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On a Rumor Propagation Model with Spatial Heterogeneity

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Abstract

Rumors or wrong information are always spread in different spatial locations by virtue of various available media. Therefore, the propagation ability of rumors or wrong information should be different in different geographical locations. In this paper, we report the dynamical behaviors of a diffusive SI (susceptible-infected) type rumor propagation model in a spatially heterogeneous environment. In view of the fact that rumor-refuting is a common phenomenon in the real world, we introduce this concern in the rumor propagation model. We first derive the properties of uniform boundedness and permanence of the rumor propagation model. These results indicate that the rumor propagation model has at least one rumor-spreading steady state. Thereafter, the asymptotic profiles of the rumor-spreading steady state are reported. We thus find that such rumor-spreading steady state exists if one of the migration rates of the rumor-infected individuals or the rumor-susceptible individuals tends to zero and infinity, respectively. Our theoretical results reveal that this rumor propagation model can admit wealthy dynamical profiles in a spatially heterogeneous environment. Some numerical results are also presented in order to check the theoretical conclusions.

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References

  1. Daley, D.J., Kendall, D.G.: Epidemics and rumours. Nature 204, 1118 (1964)

    Article  Google Scholar 

  2. Sahafizadeh, E., Ladani, B.T.: The impact of group propagation on rumor spreading in mobile social networks. Physica A 506, 412–423 (2018)

    Article  MathSciNet  Google Scholar 

  3. Pastor-Satorras, R., Castellano, C., Mieghem, P.V., et al.: Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925 (2015)

    Article  MathSciNet  Google Scholar 

  4. Wang, Y., Wei, Z.C., Cao, J.D.: Epidemic dynamics of influenza-like diseases spreading in complex networks. Nonlinear Dyn. 101, 1801–1820 (2020)

    Article  Google Scholar 

  5. Jia, F.J., Lv, G.Y., Zou, G.A.: Dynamic analysis of a rumor propagation model with Lévy noise. Math. Methods Appl. Sci. 41, 1661–1673 (2018)

    Article  MathSciNet  Google Scholar 

  6. Zhang, Y.H., Zhu, J.J.: Dynamics of a rumor propagation model with stochastic perturbation on homogeneous social networks. J. Comput. Nonlinear Dyn. 17, 031005 (2022)

    Article  Google Scholar 

  7. Indu, V., Thampi, S.M.: A nature-inspired approach based on forest fire model for modeling rumor propagation in social networks. J. Netw. Comput. Appl. 125, 28–41 (2019)

    Article  Google Scholar 

  8. Myilsamy, K., Kumar, M.S., Kumar, A.S.: Optimal control of a rumor model with group propagation over complex networks. Int. J. Mode. Phys. C 32, 2150035 (2021)

    Article  MathSciNet  Google Scholar 

  9. Dong, S., Huang, Y.C.: A class of rumor spreading models with population dynamics. Commun. Theor. Phys. 70, 795–802 (2018)

    Article  MathSciNet  Google Scholar 

  10. Tian, Y., Ding, X.J.: Rumor spreading model with considering debunking behavior in emergencies. Appl. Math. Comput. 363, 124599 (2019)

    MathSciNet  Google Scholar 

  11. Zehmakan, A.N., Galam, S.: Rumor spreading: a trigger for proliferation or fading away. Chaos 30, 073122 (2020)

    Article  MathSciNet  Google Scholar 

  12. Chen, J., Hu, M.B., Li, M.: Traffic-driven epidemic spreading dynamics with heterogeneous infection rates. Chaos Solit. Fract. 132, 109577 (2020)

    Article  MathSciNet  Google Scholar 

  13. Choi, W., Ahn, I.: Effect of prey-taxis on predator’s invasion in a spatially heterogeneous environment. Appl. Math. Lett. 98, 256–262 (2019)

    Article  MathSciNet  Google Scholar 

  14. Ni, W.J., Shi, J.P., Wang, M.X.: Global stability of spatially nonhomogeneous steady state solution in a diffusive Holling–Tanner predator-prey model. Proc. Am. Math. Soc. 149, 3781–3794 (2021)

    Article  MathSciNet  Google Scholar 

  15. Chen, M.X., Wu, R.C.: Dynamics of diffusive nutrient-microorganism model with spatially heterogeneous environment. J. Math. Anal. Appl. 511, 126078 (2022)

    Article  MathSciNet  Google Scholar 

  16. Li, S.B., Wu, J.H.: Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator-prey system. J. Differ. Equ. 265, 3754–3791 (2018)

    Article  MathSciNet  Google Scholar 

  17. Chen, J., Cao, J.D., Li, M., et al.: Optimizing protection resource allocation for traffic-driven epidemic spreading. Chaos 32, 083141 (2022)

    Article  MathSciNet  Google Scholar 

  18. Tarboush, A.K., Ge, J., Lin, Z.G.: Asymptotic periodicity in a diffusive West Nile virus model in a heterogeneous environment. Int. J. Biomath. 10, 1750110 (2017)

    Article  MathSciNet  Google Scholar 

  19. Lou, Y., Wang, B.: Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment. J. Fix. Point Theor. Appl. 19, 755–772 (2017)

    Article  MathSciNet  Google Scholar 

  20. Zhu, L.H., Huang, X.Y., Liu, Y., Zhang, Z.D.: Spatiotemporal dynamics analysis and optimal control method for an SI reaction-diffusion propagation model. J. Math. Anal. Appl. 493, 124539 (2021)

    Article  MathSciNet  Google Scholar 

  21. Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity. Longman Scientific & Technical Harlow, New York (1991)

    Google Scholar 

  22. Zhu, S.Y., Wang, J.L.: Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source, Commun. Pure. Appl. Anal. 19, 3323–3340 (2020)

    MathSciNet  Google Scholar 

  23. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equation of Second Order. Springer, New York (2001)

    Book  Google Scholar 

  24. Chen, M.X., Wu, R.C.: Patterns in the predator–prey system with network connection and harvesting policy. Math. Methods Appl. Sci. 46(2), 2433–2454 (2023)

    Article  MathSciNet  Google Scholar 

  25. Chen, M.X., Wu, R.C., Wang, X.H.: Non-constant steady states and Hopf bifurcation of a species interaction model. Commun. Nonlinear Sci. Numer. Simul. 116, 106846 (2023)

    Article  MathSciNet  Google Scholar 

  26. Zhang, J.L., Cui, R.H.: Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment. Nonlinear Anal.: RWA 55, 103115 (2020)

    Article  MathSciNet  Google Scholar 

  27. Weinberger, H.F.: Invariant sets for weakly coupled parabolic and elliptic systems. Rend. Mat. 8, 295–310 (1975)

    MathSciNet  Google Scholar 

  28. Le, D.: Dissipativity and global attractors for a class of quasilinear parabolic systems. Commun. Part Differ. Equ. 22, 413–433 (1997)

    Article  MathSciNet  Google Scholar 

  29. Peng, R., Zhao, X.Q.: A reaction–diffusion SIS epidemic model in a timeperiodic environment. Nonlinearity 25, 1451–1471 (2012)

    Article  MathSciNet  Google Scholar 

  30. Magal, X.P., Zhao, Q.: Global attractors and steady states for uniformlly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005)

    Article  MathSciNet  Google Scholar 

  31. Zhao, X.Q.: Uniform persistent and periodic coexistence states in infinite-dimensional periodic seminows with applications. Can. Appl. Math. Q. 3, 473–495 (1995)

    Google Scholar 

  32. Lou, Y., Ni, W.M.: Diffusion, self-diffusion and cross-diffusion. J. Differ. Equ. 131(1), 79–131 (1996)

    Article  MathSciNet  Google Scholar 

  33. Chen, M.X., Wu, R.C., Xu, Y.C.: Dynamics of a depletion-type Gierer–Meinhardt model with Langmuir–Hinshelwood reaction scheme. Discrete Contin. Dyn. Syst. B 27(4), 2275–2312 (2022)

    Article  MathSciNet  Google Scholar 

  34. Lei, C.X., Li, F.J., Liu, J.: Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment. Discrete Contin. Dyn. Syst. B 23(10), 4499–4517 (2018)

    MathSciNet  Google Scholar 

  35. Du, Y., Peng, R., Wang, M.X.: Effect of a protection zone in the diffusive Leslie predator–prey model. J. Differ. Equ. 246, 3932–3956 (2009)

    Article  MathSciNet  Google Scholar 

  36. Nie, Y.Y., Li, W.Y., Pan, L.M., et al.: Markovian approach to tackle competing pathogens in simplicial complex. Appl. Math. Comput. 417, 126773 (2022)

    MathSciNet  Google Scholar 

  37. Nie, Y.Y., Zhong, X.N., Lin, T., et al.: Pathogen diversity in meta-population networks. Chaos Solit. Fract. 166, 112909 (2023)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors express their sincere gratitude to the anonymous referee for her/his careful reading and helpful suggestions which led to great improvements of the presentation of this paper. This work was supported by the China Postdoctoral Science Foundation (No. 2021M701118).

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

    Mengxin Chen

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    Hari Mohan Srivastava

  3. Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Republic of Korea

    Hari Mohan Srivastava

  4. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007, Baku, Azerbaijan

    Hari Mohan Srivastava

  5. Section of Mathematics, International Telematic University Uninettuno, 00186, Rome, Italy

    Hari Mohan Srivastava

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  1. Mengxin Chen
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Contributions

MC: formal analysis, writing-original draft, review and editing, project administration. HMS: writing-original draft, review and editing. All authors reviewed the manuscript.

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Correspondence to Mengxin Chen.

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Chen, M., Srivastava, H.M. On a Rumor Propagation Model with Spatial Heterogeneity. Qual. Theory Dyn. Syst. 23, 1 (2024). https://doi.org/10.1007/s12346-023-00858-0

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