# American Institute of Mathematical Sciences

June  2020, 25(6): 2351-2372. doi: 10.3934/dcdsb.2020124

## A multi-stage SIR model for rumor spreading

 1 Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 446-701, South Korea 2 National Institute for Mathematical Sciences, Daejeon, 34047, South Korea

* Corresponding author: Hyowon Seo

Received  January 2019 Revised  February 2020 Published  June 2020 Early access  March 2020

We propose a multi-stage structured rumor spreading model that consists of ignorant, new spreader, old spreader, and stifler.We derive a mean field equation to obtain the multi-stage structured model on homogeneous networks. Since rumors spread from a few people, we consider a large population by setting the number of initial spread to one in total population $n$ and limiting $n$ to $\infty$. We investigate a threshold phenomenon of rumor outbreak in the sense of the large population limit by studying the driven multi-stage structured model. The main conclusion of this paper is that the proposed model has a threshold phenomenon in terms of a basic reproduction number which is similar to the SIR epidemic model. We present numerical simulations to show the developed theory numerically.

Citation: Sun-Ho Choi, Hyowon Seo, Minha Yoo. A multi-stage SIR model for rumor spreading. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2351-2372. doi: 10.3934/dcdsb.2020124
##### References:
 [1] C. Barril, Calsina and J. Ripoll, A practical approach to $\mathcal{R}_0$ in continuousime ecological models, Mathematical Methods in the Applied Sciences, 41 (2018), 8432-8445.  doi: 10.1002/mma.4673.  Google Scholar [2] P. Bordia and N. DiFonzo, Problem solving in social interactions on the Internet: Rumor as social cognition, Social Psychology Quarterly, 67 (2004), 33-49.  doi: 10.1177/019027250406700105.  Google Scholar [3] 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.  Google Scholar [4] J. Borge-Holthoefer, S. Meloni, B. Gonalves 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.  Google Scholar [5] J. Borge-Holthoefer and Y. Moreno, Absence of influential spreaders in rumor dynamics, Physical Review E, 85 (2012), 026116. doi: 10.1103/PhysRevE.85.026116.  Google Scholar [6] J. Borge-Holthoefer, A. Rivero and Y. Moreno, Locating privileged spreaders on an online social network, Physical Review E, 85 (2012), 066123. doi: 10.1103/PhysRevE.85.066123.  Google Scholar [7] D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0.  Google Scholar [8] D. J. Daley and D. G. Kendall, Stochastic rumours, IMA Journal of Applied Mathematics, 1 (1965), 42-55.  doi: 10.1093/imamat/1.1.42.  Google Scholar [9] 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.  Google Scholar [10] V. Driessche, Pauline 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.  Google Scholar [11] N. Fountoulakis, K. Panagiotou and T. Sauerwald, Ultra-fast rumor spreading in social networks, in Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2012, 1642–1660. doi: 10.1137/1.9781611973099.130.  Google Scholar [12] B. I. Hong, N. Hahm and S.-H. Choi, SIR rumor spreading model with trust rate distribution, Networks and Heterogeneous Media, 13 (2018), 515-530.  doi: 10.3934/nhm.2018023.  Google Scholar [13] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM Journal on Applied Mathematics, 72 (2012), 25-38.  doi: 10.1137/110826588.  Google Scholar [14] L. A. Huo, L. Wang, N. Song, C. Ma and B. He, Rumor spreading model considering the activity of spreaders in the homogeneous network, Physica A: Statistical Mechanics and its Applications, 468 (2017), 855-865.  doi: 10.1016/j.physa.2016.11.039.  Google Scholar [15] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 69-96.  doi: 10.3934/dcdsb.2006.6.69.  Google Scholar [16] H. Inaba, Threshold and stability results for an age-structured epidemic model, Journal of Mathematical Biology, 28 (1990), 411-434.  doi: 10.1007/BF00178326.  Google Scholar [17] R. H. Knapp, A psychology of rumor, Public Opinion Quarterly, 8 (1944), 22-37.  doi: 10.1086/265665.  Google Scholar [18] T. Kuniya, Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients, Applied Mathematics Letters, 27 (2014), 15-20.  doi: 10.1016/j.aml.2013.08.008.  Google Scholar [19] T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Analysis: Real World Applications, 12 (2011), 2640-2655.  doi: 10.1016/j.nonrwa.2011.03.011.  Google Scholar [20] S. Kwon, M. Cha, K. Jung, W. Chen and Y. Wang, Prominent features of rumor propagation in online social media, in 2013 IEEE 13th International Conference on Data Mining, 2013, 1103–1108. doi: 10.1109/ICDM.2013.61.  Google Scholar [21] D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.  Google Scholar [22] M. McDonald, O. Suleman, S. Williams, S. Howison and N. F. Johnson, Impact of unexpected events, shocking news, and rumors on foreign exchange market dynamics, Physical Review E, 77 (2008), 046110. doi: 10.1103/PhysRevE.77.046110.  Google Scholar [23] Y. Moreno, M. Nekovee and A. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E, 69 (2004), 066130. doi: 10.1103/PhysRevE.69.066130.  Google Scholar [24] M. Nagao, K. Suto and A. Ohuchi, A media information analysis for implementing effective countermeasure against harmful rumor, Journal of Physics, Conference Series, 221 (2010), 012004. doi: 10.1088/1742-6596/221/1/012004.  Google Scholar [25] A. Noymer, The transmission and persistence of urban legends Sociological application of agestructured epidemic models, Journal of Mathematical Sociology, 25 (2001), 299-323.  doi: 10.1080/0022250X.2001.9990256.  Google Scholar [26] J. Ripoll, M. Manzano and E. Calle, Spread of epidemic-like failures in telecommunication networks, Physica A: Statistical Mechanics and its Applications, 410 (2014), 457-469.  doi: 10.1016/j.physa.2014.05.052.  Google Scholar [27] N. Sherborne, K. B. Blyuss and I. Z. Kiss, Dynamics of Multi-stage Infections on Networks, Bulletin of Mathematical Biology, 77 (2015), 1909-1933.  doi: 10.1007/s11538-015-0109-1.  Google Scholar [28] A. Sudbury, The proportion of population never hearing a rumour, Journal of Applied Probability, 22 (1985), 443-446.  doi: 10.2307/3213787.  Google Scholar [29] S. A. Thomas, Lies, damn lies, and rumors: An analysis of collective efficacy, rumors, and fear in the wake of Katrina, Sociological Spectrum, 27 (2007), 679-703.  doi: 10.1080/02732170701534200.  Google Scholar [30] J. Wang, L. Zhao and R. Huang, 2SI2R rumor spreading model in homogeneous networks, Physica A: Statistical Mechanics and its Applications, 413 (2014), 153-161.  doi: 10.1016/j.physa.2014.06.053.  Google Scholar [31] Y.-Q. Wang, X.-Y. Yang, Y.-L. Han and X.-A. Wang, Rumor spreading model with trust mechanism in complex social networks, Communications in Theoretical Physics, 59 (2013), 510-516.   Google Scholar [32] Y. Zan, J. Wu, P. Li and Q. Yua, SICR rumor spreading model in complex networks: Counterattack and self-resistance, Physica A: Statistical Mechanics and its Applications, 405 (2014), 159-170.  doi: 10.1016/j.physa.2014.03.021.  Google Scholar [33] D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901. doi: 10.1103/PhysRevE.64.050901.  Google Scholar [34] D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908. doi: 10.1103/PhysRevE.65.041908.  Google Scholar [35] L. Zhang, Q. Zhong and W. Qi, Two-stage dynamics of crisis information dissemination on social network, in 2009 International Conference on Management Science and Engineering. IEEE, 2009, 2117–2122. doi: 10.1109/ICMSE.2009.5317652.  Google Scholar [36] L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng and H. Cui, SIHR rumor spreading model in social networks, Physica A: Statistical Mechanics and its Applications, 391 (2012), 2444-2453.   Google Scholar [37] 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.  Google Scholar

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##### References:
 [1] C. Barril, Calsina and J. Ripoll, A practical approach to $\mathcal{R}_0$ in continuousime ecological models, Mathematical Methods in the Applied Sciences, 41 (2018), 8432-8445.  doi: 10.1002/mma.4673.  Google Scholar [2] P. Bordia and N. DiFonzo, Problem solving in social interactions on the Internet: Rumor as social cognition, Social Psychology Quarterly, 67 (2004), 33-49.  doi: 10.1177/019027250406700105.  Google Scholar [3] 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.  Google Scholar [4] J. Borge-Holthoefer, S. Meloni, B. Gonalves 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.  Google Scholar [5] J. Borge-Holthoefer and Y. Moreno, Absence of influential spreaders in rumor dynamics, Physical Review E, 85 (2012), 026116. doi: 10.1103/PhysRevE.85.026116.  Google Scholar [6] J. Borge-Holthoefer, A. Rivero and Y. Moreno, Locating privileged spreaders on an online social network, Physical Review E, 85 (2012), 066123. doi: 10.1103/PhysRevE.85.066123.  Google Scholar [7] D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0.  Google Scholar [8] D. J. Daley and D. G. Kendall, Stochastic rumours, IMA Journal of Applied Mathematics, 1 (1965), 42-55.  doi: 10.1093/imamat/1.1.42.  Google Scholar [9] 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.  Google Scholar [10] V. Driessche, Pauline 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.  Google Scholar [11] N. Fountoulakis, K. Panagiotou and T. Sauerwald, Ultra-fast rumor spreading in social networks, in Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2012, 1642–1660. doi: 10.1137/1.9781611973099.130.  Google Scholar [12] B. I. Hong, N. Hahm and S.-H. Choi, SIR rumor spreading model with trust rate distribution, Networks and Heterogeneous Media, 13 (2018), 515-530.  doi: 10.3934/nhm.2018023.  Google Scholar [13] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM Journal on Applied Mathematics, 72 (2012), 25-38.  doi: 10.1137/110826588.  Google Scholar [14] L. A. Huo, L. Wang, N. Song, C. Ma and B. He, Rumor spreading model considering the activity of spreaders in the homogeneous network, Physica A: Statistical Mechanics and its Applications, 468 (2017), 855-865.  doi: 10.1016/j.physa.2016.11.039.  Google Scholar [15] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 69-96.  doi: 10.3934/dcdsb.2006.6.69.  Google Scholar [16] H. Inaba, Threshold and stability results for an age-structured epidemic model, Journal of Mathematical Biology, 28 (1990), 411-434.  doi: 10.1007/BF00178326.  Google Scholar [17] R. H. Knapp, A psychology of rumor, Public Opinion Quarterly, 8 (1944), 22-37.  doi: 10.1086/265665.  Google Scholar [18] T. Kuniya, Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients, Applied Mathematics Letters, 27 (2014), 15-20.  doi: 10.1016/j.aml.2013.08.008.  Google Scholar [19] T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Analysis: Real World Applications, 12 (2011), 2640-2655.  doi: 10.1016/j.nonrwa.2011.03.011.  Google Scholar [20] S. Kwon, M. Cha, K. Jung, W. Chen and Y. Wang, Prominent features of rumor propagation in online social media, in 2013 IEEE 13th International Conference on Data Mining, 2013, 1103–1108. doi: 10.1109/ICDM.2013.61.  Google Scholar [21] D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.  Google Scholar [22] M. McDonald, O. Suleman, S. Williams, S. Howison and N. F. Johnson, Impact of unexpected events, shocking news, and rumors on foreign exchange market dynamics, Physical Review E, 77 (2008), 046110. doi: 10.1103/PhysRevE.77.046110.  Google Scholar [23] Y. Moreno, M. Nekovee and A. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E, 69 (2004), 066130. doi: 10.1103/PhysRevE.69.066130.  Google Scholar [24] M. Nagao, K. Suto and A. Ohuchi, A media information analysis for implementing effective countermeasure against harmful rumor, Journal of Physics, Conference Series, 221 (2010), 012004. doi: 10.1088/1742-6596/221/1/012004.  Google Scholar [25] A. Noymer, The transmission and persistence of urban legends Sociological application of agestructured epidemic models, Journal of Mathematical Sociology, 25 (2001), 299-323.  doi: 10.1080/0022250X.2001.9990256.  Google Scholar [26] J. Ripoll, M. Manzano and E. Calle, Spread of epidemic-like failures in telecommunication networks, Physica A: Statistical Mechanics and its Applications, 410 (2014), 457-469.  doi: 10.1016/j.physa.2014.05.052.  Google Scholar [27] N. Sherborne, K. B. Blyuss and I. Z. Kiss, Dynamics of Multi-stage Infections on Networks, Bulletin of Mathematical Biology, 77 (2015), 1909-1933.  doi: 10.1007/s11538-015-0109-1.  Google Scholar [28] A. Sudbury, The proportion of population never hearing a rumour, Journal of Applied Probability, 22 (1985), 443-446.  doi: 10.2307/3213787.  Google Scholar [29] S. A. Thomas, Lies, damn lies, and rumors: An analysis of collective efficacy, rumors, and fear in the wake of Katrina, Sociological Spectrum, 27 (2007), 679-703.  doi: 10.1080/02732170701534200.  Google Scholar [30] J. Wang, L. Zhao and R. Huang, 2SI2R rumor spreading model in homogeneous networks, Physica A: Statistical Mechanics and its Applications, 413 (2014), 153-161.  doi: 10.1016/j.physa.2014.06.053.  Google Scholar [31] Y.-Q. Wang, X.-Y. Yang, Y.-L. Han and X.-A. Wang, Rumor spreading model with trust mechanism in complex social networks, Communications in Theoretical Physics, 59 (2013), 510-516.   Google Scholar [32] Y. Zan, J. Wu, P. Li and Q. Yua, SICR rumor spreading model in complex networks: Counterattack and self-resistance, Physica A: Statistical Mechanics and its Applications, 405 (2014), 159-170.  doi: 10.1016/j.physa.2014.03.021.  Google Scholar [33] D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901. doi: 10.1103/PhysRevE.64.050901.  Google Scholar [34] D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908. doi: 10.1103/PhysRevE.65.041908.  Google Scholar [35] L. Zhang, Q. Zhong and W. Qi, Two-stage dynamics of crisis information dissemination on social network, in 2009 International Conference on Management Science and Engineering. IEEE, 2009, 2117–2122. doi: 10.1109/ICMSE.2009.5317652.  Google Scholar [36] L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng and H. Cui, SIHR rumor spreading model in social networks, Physica A: Statistical Mechanics and its Applications, 391 (2012), 2444-2453.   Google Scholar [37] 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.  Google Scholar
The rumor spreading process of the multi-stage SIR model
Zero sets of $F_1(x,y)$ and $F_2(x,y)$ on the $xy$ plane with $\sigma = 1$
Numerical solutions of (1) when $k = 0.5$, $\sigma = 1$, and $n = 10^6$
Phase diagram $I(\infty)$ on $(x,y) = (\lambda_1,\lambda_2)$ with $\delta_1 = \delta_2 = 0.2$
Phase diagram $R(\infty)$ on $(x,y) = (\lambda_1,\lambda_2)$ with $\delta_1 = \delta_2 = 0.2$
Time evolution of the solutions $(I,S_1,S_2,R)$ with $k = 0.5$, $\sigma = 1$, $n = 10^6$, $\delta_1 = \delta_2 = 0.2$ and $\lambda_1 = 0.1$
Phase diagram $I$ on $(x,y) = (\delta_1,\delta_2)$ with $\lambda_1 = \lambda_2 = 0.2$
Time evolution of the solutions $(I,S_1,S_2,R)$ with $k = 0.5$, $\sigma = 1$, $n = 10^6$, $\lambda_1 = \lambda_2 = 0.2$, and $\delta_1 = 0.4$
Surface graph $R$ on $(x,y) = (\delta_1,\delta_2)$ with $\lambda_1 = \lambda_2 = 0.2$
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