Journal of Innovative Science and Engineering
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Year 2020, Volume 4, Issue 2, 96 - 108, 14.12.2020

Mehmet KOCABIYIK Nihal ÖZDOĞAN Mevlüde YAKIT ONGUN

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

Abstract

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 KOCABIYIK Nihal ÖZDOĞAN Mevlüde YAKIT ONGUN

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

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

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.

Details

Primary Language English
Subjects Engineering
Journal Section Research Articles
Authors

Mehmet KOCABIYIK>
MEHMET AKİF ERSOY ÜNİVERSİTESİ
0000-0002-7701-6946
Türkiye

Nihal ÖZDOĞAN> (Primary Author)
Bursa Teknik Üniversitesi
0000-0002-7551-1636
Türkiye

Mevlüde YAKIT ONGUN>
SÜLEYMAN DEMİREL ÜNİVERSİTESİ
0000-0002-7701-6946
Türkiye

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.
Publication Date December 14, 2020
Published in Issue Year 2020, Volume 4, Issue 2

Cite

Bibtex @research article { jise705728, journal = {Journal of Innovative Science and Engineering}, eissn = {2602-4217}, address = {ursa Technical University, Mimar Sinan Campus, Mimar Sinan Mah. Mimar Sinan Blv. Eflak Cad. No:177 16310 Yıldırım, Bursa / Turkey}, publisher = {Bursa Technical University}, year = {2020}, volume = {4}, number = {2}, pages = {96 - 108}, doi = {10.38088/jise.705728}, title = {Nonstandard Finite Difference Scheme for a Computer Virus Model}, key = {cite}, author = {Kocabıyık, Mehmet and Özdoğan, Nihal and Yakıt Ongun, Mevlüde} }
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 . DOI: 10.38088/jise.705728
MLA Kocabıyık, M. , Özdoğan, N. , Yakıt Ongun, M. "Nonstandard Finite Difference Scheme for a Computer Virus Model" . Journal of Innovative Science and Engineering 4 (2020 ): 96-108 <http://jise.btu.edu.tr/en/pub/issue/56253/705728>
Chicago Kocabıyık, M. , Özdoğan, N. , Yakıt Ongun, M. "Nonstandard Finite Difference Scheme for a Computer Virus Model". Journal of Innovative Science and Engineering 4 (2020 ): 96-108
RIS TY - JOUR T1 - Nonstandard Finite Difference Scheme for a Computer Virus Model AU - MehmetKocabıyık, NihalÖzdoğan, MevlüdeYakıt Ongun Y1 - 2020 PY - 2020 N1 - doi: 10.38088/jise.705728 DO - 10.38088/jise.705728 T2 - Journal of Innovative Science and Engineering JF - Journal JO - JOR SP - 96 EP - 108 VL - 4 IS - 2 SN - -2602-4217 M3 - doi: 10.38088/jise.705728 UR - https://doi.org/10.38088/jise.705728 Y2 - 2020 ER -
EndNote %0 Journal of Innovative Science and Engineering Nonstandard Finite Difference Scheme for a Computer Virus Model %A Mehmet Kocabıyık , Nihal Özdoğan , Mevlüde Yakıt Ongun %T Nonstandard Finite Difference Scheme for a Computer Virus Model %D 2020 %J Journal of Innovative Science and Engineering %P -2602-4217 %V 4 %N 2 %R doi: 10.38088/jise.705728 %U 10.38088/jise.705728
ISNAD Kocabıyık, Mehmet , Özdoğan, Nihal , Yakıt Ongun, Mevlüde . "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
AMA Kocabıyık M. , Özdoğan N. , Yakıt Ongun M. Nonstandard Finite Difference Scheme for a Computer Virus Model. JISE. 2020; 4(2): 96-108.
Vancouver Kocabıyık M. , Özdoğan N. , Yakıt Ongun M. Nonstandard Finite Difference Scheme for a Computer Virus Model. Journal of Innovative Science and Engineering. 2020; 4(2): 96-108.
IEEE M. Kocabıyık , N. Özdoğan and M. Yakıt Ongun , "Nonstandard Finite Difference Scheme for a Computer Virus Model", Journal of Innovative Science and Engineering, vol. 4, no. 2, pp. 96-108, Dec. 2020, doi:10.38088/jise.705728
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