# American Institute of Mathematical Sciences

December  2021, 16(4): 535-552. doi: 10.3934/nhm.2021016

## Rumor spreading dynamics with an online reservoir and its asymptotic stability

 Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 17104, Republic of Korea

* Corresponding author: Hyowon Seo

Received  November 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

The spread of rumors is a phenomenon that has heavily impacted society for a long time. Recently, there has been a huge change in rumor dynamics, through the advent of the Internet. Today, online communication has become as common as using a phone. At present, getting information from the Internet does not require much effort or time. In this paper, the impact of the Internet on rumor spreading will be considered through a simple SIR type ordinary differential equation. Rumors spreading through the Internet are similar to the spread of infectious diseases through water and air. From these observations, we study a model with the additional principle that spreaders lose interest and stop spreading, based on the SIWR model. We derive the basic reproduction number for this model and demonstrate the existence and global stability of rumor-free and endemic equilibriums.

Citation: Sun-Ho Choi, Hyowon Seo. Rumor spreading dynamics with an online reservoir and its asymptotic stability. Networks and Heterogeneous Media, 2021, 16 (4) : 535-552. doi: 10.3934/nhm.2021016
##### References:
 [1] P. Bordia and R. L. Rosnow, Rumor rest stops on the information highway transmission patterns in a computer-mediated rumor chain, Human Communication Research, 25 (1998), 163-179.  doi: 10.1111/j.1468-2958.1998.tb00441.x. [2] J. Borge-Holthoefer, S. Meloni, B. Gonçalves and Y. Moreno, Emergence of influential spreaders in modified rumor models, Journal of Statistical Physics, 151 (2013), 383-393.  doi: 10.1007/s10955-012-0595-6. [3] D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0. [4] J. Dhar, A. Jain and V. K. Gupta, A mathematical model of news propagation on online social network and a control strategy for rumor spreading, Social Network Analysis and Mining, 6 (2016), 57. doi: 10.1007/s13278-016-0366-5. [5] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324. [6] S. Dong, F.-H. Fan and Y.-C. Huang, Studies on the population dynamics of a rumor-spreading model in online social networks, Physica A: Statistical Mechanics and its Applications, 492 (2018), 10-20.  doi: 10.1016/j.physa.2017.09.077. [7] K. Kenney, A. Gorelik and S. Mwangi, Interactive features of online newspapers, First Monday, 5 (2000). doi: 10.5210/fm.v5i1.720. [8] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Mathematical Medicine and Biology: A Journal of the IMA, 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75. [9] S. Kwon, M. Cha, K. Jung, W. Chen and Y. Wang, Prominent features of rumor propagation in online social media, 2013 IEEE 13th International Conference on Data Mining, (2013) 1103–1108. doi: 10.1109/ICDM.2013.61. [10] J. Ma and D. Li, Rumor Spreading in Online-Offline Social Networks, PACIS 2016 Proceedings, 173 (2016). [11] J. Ma, D. Li and Z. Tian, Rumor spreading in online social networks by considering the bipolar social reinforcement, Physica A: Statistical Mechanics and its Applications, 447 (2016), 108-115.  doi: 10.1016/j.physa.2015.12.005. [12] Y. Moreno, M. Nekovee and A. F. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E, 69 (2004), 066130. doi: 10.1103/PhysRevE.69.066130. [13] O. Oh, K. H. Kwon and H. R. Rao, An Exploration of Social Media in Extreme Events: Rumor Theory and Twitter during the Haiti Earthquake 2010, Icis, 231 (2010), 7332-7336. [14] R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics, 87 (2015), 925. doi: 10.1103/RevModPhys.87.925. [15] J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72 (2010), 1506-1533.  doi: 10.1007/s11538-010-9507-6. [16] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [17] L. Zhao, Q. Wang, J. Cheng, Y. Chen, J. Wang and W. Huang, Rumor spreading model with consideration of forgetting mechanism: A case of online blogging LiveJournal, Physica A: Statistical Mechanics and its Applications, 390 (2011), 2619-2625.  doi: 10.1016/j.physa.2011.03.010. [18] L. Zhu, M. Liu and Y. Li, The dynamics analysis of a rumor propagation model in online social networks, Physica A: Statistical Mechanics and its Applications, 520 (2019), 118-137.  doi: 10.1016/j.physa.2019.01.013. [19] L. Zhu, X. Zhou and Y. Li, Global dynamics analysis and control of a rumor spreading model in online social networks, Physica A: Statistical Mechanics and its Applications, 526 (2019), 120903, 15 pp. doi: 10.1016/j.physa.2019.04.139.

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##### References:
 [1] P. Bordia and R. L. Rosnow, Rumor rest stops on the information highway transmission patterns in a computer-mediated rumor chain, Human Communication Research, 25 (1998), 163-179.  doi: 10.1111/j.1468-2958.1998.tb00441.x. [2] J. Borge-Holthoefer, S. Meloni, B. Gonçalves and Y. Moreno, Emergence of influential spreaders in modified rumor models, Journal of Statistical Physics, 151 (2013), 383-393.  doi: 10.1007/s10955-012-0595-6. [3] D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0. [4] J. Dhar, A. Jain and V. K. Gupta, A mathematical model of news propagation on online social network and a control strategy for rumor spreading, Social Network Analysis and Mining, 6 (2016), 57. doi: 10.1007/s13278-016-0366-5. [5] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324. [6] S. Dong, F.-H. Fan and Y.-C. Huang, Studies on the population dynamics of a rumor-spreading model in online social networks, Physica A: Statistical Mechanics and its Applications, 492 (2018), 10-20.  doi: 10.1016/j.physa.2017.09.077. [7] K. Kenney, A. Gorelik and S. Mwangi, Interactive features of online newspapers, First Monday, 5 (2000). doi: 10.5210/fm.v5i1.720. [8] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Mathematical Medicine and Biology: A Journal of the IMA, 21 (2004), 75-83.  doi: 10.1093/imammb/21.2.75. [9] S. Kwon, M. Cha, K. Jung, W. Chen and Y. Wang, Prominent features of rumor propagation in online social media, 2013 IEEE 13th International Conference on Data Mining, (2013) 1103–1108. doi: 10.1109/ICDM.2013.61. [10] J. Ma and D. Li, Rumor Spreading in Online-Offline Social Networks, PACIS 2016 Proceedings, 173 (2016). [11] J. Ma, D. Li and Z. Tian, Rumor spreading in online social networks by considering the bipolar social reinforcement, Physica A: Statistical Mechanics and its Applications, 447 (2016), 108-115.  doi: 10.1016/j.physa.2015.12.005. [12] Y. Moreno, M. Nekovee and A. F. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E, 69 (2004), 066130. doi: 10.1103/PhysRevE.69.066130. [13] O. Oh, K. H. Kwon and H. R. Rao, An Exploration of Social Media in Extreme Events: Rumor Theory and Twitter during the Haiti Earthquake 2010, Icis, 231 (2010), 7332-7336. [14] R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics, 87 (2015), 925. doi: 10.1103/RevModPhys.87.925. [15] J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72 (2010), 1506-1533.  doi: 10.1007/s11538-010-9507-6. [16] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [17] L. Zhao, Q. Wang, J. Cheng, Y. Chen, J. Wang and W. Huang, Rumor spreading model with consideration of forgetting mechanism: A case of online blogging LiveJournal, Physica A: Statistical Mechanics and its Applications, 390 (2011), 2619-2625.  doi: 10.1016/j.physa.2011.03.010. [18] L. Zhu, M. Liu and Y. Li, The dynamics analysis of a rumor propagation model in online social networks, Physica A: Statistical Mechanics and its Applications, 520 (2019), 118-137.  doi: 10.1016/j.physa.2019.01.013. [19] L. Zhu, X. Zhou and Y. Li, Global dynamics analysis and control of a rumor spreading model in online social networks, Physica A: Statistical Mechanics and its Applications, 526 (2019), 120903, 15 pp. doi: 10.1016/j.physa.2019.04.139.
Numerical simulations when $b = 1$, $\sigma_s = 0.5$, and $\sigma_r = 0.5$
Final densities $I(T)$, $S(T)$, $W(T)$, and $R(T)$ with $T = 10^3$
Evolution of the solution with different parameters $\sigma_s$ and $\sigma_r$
Final densities $I(T)$, $S(T)$, $W(T)$, and $R(T)$ with $T = 30$
Comparison of the SIR and SIWR models
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