Applied and Computational Mathematics
Volume 7, Issue 1-2, January 2018, Pages: 12-21
Received: Jun. 25, 2017;
Accepted: Aug. 16, 2017;
Published: Sep. 6, 2017
Views 2101 Downloads 220
Pabel Shahrear, Department of Mathematics, Shahjalal University of Science & Technology, Sylhet, Bangladesh
Amit Kumar Chakraborty, Department of Computer Science and Engineering, Metropolitan University, Sylhet, Bangladesh
Md. Anowarul Islam, Department of Mathematics, Shahjalal University of Science & Technology, Sylhet, Bangladesh
Ummey Habiba, Government Teachers Training College, Sylhet, Bangladesh
Computer viruses pose a considerable problem for users of personal computers. In order to effectively defend against a virus, this paper proposes a compartmental model SAEIQRS (Susceptible – Antidotal – Exposed - Infected – Quarantine - Recovered - Susceptible) of virus transmission in a computer network. The differential transformation method (DTM) is applied to obtain an improved solution of each compartment. We have achieved an accuracy of order O(h6) and validated the results of DTM with fourth-order Runge-Kutta (RK4) method. Based on the basic reproduction number, we analyzed the local stability of the model for virus free and endemic equilibria. Using a Lyapunov function, we demonstrated the global stability of virus free equilibria. Numerically the eigenvalues are computed using two different sets of parameter values and the corresponding dominant eigenvalues are determined by means of power method. Finally, we simulate the system in MATLAB. Based on the analysis, aspects of different compartments are investigated.
Amit Kumar Chakraborty,
Md. Anowarul Islam,
Analysis of Computer Virus Propagation Based on Compartmental Model, Applied and Computational Mathematics. Special Issue: Recurrent Neural Networks, Bifurcation Analysis and Control Theory of Complex Systems.
Vol. 7, No. 1-2,
2018, pp. 12-21.
J. Aycock. Computer viruses and malware, volume 22. Springer Science & Business Media, 2006.
J. Li and P. Knickerbocker. Functional similarities between computer worms and biological pathogens. Computers & Security, 26(4): 338-347, 2007.
J. O. Kephart, S. R. White, and D. M. Chess. Computers and epidemiology. IEEE Spectrum, 30(5):20-26, 1993.
B. Ebenezer, N. Farai, and A. A. S Kwesi. Fractional dynamics of computer virus propagation. Science Journal of Applied Mathematics and Statistics, 3(3):63-69, 2015.
J. Hruska. Computer viruses and anti-virus warfare. Ellis Horwood, 1992.
E. Al Daoud, I. H. Jebril, and B. Zaqaibeh. Computer virus strategies and detection methods. International Journal of Open Problems Compt. Math, 1(2):12-20, 2008.
A. Thengade, A. Khaire, D. Mitra, and A. Goyal. Virus detection techniques and their limitations. International Journal of Scientific & Engineering Research, 5(10):1334-1337, 2014.
J. O. Kephart and S. R. White. Measuring and modeling computer virus prevalence. In Research in Security and Privacy, Proceedings, IEEE Computer Society Symposium on, pages 2-15. IEEE, 1993.
X. Zhang, S. Chen, H. Lu, and F. Zhang. An improved computer multi-virus propagation model with user awareness. Journal of Information and Computational Science, pages 4301-4308, 2011.
S. Xu, W. Lu, and Z. Zhan. A stochastic model of multivirus dynamics. IEEE Transactions on Dependable and Secure Computing, 9(1):30-45, 2012.
J. R. C. Piqueira, A. A. De Vasconcelos, C. E. C. J. Gabriel, and V. O. Araujo. Dynamic models for computer viruses. Computers & Security, 27(7):355-359, 2008.
B. K. Mishra and A. K. Singh. Two quarantine models on the attack of malicious objects in computer network. Mathematical Problems in Engineering, 2012:1-13, 2011.
M. Kumar, B. K. Mishra, and T. C. Panda. Effect of quarantine and vaccination on infectious nodes in computer network. International Journal of Computer Networks and Applications, 2(2):92-98, 2015.
B. K. Mishra and N. Jha. SEIQRS model for the transmission of malicious objects in computer network. Applied Mathematical Modelling, 34(3):710-715, 2010.
F. Cohen. Computer viruses: theory and experiments. Computers & Security, 6(1):22-35, 1987.
W. H. Murray. The application of epidemiology to computer viruses. Computers & Security, 7(2):139-145, 1988.
W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. In Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, volume 115, pages 700-721. The Royal Society, 1927.
W. O. Kermack and A. G. McKendrick. Contributions to the mathematical theory of epidemics II, the problem of endemicity. In Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, volume 138, pages 55-83. The Royal Society, 1932.
J. O. Kephart and S. R. White. Directed-graph epidemiological models of computer viruses. In Research in Security and Privacy, 1991. Proceedings, 1991 IEEE Computer Society Symposium on, pages 343-359. IEEE, 1991.
F. Wang, Y. Yang, D. Zhao, and Y. Zhang. A worm defending model with partial immunization and its stability analysis. Journal of Communications, 10(4):276-283, 2015.
I. H. A. H. Hassan. Application to Differential Transformation Method for Solving Systems of Differential Equations. Applied Mathematical Modeling, 32 (2008), PP. 2552–2559.
A. K. Chakraborty, P. Shahrear, M. A. Islam, Analysis of Epidemic Model by Differential Transform Method. Journal of Multidisciplinary Engineering Science and Technology, 4(2): 6574-6581, 2017.
B. K. Mishra and A. Prajapati. Spread of malicious objects in computer network: A fuzzy approach. Applications & Applied Mathematics, 8(2):684-700, 2013.
A. L. V. Barber, C. C. Chavez, and E. Barany. Dynamics of an SAIQR influenza model. BIOMATH, 3(2):1409251, 2014.
J. P. LaSalle. The stability of dynamical systems, volume 25. SIAM, 1976.
F. Brauer and C. C. Chavez. Mathematical models in population biology and epidemiology, volume 40. Springer, 2001.
M. A. Khan, Z. Ali, L. C. C. Dennis, I. Khan, S. Islam, M. Ullah, and T. Gul. Stability analysis of an SVIR epidemic model with non-linear saturated incidence rate. Applied Mathematical Sciences, 9(23):1145-1158, 2015.
A. Arbi, C. Aouiti, F. Chérif, A. Touati and A. M. Alimi. Stability analysis for delayed high-order type of Hopfield neural networks with impulses. Neurocomputing, volume 165, pages 312-329, 2015
A. Arbi, C. Aouiti, F. Chérif, A. Touati and A. M. Alimi. Stability analysis of delayed Hopfield neural networks with impulses via inequality techniques. Neurocomputing, volume 158, pages 281-294, 2015.
A. Arbi, F. Chérif, C. Aouiti and A. Touati. Dynamics of new class of hopfield neural networks with time-varying and distributed delays. Acta Mathematica Scientia, 36(3), 891-912, 2016.