Fractional Dynamics of Computer Virus Propagation

Transcription

1 Science Journal of Applied Matematics and Statistics 5; (): 6-69 Publised online April, 5 (ttp://wwwsciencepublisinggroupcom/j/sjams) doi: 648/jsjams5 ISSN: (Print); ISSN: (Online) Fractional Dynamics of Computer Virus Propagation Bonya Ebenezer, *, Nyabadza Farai, Asiedu-Addo Samuel Kwesi Department of Matematics and Statistics, Kumasi Polytecnic, Kumasi, Gana Department of Matematical Science, University of Stellenbosc, Matieland, Sout Africa Department of Matematics Education, University of Education, Winneba Gana address: ebbonya@yaoocom (Bonya E) To cite tis article: Bonya Ebenezer, Nyabadza Farai, Asiedu-Addo Samuel Kwesi Fractional Dynamics of Computer Virus Propagation Science Journal of Applied Matematics and Statistics Vol, No, 5, pp 6-69 doi: 648/jsjams5 Abstract: Tis paper studies te fractional order model for computer virus in SEI model Firstly, te basic reproduction number, wic determines te tresold of te spread of te virus is determined Te stability of equilibra was also determined and studied Te Adams-Basfort-Moulton algoritm was employed to solve and simulate te system of differential equations Te results of te simulation depicts tat by small cange in led to big cange in te associated numerical results Keywords: Nonlinear System, Fractional Calculus, Computer Virus Model Introduction Wile te information tecnology revolution as significantly improve business activities and made living comparatively easy to manage troug services suc as managing ban account, travelling arrangement, and buying items online, it as also come wit a ig cost operations and manipulations troug propagation of computer virus Tis computer virus does not only propagates and leads to uge losses in terms of money to companies and customers, but is also implicated for loss of important data It is estimated tat annually, millions dollars are lost by te virtue of various infection [] In early 98(s) te idea of matematical models for te study of computer virus spread became pronounced Since 988 Epidemic models for computer viruses ave been studied Murray [] seems to be te first to propose te relationsip between epidemiology and computer viruses, altoug e did not provide any specific models Viruses were once propagated by excanging dis; now; global connectivity gives malicious code to propagate a farter and faster Bad use of computer troug networ invasion is on te increase Today, over 74 different strains of computer viruses ave been observed since in 986 te first virus was identified (Symantec Security esponse, ) Te ave been several callenges on cyber world due to cyber attac leading to a great defense to protect valuable information from certain malicious agents (Trojan orse, worms, virus) Te spread of tese dangerous agents is similar to tat of spread of endemic in biological processes Many studies ave employed biological system to understand te dynamics of spread of malicious objects in a computer networ and prescribe procedures for protecting te computer system [,] Te activity of malicious objects in entire networ can be examined by applying epidemiological models for disease spread [, 5, ] icard et al 5 design an SEI (Susceptible-Exposed- Infected) model to study te propagation of computer virus Tey owever, did not consider te lengt of latency period and consider te effect of anti-virus software Yen and Liu, 6 also proposed SEI tat assumes tat recovery osts ave a permanent immunization period wit a certain probability, wic is not consistent wit real life situation By obtaining solution to tis obstacle, [, ], also propose a SEIS model wit latent and temporary immune periods, wic can identify common worm propagation Garretto et al [9] present a model tat sees to examine te propagation of virus and worms in distinct networ topologies Zou et al [9] propose an internet worm monitoring system tat cecs a worm in its initial stage of propagation employing Kalman filter Zu et al [] apply optimal control metod to study te dynamics of computer virus Tey tae into consideration a controlled delayed model and ten use an optimal control tecnique, by maing an assumption tat tere is a trade off between central loss and te effect Carla and Teneriro also present

2 64 Bonya Ebenezer et al: Fractional Dynamics of Computer Virus Propagation fractional dynamics of computer virus propagation to study [] model In tis paper we analyze te fractional order version of te integer order model proposed by Mei et al [] for computer virus and its dynamics We simulate numerically te model for different values of te order of te fractional derivatives In tis regard, te paper is arranged as follows In Section, we describe te model proposed for computer virus dynamics In Section 4, we analyze several simulations of te model for different values of te fractional derivatives explain te implication of te results In te last Section, we sow te main conclusion and outline same future researc figures Fractional Order Calculus (FOC) Te teory of differential calculus began wen Leibniz / wrote about D f ( x ) for generalization of te derivative operator D f ( x) to fractional values of, te order of te derivative Te growt of te fractional calculus (FC) is attributable to many contributions of matematicians for example Euler, Liouville, iemann, and Letniov [7] Presently fractional calculus as been associated wit long memory in te fields suc as pysics and engineering [8, 5] Notwitstanding te FC applications in engineering tere are ongoing effort to explore new areas of applications of FC suc as te modeling of dynamical [9] We present some of te definitions of fractional calculus Te most common applied definitions of a fractional derivative of order are te iemann-liouville (L), Gr unwald-letniov (GL), and Caputo (C) formulations GL is stated as GL a [ t a/ ] Dt f ( t) lim ( ) f ( t ), t a,, > > () Definition Te Caputo fractional derivative of order ( n, n) of a continuous function f : is given by ( ) n d D f x I D f ( x), D () were Γ () is Euler s gamma function, [] means te integer part of, and is te step time increment Tese expressions contain te istory of te past dynamics, divergent to te integer counterpart tat is a local operator Tis property was observed in numerous penomena and teir modeling turns easier employing te FC formalism, wile integer order models are often loos more complicated We observe tat te definition of time-fractional derivative of a function f ( t ) at t tn deals wit integration and computing time-fractional derivative tat demands Te concept of fractional derivative, we will implement Caputo s definition wic is a variation of te iemann- Liouville definition and as te advantage of solving initial value problems Model Formulation By considering connection of computers, it is classified as external if connected to internet and external not connected We subdivide te population into four classes S( t) represents te susceptible computers, tat is, uninfected computers and new computers wic connected to networ at time t, E( t ) represents te exposed computers, tat is, infected but not yet broen-out, I( t ) denotes te infectious computers and ( t) te recovered computers, tat is, virusfree computer aving immunity Let (), (), (), () denote teir corresponding numbers at time, witout ambiguity; (), (), (), () will be abbreviated as,,,, respectively Te model is formulated as te following system of differential equations: Now we introduce fractional order into te ODE model by Mei et al [] Te new system is described by te following set of ODE ds de di d ( q) N βsi β SE ps µ S, βsi βse E σ E µ E, σ E di µ I, ps E di, () dn ds de di d () It can be obviously noticed tat te first tree equations in () are independent of te fourt equation, and ence, te fourt equation can be done away wit witout loss of generality Terefore, system () can be stated as ds de di ( p) N βsi β SE ( p µ ) S βsi βse ( σ µ ) E, σ E ( d µ ) I () were N represents te rate at wic external computers are connected to te networ; p represents te recovery rate of susceptible computer as a results of te anti-virus ability of networ; stands for te recovery rate of exposed computer due to te anti-virus ability of networ; β denotes te rate at wic, wen aving a connection to one infected computer, one susceptible computer can turn into exposed but as not broen-out; β indicates te rate of wic, wen aving connection to one exposed computer, one susceptible computer can turn into exposed; σ symbolizes te rate of te exposed computers cannot be cured by anti-virus software and broen-out; d represents te recovery rate of

3 Science Journal of Applied Matematics and Statistics 5; (): infected computers tat are cured; µ indicates te rate at wic one computer is removed from te networ were <, N S E I,( S, E, I ) Te reason for considering a fractional order system instead of its integer order counterpart is tat fractional order differential equations are generalizations of integer order differential equations Also, using fractional order differential equations can elp us to reduce te errors arising from te neglected parameters in modeling real life penomena We sould note tat te system () can be reduced to an integer order system by setting Adding up te equations given in (), we obtain Let { X : X } and X ( t) ( S( t), E( t), I ( t))t For te proof of te teorem about non-negative solutions we sall require te following Lemma [] Lemma (Generalized Mean Value Teorem) Let f ( x) C a, b and D f ( x) C( a, b] for < Ten we state [ ] f ( x) f ( a) D f ( ξ )( x a) Γ( ) wit ξ x, x ( a, b] emar Suppose f ( x) C [, b] and D f ( x) ( o, b] for < It is obvious from te Lemma tat if D f ( x) ), x ( a, b) ten te function f is nondecreasing, and if D f ( x), x ( a, b) ten te function f is nonincreasing for all x [, b] Teorem Tere is a unique solution for te initial value problem given by () - (), and te solution remains in Proof Te existence and uniqueness of te solution of ()-() in (, ) can be obtained from [, Teorem and emar ] We need to sow tat te domain positively invariant Since D S D E D I S E I A, βsi, σe, is (4) on eac yperplane bounding te nonnegative ortant, te vector field points into It is clear tat N( t) also remains nonnegative For convenience in calculations we te following system, wic can be obtained from (): ds de di ( q) N βsi β SE ( p µ ) S βsi βse ( σ µ ) E, σ E ( d µ ) I (5) wit initial conditions S() S, E() E, I() I (6) Te basic reproduction number,, of te integer order model ( ) is computed in [] to be ( β β r µ ) N( p) ( ) (7) ( p µ )( r µ )( µ ) Te basic reproduction number,, is expressed as te number of secondary infections owing to a single infection in a completely susceptible population For < te diseasefree equilibrium is globally asymptotically stable and if >, te endemic equilibrium is globally asymptotically stable [] Equilibrium Points and Stability We consider te initial value problem (5)-(6) wit satisfying < in order to estimate te equilibrium points of (5), let D S, D E, D I Ten te equilibrium points are E (,, ) and E S E I, were * * * (,, ) S E I * A a A( ), b, * A ( ) bc * Te Jacobian matrix J ( E ) for te system given by (6), computed at te disease free equilibrium is below ( p µ ) β β J ( E ) ( β ( µ )) β ( r µ ) Teorem Disease free equilibrium of te system (5) is asymptotically stable if ( β β r µ ) N( p) ( ) < ( p µ )( r µ )( µ ) Proof Disease free equilibrium is asymptotically stable if all of te eigenvalues, λ i i,,, of J ( E ) satisfy te following conditions [6, ]:

4 66 Bonya Ebenezer et al: Fractional Dynamics of Computer Virus Propagation π arg λi > (8) Tese eigenvalues can be obtained by solving te caracteristic equation det( J ( E) λi) Hence, we obtain te following algebraic equation: were ( λ ( p µ ))[ λ (( A B) β ) λ ( ABC)] A ( µ ), B r µ ), C ( ), If AB > C, ten te condition given by (8) is met We now examine te asymptotic stability of te endemic (positive) equilibrium of te system given by () Te Jacobian matrix J ( E ) determined at te endemic equilibrium is expressed as * * ( p µ ) βs βs * * ( E ) ( p µ )( ) βs ( µ ) βs ( r µ ) Tus, te caracteristic equation of te linearized system is expressed of te form c c c, λ λ λ c p S r * ( µ ) ( β ( µ ) ( µ )), c ( p µ )( µ )( r µ )( ), c ( p µ )( µ )( r µ )( ), Let D( φ) stands for te discriminant of a polynomial f If φ ( x) x cx c x c ten Denote c c c c c c D( φ ) c c 8 cc c ( cc ) 4cc 4c 7 c c c c c Following [; ] we arrive at te proposition Proposition One assume tat E exists in ) If te discriminant of φ ( x), D( φ) is positive and out- Hurwitz are satisfied, tat is, D( φ ) >, c >, c >, cc > c, ten E is locally asymptotically stable ) If D( φ) <, c >, c >, cc c, [,) ten E is locally asymptotically stable ) If D( φ) <, c <, c <, > /, ten E is unstable equation [4, 5] D y( t) f ( t, y( t)), t T t were y () y,,,, m D y( t) f ( t, y( t)), t T t y () y,,,, m y () y,,,, m Tis equation corresponds to te Volterra integral equation 4 Numerical Metods and Simulations In view of te fact tat most of te fractional-order differential equations ardly ave exact analytic solutions, approximation and numerical tecniques is te most effective way of solving suc systems Numerous analytical and numerical metods ave been developed to solve te fractional order differential equations For numerical solutions of system (), one can apply te generalized Adams-Basfort- Moulton metod In order to give te approximate solution by means of tis algoritm, we consider te following nonlinear fractional differential Γ ( ) m t t ( ) ( ) (, ( ))! Γ( ) (4) y t y t s f s y s ds Dietelm et al employed te predictor-correctors sceme [4, 5], depended on te Adams-Basfort- Moulton algoritm to integrate Eq (4) By employing tis sceme to te fractional-order model for computer virus, and putting T, tn n, n,,,, N Z, Eq N (4) can be discretized as follows [4, 5]: p p p p p (( ) β β ( µ ) ) S S q N S I S E q S n n n n n n n aj, n (( q) N βsjij β SjEj ( q µ ) Sj ), Γ ( ) j

5 Science Journal of Applied Matematics and Statistics 5; (): p p p p p En E ( βsn In βsn En ( µ ) En ) Γ ( ) n aj, n ( βsjij βsjej ( µ ) Ej ), Γ ( ) j p p In I ( σ En ( d µ ) In ) Γ ( ) n aj, n ( σ Ej ( d µ ) Ij ), Γ ( ) j were n S S b, n (( q) N βs I β S E ( q µ ) S ), Γ n j j j j j j ( ) j n E E b, n ( βs I β S E ( µ ) E ), Γ n j j j j j j ( ) j n I I b, n E ( d ) I Γ ( σ µ ) n j j j ( ) j n n n j j ( )( ),, a n n j n j n j j n n, j, ( ) ( ) ( ), (( ) ( ) ) bj, n n j n j, j n Te parameter values used for te simulations were, N, p 7, β, β, µ, σ 9, r 4 and te following set of (,95,9,85) for eac compartment Now we tae into account te initial 9 8 population include susceptible nodes S (), exposed to infected nodes E (), infected nodes () for numerical simulation Susceptible Time Figure Dynamics of te susceptible computers versus time, of system (7), for {,, } Parameter values and initial conditions are tose stated

6 68 Bonya Ebenezer et al: Fractional Dynamics of Computer Virus Propagation Exposed Time Figure Dynamics of te exposed computers versus time, of system (7), for {,, } Parameter values and initial conditions are tose stated Infected Time Figure Dynamics of te infected computers versus time, of system (7), for {,, } Parameter values and initial conditions are tose stated 5 Discussion In tis paper, we ave considered a fractional calculus model for computer virus propagation From te numerical results in Figures, and, it is obvious tat te approximate solutions depend continuously on te fractional derivative Te approximate solutions S( t ), E( t ), and I( t) are sown in Figures, and wit four different values of In eac figure four different values of are taen into account Given, system () is termed as te classical integer-order system () We sowed in Figure, te variation of S( t) versus time wit varying values of, 95, 9, 85 by fixing oter parameters It is observed tat S( t ) does not drop sarply in a relatively small period of time for small values of Figure depicts E( t) versus time and Figure also sows I( t ) versus time However, variation in for Figure and is more apparent tan tat of Figure Tis buttresses te sensitive nature of fractional order models Following [7, 8], one observes tat bot steady states of integer order and fractional order turn to point to te fixed point given a long period of time In dealing wit situations in real life, data collected can elp determine te order of te system One also needs to mention tat wen dealing wit real life problems, te order of te system can be determined by using te collected data 6 Conclusion In tis paper, it is assumed tat te virus process as a latent period and computers infected by te virus ave also infectivity An SEI compartmental model for transmission of virus in computer networ is formulated and studied Te tresold parameter is determined Te steady states of te model is also derived and analyzed in order to determine te stability of te system Adams-Basfort-Moulton metod is used to carry out te numerical simulations of te fraction order model Te simulation results sow tat by varying small result in a big cange in te associated

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