Abstract
How to accurately model and effectively suppress the spread of network viruses has been a major concern in the field of complex networks and cybersecurity. Most existing work often considers the transmission between the infected and uninfected nodes (i.e., horizontal transmission) and assumes that all new nodes connected to the Internet are susceptible, but the nodes might have been implanted with a backdoor or virus by attackers or infected nodes before connecting to the Internet. This vertical transmission also provides an important route for virus propagation. In this paper, we investigate the propagation of network viruses under the combined influence of network topology and hybrid transmission (i.e., horizontal and vertical transmissions). Through rigorous qualitative analysis, we identify the propagation threshold \(R_0\) which determines whether viruses in the network tend to become extinct or persist, and explore the impacts of vertical transmission on the viral spread. Furthermore, we consider the problem of how to dynamically contain the hybrid spread of network viruses with limited resources. By utilizing optimal control theory, we prove the existence of an optimal control strategy. Finally, a group of representative simulation experiments verify the validity of the theoretical findings. Specifically, the simulation results show that the optimal control strategy proposed in this paper reduces the value of the target generic function J by 67.69% compared with no control.
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References
Addai, E., Zhang, L., Asamoah, J.K., Essel, J.F.: A fractional order age-specific smoke epidemic model. Appl. Math. Model. 119, 99–118 (2023)
Ahmed, N., Fatima, M., Baleanu, D., Nisar, K.S., Khan, I., Rafiq, M., Ahmad, M.O., et al.: Numerical analysis of the susceptible exposed infected quarantined and vaccinated (SEIQV) reaction-diffusion epidemic model. Front. Phys. 7, 220 (2020)
Ahmed, N., Fatima, U., Iqbal, S., Raza, A., Rafiq, M., Aziz-ur Rehman, M., Saeed, S., Khan, I., Nisar, K.S.: Spatio-temporal dynamics and structure preserving algorithm for computer virus model. CMC-Comput. Mater. Con. 68(1), 201–211 (2021)
Akuno, A.O., Ramírez-Ramírez, L.L., Mehta, C., Krishnanunni, C., Bui-Thanh, T., Montoya, J.A.: Multi-patch epidemic models with partial mobility, residency, and demography. Chaos Solutions Fractals 173, 113690 (2023)
Basu, K., Krishnamurthy, P., Khorrami, F., Karri, R.: A theoretical study of hardware performance counters-based malware detection. IEEE Trans. Inf. Forens. Secur. 15, 512–525 (2020)
Dubey, V.P., Kumar, R., Kumar, D.: A hybrid analytical scheme for the numerical computation of time fractional computer virus propagation model and its stability analysis. Chaos Solutions Fractals 133, 109626 (2020)
Essouifi, M., Achahbar, A.: A mixed SIR-SIS model to contain a virus spreading through networks with two degrees. Int. J. Mod. Phys. C 28(09), 1750114 (2017)
Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. SIGCOMM Comput. Commun. Rev. 29(4), 251–262 (1999)
Gan, C., Yang, X., Zhu, Q., Jin, J.: The combined impact of external computers and network topology on the spread of computer viruses. Int. J. Comput. Math. 91(12), 2491–2506 (2014)
Ghafur, S., Kristensen, S., Honeyford, K., Martin, G., Darzi, A., Aylin, P.: A retrospective impact analysis of the Wannacry cyberattack on the NHS. NPJ Digit. Med. 2(1), 1–7 (2019)
Haimed, A., Saba, T., Albasha, A., Rehman, A., Kolivand, M.: Viral reverse engineering using artificial intelligence and big data Covid-19 infection with long short-term memory (LSTM). Environ. Technol. Innov. 4, 101531 (2021)
von den Hoff, P., Thallmair, S., Kowalewski, M., Siemering, R., de Vivie-Riedle, R.: Optimal control theory-closing the gap between theory and experiment. Phys. Chem. Chem. Phys. 14(42), 14460–14485 (2012)
Hosseini, S., Azgomi, M.A.: Dynamical analysis of a malware propagation model considering the impacts of mobile devices and software diversification. Phys. A 526, 120925 (2019)
Huo, L., Lin, T., Fan, C., Chen, L., Zhao, J.: Optimal control of a rumor propagation model with latent period in emergency event. Adv. Differ. Equ. 2015(1), 54 (2015)
Kang, H., Sun, M., Yu, Y., Fu, X., Bao, B.: Spreading dynamics of an SEIR model with delay on scale-free networks. IEEE Trans. Netw. Sci. Eng. 7(1), 489–496 (2018)
Kephart, J.O., White, S.R.: Directed-graph epidemiological models of computer viruses. In: Proceedings, 1991 IEEE Computer Society Symposium on Research in Security and Privacy, Oakland, CA, USA, pp. 343–359. https://doi.org/10.1109/RISP.1991.130801 (1991)
Kephart, J.O., White, S.R.: Measuring and modeling computer virus prevalence. In: Proceedings 1993 IEEE Computer Society Symposium on Research in Security and Privacy, pp. 2–15. IEEE (1993)
Khan, M.A., Khan, Y., Khan, T.W., Islam, S.: Dynamical system of a SEIQV epidemic model with nonlinear generalized incidence rate arising in biology. Int. J. Biomath. 10(07), 1750096 (2017)
Li, P., Yang, L.X., Yang, X., Zhong, X., Wen, J., Xiong, Q.: Energy-efficient patching strategy for wireless sensor networks. Sensors 19(2), 262 (2019)
Mishra, B.K., Pandey, S.K.: Dynamic model of worms with vertical transmission in computer network. Appl. Math. Comput. 217(21), 8438–8446 (2011)
Mishra, B.K., Pandey, S.K.: Effect of anti-virus software on infectious nodes in computer network: a mathematical model. Phys. Lett. A 376, 2389–2393 (2012)
Mishra, B.K., Pandey, S.K.: Dynamic model of worm propagation in computer network. Appl. Math. Model. 38(7–8), 2173–2179 (2014)
Moreno, Y., Pastor-Satorras, R., Vespignani, A.: Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B Condens. Matter Complex Syst. 26(4), 521–529 (2001)
Myilsamy, K., Kumar, M.S., Kumar, A.S.: Optimal control of a rumor model with group propagation over complex networks. Int. J. Mod. Phys. C 32(03), 2150035 (2021)
Nie, Y., Li, W., Pan, L., Lin, T., Wang, W.: Markovian approach to tackle competing pathogens in simplicial complex. Appl. Math. Comput. 417, 126773 (2022)
Nie, Y., Zhong, X., Lin, T., Wang, W.: Homophily in competing behavior spreading among the heterogeneous population with higher-order interactions. Appl. Math. Comput. 432, 127380 (2022)
Nie, Y., Zhong, X., Lin, T., Wang, W.: Pathogen diversity in meta-population networks. Chaos Solutions Fractals 166, 112909 (2023)
Ren, J., Xu, Y.: A compartmental model for computer virus propagation with kill signals. Physica A 486, 446–454 (2017)
Scaife, N., Carter, H., Traynor, P., Butler, K.: Cryptolock (and drop it): stopping ransomware attacks on user data. In: IEEE International Conference on Distributed Computing Systems (2016)
Seidu, B., Makinde, O.D., Asamoah, J.K.K.: Threshold quantities and Lyapunov functions for ordinary differential equations epidemic models with mass action and standard incidence functions. Chaos Solutions Fractals 170, 113403 (2023)
Shahini, M., Ebrahimzadeh, A., Khanduzi, R.: A spectral collocation method for computer virus spread case of delayed optimal control problem. Bull. Iran. Math. Soc. 48(2), 507–535 (2022)
Song, P., Lou, Y., Xiao, Y.: A spatial SEIRS reaction-diffusion model in heterogeneous environment. J. Differ. Equ. 267(9), 5084–5114 (2019)
Staniford, S., Paxson, V., Weaver, N., et al.: How to own the internet in your spare time. In: USENIX Security Symposium, vol. 2, pp. 14–15 (2002)
Szor, P.: The Art of Computer Virus Research and Defense (Symantec Press). Addison-Wesley, Boston (2013)
Taddeo, M., McCutcheon, T., Floridi, L.: Trusting artificial intelligence in cybersecurity is a double-edged sword. Nat. Mach. Intell. 1(12), 557–560 (2019)
Tang, W., Liu, Y.J., Chen, Y.L., Yang, Y.X., Niu, X.X.: Slbrs: network virus propagation model based on safety entropy. Appl. Soft Comput. 97, 106784 (2020)
Upadhyay, R.K., Singh, P.: Modeling and control of computer virus attack on a targeted network. Phys. A 538, 122617 (2020)
Yang, F., Zhang, Z.: Hopf bifurcation analysis of SEIR-KS computer virus spreading model with two-delay. Results Phys. 24, 104090 (2021)
Yang, L., Draief, M., Yang, X.: Heterogeneous virus propagation in networks: a theoretical study. Math. Methods Appl. Sci. 40(5), 1396–1413 (2017)
Yang, L.X., Yang, X.: The effect of network topology on the spread of computer viruses: a modelling study. Int. J. Comput. Math. 94(8), 1591–1608 (2017)
Yang, W.: Dynamical behaviors and optimal control problem of an SEIRS epidemic model with interventions. Bull. Malays. Math. Sci. Soc. 44, 2737–2752 (2021)
Yang, X., Yang, L.X.: Towards the epidemiological modeling of computer viruses. Discrete Dyn. Nat. Soc. 2012, 259671 (2012)
Yin, Z., Yu, Y., Lu, Z.: Stability analysis of an age-structured SEIRS model with time delay. Mathematics 8(3), 455 (2020)
Zhang, C.: Global behavior of a computer virus propagation model on multilayer networks. Secur. Commun. Netw. 2018, 2153195 (2018)
Zhang, C., Huang, H.: Optimal control strategy for a novel computer virus propagation model on scale-free networks. Phys. A 451, 251–265 (2016)
Zhang, L., Fan, X., Teng, Z.: Global dynamics of a nonautonomous SEIRS epidemic model with vaccination and nonlinear incidence. Math. Methods Appl. Sci. 44(2021), 9315–9333 (2021)
Zhang, M., Liu, K., Chen, L., Li, Z.: State feedback impulsive control of computer worm and virus with saturated incidence. Math. Biosci. Eng. 15(6), 1465 (2018)
Zhang, Z., Kundu, S., Wei, R.: A delayed epidemic model for propagation of malicious codes in wireless sensor network. Mathematics 7(5), 396 (2019)
Zhang, Z., Song, L.: Dynamics of a delayed worm propagation model with quarantine. Adv. Differ. Equ. 2017(1), 1–13 (2017)
Zhu, Q., Loke, S.W., Zhang, Y.: State-based switching for optimal control of computer virus propagation with external device blocking. Secur. Commun. Netw. 2018, 1–10 (2018)
Zhu, Q., Yang, X., Yang, L.X., Zhang, C.: Optimal control of computer virus under a delayed model. Appl. Math. Comput. 218(23), 11613–11619 (2012)
Acknowledgements
The authors are grateful to the anonymous reviewers and the editor for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 61903056), and the Chongqing Research Program of Basic Research and Frontier Technology (Nos. cstc2019jcyj-msxmX0681 and cstc2021jcyj-msxmX0761).
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Zhu, Q., Xiang, P., Cheng, K. et al. Hybrid Propagation and Control of Network Viruses on Scale-Free Networks. Bull. Iran. Math. Soc. 49, 87 (2023). https://doi.org/10.1007/s41980-023-00834-z
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DOI: https://doi.org/10.1007/s41980-023-00834-z