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Optimal Control of Rumor Spreading Model on Homogeneous Social Network with Consideration of Influence Delay of Thinkers

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Abstract

The internet has emerged as a part of life for modern generations. We use social networks to interact with others instantly and make ourselves up to date. A bunch of information and misinformation propagate quickly through social networks. It is necessary to detect the correctness of the information and create a mechanism to control rumor; otherwise, it may be drastic to the whole world in emergencies. Here, we introduce an optimal control of rumor spreading in a homogeneously mixed population considering influence delay of thinkers. Firstly, we formulate the model and obtain possible steady states of the system with their local stability conditions. It is necessary for the extinction of the rumor that the control influence number should not exceed 1. We also observe that the system bifurcates and Hopf bifurcation occurs when the influence delay of thinkers is higher than its threshold value, which may become a reason for panic in emergencies. Secondly, we recognize the most sensitive parameters for the proposed model. Moreover, using Pontryagin’s maximum principle and counter news attack mechanism, we design an optimal rumor control system to minimize the density of rumor adopters and control cost.

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References

  1. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1978)

    MATH  Google Scholar 

  2. Campbell, K.: Fake BBC Twitter Accounts Spread Queen Elizabeth II Death Hoax (2016). https://www.usmagazine.com/celebrity-news. Accessed 29 Dec 2016

  3. Daley, D., Kendall, D.: Stochastic rumours. IMA J. Appl. Math. 1(1), 42–56 (1965)

    Article  Google Scholar 

  4. Dhar, J., Jain, A., Gupta, V.K.: A mathematical model of news propagation on an online social network and a control strategy for rumor spreading. Soc. Netw. Anal. Min. 6(1), 57 (2016). (Springer)

    Article  Google Scholar 

  5. DNA Web Team: Does the new Rs 2000 note have a GPS chip? Here’s the truth! (2016). https://www.dnaindia.com/scitech/report. Accessed 09 Nov 2016

  6. Driver, R.D.: Ordinary and Delay Differential Equations, vol. 20. Springer, New York (2012)

    Google Scholar 

  7. Gao, S., Chen, L., Nieto, J., Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24(35–36), 6037–6045 (2006). (Elsevier)

    Article  Google Scholar 

  8. Halanay, A.: Optimal controls for systems with time lag. Inf. Sci. 6(2), 215–234 (1968). SIAM

    MATH  Google Scholar 

  9. Hattaf, K., Yousfi, N.: Optimal Control of a Delayed HIV Infection model with the Immune Response using an Efficient Numerical Method. Hindawi Publishing Corporation, ISRN Biomathematics, London (2012)

    Book  MATH  Google Scholar 

  10. Hattaf, K., Lashari, A., Louartassi, Y., Yousfi, N.: Electronic Journal of Qualitative Theory of Differential Equations, vol. 3, pp. 1–9. University of Szeged, Szeged (2013)

    Google Scholar 

  11. Jeelani, G.: RBI says Rs 10 coin is valid, those refusing to accept may face legal action (2016). http://www.hindustantimes.com/business-news. Accessed 20 Sep 2016

  12. Kamien, M., Schwartz, N.: Dynamic optimization: The calculus of variations and optimal control in economics and management, Courier Corporation (1991)

  13. Kandhway, K., Kuri, J.: Optimal control of information epidemics modeled as Maki Thompson rumors. Commun. Nonlinear Sci. Numer. Simul. 19(12), 4135–4147 (2014). (Elsevier)

    Article  MATH  Google Scholar 

  14. Kirk, D.E.: Optimal Control Theory: An Introduction. Courier Corporation, Chelmsford (2012)

    Google Scholar 

  15. Laarabi, H., Abta, A., Rachik, M., Bouyaghroumni, J.: Stability analysis of a delayed rumor propagation model. Differ. Equations Dyn. Syst. 24(4), 407–415 (2016). (Springer)

    Article  MATH  Google Scholar 

  16. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models. Springer International Publishing, CRC Press, New York, Boca Raton (2007)

    Book  MATH  Google Scholar 

  17. Liu, Y., Zeng, C., Luo, Y.: Dynamics of a new rumor propagation model with the spread of truth. Appl. Math. 9(5), 536 (2018). (Scientific Research Publishing)

    Article  Google Scholar 

  18. Silverman, C.: Recent research reveals false rumours really do travel faster and further than the truth (2016). https://firstdraftnews.com/. Accessed 06 May 2016

  19. Singh, H., Dhar, J., Bhatti, H.S.: Bifurcation in disease dynamics with latent period of infection and media awareness. Int. J. Bifurc. Chaos 25(6), 1650097 (2016). (World Scientific)

    Article  MATH  Google Scholar 

  20. Smith, H.L., Waltman, P.: The theory of chemostat: Dynamics of microbial competition. J. Am. Chem. Soc. 117, 11616 (1995). [Journal of the American Chemical Society, 118(43), 10678–10678, (1996); ACS Publications]

    MATH  Google Scholar 

  21. Xia, L., Jiang, G., Song, B., Song, Y.: Rumor spreading model considering hesitating mechanism in complex social networks. Phys. A 437, 295–303 (2015). (Elsevier)

    Article  MATH  Google Scholar 

  22. Yang, S., Long, B., Smola, A., Sadagopan, N., Zheng, Z., Zha, H.: Like Like Alike: Joint Friendship and Interest Propagation in Social Networks. Proceedings of the 20th International Conference on World Wide Web, pp. 537–546. ACM, New York (2011)

    Google Scholar 

  23. Zhao, L., Yin, J., Song, Y.: An exploration of rumor combating behavior on social media in the context of social crises. Comput. Hum. Behav. 58, 25–36 (2016). (Elsevier)

    Article  Google Scholar 

  24. Zhu, L., Zhao, H.: Dynamical analysis and optimal control for a malware propagation model in an information network. Neurocomputing 149, 1370–1386 (2015). (Elsevier)

    Article  Google Scholar 

  25. Zhu, L., Zhao, H.: Bifurcation and Control of a Delayed Reaction-Diffusion Rumor Spreading Model with Medium Mechanism, 2016 Chinese Control and Decision Conference (CCDC), pp. 1065–1070. IEEE (2016)

  26. Zhu, Q., Yang, X., Yang, L., Zhang, C.: Optimal control of computer virus under a delayed model. Appl. Math. Comput. 218(23), 11613–11619 (2012). (Elsevier)

    MATH  Google Scholar 

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Acknowledgements

We want to thank the referee for needy comments and suggestions on an earlier version of this paper.

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Correspondence to Ankur Jain.

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Jain, A., Dhar, J. & Gupta, V.K. Optimal Control of Rumor Spreading Model on Homogeneous Social Network with Consideration of Influence Delay of Thinkers. Differ Equ Dyn Syst 31, 113–134 (2023). https://doi.org/10.1007/s12591-019-00484-w

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