Research Article An Impulse Dynamic Model for Computer Worms

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1 Asrac and Applied Analysis Volume 23, Aricle ID 28629, 8 pages hp://dx.doi.org/.55/23/28629 Research Aricle An Impulse Dynamic Model for Compuer Worms Chunming Zhang, Yun Zhao, and Yingjiang Wu School of Informaion Engineering, Guangdong Medical College, Dongguan 52388, China Correspondence should e addressed o Chunming Zhang; [email protected] Received 3 May 23; Acceped 2 June 23 Academic Edior: Luca Guerrini Copyrigh 23 Chunming Zhang e al. This is an open access aricle disriued under he Creaive Commons Ariuion License, which permis unresriced use, disriuion, and reproducion in any medium, provided he original work is properly cied. A worm spread model concerning impulsive conrol sraegy is proposed and analyzed. We prove ha here exiss a gloally aracive virus-free periodic soluion when he vaccinaion rae is larger han θ. Moreover, we show ha he sysem is uniformly persisen if he vaccinaion rae is less han θ. Some numerical simulaions are also given o illusrae our main resuls.. Inroducion Compuervirusisakindofcompuerprogramhacan replicae iself and spread from one compuer o ohers including viruses, worms, and rojan horses. Worms use sysem vulnerailiy o search and aack compuers. As hardware and sofware echnologies develop and compuer neworks ecome an essenial ool for daily life, worms sar o e a major hrea. In June 2, he Belarusian securiy firm Virus Block Ada discovered deadly Suxne worm. The Suxne worm is he firs known example of a cyer-weapon ha is designed no jus o seal and manipulae daa u o aack a processing sysem and cause physical damage. The Suxne worm is he firs cyer-aack of is kind and has infeced housands of compuer sysems worldwide. Consequenly, he rial on eer undersanding he worm propagaion dynamics is an imporan maer for improving he safey and reliailiy in compuer sysems and neworks. Similaroheiologicalviruses,herearewowaysosudy his prolem: microscopic and macroscopic. Following a macroscopic approach, since [, 2] ook he firs sep owards modelinghespreadehaviorofworms,muchefforhas een done in he area of developing a mahemaical model for he worms propagaion [3 3]. These models provide a reasonale qualiaive undersanding of he condiions under which viruses spread much faser han ohers and why. In [7], he auhors invesigaed a differenial SEIR model y making he following assumpions (Figure ). A populaion size N(), ha is, he oal nodes a any ime in he compuer nework, is pariioned ino suclasses of nodes which are suscepile, exposed (infeced u no ye infecious), infecious, and recovered wih sizes denoed y S(), E(), I(),andR(),respecively. One has N () =S() +E() +I() +R(), S () = λis pe qi ds+ζr, E () =λis+pe+qi εe de, I () =εe γi di ηi, R () =γi ζr dr, where, d, andλ are posiive consans and ε, η, γ, ζ are nonnegaive consans. The consan is he recruimen rae of suscepile nodes o he compuer nework, d is he per capia naural moraliy rae (i.e., he crashing of nodes due o he reason oher han he aack of worms), ε is he rae consan for nodes leaving he exposed class E for infecive class I, γ is he rae consan for nodes leaving he infecive class I for recovered class R, η is he disease relaed deah rae (i.e., crashing of nodes due o he aack of worms) in he class I, and ζ is he rae consan for nodes ecoming suscepile again afer recovering. In he SEIRS model, he flow is from class S o class E, class E o class I, classi o class R, andagainclassr o class S. For he verical ransformaion, we assume ha a fracion p and a fracion q of he new nodes from he exposed and he infecious classes, respecively, are inroduced ino ()

2 2 Asrac and Applied Analysis ζr θs λis εe γi S E I R qi+ ds pe de di dr Figure : Original model. ζr λis εe γi S E I R qi+ ds pe de di dr Figure 2: Impulse model. he exposed class E. Consequenly, he irh flux ino he exposed class is given y pe + qi, andheirhfluxino he suscepile class is given y pe qi. As we know, anivirus sofware is a kind of compuer program which can deec and eliminae known worm. There are wo common mehods o deec worms: using a lis of worm signaure definiion and using a heurisic algorihm o find worm ased on common ehaviors. I has een oserved ha i does no always work in deecing a novel worm y using he heurisic algorihm. On he oher hand, oviously, i is impossile for anivirus sofware o find a new worm signaure definiion on he daed lis. So, o keep he anivirus sofware in high efficiency, i is imporan o ensure ha i is updaed. Based on he previous facs, we propose an impulsive sysem o model he process of periodic insalling or updaing anivirus sofware on suscepile compuers a fixed ime for conrolling he spread of worm. Based on he previous facs, we propose he following assumpions: (H)heanivirussofwareisinsalledorupdaedaime =kτ(k N),whereτ is he period of he impulsive effec; (H2) S compuers are successfully vaccinaed from S class o R class wih rae θ(<θ<). According o he previous assumpions (H)-(H2) and for he reason of simpliciy, we propose he following model (Figure 2): S () = λis pe qi ds+ζr, E () =λis+pe+qi εe de, =kτ, k Z +, I () =εe γi di ηi, R () =γi ζr dr, S( + )=( θ) S (), E( + )=E(), I( + =kτ, k Z +. )=I(), R( + )=R() +θs(), (2) The oal populaion size N() can e deermined y N() = S() + E() + I() + R() o form he differenial equaion N () = dn() ηi(), (3) which is derived y adding he equaions in sysem (). Thus he oal populaion size N may vary in ime. From (2), we have (d+η) N () N () dn(). (4) I follows ha (d + η) lim inf N () lim sup N () x x d. (5) The sysem (2) can e reduced o he equivalen sysem S () = λis pe qi ds +ζ(n S E I), E () =λis+pe+qi εe de, I () =εe γi di ηi, N () = dn() ηi(), S( + )=( θ) S (), E( + )=E(), I( + )=I(), N( + )=N(). The iniial condiions for (6)are =kτ, k Z +, =kτ, k Z +, S( + )>, E( + )>, I( + )>, N( + )>. From physical consideraions, we discuss sysem (6) in he closed se Ω={(S, E, I, N) R 4 + S+E+I d, N d }. (8) The organizaion of his paper is as follows. In Secion 2, we esalish sufficien condiion for he local and gloal araciviy of virus-free periodic soluion. The sufficien condiion for he permanence of he model is oained in Secion3. Some numerical simulaions are performed in Secion 4. In he final secion, a rief conclusion is given, and some fuure research direcions are also poined ou. (6) (7)

3 Asrac and Applied Analysis 3 2. Gloal Araciviy of Virus-Free Periodic Soluion To prove our main resuls, we sae hree lemmas which will e essenial o our proofs. Lemma (see [4]). Consider he following impulsive differenial equaions: u () =a u(), =kτ, u( + )=( θ) u (), = kτ, where a>, >,and<θ<. Then sysem (9) has a unique posiive periodic soluion u e () = a +(u a )e ( kτ), kτ < (k+) τ, (9) () which is gloally asympoically sale; here u = a( θ)( e τ )/( ( θ)e τ ). If I(), we have he following limi sysems: S () = λis pe qi ds+ζ(n S E), E () =pe εe de, N () = dn, S( + )=( θ) S (), E( + )=E(), N( + )=N(). =kτ, k Z +, =kτ, k Z +, () When p d ε <, here exiss when >,lim E() =. From he hird and sixh equaions of sysem (), we have lim N() = /d. We have he following limi sysems: ds d =( p)+v μ (μ+v)s, =kt, k Z+, S( + )=( θ) S (), = kt. (2) According o Lemma, we know ha periodic soluion of sysem (2)isofheform S e () = (μ ( p) + V) μ(μ+v) +(S (μ ( p) + V) )e (μ+v)( kτ), μ(μ+v) kτ < (k+) τ, (3) and i is gloally asympoically sale, where S = (( p)+ (V/μ))( θ)(e (μ+v)τ )/(μ + V)(e (μ+v)τ +θ). Theorem 2. Le (S(), E(), I(), N()) e any soluion of sysem (6) wih iniial values S( + )>, E( + )>, I( + )>, and N( + )>;hen(s e (),,, /d) is locally asympoically sale, provided ha p d ε<and R <,where R = (d+ε p)(γ+d+η)τ εqτ ελ d [τ + θ( e (ζ+d)τ ) (ζ+d) (e (ζ+d)τ +θ) ]. (4) Proof. The local sailiy of virus-free periodic soluion may e deermined y considering he ehaviors of a small ampliude peruraion of he soluion. Define w() = S() S e (), x() = E(), y() = I(),andz() = N() /d,andhen he linearized sysem of sysem (6)reads as w () = (d+ζ) w () (p+ζ)x() (λs e () +ζ+q)y() +ζz(), x () =(p d ε)x() +(q+λs e ())y(), y () =εx() (r+d+η)y(), z () = ηy() dz(), w( + )=w(), x( + )=x(), y( + )=y(), z( + )=z(). =kτ, k Z +, =kτ,k Z +, (5) Le Φ() e he fundamenal soluion marix of sysem (5), and hen Φ() mus saisfy dφ () d d ζ p ζ λs e () ζ q ζ =( p d ε q+λs e () )Φ() ε r d η η d =AΦ(), (6) and Φ() = I, he ideniy marix. We can easily see ha wo eigenvalues of he marix A are d ζ and d, andhe oher wo eigenvalues are deermined y he 2 2marixB= ( p d ε q+λs e() ε r d η ). Denoe he eigenvalues of B as λ, λ 2,and hen as p d ε <,wehaveλ +λ 2 = p d ε r d η <, λ λ 2 =(p d ε)( r d η) ε(q+λs e ()). (7)

4 4 Asrac and Applied Analysis Therefore, y he Floque heorem [5], (S e (),,, /d) is locally asympoically sale, provided ha τ G= [(p d ε)( γ d η) ε(q+λs e ())] d = (p d ε)( γ d η)τ εqτ ελ d [τ + θ( e (ζ+d)τ ) (ζ+d) (e (ζ+d)τ +θ) ]>. (8) When R = (/((d+ε p)(γ+d+η)τ εqτ))ελ(/d)[τ+θ( e (ζ+d)τ )/(ζ + d)(e (ζ+d)τ +θ)], he previous inequaliy is saisfied for p d ε<. The proof is complee. Theorem 3. If R,andp d ε < hen (S e (),,, /d) is gloally asympoically sale for sysem (),where R = (d+ε p)(γ+d+η)τ εqτ ελτ d [ θ (e (ζ+d)τ +θ) ]. Proof. Because e x >x,forx>,wehave (9) R <R. (2) By Theorem 2,weknowha(S e (),,, /d) is locally asympoically sale. In he following, we will prove he gloal aracion of (S e (),,, /d). Le Then L=εE (p ε d)i. (2) Corollary 4. The virus-free periodic soluion (S e (),,, /d) of sysem (6) is gloally aracive, if θ>θ,whereθ = e (ζ+d)τ + ελ(e (ζ+d)τ )/d[(d + ε p)(r + d + η) εq]. Theorem 2 deermines he gloal araciviy of (6) in Ω for he case R <. Is realisic implicaion is ha he infeced compuers vanish, so he worms are removed from he nework. Corollary 4 implies ha he compuer virus will disappear if he vaccinaion rae is less han θ. 3. Permanence In his secion, we say ha he worm is local if he infeced populaion persiss aove a cerain posiive level for sufficienlylargeime.thelocalofwormcanewellcapuredand sudied hrough he noion of permanence. Definiion 5. Sysem (6) is said o e uniformly persisen if here is an φ>(independen of he iniial daa) such ha every soluion (S(), I(), R(), N()) wih iniial condiions (8) ofsysem(6) saisfies lim inf S () φ, lim lim inf R () φ, lim inf I () φ, inf N () φ. (24) Theorem 6. Suppose ha R >and p d ε <.Then here is a posiive consan m I such ha each posiive soluion (S(), E(), I(), R()) of sysem (6) saisfies I() m I,for large enough. Proof. Now, we will prove ha here exis m I > and a sufficienly large p such ha I() m I holds for all > p. Suppose ha I() < I for all >.Fromheforhequaion of (6), we have L =εe (p ε d)i, L =[ελs+εq+(p+ε d)(r+d+η)]i. (22) Therefore, (S e (),,, /d) is gloally asympoically sale, provided ha L τ = [ελs+εq+(p+ε d)(r+d+η)]id<, L τ = [ελs+εq+(p+ε d)(r+d+η)]d < [εq + (p + ε d) (r + d + η)] τ +ελ d ( θ e (d+ζ)τ +θ )τ<. (23) dn() >N () > dn() ηi, d ηi >N() >. d From he second equaion of (6), we have As p d ε<, E () = λis + pe + qi εe de <λi S+qI +(p ε d)e <λi d +qi +(p ε d)e. (25) (26) When R = (/((d + ε p)(γ + d + η)τ εqτ))ελ(τ/d)[ θ/(e (ζ+d)τ +θ)], he previous inequaliy is saisfied. The proof is complee. E () < (q + λ) I d(ε+d p). (27)

5 Asrac and Applied Analysis 5 From he firs equaion of (6), we have S () = λis pe qi ds+ζ(n S E I) >+ζ ηi d (p+ζ) qi ζi (q + λ) I d(ε+d p) (λi +d+ζ)s(). Consider he following comparison sysem: V () =+ζ ηi d (p+ζ) qi ζi (28) (q + λ) I d(ε+d p) (λi +d+ζ)v (), V ( + )=( θ) V (), V ( + )=S( + ). = kτ, =kτ, (29) le A = +ζ(( ηi )/d) qi ζi (p+ζ)((q+λ)i /d(ε+ d p)), B =λi +d+ζ. By Lemma, we know ha here exiss > such ha V () = A B +(V A B )e B( kτ), V = A ( θ) ( e Bτ ) B( ( θ) e Bτ ), S () > V () kτ < (k+) τ, =S >, for >. From he second equaion of (6), we have E () = λis + pe + qi εe de >λi S +qi (ε+d p)e, E () > λi S +qi ε+d p. From he hird equaion of (6); we have I () =εe γi di ηi >ε λi S +qi (γ + d + η) I. ε+d p Noe ha R >,wehave (3) (3) (32) ε λi S +qi I () > >I. (33) γ+d+η ε+d p This conradics I() I. Hence, we can claim ha for any >,iisimpossileha I () <I. (34) By he claim, we are lef o consider wo cases. Firs, I() I for large enough. Second, I() oscillaes aou I for large enough. Oviously, here is nohing o prove for he firs case. For he second case, we can choose 2 >,andξ>saisfy I ( 2 ) =I( 2 +ξ) =I, for 2 << 2 +ξ. (35) I() is uniformly coninuous since he posiive soluions of (6) are ulimaely ounded, and I() is no affeced y impulses. Therefore, i is cerain ha here exiss a η ( <η<τ,and η is independen of he choice of 2 )suchha I () I 2, for 2 2 +η. (36) In his case, we consider he following hree possile cases in erm of he sizes of η, ξ,andτ. Case. Ifξ η<τ,heniisovioushai() I /2, for [ 2, 2 +ξ]. Case 2.Ifη<ξ τ, hen from he second equaion of sysem (6), we oain I() > (d + γ + η)i(). Since I( 2 )=I,iisovioushaI() > I e (d+γ+η)τ =I, for [ 2, 2 +ξ]. Case 3. Ifη<τ ξ,iiseasyooainhai() > I for [ 2, 2 +τ]. Then, proceeding exacly as he proof for he previous claim, we have ha I() > I for [ 2 +τ, 2 +ξ]. Owing o he randomiciy of 2, we can oain ha here exiss m I = min{i /2, I } such I() > m I holds for all > p. The proof of Theorem 6 is compleed. Theorem 7. Suppose R >. Then sysem (6) is permanen. Proof. Le (S(), E(), I(), N()) e any soluion of sysem (6). Firs, from he firs equaion of sysem (6), we have S () > p 2 q 2 (d+λ) S. (37) Consider he following comparison sysem: z () = p 2 q 2 (d+λ) z (), =kτ, z ( + )=( θ) z (), = kτ. (38) By Lemma, weknowhaforanysufficienlysmallε>, here exiss a ( is sufficienly large) such ha S () z () >z () ε ( p q)(e(d+λ)τ ) (d+λ) (e (d+λ)τ +θ) From (3), we have E () > λm Im S +qm I ε+d p ε=m S >, kτ < (k+) τ. (39) ε=m E. (4)

6 6 Asrac and Applied Analysis 8.5 The populaion of suscepile compuers The populaion of exposed compuers.5 2 (a) () 9 The populaion of infeced compuers The populaion of recovered compuers (c) (d) Figure 3: Gloal araciviy of virus-free periodic soluion of sysem. From he hird equaion of (6), we have I is easy o see ha N () > μn() αn(). (4) N () > μ+α ε=m N. (42) We le Ω = {(S, I, N) R 3 + m S S, m I I, m N N /μ, S + I /μ}.bytheorem 6 and he previous discussions, we know ha he se Ω is a gloal aracor in Ω,andofcourse, every soluion of sysem (6) wih iniial condiions (8) will evenually ener and remain in region Ω. Therefore, sysem (6)ispermanen. The proof of Theorem 7 is compleed. Corollary 8. I follows from Theorem 6 ha he sysem (6) is uniformly persisen, provided ha θ < θ,whereθ = e (ζ+d)τ + ελ(e (ζ+d)τ )/d[(d+ε p)(r+d+η) εq]. 4. Numerical Simulaions In his secion we have performed some numerical simulaions o show he geomeric impression of our resuls. To demonsrae he gloal araciviy of virus-free periodic soluion of sysem (6), we ake following se parameer values: = 2., λ =.2, p =., q =.5, d =.3, ζ =.6, ε =.4, γ=.6, τ=.5, η =.3, andθ =.4. Inhis case, we have R =.6886 <.InFigures3(a), 3(), 3(c),and 3(d),wehavedisplayed,respecively,hesuscepile,exposed, infeced and recovered populaion of sysem (6) wihiniial condiions: S() = 6, E() =, I() =,andr() = 8. To demonsrae he permanence of sysem (6), we ake he following se parameer values: =2.7, λ =.2, p =., q =.5, d =.3, ζ =.6, ε =.4, γ =.6, τ=.5, η=.3 and θ =..Inhiscase,wehaveR =.6496 >.InFigures 4(a), 4(), 4(c), and 4(d), we have displayed, respecively, he suscepile, exposed, infeced, and recovered populaions of sysem (6)wihiniialcondiions:S() = 6, E() =, I() = and R() = 8.

7 Asrac and Applied Analysis 7 The populaion of suscepile compuers (a) The populaion of exposed compuers () 8 The populaion of infeced compuers The populaion of recovered compuers (c) (d) Figure 4: Permanence of sysem. 5. Conclusion We have analyzed he SEIRS model wih pulse vaccinaion and varying oal populaion size. We have shown ha R > or θ<θ implies ha he worm will e local, whereas R < or θ>θ implies ha he worm will fade ou. We have also esalished sufficien condiion for he permanence of he model. Our resuls indicae ha a large pulse vaccinaion rae will lead o eradicaion of he worm. References [] J. O. Kephar and S. R. Whie, Direced-graph epidemiological models of compuer viruses, in Proceedings of IEEE Symposium on Securiy and Privacy, pp , 99. [2] J.O.Kephar,S.R.Whie,andD.M.Chess, Compuersand epidemiology, IEEE Specrum,vol.3,no.5,pp.2 26,993. [3] L. Billings, W. M. Spears, and I. B. Schwarz, A unified predicion of compuer virus spread in conneced neworks, Physics Leers A,vol.297,no.3-4,pp ,22. [4] X. Han and Q. Tan, Dynamical ehavior of compuer virus on Inerne, Applied Mahemaics and Compuaion, vol. 27, no. 6, pp , 2. [5] B. K. Mishra and N. Jha, Fixed period of emporary immuniy afer run of ani-malicious sofware on compuer nodes, Applied Mahemaics and Compuaion,vol.9,no.2,pp.27 22, 27. [6] B. K. Mishra and D. Saini, Mahemaical models on compuer viruses, Applied Mahemaics and Compuaion,vol.87,no.2, pp , 27. [7] B. K. Mishra and S. K. Pandey, Dynamic model of worms wih verical ransmission in compuer nework, Applied Mahemaics and Compuaion,vol.27,no.2,pp ,2. [8] J. R. C. Piqueira and V. O. Araujo, A modified epidemiological model for compuer viruses, Applied Mahemaics and Compuaion,vol.23,no.2,pp ,29. [9] J. R. C. Piqueira, A. A. Vasconcelos, C. E. C. J. Gariel, and V. O. Araujo, Dynamic models for compuer viruses, Compuers & Securiy, vol. 27, no. 7-8, pp , 28.

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