A simple approach to the study of global asymptotic stability of some modified continuous-time epidemiological models for distributed denial of service attacks
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A. Ahmad, Y. AbuHour, F. Alghanim, A Novel Model for Distributed Denial of Service Attack Analysis and Interactivity, Symmetry 13 (2021), 2443.
L. J. S. Allen, An Introduction to Mathematical Biology, Prentice Hall, 2007.
U. M. Ascher, L. R. Petzold, Computer Methods for Ordinary Differential Equations and Di_erential-Algebraic Equations, Society for Industrial and Applied Mathematics, SIAM Philadelphia,1998.
R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company Inc., 1953.
F. Brauer, P. Driessche, J. Wu, Mathematical Epidemiology, Springer, Berlin, 2008.
A. M. del Rey, Mathematical modeling of the propagation of malware: a review, Security and Communication Networks 8 (2015), 2561-2579.
C. Gan, X. Yang, Q. Zhu, J. Jin, L. He, The spread of computer virus under the effect of external computers, Nonlinear Dynamics 73 (2013), 1615-1620.
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.
J. D. H. Guillén, A. Martín del Rey, Modeling malware propagation using a carrier compartment, Communications in Nonlinear Science and Numerical Simulation 56 (2018), 217-226.
K. Haldar, B. K. Mishra, A mathematical model for a distributed attack on targeted resources in a computer network, Communications in Nonlinear Science and Numerical Simulation 19 (2014), 3149-3160.
M. T. Hoang, Global asymptotic stability of some epidemiological models for computer viruses and malware using nonlinear cascade systems, Boletín de la Sociedad Matemática Mexicana 28 (2022), Art. 39.
W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London-Series A 115 (1927), 700-721.
W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. -The problem of endemicity, Proceedings of the Royal Society of London-Series A 138 (1932), 55-83.
W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics. III.-Further studies of the problem of endemicity, Proceedings of the Royal Society of London-Series A 141 (1933), 94-122.
H. K. Khalil, Nonlinear systems (Third Edition), Prentice Hall, 2002.
A. Korobeinikov, Global properties of basic virus dynamics models, Bulletin of Mathematical Biology 66 (2004), 879-883.
A. Korobeinikov, G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Applied Mathematics Letters 15 (2002), 955-960.
A. Korobeinikov, Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non{Linear Transmission, Bulletin of Mathematical Biology 30 (2006), 615-626.
A. Korobeinikov, Global Properties of Infectious Disease Models with Nonlinear Incidence, Bulletin of Mathematical Biology 69 (2007), 1871-1886.
A. Korobeinikov, P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Mathematical Biosciences & Engineering 1 (2004), 57-60.
S. Kumari, P. Singh, R. K. Upadhyay, Virus dynamics of a distributed attack on a targeted network: E_ect of firewall and optimal control, Communications in Nonlinear Science and Numerical Simulation 73 (2019), 74-91.
J. La Salle, S. Lefschetz, Stability by Liapunovs Direct Method, Academic Press, New York, 1961.
M. Y. Li, J. S. Muldowney, A geometric approach to global-stability problems, SIAM Journal on Mathematical Analysis 27 (1996), 1070-1083.
A. M. Lyapunov, The Geneml Problem of the Stability of Motion, Taylor & Francis, London, 1992.
M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015.
B. K. Mishra, A. K. Keshri, D. K. Mallick, B. K. Mishra, Mathematical model on distributed denial of service attack through Internet of things in a network, Nonlinear Engineering 8 (2019), 486-495.
B. K. Mishra, S. K. Pandey, Fuzzy epidemic model for the transmission of worms in computer network, Nonlinear Analysis: Real World Applications 11 (2010), 4335-4341.
B. K. Mishra, S. K. Pandey, Dynamic model of worms with vertical transmission in computer network, Applied Mathematics and Computation217 (2011), 8438-8446.
J. R. C. Piqueira, A. A. de Vasconcelos, C. E. C. J. Gabriel, V. O. Araujo, Dynamic models for computer viruses, Computers & Security 27 (2008), 355-359.
J. R. C. Piqueira, V. O. Araujo, A modified epidemiological model for computer viruses, Elsevier Applied Mathematics and Computation 213 (2009), 355-360.
Y. S. Rao, A. K. Keshri, B. K. Mishra, T. C. Panda, Distributed denial of service attack on targeted resources in a computer network for critical infrastructure: A differential e-epidemic model, Physica A: Statistical Mechanics and its Applications 540 (2020), 123240.
J. Ren, X. Yang, Q. Zhu, L. -X. Yang, C. Zhang, A novel computer virus model and its dynamics, Nonlinear Analysis: Real World Applications 13 (2012), 376-384.
P. Seibert, R. Suarez, Global stabilization of nonlinear cascade systems, Systems & Control Letters 14 (1990), 347-352.
A. Stuart, A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1998.
O.A. Toutonji, S.-M. Yoo, M. Park, Stability analysis of VEISV propagation modeling for network worm attack, Applied Mathematical Modelling 36 (2012), 2751-2761.
R. K. Upadhyay, P. Singh, Modeling and control of computer virus attack on a targeted network, Physica A: Statistical Mechanics and its Applications 538 (2020), 122617.
C. Vargas-De-León, Volterra-type Lyapunov functions for fractional-order epidemic systems, Communications in Nonlinear Science and Numerical Simulation 24 (2015), 75-85.
C. Vargas-De-León, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos, Solitons & Fractals 44 (2011), 1106-1110.
C. Vargas-De-León, Lyapunov functions for two-species cooperative systems, Elsevier Applied Mathematics and Computation 219 (2012), 2493-2497.
F. Wang, Y. Zhang, C. Wang, J. Ma, S. Moon, Stability analysis of a SEIQV epidemic model for rapid spreading worms, Computers & Security 29 (2010), 410-418.
R. Wang, Y. Xue, Stability analysis and optimal control of worm propagation model with saturated incidence rate, Computers & Security 125 (2023), 103063.
X. Xiao, P. Fu, C. Dou, Q. Li, G. Hu, S. Xia, Design and analysis of SEIQR worm propagation model in mobile internet, Communications in Nonlinear Science and Numerical Simulation 143 (2017), 341-350.
L.-X. Yang, X. Yang, The effect of infected external computers on the spread of viruses: a compartment modeling study, Physica A: Statistical Mechanics and its Applications 392 (2013), 6523-6535.
L. X. Yang, X. Yang, A new epidemic model of computer viruses, Communications in Nonlinear Science and Numerical Simulation 19 (2014), 935-1944.
L. X. Yang, X. Yang, Q. Zhu, L. Wen, A computer virus model with graded cure rates, Nonlinear Analysis: Real-World Applications 14 (2013), 414-422.
Q. Zhu, X. Yang, L. Yang, X. Zhang, A mixing propagation model of computer viruses and countermeasures, Nonlinear Dynamics 73 (2013), 1433-1441.
DOI: https://doi.org/10.52846/ami.v52i1.1925