Confirm
Archive
Special Issues
Fractional Dynamics of Computer Virus Propagation
Science Journal of Applied Mathematics and Statistics
Volume 3, Issue 3, June 2015, Pages: 63-69
Received: Mar. 9, 2015; Accepted: Apr. 3, 2015; Published: Apr. 14, 2015
Authors
Bonyah Ebenezer, Department of Mathematics and Statistics, Kumasi Polytechnic, Kumasi, Ghana
Nyabadza Farai, Department of Mathematical Science, University of Stellenbosch, Matieland, South Africa
Asiedu-Addo Samuel Kwesi, Department of Mathematics Education, University of Education, Winneba Ghana
Article Tools
Abstract
This paper studies the fractional order model for computer virus in SEIR model. Firstly, the basic reproduction number R0, which determines the threshold of the spread of the virus is determined. The stability of equilibra was also determined and studied. The Adams-Bashforth-Moulton algorithm was employed to solve and simulate the system of differential equations. The results of the simulation depicts that by small change in α led to big change in the associated numerical results.
Keywords
Nonlinear System, Fractional Calculus, Computer Virus Model
Bonyah Ebenezer, Nyabadza Farai, Asiedu-Addo Samuel Kwesi, Fractional Dynamics of Computer Virus Propagation, Science Journal of Applied Mathematics and Statistics. Vol. 3, No. 3, 2015, pp. 63-69. doi: 10.11648/j.sjams.20150303.11
References
[1]
Zhu, Q.,Yang X., Yang L.X., Zhang C. Optimal control of computer virus under a delayed model,” Applied Mathematics and Computation, vol. 218, no. 23,(2012) pp 11613–11619.
[2]
Mishra B.K., Saini D. K. SEIRS epidemic model with delay for transmission of malicious objects in computer network," Applied Mathematics, and Computation, vol. 188, no. 2,(2007) pp. 1476-1482.
[3]
Mishra B.K., Saini D.K. SEIRS epidemic model with delay for transmission of malicious objects in computer network," Applied Mathematics, and Computation, vol. 188, no. 2,(2007) pp. 1476-1482.
[4]
Mishra B.K., Saini D. Mathematical models on computer virus, Applied Mathematics, and Computation," vol. 187, no. 2,(2007) pp. 929-936.
[5]
Mishra B.K., N. Jha N. Fixed period of temporary immunity after run of anti-malicious software on computer nodes," Applied Mathematics and Computation, vol. 190, no. 2,(2007) pp. 1207-1212.
[6]
Forest S., Hofmeyr S., Somayaji A., Longsta T. Self-nonself discrimination in a computer," Proceedings of IEEE Symposium on Computer Security, and Privacy,(1994) pp. 202-212.
[7]
Symantec Security Response-Definitions, (http://www.symantec.com/avcenter/d( 2010).
[8]
Richard W. T., Mark J. C. Modeling virus propagation in peer-to-peer networks," IEEE International Conference on Information, Communications, and Signal Processing, ICICS 2005, pp. 981- 985.
[9]
Zou C.C., Gong, W., Towsley D., Gao L. Themonitoring and early detection of internet worms,” IEEE/ACMTransactions on Networking, vol. 13, no. 5,(2005) pp. 961–974.
[10]
ChenT., Jamil N. Effectiveness of quarantine in worm epidemics," IEEE International Conference on Communications,(2006) pp. 2142-2147.
[11]
Cohen F. Computer virus, theory, and experiments," Proceedings of the 7th DOD/NBS Computer and Security Conference,(1987) pp. 22-35.
[12]
Mishra B.K., Saini D.K. SEIRS epidemic model with delay for transmission of malicious objects in computer network," Applied Mathematics and Computation, vol. 188, no. 2, (2007) pp. 1476-1482.
[13]
Bimal K. M., Gholam M. A. Differential Epidemic Model of Virus and Worms in Computer Network, International Journal of Network Security, Vol.14, No.3,(2012) PP. 149-155.
[14]
Murray, W. The application of epidemiology to computer viruses. Comput. Security 7,(1988) 139–150.
[15]
Podlubny. Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA (1999).
[16]
Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential
[17]
Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands (2006).
[18]
Oldham K. B., Spanier J. TheFractionalCalculus:Theory and Application of Differentiation and Integration to ArbitraryOrder, Academic Press, New York, NY, USA (1974).
[19]
Ionescu.C.M. The Human Respiratory System: An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics, Series in BioEngineering, Springer, London, UK (2013).
[20]
Tenreiro J.A., Machado Galhano Oliveira A.M., Tar J.K. Approximating fractional derivatives through the generalized mean, Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11,(2009) pp. 3723–3730.
[21]
Odibat, Z.M., Shawagfeh, N.T. Generalized Taylor’s formula, Applied Mathematics and Computation 186,(2007) 286–293.
[22]
Q. Zhu Q., Yang X., Ren. J. Modeling and analysis of the spread of computer virus,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12,(2012) pp. 5117–5124.
[23]
Ahmed E., El-Sayed A. M. A., El-Saka H. A. A. On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R¨ossler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1,(2006) pp. 1–4.
[24]
DingY., H. Ye H. A fractional-order differential equation model of HIV infection of CD4_T-cells,”Mathematical and Computer Modelling, vol. 50, no. 3-4,(2009) pp. 386–392.
[25]
Diethelm, K., Ford. N.J. Analysis of fractional differential equations,J Math Anal Appl, 256 (2002) 229–248.
[26]
Diethelm K., Ford N. J., Freed A.D. A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn, 29 (2002), 3–22.
[27]
Matignon D . Stability results for fractional differential equations with applications to control processing, in: Computational Eng. in Sys. Appl. 2 (Lille, France), (1996) pp. 963.
[28]
E. Demirci, A. Unal, and N. ¨Ozalp, “A fractional order SEIR model with density dependent death rate,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 2,(2011) pp. 287–295.
[29]
N. ¨ Ozalp and E.Demi¨orci¨o, “A fractional order SEIR model with vertical transmission,” Mathematical and Computer Modelling, vol. 54, no. 1-2,(2011) pp. 1–6.
[30]
M. Garetto, W. Gong, and D. Towsley, “Modeling malware spreading dynamics,” in Proc. IEEE Infocom, San Francisco, CA, Mar. 2003.
[31]
P. Mei X. He, J. Huang, and T. Dong. Modeling Computer Virus and Its Dynamics. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, 5 (842614).
PUBLICATION SERVICES