Figure 1.
The rumor spreading process of the multi-stage SIR model
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An adaptative model for a multistage structured population under fluctuating environment
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 |
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.
Keywords:
Multi-stage structured SIR model,
large population limit,
rumor spreading model,
asymptotic stability,
threshold phenomena.
Mathematics Subject Classification:
Primary: 34A34, 34D05; Secondary: 00A69.
Citation:
Sun-Ho Choi, Hyowon Seo, Minha Yoo. A multi-stage SIR model for rumor spreading. Discrete and Continuous Dynamical Systems - B,
2020, 25
(6)
: 2351-2372.
doi: 10.3934/dcdsb.2020124
![]()
References:
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118.
doi: 10.1038/2041118a0. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
R. H. Knapp,
A psychology of rumor, Public Opinion Quarterly, 8 (1944), 22-37.
doi: 10.1086/265665. |
| [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. |
| [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. |
| [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. |
| [21] |
D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
A. Sudbury,
The proportion of population never hearing a rumour, Journal of Applied Probability, 22 (1985), 443-446.
doi: 10.2307/3213787. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [33] |
D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901.
doi: 10.1103/PhysRevE.64.050901. |
| [34] |
D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908.
doi: 10.1103/PhysRevE.65.041908. |
| [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. |
| [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.
|
| [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. |
show all references
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [7] |
D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118.
doi: 10.1038/2041118a0. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
R. H. Knapp,
A psychology of rumor, Public Opinion Quarterly, 8 (1944), 22-37.
doi: 10.1086/265665. |
| [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. |
| [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. |
| [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. |
| [21] |
D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
A. Sudbury,
The proportion of population never hearing a rumour, Journal of Applied Probability, 22 (1985), 443-446.
doi: 10.2307/3213787. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [33] |
D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901.
doi: 10.1103/PhysRevE.64.050901. |
| [34] |
D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908.
doi: 10.1103/PhysRevE.65.041908. |
| [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. |
| [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.
|
| [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. |
Figure 4.
Phase diagram $ I(\infty) $ on $ (x,y) = (\lambda_1,\lambda_2) $ with $ \delta_1 = \delta_2 = 0.2 $
Figure 5.
Phase diagram $ R(\infty) $ on $ (x,y) = (\lambda_1,\lambda_2) $ with $ \delta_1 = \delta_2 = 0.2 $
Figure 6.
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 $
Figure 7.
Phase diagram $ I $ on $ (x,y) = (\delta_1,\delta_2) $ with $ \lambda_1 = \lambda_2 = 0.2 $
Figure 8.
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 $
Figure 9.
Surface graph $ R $ on $ (x,y) = (\delta_1,\delta_2) $ with $ \lambda_1 = \lambda_2 = 0.2 $
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