April  2020, 7(2): 163-174. doi: 10.3934/jdg.2020010

A stochastic model for computer virus propagation

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

Received  February 2020 Revised  March 2020 Published  April 2020

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.

Citation: Mario Lefebvre. A stochastic model for computer virus propagation. Journal of Dynamics & Games, 2020, 7 (2) : 163-174. doi: 10.3934/jdg.2020010
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.  Google Scholar

[2]

A. IonescuM. 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.  Google Scholar

[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.   Google Scholar

[4]

M. Lefebvre, Optimally ending an epidemic, Optimization, 67 (2018), 399-407.  doi: 10.1080/02331934.2017.1397147.  Google Scholar

[5]

M. Lefebvre, Computer virus propagation modelled as a stochastic differential game, Submitted for publication. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X.-J. TongM. 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.  Google Scholar

[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.  Google Scholar

[14]

P. Whittle, Risk-sensitive optimal control, in Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. IonescuM. 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.  Google Scholar

[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.   Google Scholar

[4]

M. Lefebvre, Optimally ending an epidemic, Optimization, 67 (2018), 399-407.  doi: 10.1080/02331934.2017.1397147.  Google Scholar

[5]

M. Lefebvre, Computer virus propagation modelled as a stochastic differential game, Submitted for publication. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X.-J. TongM. 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.  Google Scholar

[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.  Google Scholar

[14]

P. Whittle, Risk-sensitive optimal control, in Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 1990.  Google Scholar

[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.  Google Scholar

Figure 1.  Function $ X(t) $ in the interval $ [0, 80] $ when $ k_1 = k_3 = $ 0.1 and $ k_2 = $ 0.001
Figure 2.  Function $ Y(t) $ in the interval $ [0, 80] $ when $ k_1 = k_3 = $ 0.1 and $ k_2 = $ 0.001
Figure 3.  Function $ Z(t) $ in the interval $ [0, 80] $ when $ k_1 = k_3 = $ 0.1 and $ k_2 = $ 0.001
Figure 4.  Function $ Y(t) $ in the interval $ [0, 2] $ when $ k_i = $ 0.1 for $ i = 1, 2, 3 $
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