Nonstandard Finite Difference Scheme for a Computer Virus Model

Mehmet Kocabıyık; Nihal Özdoğan; Mevlüde Yakıt Ongun

doi:10.38088/jise.705728

Research Article

Year 2020, Volume: 4 Issue: 2, 96 - 108, 14.12.2020

Mehmet Kocabıyık Nihal Özdoğan Mevlüde Yakıt Ongun

https://doi.org/10.38088/jise.705728

Cited By: 4

Abstract

Project Number

TUBITAK 2211

References

[1] Cohen, F. (1987). Computer Viruses Theory and experiments, Computers and Security, vol.6, no.1, pp. 22-35.
[2] Szor, P. (2005). The Art of Computer Virus Research and Defense, Addison-Wesley
[3] Chen, L., Hattaf, K. and Sun, J. (2015). Optimal control of a delayed SLBS computer virus model. Physica A: Statistical Mechanics and its Applications, 427: 244-250.
[4] Yang, X. and Yang, L. X. (2012). Towards the epidemiological modeling of computer viruses. Discrete Dynamics in Nature and Society, (Article ID 259671).
[5] Yang, L. X. and Yang, X. (2014). A new epidemic model of computer viruses. Communications in Nonlinear Science and Numerical Simulation, 19(6): 1935-1944.
[6] Zhu, Q., Yang, X. and Ren, J. (2012). Modeling and analysis of the spread of computer virus. Communications in Nonlinear Science and Numerical Simulation, 17(12): 5117-5124.
[7] Lijuan, C., Khalid, H. and Jitao, S. (2015). Optimal control of a delayed SLBS computer virus model, Physica A: Statistical Mechanics and its Applications, 427: 244-250.
[8] Hoang, T. M., Dang, A. Q., and Dang, L. Q. (2018). Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses. Journal of Computer Science and Cybernetics, 34(2): 171-185.
[9] Barthelemy, M., Barrat, A., Pastor-Satorras, R. and Vespignani, A. (2004). Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys Re Lett; 92(7) (Article ID 178701).
[10] Karsai, M., Kivela, M., Pan, R.K., Kaski, K., Kertesz, J., Barabasi, A.L. and Saramaki, J. (2011). Small but slow world: How network topology and burstiness slow downspreading. Phys Rev E; 83(2) (Article ID 025102).
[11] Billings, L., Spears, W.M. and Schwartz, I.B. (2002). A unied prediction of computer virus spread in connected networks. Phys Lett A; 297(3-4):261-6.
[12] Boguna ,M., Pastor-Satorras, R. and Vespignani, A. (2003). Absence of epidemic threshold in scale-free networks with degree correlations. Phys Rev Lett ;90(2)(Article ID 028701).
[13] Castellano, C. and Pastor-Satorras, R.. (2010). Thresholds for epidemic spreading in networks. Phys Rev Lett; 105(21) (Article ID 218701).
[14] Dezso, Z. and Barabasi, A.L. (2002). Halting viruses in scale-free networks. Phys Rev E; 65(5) (Article ID 055103).
[15] Fu, X., Small, M., Walker, D.M. and Zhang, H. (2008). Epidemic dynamics on scale- free networks with piecewise linear infectivity and immunization. Phys Rev E;77(3) (Article ID 036113).
[16] Lloyd, A.L. and May, R.M. (2001). How viruses spread among computers and people. Science; 292(5520):1316-7.
[17] Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic spreading in scale-free networks. Phys Rev Lett ;86(14): 3200-3.
[18] Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic dynamics and endemic states in complex networks. Phys Rev E ; 63(6) (Article ID 066117).
[19] Pastor-Satorras, R. and Vespignani, A. (2002). Immunization of complex networks. Phys Rev E; 65(3) (Article ID 036104).
[20] Wierman, J.C. and Marchette, D.J. (2004). Modeling computer virus prevalence with a susceptibleinfected- susceptible model with reintroduction. Comput Stat Data Anal; 45(1):3-23.
[21] Chen, L.C. and Carley, K.M. (2004). The impact of countermeasure propagation on the prevalence of computer viruses. IEEE Trans Syst Man Cybern -Part B; 34(2): 823-33.
[22] Draief, M., Ganesh, A. and Massouilie, L. (2008). Thresholds for virus spread on networks. Ann Appl Probab; 18(2): 359-78.
[23] Griffin, C. and Brooks, R. (2006). A note on the spread of worms in scale-free networks. IEEE Trans Syst Man Cybern- Part B: Cybern; 36(1): 198-202.
[24] Moreno, Y., Pastor-Satorras, R. and Vespignani, A. (2002). Epidemic outbreaks in comple heterogeneus networks. Eur Phys J B; 26(4): 5219.
[25] Mickens,R.E. (1989). Exact solutions to a finite difference model of a nonlinear reaction advection equation: implications for numerical analysis. Numerical Methods for Partial Diferential Equations, vol. 5, no. 4, pp. 313-325.
[26] Mickens, R.E. (1990) Difference Equations Theory and Applications,Chapman Hall, Atlanta, Ga, USA.
[27] Mickens, R.E. (1993). Nonstandard Finite Difference Models of Differential Equations,World Scientic Publishing, Atlanta, Ga,USA.
[28] Mickens, R.E. (1999). Discretizations of nonlinear differential equations using explicit nonstandard methods. Journal of Computational and AppliedMathematics, 110(1): 181-185.
[29] Mickens, R.E. (1999). Applications of Nonstandard Finite Difference Schemes, World Scientic Publishing, Atlanta, Ga, USA.
[30] Mickens, R.E. (2005). Advances in the Applications of Nonstandard Finite Difference Schemes,Wiley-Interscience, Singapore.
[31] Mickens, R.E. (2007). Calculation of denominator functions for nonstandard finite difference schemes for dierential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations, 23(3): 672-691, 2007.
[32] Mickens, R.E., Talitha M. W. (2013). NSFD discretizations of interacting population models satisfying conservation laws, Computers & Mathematics with Applications.
[33] Zibaei, S. and Namjoo, M. (2014). A NSFD scheme for Lotka Volterra food web model. Iranian Journal of Science and Technology (Sciences), 38(4), 399-414.
[34] Cohn, A. (1922). Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Mathematische Zeitschrift, 14(1):110-148.
[35] Duffin, R. J. (1969). Algorithms for classical stability problems. Siam Review, 11(2):196-213.
[36] Elaydi, S.N. (1999). An Introduction to Difference Equations, Springer, New York, USA.
[37] Jury, E. (1964). I.Theory and Application of the z-Transform Method.
[38] Miller, J. J. (1971). On the location of zeros of certain classes of polynomials with applications to numerical analysis. IMA Journal of Applied Mathematics, 8(3), 397-406.
[39] Ongun, M. Y. and Turhan, I. (2013). A numerical comparison for a discrete HIV Infection of CD4+ T-Cell model derived from nonstandard numerical scheme. Journal of Applied Mathematics.
[40] Ongun, M.Y. and Özdoğan N. (2017). A nonstandard numerical scheme for a predator-prey model with Allee effect, J. Nonlinear Sci. Appl., 10:713-723.
[41] Schur, I. (1917). Uber Potenzreihen, die im Innern des Einheitskreises beschrankt sind. Journal für die reine und angewandte Mathematik, 147:205-232.
[42] Schur, I. (1918). Uber endliche Gruppen und Hermitesche Formen. Mathematische Zeitschrift, 1(2-3):184-207.

Nonstandard Finite Difference Scheme for a Computer Virus Model

Year 2020, Volume: 4 Issue: 2, 96 - 108, 14.12.2020

Mehmet Kocabıyık Nihal Özdoğan Mevlüde Yakıt Ongun

https://doi.org/10.38088/jise.705728

Cited By: 4

Abstract

This study introduces us to a new model developed for computer viruses. The model is presented to remove the protective restriction on the total number of computers connected to the Internet. This model is nonlinear differential equation system. Therefore, finding analytical solutions is very difficult. This means that we have to apply numerical methods in order to find the solution. The behavior of numerical solution has been investigated for the discretized system. By using Nonstandard Finite Difference Scheme (NSFD), it is aimed to preserve both the positivity of the solutions for positive initial points and the local asymptotic stability of the equilibrium point.

Keywords

Computer virus, Nonstandard Finite Difference Scheme, Discretization, Internet

Project Number

TUBITAK 2211

Thanks

The author Mehmet KOCABIYIK would like to thank the Scientific and Technological Research Council of Turkey (TUBITAK 2211 programme) for the financial support.

References

[1] Cohen, F. (1987). Computer Viruses Theory and experiments, Computers and Security, vol.6, no.1, pp. 22-35.
[2] Szor, P. (2005). The Art of Computer Virus Research and Defense, Addison-Wesley
[3] Chen, L., Hattaf, K. and Sun, J. (2015). Optimal control of a delayed SLBS computer virus model. Physica A: Statistical Mechanics and its Applications, 427: 244-250.
[4] Yang, X. and Yang, L. X. (2012). Towards the epidemiological modeling of computer viruses. Discrete Dynamics in Nature and Society, (Article ID 259671).
[5] Yang, L. X. and Yang, X. (2014). A new epidemic model of computer viruses. Communications in Nonlinear Science and Numerical Simulation, 19(6): 1935-1944.
[6] Zhu, Q., Yang, X. and Ren, J. (2012). Modeling and analysis of the spread of computer virus. Communications in Nonlinear Science and Numerical Simulation, 17(12): 5117-5124.
[7] Lijuan, C., Khalid, H. and Jitao, S. (2015). Optimal control of a delayed SLBS computer virus model, Physica A: Statistical Mechanics and its Applications, 427: 244-250.
[8] Hoang, T. M., Dang, A. Q., and Dang, L. Q. (2018). Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses. Journal of Computer Science and Cybernetics, 34(2): 171-185.
[9] Barthelemy, M., Barrat, A., Pastor-Satorras, R. and Vespignani, A. (2004). Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys Re Lett; 92(7) (Article ID 178701).
[10] Karsai, M., Kivela, M., Pan, R.K., Kaski, K., Kertesz, J., Barabasi, A.L. and Saramaki, J. (2011). Small but slow world: How network topology and burstiness slow downspreading. Phys Rev E; 83(2) (Article ID 025102).
[11] Billings, L., Spears, W.M. and Schwartz, I.B. (2002). A unied prediction of computer virus spread in connected networks. Phys Lett A; 297(3-4):261-6.
[12] Boguna ,M., Pastor-Satorras, R. and Vespignani, A. (2003). Absence of epidemic threshold in scale-free networks with degree correlations. Phys Rev Lett ;90(2)(Article ID 028701).
[13] Castellano, C. and Pastor-Satorras, R.. (2010). Thresholds for epidemic spreading in networks. Phys Rev Lett; 105(21) (Article ID 218701).
[14] Dezso, Z. and Barabasi, A.L. (2002). Halting viruses in scale-free networks. Phys Rev E; 65(5) (Article ID 055103).
[15] Fu, X., Small, M., Walker, D.M. and Zhang, H. (2008). Epidemic dynamics on scale- free networks with piecewise linear infectivity and immunization. Phys Rev E;77(3) (Article ID 036113).
[16] Lloyd, A.L. and May, R.M. (2001). How viruses spread among computers and people. Science; 292(5520):1316-7.
[17] Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic spreading in scale-free networks. Phys Rev Lett ;86(14): 3200-3.
[18] Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic dynamics and endemic states in complex networks. Phys Rev E ; 63(6) (Article ID 066117).
[19] Pastor-Satorras, R. and Vespignani, A. (2002). Immunization of complex networks. Phys Rev E; 65(3) (Article ID 036104).
[20] Wierman, J.C. and Marchette, D.J. (2004). Modeling computer virus prevalence with a susceptibleinfected- susceptible model with reintroduction. Comput Stat Data Anal; 45(1):3-23.
[21] Chen, L.C. and Carley, K.M. (2004). The impact of countermeasure propagation on the prevalence of computer viruses. IEEE Trans Syst Man Cybern -Part B; 34(2): 823-33.
[22] Draief, M., Ganesh, A. and Massouilie, L. (2008). Thresholds for virus spread on networks. Ann Appl Probab; 18(2): 359-78.
[23] Griffin, C. and Brooks, R. (2006). A note on the spread of worms in scale-free networks. IEEE Trans Syst Man Cybern- Part B: Cybern; 36(1): 198-202.
[24] Moreno, Y., Pastor-Satorras, R. and Vespignani, A. (2002). Epidemic outbreaks in comple heterogeneus networks. Eur Phys J B; 26(4): 5219.
[25] Mickens,R.E. (1989). Exact solutions to a finite difference model of a nonlinear reaction advection equation: implications for numerical analysis. Numerical Methods for Partial Diferential Equations, vol. 5, no. 4, pp. 313-325.
[26] Mickens, R.E. (1990) Difference Equations Theory and Applications,Chapman Hall, Atlanta, Ga, USA.
[27] Mickens, R.E. (1993). Nonstandard Finite Difference Models of Differential Equations,World Scientic Publishing, Atlanta, Ga,USA.
[28] Mickens, R.E. (1999). Discretizations of nonlinear differential equations using explicit nonstandard methods. Journal of Computational and AppliedMathematics, 110(1): 181-185.
[29] Mickens, R.E. (1999). Applications of Nonstandard Finite Difference Schemes, World Scientic Publishing, Atlanta, Ga, USA.
[30] Mickens, R.E. (2005). Advances in the Applications of Nonstandard Finite Difference Schemes,Wiley-Interscience, Singapore.
[31] Mickens, R.E. (2007). Calculation of denominator functions for nonstandard finite difference schemes for dierential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations, 23(3): 672-691, 2007.
[32] Mickens, R.E., Talitha M. W. (2013). NSFD discretizations of interacting population models satisfying conservation laws, Computers & Mathematics with Applications.
[33] Zibaei, S. and Namjoo, M. (2014). A NSFD scheme for Lotka Volterra food web model. Iranian Journal of Science and Technology (Sciences), 38(4), 399-414.
[34] Cohn, A. (1922). Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Mathematische Zeitschrift, 14(1):110-148.
[35] Duffin, R. J. (1969). Algorithms for classical stability problems. Siam Review, 11(2):196-213.
[36] Elaydi, S.N. (1999). An Introduction to Difference Equations, Springer, New York, USA.
[37] Jury, E. (1964). I.Theory and Application of the z-Transform Method.
[38] Miller, J. J. (1971). On the location of zeros of certain classes of polynomials with applications to numerical analysis. IMA Journal of Applied Mathematics, 8(3), 397-406.
[39] Ongun, M. Y. and Turhan, I. (2013). A numerical comparison for a discrete HIV Infection of CD4+ T-Cell model derived from nonstandard numerical scheme. Journal of Applied Mathematics.
[40] Ongun, M.Y. and Özdoğan N. (2017). A nonstandard numerical scheme for a predator-prey model with Allee effect, J. Nonlinear Sci. Appl., 10:713-723.
[41] Schur, I. (1917). Uber Potenzreihen, die im Innern des Einheitskreises beschrankt sind. Journal für die reine und angewandte Mathematik, 147:205-232.
[42] Schur, I. (1918). Uber endliche Gruppen und Hermitesche Formen. Mathematische Zeitschrift, 1(2-3):184-207.

There are 42 citations in total.

Details

Primary Language	English
Subjects	Engineering
Journal Section	Research Articles
Authors	Mehmet Kocabıyık 0000-0002-7701-6946 Nihal Özdoğan 0000-0002-6936-3454 Mevlüde Yakıt Ongun 0000-0002-7701-6946
Project Number	TUBITAK 2211
Publication Date	December 14, 2020
Published in Issue	Year 2020Volume: 4 Issue: 2

Cite

APA	Kocabıyık, M., Özdoğan, N., & Yakıt Ongun, M. (2020). Nonstandard Finite Difference Scheme for a Computer Virus Model. Journal of Innovative Science and Engineering, 4(2), 96-108. https://doi.org/10.38088/jise.705728
AMA	Kocabıyık M, Özdoğan N, Yakıt Ongun M. Nonstandard Finite Difference Scheme for a Computer Virus Model. JISE. December 2020;4(2):96-108. doi:10.38088/jise.705728
Chicago	Kocabıyık, Mehmet, Nihal Özdoğan, and Mevlüde Yakıt Ongun. “Nonstandard Finite Difference Scheme for a Computer Virus Model”. Journal of Innovative Science and Engineering 4, no. 2 (December 2020): 96-108. https://doi.org/10.38088/jise.705728.
EndNote	Kocabıyık M, Özdoğan N, Yakıt Ongun M (December 1, 2020) Nonstandard Finite Difference Scheme for a Computer Virus Model. Journal of Innovative Science and Engineering 4 2 96–108.
IEEE	M. Kocabıyık, N. Özdoğan, and M. Yakıt Ongun, “Nonstandard Finite Difference Scheme for a Computer Virus Model”, JISE, vol. 4, no. 2, pp. 96–108, 2020, doi: 10.38088/jise.705728.
ISNAD	Kocabıyık, Mehmet et al. “Nonstandard Finite Difference Scheme for a Computer Virus Model”. Journal of Innovative Science and Engineering 4/2 (December 2020), 96-108. https://doi.org/10.38088/jise.705728.
JAMA	Kocabıyık M, Özdoğan N, Yakıt Ongun M. Nonstandard Finite Difference Scheme for a Computer Virus Model. JISE. 2020;4:96–108.
MLA	Kocabıyık, Mehmet et al. “Nonstandard Finite Difference Scheme for a Computer Virus Model”. Journal of Innovative Science and Engineering, vol. 4, no. 2, 2020, pp. 96-108, doi:10.38088/jise.705728.
Vancouver	Kocabıyık M, Özdoğan N, Yakıt Ongun M. Nonstandard Finite Difference Scheme for a Computer Virus Model. JISE. 2020;4(2):96-108.