Dynamic analysis and optimal control of rumor propagation models considering different education levels and hesitation mechanisms

Hongshuang Wang; Sida Kang; Yuhan Hu; Hongshuang Wang; Sida Kang; Yuhan Hu

doi:10.3934/math.2024979

AIMS Mathematics

2024, Volume 9, Issue 8: 20089-20117. doi: 10.3934/math.2024979

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Dynamic analysis and optimal control of rumor propagation models considering different education levels and hesitation mechanisms

1.
School of Science, University of Science and Technology Liaoning, Anshan 114051, China
2.
School of Business Administration, University of Science and Technology Liaoning, Anshan 114051, China

Received: 08 April 2024 Revised: 21 May 2024 Accepted: 07 June 2024 Published: 20 June 2024
MSC : 34D20, 49J15, 37D35

The spread of rumors has an important impact on the production and life of human society. Moreover, in the process of rumor propagation, individuals with different educational levels show different degrees of trust and ability to spread rumors. Therefore, a new rumor propagation model was established, which considers the influence of education level on rumor propagation. Initially, the basic reproduction number of the model was calculated. Then, we analyzed the existence and stability of the rumor equilibrium point. Next, based on the principle of Pontryagin's maximum value, we obtained a control strategy, which effectively reduced the spread of rumors. Numerical simulations verified the results of theoretical analysis. The results showed that the higher the education level of the population, the slower the spread of rumors to a certain extent, but it could not prevent the spread of rumors. In addition, through the support of the government and the propaganda of the official media, strengthening education can improve people's education level to a certain extent, and then minimize the speed of rumor propagation.
- rumor propagation model,
- basic reproduction number,
- education level,
- stability,
- optimal control
Citation: Hongshuang Wang, Sida Kang, Yuhan Hu. Dynamic analysis and optimal control of rumor propagation models considering different education levels and hesitation mechanisms[J]. AIMS Mathematics, 2024, 9(8): 20089-20117. doi: 10.3934/math.2024979

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Abstract

The spread of rumors has an important impact on the production and life of human society. Moreover, in the process of rumor propagation, individuals with different educational levels show different degrees of trust and ability to spread rumors. Therefore, a new rumor propagation model was established, which considers the influence of education level on rumor propagation. Initially, the basic reproduction number of the model was calculated. Then, we analyzed the existence and stability of the rumor equilibrium point. Next, based on the principle of Pontryagin's maximum value, we obtained a control strategy, which effectively reduced the spread of rumors. Numerical simulations verified the results of theoretical analysis. The results showed that the higher the education level of the population, the slower the spread of rumors to a certain extent, but it could not prevent the spread of rumors. In addition, through the support of the government and the propaganda of the official media, strengthening education can improve people's education level to a certain extent, and then minimize the speed of rumor propagation.

References

[1]	X. Zhao, J. Wang, Dynamical model about rumor spreading with medium, Discrete Dyn. Nat. Soc., 2013 (2013). http://dx.doi.org/10.1155/2013/586867 doi: 10.1155/2013/586867
[2]	D. J. Daley, D. G. Kendall, Stochastic rumors, IMA J. Appl. Math., 1 (1965), 42–55. https://doi.org/10.1093/imamat/1.1.42 doi: 10.1093/imamat/1.1.42
[3]	D. P. Maki, Mathematical models and applications, 1973.
[4]	J. Gu, W. Li, X. Cai, The effect of the forget-remember mechanism on spreading, Eur. Phys. J. B, 62 (2008), 247–255. https://doi.org/10.1140/epjb/e2008-00139-4 doi: 10.1140/epjb/e2008-00139-4
[5]	L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng, H. Cui, Sihr rumor spreading model in social networks, Physica A, 391 (2012), 2444–2453. https://doi.org/10.1016/j.physa.2011.12.008 doi: 10.1016/j.physa.2011.12.008
[6]	W. Jing, L. Min, W. Y. Qi, Z. Z. Chen, Z. L. Qiong, The influence of oblivion-recall mechanism and loss-interest mechanism on the spread of rumors in complex networks, Int. J. Mod. Phys. C, 30 (2019), 1950075. https://doi.org/10.1142/S012918311950075X doi: 10.1142/S012918311950075X
[7]	L. L. Xia, G. P. Jiang, B. Song, Y. R. Song, Rumor spreading model considering hesitating mechanism in complex social networks, Physica A, 437 (2015), 295–303. https://doi.org/10.1016/j.physa.2015.05.113 doi: 10.1016/j.physa.2015.05.113
[8]	Y. H. Xing, G. X. Yang, SE2IR invest market rumor spreading model considering hesitating mechanism, J. Syst. Sci. Inform., 7 (2018), 54–69. https://doi.org/10.21078/JSSI-2019-054-16 doi: 10.21078/JSSI-2019-054-16
[9]	Y. Zan, J. Wu, P. Li, Q. Yu, Sicr rumor spreading model in complex networks: Counterattack and self-resistance, Physica A, 405 (2014), 159–170. https://doi.org/10.1016/j.physa.2014.03.021 doi: 10.1016/j.physa.2014.03.021
[10]	A. Jain, J. Dhar, V. K. Gupta, Optimal control of rumor spreading model on homogeneous social network with consideration of influence delay of thinkers, Differ. Equat. Dyn. Syst., 31 (2023), 113–134. https://doi.org/10.1007/s12591-019-00484-w doi: 10.1007/s12591-019-00484-w
[11]	H. Yao, Y. Zou, Research on rumor spreading model with time delay and control effect, J. Syst. Sci. Inform., 7 (2019), 373–389. https://doi.org/10.21078/JSSI-2019-373-17 doi: 10.21078/JSSI-2019-373-17
[12]	Y. Zhang, Y. Su, L. Weigang, H. Liu, Interacting model of rumor propagation and behavior spreading in multiplex networks, Chaos Soliton. Fract., 121 (2019), 168–177. https://doi.org/10.1016/j.chaos.2019.01.035 doi: 10.1016/j.chaos.2019.01.035
[13]	L. Yang, Z. Li, A. Giua, Containment of rumor spread in complex social networks, Inform. Sci., 506 (2020), 113–130. https://doi.org/10.1016/j.ins.2019.07.055 doi: 10.1016/j.ins.2019.07.055
[14]	K. Afassinou, Analysis of the impact of education rate on the rumor spreading mechanism, Physica A, 414 (2014), 43–52. https://doi.org/10.1016/j.physa.2014.07.041 doi: 10.1016/j.physa.2014.07.041
[15]	Y. Hu, Q. Pan, W. Hou, M. He, Rumor spreading model considering the proportion of wisemen in the crowd, Physica A, 505 (2018), 1084–1094. https://doi.org/10.1016/j.physa.2018.04.056 doi: 10.1016/j.physa.2018.04.056
[16]	D. Li, Y. Zhao, Y. Deng, Rumor spreading model with a focus on educational impact and optimal control, Nonlinear Dynam., 112 (2024), 1575–1597. https://doi.org/10.1007/s11071-023-09102-5 doi: 10.1007/s11071-023-09102-5
[17]	W. Pan, W. Yan, Y. Hu, R. He, L. Wu, G. Rasool, Dynamic analysis and optimal control of rumor propagation model with reporting effect, Adv. Math. Phys., 2022 (2022), 1–14. https://doi.org/10.1155/2022/5503137 doi: 10.1155/2022/5503137
[18]	O. Diekmann, J. Heesterbeek, M. G. Roberts, The construction of next generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010), 873–885. https://doi.org/10.1098/rsif.2009.0386 doi: 10.1098/rsif.2009.0386
[19]	P. Van den Driessche, J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02)00108-6 doi: 10.1016/s0025-5564(02)00108-6
[20]	J. J. Anagnost, C. A. Desoer, An elementary proof of the routh-hurwitz stability criterion, Circ. Syst. Signal Pr., 10 (1991), 101–114. https://doi.org/10.1007/BF01183243 doi: 10.1007/BF01183243
[21]	S. F. Marinósson, Lyapunov function construction for ordinary differential equations with linear programming, Dynam. Syst., 17 (2002), 137–150. https://doi.org/10.1080/0268111011011847 doi: 10.1080/0268111011011847
[22]	J. P. Lasalle, The stability of dynamical systems, Soc. Ind. Appl. Math., 1976. https://doi.org/10.1137/1.9781611970432
[23]	S. Kang, X. Hou, Y. Hu, H. Liu, Dynamical analysis and optimal control of the developed information transmission model, Plos One, 17 (2022), e0268326. https://doi.org/10.1371/journal.pone.0268326 doi: 10.1371/journal.pone.0268326
[24]	F. L. Lewis, D. Vrabie, V. L. Syrmos, Optimal control, John Wiley Sons, 2012. https://doi.org/10.1002/9781118122631
[25]	S. Kang, X. Hou, Y. Hu, H. Liu, Dynamic analysis and optimal control considering cross transmission and variation of information, Sci. Rep., 12 (2022), 18104. https://doi.org/10.1038/s41598-022-21774-4 doi: 10.1038/s41598-022-21774-4
[26]	O. Sharomi, T. Malik, Optimal control in epidemiology, Ann. Oper. Res., 251 (2017), 55–71. https://doi.org/10.1007/s10479-015-1834-4 doi: 10.1007/s10479-015-1834-4
[27]	W. H. Fleming, R. W. Rishel, Deterministic and stochastic optimal control, Springer Science & Business Media, 1 (2012).
[28]	S. Kang, X. Hou, Y. Hu, H. Liu, Dynamic analysis and optimal control of a stochastic information spreading model considering super-spreader and implicit exposer with random parametric perturbations, Front. Phys., 11 (2023), 377. https://doi.org/10.3389/fphy.2023.1194804 doi: 10.3389/fphy.2023.1194804
[29]	M. Hale, Y. Wardi, H. Jaleel, M. Egerstedt, Hamiltonian-based algo rithm for optimal control, arXiv: 1603.02747, 2016. https://doi.org/10.48550/arXiv.1603.02747
[30]	R. E. Kopp, Pontryagin maximum principle, Math. Sci. Eng., Elsevier, 5 (1962), 255–279. https://doi.org/10.1016/S0076-5392(08)62095-0
[31]	L. Bittner, L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, New York, ZAMM-J. Appl. Math. Mech. Z. Angew. Math. Mech., 43 (1962), 514–515. https://doi.org/10.1002/zamm.19630431023 doi: 10.1002/zamm.19630431023

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