A stochastic model for computer virus propagation

Mario Lefebvre

doi:10.3934/jdg.2020010

Article Contents

2020, Volume 7, Issue 2: 163-174. Doi: 10.3934/jdg.2020010

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A stochastic model for computer virus propagation

Mario Lefebvre

Department of Mathematics and Industrial Engineering, Polytechnique Montréal, C.P. 6079, Succursale Centre-ville, Montréal, H3C 3A7, Canada

Received: January 31, 2020

Revised: February 29, 2020

Published: April 2020

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Abstract

A three-dimensional continuous-time stochastic model based on the classic Kermack-McKendrick model for the spread of epidemics is proposed for the propagation of a computer virus. Moreover, control variables are introduced into the model. We look for the controls that either minimize or maximize the expected time it takes to clean the infected computers, or to protect them from the virus. Using dynamic programming, the equations satisfied by the value functions are derived. Particular problems are solved explicitly.

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Mathematics Subject Classification: 49N90.

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Figure 1. Function $ X(t) $ in the interval $ [0, 80] $ when $ k_1 = k_3 = $ 0.1 and $ k_2 = $ 0.001

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Figure 2. Function $ Y(t) $ in the interval $ [0, 80] $ when $ k_1 = k_3 = $ 0.1 and $ k_2 = $ 0.001

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Figure 3. Function $ Z(t) $ in the interval $ [0, 80] $ when $ k_1 = k_3 = $ 0.1 and $ k_2 = $ 0.001

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Figure 4. Function $ Y(t) $ in the interval $ [0, 2] $ when $ k_i = $ 0.1 for $ i = 1, 2, 3 $

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References

[1]	C. Gan, X. Yang, W. Liu, Q. Zhu and X. Zhang, Propagation of computer virus under human intervention: A dynamical model, Discrete Dyn. Nat. Soc., 2012, Art. ID 106950, 8 pp. doi: 10.1155/2012/106950.
[2]	A. Ionescu, M. Lefebvre and F. Munteanu, Feedback linearization and optimal control of the Kermack-McKendrick model for the spread of epidemics, Advances in Analysis, 2 (2017), 157-166. doi: 10.22606/aan.2017.23003.
[3]	W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1997), 700-721.
[4]	M. Lefebvre, Optimally ending an epidemic, Optimization, 67 (2018), 399-407. doi: 10.1080/02331934.2017.1397147.
[5]	M. Lefebvre, Computer virus propagation modelled as a stochastic differential game, Submitted for publication.
[6]	B. K. Mishra and S. K. Pandey, Dynamic model of worms with vertical transmission in computer network, Appl. Math. Comput., 217 (2011), 8438-8446. doi: 10.1016/j.amc.2011.03.041.
[7]	B. K. Mishra and D. Saini, Mathematical models on computer viruses, Appl. Math. Comput., 187 (2007), 929-936. doi: 10.1016/j.amc.2006.09.062.
[8]	M. Peng, X. He, J. Huang and T. Dong, Modeling computer virus and its dynamics, Math. Probl. Eng., 2013, Art. ID 842614, 5 pp. doi: 10.1155/2013/842614.
[9]	P. Qin, Analysis of a model for computer virus transmission, Math. Probl. Eng., 2015, Art. ID 720696, 10 pp. doi: 10.1155/2015/720696.
[10]	A. Rachah and D. F. M. Torres, Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa, Discrete Dyn. Nat. Soc., 2015, Art. ID 842792, 9 pp. doi: 10.1155/2015/842792.
[11]	H. Song, Q. Wang and W. Jiang, Stability and Hopf bifurcation of a computer virus model with infection delay and recovery delay, J. Appl. Math., 2014, Art. ID 929580, 10 pp. doi: 10.1155/2014/929580.
[12]	X.-J. Tong, M. Zhang and Z. Wang, The cost optimal control system based on the Kermack-Mckendrick worm propagation model, J. Algorithms Comput. Technol., 10 (2016), 82-89. doi: 10.1177/1748301816640704.
[13]	P. Whittle, Optimization over time, in Dynamic Programming and Stochastic Control, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, 1, John Wiley & Sons, Ltd., Chichester, 1982.
[14]	P. Whittle, Risk-sensitive optimal control, in Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1990.
[15]	Y. Xu and J. Ren, Propagation effect of a virus outbreak on a network with limited anti-virus ability, PLoS ONE, 11 (2016), e0164415. doi: 10.1371/journal.pone.0164415.