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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Numerical Simulation of Computer Virus Reaction-Diffusion Model using Cubic B-splines Collocation

Discontinuity, Nonlinearity, and Complexity 12(3) (2023) 673--684 | DOI:10.5890/DNC.2023.09.013

R.C. Mittal$^1$, Rohit Goel$^{1,2}$, N. Ahlawat$^1$

$^1$ Department of Mathematics, Jaypee Institute of Information Technology, Noida (U.P.), India

$^2$ Department of Mathematics, Deshbandhu College (University of Delhi), Kalkaji, New Delhi 110019, India

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Abstract

A reaction-diffusion model characterizing the dynamics of computer virus epidemic is considered in this paper. The propagation of viruses in computers is similar to the case of many infectious diseases so that the consideration of reaction-diffusion terms becomes necessary to look into the deep insights. The structure preserving analysis of virus propagation in the computers connected globally is performed through the extended reaction-diffusion mathematical model. A numerical scheme based on the collocation of cubic B-splines is proposed to investigate the computer virus epidemic model. The numerical results obtained are compared and validated by performing stability analysis and are found in good agreement with those already available in the literature. Due to the unavailability of the analytic solutions of these models, such a numerical simulation scheme can be of prime interest for biologists to interpret the results theoretically.

References

  1. [1]  Schneider, W. (1989), Computer viruses: What they are, how they work, how they might get you, and how to control them in academic institutions, Behaviour Research Methods, Instruments \& Computers, 21, 334-340.
  2. [2]  Ozt\"{u}rk, Y. and Gulsu, M. (2015), Numerical solution of a modified epidemiological model for computer viruses, Applied Mathematical Modeling, 39(24), 7600-7610.
  3. [3]  Rey, A.M.D. (2015), Mathematical modeling of the propagation of malware: A review, Security and Communication Networks, 8(15), 2561-2579.
  4. [4]  Lanz, A., Rogers, D., and Alford, T.L. (2019), An epidemic model of malware virus with quarantine, Journal of Advances in Mathematics and Computer Science, 33(4), 1-10.
  5. [5]  Xu, Y.H., Ren, J.G., and Sun, G.Q. (2016), Propagation effect of a virus outbreak on a network with limited anti-virus ability, Plos One, 11(10), 1-18.
  6. [6]  Liu, J., Bianca, G., and Guerrini, L. (2016), Dynamical analysis of a computer virus model with delays, Discrete Dynamics in Nature and Society, 16(1), 1-19.
  7. [7]  Yuan, H. and Chen, G. (2008), Network virus-epidemic model with the point-to-group information propagation, AppliedMathematics and Computation, 206(1), 357-367.
  8. [8]  Yang, X. and Yang, L. (2012), Towards the epidemiological modeling of computer viruses, Discrete Dynamics in Natureand Society, 12(1), 1-20.
  9. [9]  Cohen, F. (1987), Computer viruses: theory and experiments, Computers and Security, 6(1), 22-35.
  10. [10]  Kephart, J. and White, S. (1991), Directed-graph epidemiological models of computer viruses, in Proceedings of IEEE Computer Society Symposium on Research in Security and Privacy, Oakland CA, USA, 343-359.
  11. [11]  Ren, J., Yang, X., Zhu, Q., Yang, L.-X., and Zhang, C. (2012), A novel computer virus model and its dynamics, Nonlinear Analysis: Real World Applications, 13(1), 376-384.
  12. [12]  Zhu, Q., Yang, X., and Ren, J. (2012), Modelling and analysis of the spread of computer virus, Communications in Nonlinear Science and Numerical Simulation, 17(12), 5117-5124.
  13. [13]  Gan, C. and Yang, X. (2015), Theoretical and experimental analysis of the impacts of removable storage media and antivirus software on viral spread, Communications in Nonlinear Science and Numerical Simulation, 22(1-3), 167-174.
  14. [14]  Mishra, B.K. and Pandey, S.K. (2011), Dynamic model of worms with vertical transmission in computer network, Applied Mathematics and Computation, 217(21), 8438-8446.
  15. [15]  Yang, L.-X., Yang, X., and Wu, Y. (2017), the impact of patch forwarding on the prevalence of computer virus: A theoretical assessment approach, Applied Mathematical Modelling, 43, 110-125.
  16. [16]  Gan, C., Yang, X., Zhu, Q., Jin, J., and He, L. (2013), The spread of computer virus under the effect of external computers, Nonlinear Dynamics, 73(3), 1615-1620.
  17. [17]  Yang, L.X., Yang, X., Wen, L., and Liu, J. (2012), A novel computer virus propagation model and its dynamics, International Journal of Computer Mathematics, 89(17), 2307-2314.
  18. [18]  Kumar, U. (2016), A dynamic model on computer virus, Recent Advances in Mathematics Statistics and Computer Science, 2(1), 504-510.
  19. [19]  Doud, E.A., Jebril, I.H., and Zaqaibeh, B. (2008), Computer virus strategies and detection method, International Journalof Open Problems in Computer Science and Mathematics, 1(2), 1-18.
  20. [20]  Ebenezer, B., Farai, N., and Kwesi, A.S. (2015), Fractional dynamics of computer virus propagation, Science Journal of Applied Mathematics and Statistics, 3(3), 63-69.
  21. [21]  Ozdemir, N., U\c{c}ar, S., and Eroglu, B.I. (2019), Dynamical analysis of fractional order model for computer virus propagation with kill signals, International Journal of Nonlinear Sciences and Numerical Simulation, 21(3), 239-247.
  22. [22]  Dubey, V.P., Kumar, R., and Kumar, D. (2020), A hybrid analytical scheme for the numerical computation of time fractional computer virus propagation model and its stability analysis, Chaos Solutions \& Fractals, 133(1), 109-118.
  23. [23]  Arif, M.S., Raza, A., Rafiq, M., Bibi, M., Abbasi, J.N., Nazeer, A., and Javed, U. (2020), Numerical simulations for stochasticcomputer virus propagation model, Computers Materials and Continua, 62(1), 61-77.
  24. [24]  Parsaei, M.R., Javidan, R., Shayegh Kargar, N., and Saberi Nik, H. (2017), On the global stability of an epidemic model of computer viruses, Theory in Biosciences, 136 (3-4), 169-178.
  25. [25]  Lefebvre, M. (2020), A Stochastic model for Computer virus propagation, Journal of Dynamics \& Games, 7(2), 163-174.
  26. [26]  Cheema, A.R. (2020), Numerical treatment for stochastic computer virus model, Computer Modeling in Engineering and Sciences, 120(2), 445-466.
  27. [27]  Madhusudan, M. and Rg, G. (2020), Dynamics of epidemic computer virus spreading model with delays, Wireless Personal Communications, 115(9), 2047-2061.
  28. [28]  Ahmed, N., Fatima, U., Iqbal, S., Raza, A., Rafiq, M., Aziz-ur-Rehman, M., Saeed, S., Khan, I., and Nisar, K.S. (2021), Spatio-temporal dynamics and structure preserving algorithm for computer virus model, Computers, Materials and Continua, 680(1), 201-212.
  29. [29] Fatima, U., Baleanu, D., Ahmed, N., Azam, S., Raza, A., Rafiq, M., and Rehman, M.A.U. (2021), Numerical study of computer virus reaction diffusion epidemic model, Computers, Materials \& Continua, 66(3).
  30. [30]  Unser, M. and Splines, A. (2002), A perfect fit for medical imaging, The International Society for Optical Engineering, 4684.
  31. [31]  Mittal, R.C. and Jain, R.K. (2012), Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions, Communications in Nonlinear Science and Numerical Simulation, 17, 4616-4625.
  32. [32]  Mittal, R.C. and Tripathi, A. (2014), A collocation method for numerical solutions of coupled Burger's equations, International Journal for Computational Methods in Engineering Science and Mechanics, 15, 457-471.
  33. [33]  Rohila, R. and Mittal, R.C. (2018), Numerical study of reaction diffusion Fisher's equation by fourth order cubic B-splines collocation method, Mathematical Sciences, Cross Mark (Elsevier), 12, 79-89.
  34. [34]  Mittal, R.C., Goel, R., and Ahlawat, N. (2021), An efficient numerical simulation of a reaction-diffusion malaria infection model using b-splines collocation, Chaos, Solitons, Fractals, 143, 110566.
  35. [35]  Martin, A. and Boyd, I. (2010), Variant of the Thomas algorithm for opposite-bordered tri-diagonal system of equations, International Journal for Numerical Methods in Biomedical Engineering, 26(6), 752-9.
  36. [36]  Conte, S.D. and De Boor, C. (2017), Elementary Numerical Analysis: An Algorithmic Approach, McGraw-Hill Book Company.
  37. [37]  Sun, Z. and Shu, C.W. (2017), Stability of the fourth order Runge–Kutta method for time-dependent partial differential equations, Annals of Mathematical Sciences and Applications, 2(2), 255-284.

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