Long time behavior of a rumor model with Ornstein-Uhlenbeck process
Authors:
Xiaohuan Wang, Xinyao Wang and Wanli Yang
Journal:
Quart. Appl. Math.
MSC (2010):
Primary 60H10, 60H30, 92D30, 93E15
DOI:
https://doi.org/10.1090/qam/1701
Published electronically:
October 21, 2024
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Additional Information
Abstract: In order to study the propagation of rumors under the influence of media, this paper analyzes a random rumor propagation system with Ornstein-Uhlenbeck process. By constructing the Lyapunov function, we get that the established model has a stationary distribution, which means that rumors will persist under the side effects of the media. In addition, we solve the corresponding matrix and get the exact expression of the probability density near the positive equilibrium. At the end of this paper, numerical simulations verify our results.
References
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- Baoquan Zhou, Bingtao Han, and Daqing Jiang, Ergodic property, extinction and density function of a stochastic SIR epidemic model with nonlinear incidence and general stochastic perturbations, Chaos Solitons Fractals 152 (2021), Paper No. 111338, 20. MR 4307947, DOI 10.1016/j.chaos.2021.111338
- Baoquan Zhou, Daqing Jiang, Bingtao Han, and Tasawar Hayat, Threshold dynamics and density function of a stochastic epidemic model with media coverage and mean-reverting Ornstein-Uhlenbeck process, Math. Comput. Simulation 196 (2022), 15–44. MR 4376473, DOI 10.1016/j.matcom.2022.01.014
References
- Komi Afassinou, Analysis of the impact of education rate on the rumor spreading mechanism, Phys. A 414 (2014), 43–52. MR 3251780, DOI 10.1016/j.physa.2014.07.041
- Tawfiqullah Ayoubi and Haibo Bao, Persistence and extinction in stochastic delay logistic equation by incorporating Ornstein-Uhlenbeck process, Appl. Math. Comput. 386 (2020), 125465, 19. MR 4117312, DOI 10.1016/j.amc.2020.125465
- Junyan Ge, Wenjie Zuo, and Daqing Jiang, Stationary distribution and density function analysis of a stochastic epidemic HBV model, Math. Comput. Simulation 191 (2022), 232–255. MR 4306943, DOI 10.1016/j.matcom.2021.08.003
- Bingtao Han, Baoquan Zhou, Daqing Jiang, Tasawar Hayat, and Ahmed Alsaedi, Stationary solution, extinction and density function for a high-dimensional stochastic SEI epidemic model with general distributed delay, Appl. Math. Comput. 405 (2021), Paper No. 126236, 21. MR 4249728, DOI 10.1016/j.amc.2021.126236
- Bingtao Han and Daqing Jiang, Stationary distribution and extinction of a hybrid stochastic vegetation model with Markov switching, Appl. Math. Lett. 139 (2023), Paper No. 108549, 7. MR 4525469, DOI 10.1016/j.aml.2022.108549
- Desmond J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001), no. 3, 525–546. MR 1872387, DOI 10.1137/S0036144500378302
- Fangju Jia and Guangying Lv, Dynamic analysis of a stochastic rumor propagation model, Phys. A 490 (2018), 613–623. MR 3716391, DOI 10.1016/j.physa.2017.08.125
- Fangju Jia, Guangying Lv, and Guang-an Zou, Dynamic analysis of a rumor propagation model with Lévy noise, Math. Methods Appl. Sci. 41 (2018), no. 4, 1661–1673. MR 3767259, DOI 10.1002/mma.4694
- Fangju Jia, Guangying Lv, Shuangfeng Wang, and Guang-an Zou, Dynamic analysis of a stochastic delayed rumor propagation model, J. Stat. Mech. Theory Exp. 2 (2018), 023502, 20. MR 3772438, DOI 10.1088/1742-5468/aaa798
- D. Jiang, X. Mu, T. Hayat, A. Alsaedi, and Y. Liao, A stochastic turbidostat model with Ornstein–Cuhlenbeck process: dynamics analysis and numerical simulations, Nonlinear Dyn. 107 (2021), 2805–2817, DOI 10.1007/s11071-021-07093-9.
- Daqing Jiang, Jiajia Yu, Chunyan Ji, and Ningzhong Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Modelling 54 (2011), no. 1-2, 221–232. MR 2801881, DOI 10.1016/j.mcm.2011.02.004
- R. Khasminskii, Stochastic stability of differential equations, sijthoff and noordhoff, Alphen aan den Rijn, The Netherlands, 1980, ISBN 978-3-642-23279-4.
- Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, and Bashir Ahmad, Stationary distribution and extinction of a stochastic SEIR epidemic model with standard incidence, Phys. A 476 (2017), 58–69. MR 3623602, DOI 10.1016/j.physa.2017.02.028
- Qun Liu and Daqing Jiang, Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation, Appl. Math. Lett. 73 (2017), 8–15. MR 3659901, DOI 10.1016/j.aml.2017.04.021
- Q. Liu, A stochastic predator-prey model with two competitive preys and Ornstein-Uhlenbeck process, J. Biol. Dyn. 17 (2023), 2193211, DOI 10.1080/17513758.2023.2193211.
- Q. Liu, Dynamical behavior of a stochastic dengue model with Ornstein-Uhlenbeck process, J. Math. Phys. 64 (2023), no. 9, paper no. 092705, 33 pp., DOI 10.1063/5.0147354.
- Qun Liu, Stationary distribution and probability density for a stochastic SISP respiratory disease model with Ornstein-Uhlenbeck process, Commun. Nonlinear Sci. Numer. Simul. 119 (2023), Paper No. 107128, 17. MR 4536714, DOI 10.1016/j.cnsns.2023.107128
- Z. Ma, Y. Zhou, and C. Li, Qualitative and stability methods for ordinary differential equations, Science Press, Beijing, 2015 (Chinese).
- J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101 (2002), no. 2, 185–232. MR 1931266, DOI 10.1016/S0304-4149(02)00150-3
- Andrew J. Majda and Xin T. Tong, Simple nonlinear models with rigorous extreme events and heavy tails, Nonlinearity 32 (2019), no. 5, 1641–1674. MR 3942595, DOI 10.1088/1361-6544/aafbda
- D. Maki and M. Thomson, Mathematical models and applications, with emphasis on social, life, and management sciences, Prentice-Hall, Englewood Cliffs, NJ, 1973.
- X. Mao, Stochastic differential equations and their applications, Horwood Publishing, Chichester, 1997, ISBN 1-898563-26-8.
- Wenqi Pan, Weijun Yan, Yuhan Hu, Ruimiao He, and Libing Wu, Dynamic analysis and optimal control of rumor propagation model with reporting effect, Adv. Math. Phys. , posted on (2022), Art. ID 5503137, 14. MR 4483145, DOI 10.1155/2022/5503137
- L. C. G. Rogers and David Williams, Diffusions, Markov processes, and martingales. Vol. 2, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000. Itô calculus; Reprint of the second (1994) edition. MR 1780932, DOI 10.1017/CBO9781107590120
- H. Roozen, An asymptotic solution to a two-dimensional exit problem arising in population dynamics, SIAM J. Appl. Math. 49 (1989), 1793–1810, DOI 10.1137/0149110.
- Tan Su, Qing Yang, Xinhong Zhang, and Daqing Jiang, Stationary distribution, extinction and probability density function of a stochastic SEIV epidemic model with general incidence and Ornstein-Uhlenbeck process, Phys. A 615 (2023), Paper No. 128605, 20. MR 4555778, DOI 10.1016/j.physa.2023.128605
- Xinru Tong, Haijun Jiang, Xiangyong Chen, Jiarong Li, and Zhen Cao, Deterministic and stochastic evolution of rumor propagation model with media coverage and class-age-dependent education, Math. Methods Appl. Sci. 46 (2023), no. 6, 7125–7139. MR 4564148, DOI 10.1002/mma.8959
- Baoquan Zhou, Bingtao Han, and Daqing Jiang, Ergodic property, extinction and density function of a stochastic SIR epidemic model with nonlinear incidence and general stochastic perturbations, Chaos Solitons Fractals 152 (2021), Paper No. 111338, 20. MR 4307947, DOI 10.1016/j.chaos.2021.111338
- Baoquan Zhou, Daqing Jiang, Bingtao Han, and Tasawar Hayat, Threshold dynamics and density function of a stochastic epidemic model with media coverage and mean-reverting Ornstein-Uhlenbeck process, Math. Comput. Simulation 196 (2022), 15–44. MR 4376473, DOI 10.1016/j.matcom.2022.01.014
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Additional Information
Xiaohuan Wang
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
ORCID:
0000-0003-4073-9347
Email:
xiaohuanw@nuist.edu.cn
Xinyao Wang
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Email:
xinxin@nuist.edu.cn
Wanli Yang
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, 410083, China
MR Author ID:
1416159
Email:
prongs8fin@163.com
Keywords:
Ornstein-Uhlenbeck process,
Rumor model,
Itô’s formula,
stationary distribution,
probability density
Received by editor(s):
July 12, 2024
Received by editor(s) in revised form:
September 17, 2024
Published electronically:
October 21, 2024
Additional Notes:
The first author is the corresponding author.
This work was supported by NSFC of China grants 12171247, 11901158, and the Startup Foundation for Introducing Talent of NUIST.
Article copyright:
© Copyright 2024
Brown University
