Figure 4 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 2949 gt284522 = 12059110
0 0102
0304
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 5 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205912 = 5848 lt959606 = 12059120
0 01 02 0304
0204
0608
S(t)A(t)
010150202503035
L(t)
Figure 6 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205912 = 5848 lt959606 = 12059120
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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
Advances in Mathematical Physics 11
0 0102 03
0408
S(t)
06A(t)
0402
0
0
01
02
03
04
R(t)
Figure 7The phase plot of states 119878lowast119860lowast and119877lowast with 1205912 = 11096 gt959606 = 12059120
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
L(t)
Figure 8The phase plot of states 119878lowast 119871lowast and119860lowast with 1205912 = 11096 gt959606 = 12059120
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
01
02
03
04
R(t)
Figure 9 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2135 lt242508 = 1205910
005 01 015 02 025 03 03502
0406
08
S(t)
A(t)
00102030405
L(t)
Figure 10 The phase plot of states 119878lowast 119871lowast and 119860lowast with 120591 = 2135 lt242508 = 1205910
0 01 0203
04
002
0406
08
S(t)A(t)
01
02
03
04
R(t)
Figure 11 The phase plot of states 119878lowast 119860lowast and 119877lowast with 120591 = 2547 gt242508 = 1205910
0 0102
03 04
00204
0608
S(t)A(t)
L(t)
0
02
04
06
Figure 12 The phase plot of the states 119878lowast 119871lowast and 119860lowast with 120591 =2547 gt 242508 = 1205910
0 01 0203
04
0204
0608
S(t)
A(t)
01
02
03
04
R(t)
Figure 13 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
0 01 02 0304
0204
0608
S(t)
A(t)
01
02
03
04
L(t)
Figure 14 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 = 865 lt186255 = 120591101584010 and 1205912 = 6525
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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
12 Advances in Mathematical Physics
0 0102
0304
005
1
S(t)A(t)
01
02
03
04
R(t)
Figure 15 The phase plot of states 119878lowast 119860lowast and 119877lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 6525
0 0 01005 015 025035
02 0304
020406
08
S(t)A(t)
minus010
010203040506
L(t)
1
Figure 16 The phase plot of states 119878lowast 119871lowast and 119860lowast with 1205911 =224057 gt 186255 = 120591101584010 and 1205912 = 65255 Conclusions
In the present paper an improved model for propagationof computer virus propagation model in the network isintroduced and studied by incorporating the delay due to thelatent period of the computer viruses and the delay due to theperiod that the antivirus software needs to clean the virusesin the active computers into the model proposed in [19] Wemainly investigate effect of the two delays on the model
By choosing different combination of the two delays asa bifurcation parameter it has been found that both thetwo delays can change the stability of the viral equilibriumof the model under some conditions When the value ofthe delay is below corresponding critical value the modelis locally asymptotically stable which indicates that the lawof propagation of the computer viruses in system (2) canbe predicted However when the value of the delay isabove the corresponding critical value a Hopf bifurcationoccurs and a family of periodic solutions bifurcate fromthe viral equilibrium which suggests that the percentagesof susceptible latent active and recovered computers insystem (2) will fluctuate periodically in a range This is nothelpful in predicting the law of propagation of the computerviruses Therefore we should control the occurrence of theHopf bifurcation by using some bifurcation control strategiesand we leave this as our near future work Furthermorethe properties of the Hopf bifurcation when 1205911 gt 0 and1205912 isin (0 12059120) have been investigated in detail Finally
some numerical simulations are also included to support thetheoretical results obtained in the paper
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
Theresearchwas supported byNatural Science Foundation ofAnhui Province (no 1608085QF151 and no 1608085QF145)
References
[1] Y Ozturk and M Gulsu ldquoNumerical solution of a modifiedepidemiological model for computer virusesrdquo Applied Mathe-matical Modelling vol 39 no 23-24 pp 7600ndash7610 2015
[2] F Cohen ldquoComputer viruses theory and experimentsrdquo Com-puters and Security vol 6 no 1 pp 22ndash35 1987
[3] J O Kephart and S R White ldquoDirected-graph epidemiologicalmodels of computer virusesrdquo in Proceedings of the IEEE Com-puter Society Symposium on Research in Security and Privacypp 343ndash358 May 1991
[4] J O Kephart and S R White ldquoMeasuring and modeling com-puter virus prevalencerdquo in Proceedings of the IEEE ComputerSociety Symposium on Research in Security and Privacy (SP rsquo93)pp 2ndash15 Oakland Calif USA 1993
[5] L Chen K Hattaf and J T Sun ldquoOptimal control of a delayedSLBS computer virus modelrdquo Physica A vol 427 pp 244ndash2502015
[6] B K Mishra and N Jha ldquoFixed period of temporary immu-nity after run of anti-malicious software on computer nodesrdquoAppliedMathematics and Computation vol 190 no 2 pp 1207ndash1212 2007
[7] J R C Piqueira and V O Araujo ldquoA modified epidemiologicalmodel for computer virusesrdquoApplied Mathematics and Compu-tation vol 213 no 2 pp 355ndash360 2009
[8] B K Mishra and S K Pandey ldquoFuzzy epidemic model forthe transmission of worms in computer networkrdquo NonlinearAnalysis Real World Applications vol 11 no 5 pp 4335ndash43412010
[9] J G Ren X F 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
[10] Z Zhang and H Yang ldquoDynamical analysis of a viral infectionmodel with delays in computer networksrdquo Mathematical Prob-lems in Engineering vol 2015 Article ID 280856 15 pages 2015
[11] L P Feng X F Liao H Q Li and Q Han ldquoHopf bifurcationanalysis of a delayed viral infection model in computer net-worksrdquoMathematical and Computer Modelling vol 56 no 7-8pp 167ndash179 2012
[12] H Yuan andG Chen ldquoNetwork virus-epidemicmodel with thepoint-to-group information propagationrdquoAppliedMathematicsand Computation vol 206 no 1 pp 357ndash367 2008
[13] T Dong X F Liao andH Q Li ldquoStability andHopf bifurcationin a computer virus model with multistate antivirusrdquo AbstractandAppliedAnalysis vol 2012 Article ID 841987 16 pages 2012
[14] F Wang Y Zhang C Wang J Ma and S Moon ldquoStabilityanalysis of a SEIQV epidemic model for rapid spreading
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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
Advances in Mathematical Physics 13
wormsrdquo Computers and Security vol 29 no 4 pp 410ndash4182010
[15] B KMishra andN Jha ldquoSEIQRS model for the transmission ofmalicious objects in computer networkrdquo Applied MathematicalModelling vol 34 no 3 pp 710ndash715 2010
[16] Y Yao X-W Xie H Guo G Yu F-X Gao and X-J TongldquoHopf bifurcation in an Internet worm propagation modelwith time delay in quarantinerdquo Mathematical and ComputerModelling vol 57 no 11-12 pp 2635ndash2646 2013
[17] Y Muroya H X Li and T Kuniya ldquoOn global stability of anonresident computer virusmodelrdquoActaMathematica Scientiavol 34 no 5 pp 1427ndash1445 2014
[18] C Q Gan X F Yang and Q Y Zhu ldquoPropagation of computervirus under the influences of infected external computers andremovable storage media modeling and analysisrdquo NonlinearDynamics vol 78 no 2 pp 1349ndash1356 2014
[19] M B Yang X F Yang Q Y Zhu and L Yang ldquoA four-compartment computer virus propagation model with gradedinfection raterdquo Journal of Chongqing University vol 35 no 12pp 112ndash119 2012 (Chinese)
[20] C Bianca M Ferrara and L Guerrini ldquoQualitative analysis ofa retarded mathematical framework with applications to livingsystemsrdquo Abstract and Applied Analysis vol 2013 Article ID736058 7 pages 2013
[21] L Gori L Guerrini and M Sodini ldquoHopf bifurcation in acobweb model with discrete time delaysrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 137090 8 pages 2014
[22] C Bianca M Ferrara and L Guerrini ldquoThe Cai model withtime delay existence of periodic solutions and asymptoticanalysisrdquoAppliedMathematicsamp Information Sciences vol 7 no1 pp 21ndash27 2013
[23] C Xu and Q Zhang ldquoQualitative analysis for a Lotka-Volterramodel with time delaysrdquoWSEAS Transactions on Mathematicsvol 13 pp 603ndash614 2014
[24] C Bianca M Ferrara and L Guerrini ldquoHopf bifurcations in adelayed-energy-based model of capital accumulationrdquo AppliedMathematics amp Information Sciences vol 7 no 1 pp 139ndash1432013
[25] X Li and J Wei ldquoOn the zeros of a fourth degree exponentialpolynomial with applications to a neural network model withdelaysrdquo Chaos Solitons and Fractals vol 26 no 2 pp 519ndash5262005
[26] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge Uk 1981
Submit your manuscripts athttpswwwhindawicom
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 athttpswwwhindawicom
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