The stationary distribution of a stochastic rumor spreading model

Chaodong Chen; Dapeng Gao; Peng Guo; Chaodong Chen; Dapeng Gao; Peng Guo

doi:10.3934/math.2021076

AIMS Mathematics

2021, Volume 6, Issue 2: 1234-1248. doi: 10.3934/math.2021076

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Research article

The stationary distribution of a stochastic rumor spreading model

1.
School of Public Administration, Sichuan University, Chengdu, Sichuan 610065, China
2.
College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China
3.
School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637002, China
4.
Internet of Things Perception and Big Data Analysis Key Laboratory of Nanchong, Nanchong, Sichuan 637002, China

Received: 21 August 2020 Accepted: 29 October 2020 Published: 12 November 2020
MSC : 60G10, 60H10, 60J65, 60J70

In this paper, we develop a rumor spreading model by introducing white noise into the model. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. Finally, we provide some numerical simulations to illustrate the analytical results.
- rumor spreading,
- stationary distribution,
- threshold
Citation: Chaodong Chen, Dapeng Gao, Peng Guo. The stationary distribution of a stochastic rumor spreading model[J]. AIMS Mathematics, 2021, 6(2): 1234-1248. doi: 10.3934/math.2021076

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Abstract

In this paper, we develop a rumor spreading model by introducing white noise into the model. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. Finally, we provide some numerical simulations to illustrate the analytical results.

References

[1]	K. Afassinou, Analysis of the impact of education rate on the rumor spreading mechanism, Physica A, 414 (2014), 43-52. doi: 10.1016/j.physa.2014.07.041
[2]	D. J. Daley, D. G. Kendall, Epidemics and rumors, Nature, 204 (1964), 1118.
[3]	F. J. Jia, G. Y. Lv, Dynamics analysis of a stochastic rumor propagation model, Physica A, 490 (2018), 613-623. doi: 10.1016/j.physa.2017.08.125
[4]	D. H. He, L. G. Xu, Boundedness analysis of stochastic integro-differential systems with Lévy noise, J. Taibah Univ. Sci., 14 (2019), 1-7.
[5]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302
[6]	H. X. Hu, L. G. Xu, Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations, J. Math. Anal. Appl., 466 (2018), 896-926. doi: 10.1016/j.jmaa.2018.06.025
[7]	R. Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2012.
[8]	D. P. Maki, M. Thomson, Mathematical Models and Applications, with Emphasis on Social Life, and Management Sciences, 1973.
[9]	X. R. Mao, G. Marion, E. Renshaw, Environmental Brownian noise supresses expolosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0
[10]	B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 2007.
[11]	Y. Tian, X. J. Ding, Rumor spreading model with considering debunking behavior in emergencies, Appl. Math. Comput., 363 (2019), 124599.
[12]	D. Y. Xu, B. Li, S. J. Long, L. Y. Teng, Moment estimate and existence for soutions of stochastic functional differential equations, Nonlinear Anal-Theor., 108 (2014), 128-143. doi: 10.1016/j.na.2014.05.004
[13]	L. G. Xu, Z. L. Dai, D. H. He, Exponential ultimate boundedness of impulsive stochastic delay differential equations, Appl. Math. Lett., 85 (2018), 70-76. doi: 10.1016/j.aml.2018.05.019
[14]	D. H. Zanette, Critical behaviour of propagation on small-world networks, Phys. Rev. E, 64 (2001), 050901. doi: 10.1103/PhysRevE.64.050901
[15]	L. J. Zhao, J. J. Wang, Y. C. Chen, Q. Wang, J. J. Cheng, H. X. Cui, SIHR rumor spreading model in social networks, Physica A: Statistical Mechanics and its Applications, 391 (2012), 2444-2453. doi: 10.1016/j.physa.2011.12.008
[16]	L. Zhu, Y. G. Wang, Rumor spreading model with noise interference in complex social networks, Physica A, 469 (2017), 750-760. doi: 10.1016/j.physa.2016.11.119

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