# American Institute of Mathematical Sciences

March  2022, 27(3): 1827-1851. doi: 10.3934/dcdsb.2021111

## Phase transitions of the SIR Rumor spreading model with a variable trust rate

 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  November 2020 Published  March 2022 Early access  April 2021

We study a threshold phenomenon of rumor outbreak on the SIR rumor spreading model with a variable trust rate depending on the populations of ignorants and spreaders. Rumor outbreak in the SIR rumor spreading model is defined as a persistence of the final rumor size in the large population limit or thermodynamics limit $(n\to \infty)$, where $1/n$ is the initial population of spreaders. We present a rigorous proof for the existence of threshold on the final size of the rumor with respect to the basic reproduction number $\mathcal{R}_0$. Moreover, we prove that a phase transition phenomenon occurs for the final size of the rumor (as an order parameter) with respect to the basic reproduction number and provide a criterion to determine whether the phase transition is of first or second order. Precisely, we prove that there is a critical number $\mathcal{R}_1$ such that if $\mathcal{R}_1>1$, then the phase transition is of the first order, i.e., the limit of the final size is not a continuous function with respect to $\mathcal{R}_0$. The discontinuity is a jump-type discontinuity and it occurs only at $\mathcal{R}_0 = 1$. If $\mathcal{R}_1<1$, then the phase transition is second order, i.e., the limit of the final size is continuous with respect to $\mathcal{R}_0$ and its derivative exists, except at $\mathcal{R}_0 = 1$, and the derivative is not continuous at $\mathcal{R}_0 = 1$. We also present numerical simulations to demonstrate our analytical results for the threshold phenomena and phase transition order criterion.

Citation: Sun-Ho Choi, Hyowon Seo, Minha Yoo. Phase transitions of the SIR Rumor spreading model with a variable trust rate. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1827-1851. doi: 10.3934/dcdsb.2021111
##### References:
 [1] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8. [2] D.J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0. [3] 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. [4] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, 20 (1919), 292-296.  doi: 10.2307/1967124. [5] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Analysis: Real World Applications, 13 (2012), 1866-1872.  doi: 10.1016/j.nonrwa.2011.12.015. [6] B. I. Hong, N. Hahm and S.-H. Choi, SIR rumor spreading model with trust rate distribution, Netwworks & Heterogeneous Media, 13 (2018), 515-530.  doi: 10.3934/nhm.2018023. [7] A. Mavragani and G. Ochoa, The internet and the anti-vaccine movement: Tracking the 2017 EU measles outbreak, Big Data and Cognitive Computing, 2 (2018), 2. doi: 10.3390/bdcc2010002. [8] Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical Journal B-Condensed Matter and Complex Systems, 26 (2002), 521-529.  doi: 10.1140/epjb/e20020122. [9] J. D. Murray, Mathematical Biology, Springer Verlag, Berlin, 1993. doi: 10.1007/b98869. [10] M. Nekovee, Y. Moreno, G. Bianconi and M. Marsili, Theory of rumour spreading in complex social networks, Physica A: Statistical Mechanics and its Applications, 374 (2007), 457-470.  doi: 10.1016/j.physa.2006.07.017. [11] J. K. Olive, P. J. Hotez, A. Damania and M. S. Nolan, The state of the antivaccine movement in the United States: A focused examination of nonmedical exemptions in states and counties, PLoS medicine, 15 (2018), e1002578. doi: 10.1371/journal.pmed.1002578. [12] M. V. Pezzo and J. W. Beckstead, A multilevel analysis of rumor transmission: Effects of anxiety and belief in two field experiments, Basic and Applied Social Psychology, 28 (2006), 91-100.  doi: 10.1207/s15324834basp2801_8. [13] 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. [14] C. Wang, Z. X. Tan, Y. Ye, L. Wang, K. H. Cheong and N. G. Xie, A rumor spreading model based on information entropy, Scientific reports, 7 (2017), 1-14.  doi: 10.1038/s41598-017-09171-8. [15] 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.  doi: 10.1088/0253-6102/59/4/21. [16] D. B. Wright, K. London and M. Waechter, Social anxiety moderates memory conformity in adolescents, Applied Cognitive Psychology, 24 (2010), 1034-1045.  doi: 10.1002/acp.1604. [17] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025. [18] Y. Zan, J. Wu, P. Li and Q. Yu, 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. [19] 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.  doi: 10.1016/j.physa.2011.12.008. [20] L. Zhao, H. Cui, X. Qiu, X. Wang and J. Wang, SIR rumor spreading model in the new media age, Physica A: Statistical Mechanics and its Applications, 392 (2013), 995-1003.  doi: 10.1016/j.physa.2012.09.030.

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##### References:
 [1] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8. [2] D.J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0. [3] 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. [4] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, 20 (1919), 292-296.  doi: 10.2307/1967124. [5] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Analysis: Real World Applications, 13 (2012), 1866-1872.  doi: 10.1016/j.nonrwa.2011.12.015. [6] B. I. Hong, N. Hahm and S.-H. Choi, SIR rumor spreading model with trust rate distribution, Netwworks & Heterogeneous Media, 13 (2018), 515-530.  doi: 10.3934/nhm.2018023. [7] A. Mavragani and G. Ochoa, The internet and the anti-vaccine movement: Tracking the 2017 EU measles outbreak, Big Data and Cognitive Computing, 2 (2018), 2. doi: 10.3390/bdcc2010002. [8] Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical Journal B-Condensed Matter and Complex Systems, 26 (2002), 521-529.  doi: 10.1140/epjb/e20020122. [9] J. D. Murray, Mathematical Biology, Springer Verlag, Berlin, 1993. doi: 10.1007/b98869. [10] M. Nekovee, Y. Moreno, G. Bianconi and M. Marsili, Theory of rumour spreading in complex social networks, Physica A: Statistical Mechanics and its Applications, 374 (2007), 457-470.  doi: 10.1016/j.physa.2006.07.017. [11] J. K. Olive, P. J. Hotez, A. Damania and M. S. Nolan, The state of the antivaccine movement in the United States: A focused examination of nonmedical exemptions in states and counties, PLoS medicine, 15 (2018), e1002578. doi: 10.1371/journal.pmed.1002578. [12] M. V. Pezzo and J. W. Beckstead, A multilevel analysis of rumor transmission: Effects of anxiety and belief in two field experiments, Basic and Applied Social Psychology, 28 (2006), 91-100.  doi: 10.1207/s15324834basp2801_8. [13] 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. [14] C. Wang, Z. X. Tan, Y. Ye, L. Wang, K. H. Cheong and N. G. Xie, A rumor spreading model based on information entropy, Scientific reports, 7 (2017), 1-14.  doi: 10.1038/s41598-017-09171-8. [15] 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.  doi: 10.1088/0253-6102/59/4/21. [16] D. B. Wright, K. London and M. Waechter, Social anxiety moderates memory conformity in adolescents, Applied Cognitive Psychology, 24 (2010), 1034-1045.  doi: 10.1002/acp.1604. [17] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025. [18] Y. Zan, J. Wu, P. Li and Q. Yu, 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. [19] 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.  doi: 10.1016/j.physa.2011.12.008. [20] L. Zhao, H. Cui, X. Qiu, X. Wang and J. Wang, SIR rumor spreading model in the new media age, Physica A: Statistical Mechanics and its Applications, 392 (2013), 995-1003.  doi: 10.1016/j.physa.2012.09.030.
Time evolutions of $(I(t), S(t), R(t))$ with initial data (4.1) when $\mathcal{R}_0 = 0.5$
Time evolutions of $(I(t), S(t), R(t))$ with initial data (4.1) when $\mathcal{R}_0 = 2$
Numerical simulations for $\phi^\infty_n$ when $T = 10^4$
Numerical simulations for $\phi^\infty_n$ when $T = 10^6$
Phase transition diagrams for $\phi^e$ when $n = 10^{10}$ and final time $T = 10^6$
Phase portrait for $\sigma_R>\sigma_S$ or $\lambda_S^0>0$
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