# American Institute of Mathematical Sciences

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 and 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. [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.

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. [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.
Function $X(t)$ in the interval $[0, 80]$ when $k_1 = k_3 =$ 0.1 and $k_2 =$ 0.001
Function $Y(t)$ in the interval $[0, 80]$ when $k_1 = k_3 =$ 0.1 and $k_2 =$ 0.001
Function $Z(t)$ in the interval $[0, 80]$ when $k_1 = k_3 =$ 0.1 and $k_2 =$ 0.001
Function $Y(t)$ in the interval $[0, 2]$ when $k_i =$ 0.1 for $i = 1, 2, 3$
 [1] Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934/mbe.2017065 [2] Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065 [3] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [4] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [5] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [6] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [7] Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure and Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211 [8] Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199 [9] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [10] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047 [11] Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084 [12] Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015 [13] Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021020 [14] Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025 [15] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [16] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [17] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [18] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [19] Hassan Allouba. Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 413-463. doi: 10.3934/dcds.2013.33.413 [20] Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

Impact Factor: