×
Zhu, Linhe; Wang, Xuewei; Zhang, Zhengdi; Shen, Shuling

Global stability and bifurcation analysis of a rumor propagation model with two discrete delays in social networks. (English) Zbl 1444.91181

Int. J. Bifurcation Chaos Appl. Sci. Eng. 30, No. 12, Article ID 2050175, 25 p. (2020).
Summary: In this paper, we improve an ignorant-lurker-spreader-removal (ILSR) rumor propagation model as in [A. Yang et al. “ILSR rumor spreading model with degree in complex network”, Phys. A 531, Document ID 121807, 12 p. (2019; doi:10.1016/j.physa.2019.121807)] in social networks with consideration to Logistic growth and two discrete delays. First, we prove the existence of equilibrium points by calculating the basic reproduction number according to the next generation matrix. Regarding the two discrete delays as bifurcating parameters, the local asymptotic stability and Hopf bifurcation of the positive equilibrium point are discussed for six different scenarios by analyzing the characteristic equations of linearized systems. Applying the normal form theory and the center manifold theorem, some important conclusions about the stability and direction of bifurcating periodic solution are given when the two time delays are equal. Subsequently we study the global stability of the equilibrium points by constructing Lyapunov functions when the two delays disappear. Finally, we verify the conclusions through numerical simulations and perform sensitivity analysis on the basic reproduction numbers.

Cited in 6 Documents

MSC:

91D30 Social networks; opinion dynamics
34D23 Global stability of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations

Keywords:

rumor propagation; time delay; Hopf bifurcation; stability

Software:

Python Web Graph Generator
Cite Review PDF
Full Text: DOI

References:

[1] Afassinou, K. [2014] “ Analysis of the impact of education rate on the rumor spreading mechanism,” Physica A414, 43-52. · Zbl 1402.91591
[2] Avila-Vales, E. & Péez, Á. G. C. [2019] “ Dynamics of a time-delayed SIR epidemic model with logistic growth and saturated treatment,” Chaos Solit. Fract.127, 55-69. · Zbl 1448.92273
[3] Berezansky, L., Braverman, E. & Idels, L. [2004] “ Delay differential logistic equation with harvesting,” Math. Comput. Model.40, 1509-1525.
[4] Daley, D. J. & Kendall, D. G. [1964] “ Epidemics and rumours,” Nature204, 1118.
[5] Daley, D. J. & Kendall, D. G. [1965] “ Stochastic rumours,” IMA J. Appl. Math.1, 42-55.
[6] Dong, S., Fan, F. H. & Huang, Y. C. [2018] “ Studies on the population dynamics of a rumor-spreading model in online social networks,” Physica A429, 10-20. · Zbl 1514.91120
[7] Hassard, B., Kazarino, D. & Wan, Y. [1981] Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge). · Zbl 0474.34002
[8] Hua, J., An, L. X. & Li, Y. M. [2015] “ Bionic fuzzy sliding mode control and robustness analysis,” Appl. Math. Model.39, 4482-4493. · Zbl 1443.93072
[9] Hua, Y. H., Pan, Q. H., Hou, W. B. & He, M. F. [2018] “ Rumor spreading model with the different attitudes towards rumors,” Physica A502, 331-344. · Zbl 1514.91143
[10] Huang, L., Wu, G. Q., Zhang, Z. D. & Bi, Q. S. [2019] “ Fast-slow dynamics and bifurcation mechanism in a novel chaotic system,” Int. J. Bifurcation and Chaos29, 1930028-1-17. · Zbl 1448.34037
[11] Huo, L. A. & Song, N. X. [2016] “ Dynamical interplay between the dissemination of scientific knowledge and rumor spreading in emergency,” Physica A461, 73-84. · Zbl 1400.91470
[12] Huo, L. A., Wang, L. & Zhao, X. M. [2019] “ Stability analysis and optimal control of a rumor spreading model with media report,” Physica A517, 551-562. · Zbl 1514.91123
[13] Jia, P. Q., Wang, C., Zhang, G. Y. & Ma, J. F. [2019] “ A rumor spreading model based on two propagation channels in social networks,” Physica A524, 342-353. · Zbl 1535.91024
[14] Li, H. L., Zhang, L., Teng, Z. D., Jiang, Y. L. & Muhammadhaji, A. [2018] “ Global stability of an SI epidemic model with feedback controls in a patchy environment,” Appl. Math. Comput.321, 382-384. · Zbl 1426.92077
[15] Li, J. R., Jiang, H. J., Yu, Z. Y. & Hu, C. [2019] “ Dynamical analysis of rumor spreading model in homogeneous complex networks,” Appl. Math. Comput.359, 374-385. · Zbl 1429.91264
[16] Liu, Q., Jiang, D. Q., Shi, N. Z., Hayat, T. & Alsaedi, A. [2016] “ Periodic solution for a stochastic nonautonomous SIR epidemic model with logistic growth,” Physica A462, 816-826. · Zbl 1400.92518
[17] Liu, Q. M., Li, T. & Sun, M. C. [2017] “ The analysis of an SEIR rumor propagation model on heterogeneous network,” Physica A469, 372-380. · Zbl 1400.91479
[18] Miao, P., Zhang, Z. D., Lim, C. M. & Wang, X. D. [2018] “ Hopf bifurcation and hybrid control of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type II functional response,” Math. Probl. Eng.2018, 6052503. · Zbl 1427.37069
[19] Qian, Z., Tang, S. T., Zhang, X. & Zheng, Z. M. [2015] “ The independent spreaders involved SIR rumor model in complex networks,” Physica A429, 95-102. · Zbl 1400.91435
[20] Song, Y. L. & Yuan, S. L. [2006] “ Bifurcation analysis in a predator-prey system with time delay,” Nonlin. Anal.: Real World Appl.7, 265-284. · Zbl 1085.92052
[21] Upadhyay, R. K., Pal, A. K., Kumari, S. & Roy, P. [2019] “ Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates,” Nonlin. Dyn.96, 2351-2368. · Zbl 1468.92060
[22] Wang, J. X., Wang, C. R., Guo, Z. M., Dong, X. F., Xiao, Y. S., Xue, X. Y., Zhang, X. L. & Liang, X. M. [2014] “ A novel method for characterization and comparison of reversed-phase column selectivity,” J. Chromatogr. A1361, 153-161.
[23] Wang, Z. X., Zhang, Z. D. & Bi, Q. S. [2019] “ Relaxation oscillations in a nonsmooth oscillator with slow-varying external excitation,” Int. J. Bifurcation and Chaos29, 1930019-1-17. · Zbl 1427.34066
[24] Xia, L. L., Jiang, G. P., Song, B. & Song, Y. R. [2015] “ Rumor spreading model considering hesitating mechanism in complex social networks,” Physica A437, 295-303. · Zbl 1400.91439
[25] Yang, A. Z., Hang, X., Cai, X., Zhu, X. & Lu, L. [2019] “ ILSR rumor spreading model with degree in complex network,” Physica A531, 121807. · Zbl 07569487
[26] Yu, Y., Wang, Q. Q., Bi, Q. S. & Lim, C. W. [2019] “ Multiple-s-shaped critical manifold and jump phenomena in low frequency forced vibration with amplitude modulation,” Int. J. Bifurcation and Chaos29, 1930012-1-13. · Zbl 1419.34144
[27] Zhang, R., Wang, Y., Zhang, Z. D. & Bi, Q. S. [2015] “ Nonlinear behaviors as well as the bifurcation mechanism in switched dynamical systems,” Nonlin. Dyn.79, 465-471.
[28] Zhang, Y. H. & Zhu, J. J. [2018] “ Stability analysis of I2S2R rumor spreading model in complex networks,” Physica A503, 862-881. · Zbl 1514.91164
[29] Zhao, L. J., Wang, Q., Cheng, J. J., Chen, Y. C., Wang, J. J. & Huang, W. [2011a] “ Rumor spreading model with consideration of forgetting mechanism,” Physica A390, 2619-2625.
[30] Zhao, L. J., Wang, Q., Cheng, J. J., Chen, Y. C., Wang, J. J. & Huang, W. [2011b] “ Rumor spreading model with consideration of forgetting mechanism: A case of online blogging LiveJournal,” Physica A390, 2619-2625.
[31] Zhao, L. J., Wang, J. J., Chen, Y. C., Wang, Q., Cheng, J. J. & Cui, H. X. [2012] “ SIHR rumor spreading model in social networks,” Physica A391, 2444-2453.
[32] Zhao, L. J., Qiu, X. Y., Wang, X. L. & Wang, J. J. [2013] “ Rumor spreading model considering forgetting and remembering mechanisms in homogeneous networks,” Physica A392, 987-994.
[33] Zhao, H. Y. & Zhu, L. H. [2016] “ Dynamic analysis of a reaction-diffusion rumor propagation model,” Int. J. Bifurcation and Chaos26, 1650101-1-26. · Zbl 1343.35235
[34] Zhao, Z. J., Liu, Y. M. & Wang, K. X. [2016] “ An analysis of rumor propagation based on propagation force,” Physica A443, 263-271. · Zbl 1400.91440
[35] Zhu, L. H., Zhou, X. & Li, Y. M. [2019a] “ Global dynamics analysis and control of a rumor spreading model in online social networks,” Physica A526, 120903. · Zbl 1529.91059
[36] Zhu, L. H., Zhao, H. Y. & Wang, H. Y. [2019b] “ Stability and spatial patterns of an epidemic-like rumor propagation model with diffusions,” Phys. Scr.94, 085007.
[37] Zhu, L. H., Guan, G. & Li, Y. M. [2019c] “ Nonlinear dynamical analysis and control strategies of a network-based SIS epidemic model with time delay,” Appl. Math. Model.70, 512-531. · Zbl 1464.92255
[38] Zhu, L. H., Liu, M. X. & Li, Y. M. [2019d] “ The dynamics analysis of a rumor propagation model in online social networks,” Physica A520, 118-137. · Zbl 1514.91136
[39] Zhu, L. H. & Guan, G. [2019] “ Dynamical analysis of a rumor spreading model with self-discrimination and time delay in complex networks,” Physica A533, 121953. · Zbl 07570020
[40] Zhu, L. H., Liu, W. S. & Zhang, Z. D. [2020] “ Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function,” Appl. Math. Comput.370, 124925. · Zbl 1433.91122
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.