Extrapolated nonstandard numerical schemes for solving an epidemiological model for computer viruses
DOI:
https://doi.org/10.54654/isj.v3i17.896Keywords:
NSFD schemes, dynamic consistency, computer viruses, high-order, Richardson’s extrapolationTóm tắt
Abstract— The aim of this work is to construct high-order numerical schemes that preserve the dynamical properties of a mathematical model describing the spread of computer viruses on the Internet. For this purpose, we first apply Mickens’ methodology to formulate a dynamically consistent nonstandard finite difference (NSFD) scheme for the model under consideration. After that, the constructed NSFD scheme is combined with Richardson’s extrapolation method to generate higher-accuracy numerical approximations. The result is that we obtain extrapolated numerical schemes that not only preserve the dynamical properties of the computer virus propagation model but also provide higher-accuracy numerical approximations. In addition, a set of numerical examples is conducted to illustrate and support the theoretical findings and to show the advantages of the proposed numerical schemes.
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References
J. Amador, The stochastic SIRA model for com¬puter viruses, Applied Mathematics and Computation 232(2014) 1112-1124.
U. M. Ascher, L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Ap¬plied Mathematics, PA, United States, 1998.
R. L. Burden, J. Douglas Faires, Numerical Analysis, Ninth edition, Cengage Learning, 2015.
B. M. Chen-Charpentier, D. T. Dimitrov, H. V. Ko¬jouharov, Combined nonstandard numerical methods for ODEs with polynomial right-hand sides, Mathe¬matics and Computers in Simulation 73(2006) 105- 113.
Q. A. Dang, M. T. Hoang, Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses, Journal of Computational and Applied Mathematics 374(2020) 112753.
Quang A Dang, Manh Tuan Hoang, Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical sys¬tems, International Journal of Computer Mathematics 97 (2020) 2036-2054.
Q. A. Dang, M. T. Hoang, Q. L. Dang, Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses, Journal of Computer Science and Cybernetics 32(2018) 171- 185.
S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 2005.
C. Gan, X. Yang, W. Liu, Q. Zhu, A propagation model of computer virus with nonlinear vaccination probability, Communications in Nonlinear Science and Numerical Simulation 19(2014) 92-100.
M. Gupta, J. M. Slezak, F. Alalhareth, S. Roy, H. V. Kojouharov, Second-order Nonstandard Explicit Eu¬ler Method, AIP Conference Proceedings 2302(2020) 110003.
G. Gonzalez-Parra, A. J. Arenas, B. M. Chen¬Charpentier, Combination of nonstandard schemes and Richardson’s extrapolation to improve the nu¬merical solution of population models, Mathematical and Computer Modelling 52(2010) 1030-1036.
M. T. Hoang, Lyapunov functions for investigating stability properties of a fractional-order computer virus propagation model, Qualitative Theory of Dy¬namical Systems volume 20, Article number: 74 (2021).
M. T. Hoang, Dynamically consistent nonstandard finite difference schemes for a virus-patch dynamic model, Journal of Applied Mathematics and Com¬puting (2021). https://doi.org/10.1007/s12190-021- 01673-z.
M. T. Hoang, Dynamical analysis of two fractional-order SIQRA malware propagation models and their discretizations, Rendiconti del Circolo Matematico di Palermo Series 2 (2022). https://doi.org/10.1007/s12215-021-00707-6.
M. T. Hoang, Global asymptotic stability of some epidemiological models for computer viruses and malware using nonlinear cascade systems, Bolet´ın de la Sociedad Matemática Mexicana volume 28, Article number: 39 (2022).
M. T. Hoang, A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems, Communications in Nonlinear Science and Numerical Simulation Volume 114, November 2022, 106654.
M. T. Hoang, Reliable approximations for a hepati¬tis B virus model by nonstandard numerical schemes, Mathematics and Computers in Simulation Volume 193, March 2022, Pages 32-56.
D. C. Joyce, Survey of extrapolation processes in numerical analysis, SIAM Review 13(1971) 435-490.
H. K. Khalil, Nonlinear Systems, 3rd Edition, Pear¬son, 2002.
H. V. Kojouharov, S. Roy, M. Gupta, F. Alalhareth, J. M. Slezak, A second-order modified nonstan¬dard theta method for one-dimensional autonomous differential equations, Applied Mathematics Letters 112(2021) 106775.
J. Martin-Vaquero, A. Martin del Rey, A. H. Enci¬nas, J. D. Hernandez Guillen, A. Queiruga-Dios, G. Rodriguez Sanchez, Higher-order nonstandard fi¬nite difference schemes for a MSEIR model for a malware propagation, Journal of Computational and Applied Mathematics 317(2017) 146-156.
J. Martin-Vaquero, A. Queiruga-Dios, A. Martin del Rey, A. H. Encinas, J. D. Hernandez Guillen, G. Rodriguez Sanchez, Variable step length algorithms with high-order extrapolated non-standard finite dif¬ference schemes for a SEIR model, Journal of Com¬putational and Applied Mathematics 330(2018) 848- 854.
R. E. Mickens, Nonstandard Finite Difference Mod¬els of Differential Equations, World Scientific, 1993.
R. E. Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific, 2000.
R. E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, World Sci¬entific, 2005.
R. E. Mickens, Nonstandard Finite Difference Schemes for Differential Equations, Journal of Dif¬ference Equations and Applications 8(2002) 823-847.
R. E. Mickens, Nonstandard Finite Difference Schemes: Methodology and Applications, World Sci¬entific, 2020.
W. H. Murray, The application of epidemiology to computer viruses, Computers & Security 7(1988) 139-145.
K. C. Patidar, On the use of nonstandard finite difference methods, Journal of Difference Equations and Applications 11(2005) 735-758.
K. C. Patidar, Nonstandard finite difference meth¬ods: recent trends and further developments, Journal of Difference Equations and Applications 22(2016) 817-849.
J. R. C. Piqueira, V. O. Araujo, A modified epi-demiological model for computer viruses, Applied Mathematics and Computation 213(2009) 355-360.
L. F. Richardson, J. A. Gaunt, The deferred ap-proach to the limit, Philosophical Transactions of the Royal Society of London 226A(1927) 299-361.
J. Singh, D. Kumar, Z. Hammouch, A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation 316(2018) 504-515
L. X. Yang, X. Yang, The impact of nonlinear infection rate on the spread of computer virus, Non¬linear Dynamics 82(2015)85-95.
L. X. Yang, X. Yang, Q. Zhu, L. Wen, A computer virus model with graded cure rates, Nonlinear Anal¬ysis: Real World Applications 14(2013) 414-422.
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