A Novel Computer Virus Model and Its Stability

Transcription

1 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 67 A ovel Computer Virus Moel an Its Stability Mei eng an Huajian Mou College of Mathematics an Computer Science, Yangtze ormal University, Chongqing, China pmgs@qqcom, Mouhuajian@6com Abstract Computer virus is a malicious coe which can causes great amage A SEIR moel is propose which consiere that the newly entere computer in the network has been infecte by other ways It is concerne with the constant immigration, which inclues susceptible, expose an infectious, the threshol an equilibrium of this moel are investigate The locally an globally asymptotical stable results of virus-free equilibrium an viral equilibrium were being prove Finally, some numerical examples are given to emonstrate the analytical results Inex Terms Computer Virus; Equilibrium; Locally Asymptotically; Simulation I ITRODUCTIO Computer virus is a malicious coe which incluing virus, Trojan horses, worm, logic bomb an so on It is a program that can copy itself an attack other computers An they are resiing by erasing ata, amage files, or moifying the normal operation In recent years, some moels of the computer virus is propose [-, 5-] The action of computer virus throughout a network can be stuie by using epiemiological moels for isease propagation [4] Base on the Kermack an Mckenrick SIR classical epiemic moel, ynamical moels for malicious objects propagation were propose [5-] These ynamical moeling of the sprea process of computer virus is an effective approach to the unerstaning of behavior of computer viruses because on this basis, some effective measures can be pose to prevent infection The computer virus has latent ability An infecte computer which is in latency, calle expose computer Base on these characteristics, elay is use in some moels to escribe the latent ability of computer virus [7-9] An some moels propose SEIR or SLBS to escribe the latent of the computer virus [5-6, -] They are all assuming that the computer which is newly entere in the network are all infectious In this paper, a novel moel of computer virus, known as SEIR moel, is put forwar to escribe before enter in the network, the computer has been infecte computer virus by other ways such as removable meia etc S( t), E( t), I( t), R( t ) enote the classes of S, E, I, R at time t The total number of new infectious at a time is given by SI, with as the mass action coefficient an is use to incience In this paper, we assume that a fraction a an b of the computers which newly entere in the network flux into the expose class E an the infecte class I The computers which have been infecte computer virus flux into expose class E is given by a an flux into infectious class I is given by b The susceptible computer which flux into the susceptible class S is given by ( a b) If ab, the moel has only one viral equilibrium an it is globally asymptotically stable; If ab, the moel has only one virus-free equilibrium if R, an if R the system has another viral equilibrium an they are globally asymptotically stable; If ab, the moel has only one viral equilibrium an it is globally asymptotically In this paper, the ynamics of this moel has been fully stuie through some numerical examples This paper is organize as follows Section formulates a novel computer virus moel Section, investigate the stability of the equilibriums Section 4 some numerical examples are given to present the effectiveness of the theoretic results Finally, Section 5 summarizes this work II MODEL FORMULATIOS At any time, a computer is classifie as internal an external epening on weather it is connecte to internet or not At that time all to the internet computers are further categorize into four classes: () Susceptible computers, that is, uninfecte computers an new computers which connecte to network; () Expose computers, that is, infecte but not yet broken-out; () Infectious computers; (4) Recovere computers, that is, virus-free computer having immunity Let S( t), E( t), I( t), R( t ) enote their corresponing numbers at time t, without ambiguity, S( t), E( t), I( t), R( t ) will be abbreviate as S, E, I, R, respectively Our assumption on the ynamical transfer of the computers is epicte in Fig, an the moel is formulate as following system of ifferential equations: S ( a b) SI S, E a SI ( r ) E, I b re ( ) I, R I R () ( t) S( t) E( t) I( t) R( t) () We may see that first three equations in () are inepenent of the fourth equation, an therefore, the 4 ACADEMY UBLISHER oi:44/jnw967-74

2 68 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 fourth equation can be omitte without loss of generality Hence, system () can be rewritten as S ( a b) SI S, E a SI ( r ) E, I b re ( ) I Figure Schematic iagram for the flow of virus in computer network where enotes the rate at which external computers are connecte to the network; a enotes the rate of the new computer which has been infecte computer virus but hasn t been broken; b enotes the rate of the new computer which has been infecte computer virus an it is has been broken; enotes the rate at which, when having a connection to one infecte computer, one susceptible computer can be become expose; r enotes the rate of which, the expose computer is broken an come into the infectious state; enotes the rate recovery rate of infecte computers are cure; enotes the rate at which one computer is remove from the network All the parameters are nonnegative When having connection to one expose computer, one susceptible computer can be become expose, an its infectious rate is SI ; a is the number of the new enter the network has been infecte but hasn t broken; b is the number of the new enter the network has been infecte; Moreover, all feasible solutions of the system () are boune an enter the region D, where D {( S, E, I) R S, E, I, S E I } Refer to [], we analysis the threshol R of the system () Let T x ( E, I, S), then, the system () can be written as () SI x Fi( x) Vi( x), where Fi ( x), ( r ) E a Vi ( x) re ( ) I b The Jacobin matrices of Fx ( ) an V( x ) at the ( a b) (, a, b) are F DF( ), r V, r then V, ( )( r ) r r Thus Sr R ( FV ) ( r )( ) r( a b) ( )( r ) Especially, when ab, obviously ab, the system () can be revise as An there r R ( )( r ) S SI S, E SI ( r ) E, I re ( ) I (4) For the system (4), if R, there always exists the virus-free equilibrium is R (,,) (,,) (,,) ; if S, there also exists an virus-free equilibrium is an a viral equilibrium is ( r )( ) ( ) (, I, I ), r r there r ( r )( ) I r( r )( ) When ab, the system () can be revise as S SI S, E a SI ( r ) E, I b re ( ) I (5) For the system (5), there has only one viral equilibrium a b( r ) ar (,, ) ( r ) ( r )( ) When ab, the system () has only a viral equilibrium a SI [ ar br b] ( S,, ) ( r ) ( )( r ) rs Let G( S ) ( a b) S I S, (6) 4 ACADEMY UBLISHER

3 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 69 ( a b) when S (, ), iscuss the roots of equation GS ( ) For system (), we have G( S ) I [ ar br b] ( )( r ) rs [ ar br b] rs, [( )( r ) rs ] ( a b) we get GS ( ) monotone ecreasing in (, ) r( a b) Case (): If R, from (6), we get ( )( r ) G() ( a b), ( a b) G( ) S I, there [ ar br b] I, ( )( r ) rs ( a b) G( ) > ( a b) GS ( ) has no real root in (, ) R r( a b) Case (): If R, ( )( r ) I [ ar br b] ( )( r ) rs G() ( a b),, ( a b) G( ) For that GS ( ) monotone ecreasing in ( a b) (, ), the equation GS ( ) has only one positive root if R From the case, the system () has only one viral a SI [ ar br b] equilibrium ( S,, ) ( r ) ( )( r ) rs if R III Theorem if R Whereas STABILITY OF THE EQUILIBRIUMS is locally asymptotically stale is unstable if R roof The Jacobin matrix of system (4) about given by if is S J ( r ) S, r ( ) which equals to f ( ) ( )( a a S r), (7) where a ( r ) ( ), a ( r )( ) then, Eq (7) has negative real parts roots, a a 4 a ( R ), a a 4 a ( R ), when R,, an When R, Then there are no positive real roots of (7), is a locally asymptotically stable equilibrium if R, an is unstable if R The proof is complete Theorem is globally asymptotically stale with respect to D if R r roof Let L, ( ) obviously L, r L ( ) r [ SI ( r ) E] re ( ) I ( r ) r ( r )( ) ( r ) ( )( R ) The proof is complete Theorem is globally asymptotically stable roof Let L S, obviously L, then L SI S The proof is complete Theorem 4 The viral equilibrium ( S, E, I ) an ( S, E, I ) is locally asymptotically stable roof Denote ( S, E, I ) an ( S, E, I ) as The Jacobin matrix of the system () about by is given 4 ACADEMY UBLISHER

4 7 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 I S J I ( r ) S, r ( ) which equals is where A, f ( ) A A A A, (8) A {( I ) [( r ) ( )]}, A [( r ) ( )]( I ) [( r )( ) S r], ( )( )( ) A r I S r Case (): When ab an R, A, A {( I ) [( r ) ( )]}, A ( I )[( r ) ( )], A ( r )( ) I, then A A A Accoring to the Hurwitz criterion, all roots of Eq (8) have negative real parts Thus the equilibrium is locally asymptotically stable Case (): When aban R, A, A {( I ) [( r ) ( )]}, A [( r ) ( )]( I ) [( r )( ) S ], A ( r )( ) I [( r )( ) S r], then A A A Accoring to the Hurwitz criterion, all roots of Eq (8) have negative real parts Thus the equilibrium is locally asymptotically stable The proof is complete ow, let us examine the global stability of with respect to the D by means of a classical geometric approach [] For our propose, the following obvious result will be useful Lemma If R, the system () is uniformly persistent, ie, there exists c (inepenent of initial conitions), such that liminf S( t) c, liminf E( t) c, liminf I( t) c t t t Remark We have prove that the virus-free equilibrium is unstable if R when ab an the system () has no virus-free equilibrium when ab Furthermore, the instability of, together with D ( D enotes the bounary of D ), imply the uniform persistence of the satiate variable The uniform persistence of system () is to show the existence of a compact absorbing set in int D (int D enotes the interior of D ), Which is a necessary conition for proving the global stability by using the geometric approach Theorem 4 is globally asymptotically stable if R roof The secon compoun matrix of the Jacobin matrix J can be calculate as follows [4-6]: ( I ) ( r ) S S ( ) ( ) I ( r ) ( ) [] J r I Set as the following iagonal matrix E E x ( ) (,, ) I I Denote f iag(,, ), [] Therefore, the matrix B AJ A can be written in the following block form: B B B, B B with B ( I ) ( r ), I B S (,), E E T B ( r,), I E I ( I ) ( r ) B, E I I ( r ) ( ) g ( B ) B, g ( B ) B, there ( B ) ( I ) ( r ), E I E I ( r ) ( )} E I = ( r ) ( B ) max{ ( I ) ( r ) I, The vector norm in R ( u, v, W) {max u, v w} f R is chose as (9) The Lozinskii measure ( B) with respecte to as follows (7) ( B) sup{ g, g}, there I g I r S E ( ) ( ) (,), E I E T g ( ) ( r,) I () 4 ACADEMY UBLISHER

5 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 7 Figure Dynamical behavior of system (4) with ab Time series of S( t), E( t), I( t ) with R Figure Dynamical behavior of system (4) with ab Time series of S( t), E( t), I( t ) with R Figure 4 Dynamical behavior of system (5) with ab Time series of S( t), E( t), I( t ) 4 ACADEMY UBLISHER

6 7 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 Figure 5 Dynamical behavior of system (5) with ab, Time series of S( t), E( t), I( t ) if a, b,,, r 8,, n 8 an then R Figure 6 Dynamical behavior of system (5) with ab, Time series of S( t), E( t), I( t ) if a, b,,, r 8,, n 8 an then R 9975 Figure 7 Dynamical behavior of system (5) with ab, Time series of S( t), E( t), I( t ) if a, b,,, r 6,, n 6 an then R 89 4 ACADEMY UBLISHER

7 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 7 From the system (), we fin that Thus Relations (9)-() imply E ( B), E I E a S ( r ), E E E () I b E r ( ) I I I E a g E E I E b g ( ), t ( ) ( ) t ( E B ) ln E t t t E t E() () () If R, then the virus-frees equilibrium is unstable by the Theorem, Moreover, the behavior of the local near D as escribe in Theorem implies that the system () is uniformly persistent in D, ie there exists a constant c an T, such that t T implies liminf S( t) c, t liminf E( t) c, t liminf I( t) c, t An liminf[ S( t) E( t) I( t)] c t For all ( ( S(), E(), I()) D [8, 9] t q limsupsup ( B) t x Kt (4) The claime result follows by combining Theorem, Lemma an the negativity of q The proof is complete IV SIMULATIO ow, we woul analysis the locally asymptotically stable of the equilibriums through some numerical examples Case : For the system (4), if R, there exists the virus-free equilibrium Let a, b,,, r 8,, n 8, an then R, Fig shows the solution of system (4) if R Case : For the system (4), if R, there exists the viral equilibrium Let a, b,,, r 5,, n 5, an then R 99 Fig shows the solution of system (4) if R Case : For the system (5), there exists the viral equilibrium, Let a 5, b 5,, r 5,, n, Fig 4 shows the solution of system (5) if ab Case 4: For the system (), there exists the viral equilibrium Fig 5-Fig 7 shows the solution of system () if ab From the Fig, We can see that the virus-free equilibrium of system (4) is locally asymptotically stable if R From the Fig, We can see that the viral equilibrium of the system (4) is locally asymptotically stable R From the Fig 4, We can see that the viral equilibrium of the system (5) is locally asymptotically stable From the Fig 5-Fig 7, We can see that the viral equilibrium of the system () is locally asymptotically stable For the system (), from the Fig 5 an Fig 6, we can see that the number of computer virus is not affecte by the parameters a an b If we re keeping a an b the same, from the Fig 6 an Fig 7, we can see that the number of computer virus actually ecreases as we reuce the R V COCLUSIO We consiering the fact, the computers which is newly entere in the network computer has been infecte by other ways such as removable meia, a novel computer virus moel is propose Denotes a is the rate of the new computer which has been infecte computer virus but hasn t been broken; b enotes the rate of the new computer which has been infecte computer virus an it is has been broken If ab, the system () has only one viral equilibrium an it is locally asymptotically stable; If ab, the system () can rewritten as system (4) an it has only one virus-free equilibrium if R, an if R the system has a viral equilibrium an which are locally asymptotically stable; If ab, the System () can be rewritten as system (5) an it has only one viral equilibrium which is locally asymptotically In this paper, the ynamics of this moel has been fully stuie through some numerical examples REFERECES [] Wei Yang, Gui-RA Cheng, Yu Yao, Xiao-meng Shen, Stability Analysis of Worm ropagation Moel with Dynamic Quarantine Defence, Journal of networks, Vol 6, o (), pp 5-6, Jan [] Juan Du, Su-li Chen, Research of Improve etwork Malicious Information Filtering moel Base on SVM, Journal of etworks, Vol 8, o 5 (), pp -8, May 4 ACADEMY UBLISHER

8 74 JOURAL OF ETWORKS, VOL 9, O, FEBRUARY 4 [] Hui Xu, Xiang Gu, Extensive Design for Attack s Recognition an Resistance of Survivable etwork Journal of etworks, Vol 7, o (), -8, Feb [4] C Sun, YH Hsieh Global analysis of an SEIR moel with varying population size an vaccination In: Applie Mathematical Moelling 4 () pp [5] L Song, Z Jin, Gq Sun, Moelling an analysing of botnet interactions In: hysical A 9 () pp [6] BK Mishra, SK aney Dynamic moel of worms with vertical transmission in computer network In: Applie Mathematics an Computation 7 () pp [7] JG Ren, XF Yang, LX Yang, YH Xu, FZ Yang, A elaye computer virus propagation moel an its ynamics, Chaos, Solutions& Fractals 45 () pp [8] X Han, Q Tan, Dynamical behavior of computer virus on Internet, Applie Mathematics an Computation xxx () xxx-xxx [9] Q Zhu, X Yang, J Ren, Moeling an analysis of the sprea of computer virus, Communications onlinear Science umerical Simulation 7 () pp [] LX Yang, X Yang, Q Zhu, L Wen A computer virus moel with grae cure rats, onlinear Analysis: Real Worl Applications [] LX Yang, X Yang, L W, J L, A novel computer virus propagation moel an its ynamics, International Journal of Computer Mathematic, Vol, o, Month x, - 8 [] Van en Driessche, J watmough, Reprouction numbers an sub-threshol enemic equilibrium for compartmental moels of isease transmission, Mathematical Biosciences 8 () pp 9-48 [] Michael Y Li an James S Mulowney, A Geometric Approach to Global Stability roblems, SIAM J Math Anal, Vol 7, o 4, pp 7-8, July 996 [4] M Fieler, Aitive compoun matrices an inequality for eigenvalues of stochastic matrices, Czechoslovak Math J 99 (974) pp 9-4 [5] J S Mulowney, Compoun matrices an orinary ifferential equations, Rocky Mountain J Math (99) pp [6] Martin Jr R H, Logarithmic norms an projections applie to linear ifferential system, J Math Anal App, 974(45) pp [7] G J Butler, Waltman, ersistence in ynamical systems, roc Amer Math, Soc 96 (986) pp 45-4 [8] H I Freeman, M X Tang, S G Ruan, Uniform persistence an flows near a close positive invariant set, Journal of Dynamics an Differential Equations 6 (994) pp 58-6 [9] Waltman, A brief survey of persistence in ynamical systems, In: S Busenberg, M Martelli (Es), Delay Differential Equations an Dynamical Systems, Springer- Verlag, ew York, 99, pp -4 Mei eng is a lecturer in College of Mathematics an Computer Science, Yangtze ormal University, Chongqing, China She receive her master egree in stochastic system analysis from the Chongqing ormal University, China Her interest in the area of the computer virus propagation moel Huajian Mou is a assistant in College of Mathematics an Computer Science, Yangtze ormal University, Chongqing, China He interest in the area of the computer Software rogramming 4 ACADEMY UBLISHER