Rumor spreading is the circulation of doubtful messages on the social network. Fact retrieving process that aims at preventing the spread of the rumor, appears to have a significant global impact. In this research, we have investigated a mathematical model projecting rumor spread by considering six groups of individuals namely ignorant, exposed, intentional rumor spreader, unintentional rumor spreader, stifler, and fact retriever. To represent the current abnormal and fast pattern of the message spread around various platforms, in the projected model, we have implemented the fractional derivative in the Caputo context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the rumor-free and rumor equilibrium points, and the global stability of the rumor-free equilibrium point. Computing basic reproduction numbers, we have analyzed the existence and stability of points of equilibrium. The sensitivity of basic reproduction numbers is also examined. Importance of the fact retrieving drive is highlighted by relating it to the basic reproduction number. Finally, by applying the Adams-Bashforth-Moulton method, we have presented the numerical results by capturing the profile of each of the groups under the influence of fractional derivative and investigated the impact of rumor verification rate and contact rate in controlling and preventing the rumor. With the Caputo fractional operator in the projected model, the current research highlights the significance of the fact retriever and the curb in individual contact and captures the relevant consequences.
[1] S. J. Achar, C. Baishya, P. Veeresha, and L. Akinyemi, Dynamics of fractional model of biological pest control in tea plants with Beddington-DeAngelis functional response, Fractal fract., 6(1) (2022).
[2] D. M. Ali, E. Celik, H. Bulut, and H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, The Eur. Phys. J. Plus, 133 (2018), 1-6.
[3] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.
[4] A.O. Atede, A. Omame, and S. C. Inyama, fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data, Bulletin of Biomathematics, 1(1) (2023), 78-110.
[5] C. Baishya, S. J. Achar, P. Veeresha, and D. G. Prakasha, Dynamics of a fractional epidemiological model with disease infection in both the populations, Chaos, 31(4) (2021).
[6] C. Baishya and P. Veeresha, Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel, Proc. R. Soc., A, 477 (2021).
[7] C. Baishya, Dynamics of Fractional Holling Type-II Predator-Prey Model with Prey Refuge and Additional Food to Predator, J. Appl. Nonlinear Dyn., 10(2) (2021), 315-328.
[8] C. Baishya, An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative, SeMA Journal, 79(4) (2021), 699-717.
[9] B. Beisner, D. Haydon, and K. Cuddington, Hysteresis, Encyclopedia of Ecology, (2008), 1930-1935.
[10] E. A. Bhih, R. Ghazzali, R. S. Ben, M. Rachik, and L. A. E. Alami, A Discrete Mathematical Modeling and Optimal Control of the Rumor Propagation in Online Social Network, Discrete Dyn. Nat. Soc., (2020), 1-12.
[11] M. Caputo , Linear models of dissipation whose q is almost frequency independent-II, Geophys. J. Int., 13(5) (1967), 529-539.
[12] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1(2) (2015), 73-85.
[13] D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204:1118 (1964).
[14] J. Dhar, A. Jain, and K. V. Gupta, A mathematical model of news propagation on online social network and a control strategy for rumor spreading, Soc. Netw. Anal. Min., 6(1) (2016), 1-9.
[15] P. M. Daniel, Mathematical Models and Applications, With Emphasis on the Social, Life, and Management Sciences, Prentice Hall College Div, Englewood Cliffs, NJ, 1973.
[16] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5(1) (1997), 1-6.
[17] K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265(2) (2002), 229-248.
[18] K. Diethelm, N. J. Ford, and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3-22.
[19] X. Ding, X. Zhang, R. Fan, Q. Xu, K. Hunt, and J. Zhuang, Rumor recognition behavior of social media users in emergencies, J. Manag. Sci. Eng., 7(1) (2022), 36-47.
[20] F. Evirgen, E. Ucar, S. Ucar, and N. Ozdemir, Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates, Mathematical Modelling and Numerical Simulation with Applications, 3(1) (2023), 58-72.
[21] R. Escalante and M. Odehnal, A deterministic mathematical model for the spread of two rumor, Afrika Matematika, 31(2) (2020), 315-331.
[22] B. Ghanbari, H. Gunerhan, and H. M. Srivastava, An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model, Chaos, Solitons & Fractals, 138 (2020).
[23] M. Gholami, R. K. Ghaziani, and Z. Eskandari, Three-dimensional fractional system with the stability condition and chaos control, Mathematical Modelling and Numerical Simulation with Applications, 2(1) (2022), 41-47.
[27] F. Haq, K. Shah, G. U. Rahman, and M. Shahzad, Numerical analysis of fractional order model of HIV-1 infection of CD4+ T-cells, Comput. methods differ. equ. , 5(1) (2017), 1-11.
[28] H. Joshi and B. K. Jha, Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative., Mathematical Modelling and Numerical Simulation with Applications, 1(2) (2021), 84-94.
[29] H. Joshi, M. Yavuz, S. Townley, and B. K. Jha, Stability analysis of a non-singular fractional-order COVID-19 model with nonlinear incidence and treatment rate, Physica Scripta, 98(4) (2023).
[30] H. Joshi and M. Yavuz, Transition dynamics between a novel coinfection model of fractional-order for COVID-19 and tuberculosis via a treatment mechanism, The Eur. Phys. J. Plus, 138(5) (2023).
[31] H. Joshi, B. K. Jha, and M. Yavuz, Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data, Math. Biosci. Eng., 20(1) (2023), 213-240.
[32] M. A. Khan, Z. Hammouch, and D. Baleanu, Modelling the dynamics of hepatitis E via the CaputoFabrizio derivative, Math. Model. Nat. Phenom., 14(3) (2019).
[33] A. J. Kimmel, Rumors and Rumor Control: A Managers Guide to Understanding and Combatting Rumors, Taylor & Francis, New York 2004.
[34] I. Koca, Analysis of rubella disease model with non-local and non-singular fractional derivatives, Int. J. Optim. Control: Theor. Appl., 8(1) (2018), 17-25.
[35] M. Kosfeld, Rumours and markets, J. Math. Econ., 41(6) (2005), 646-664.
[36] C. Li and C. Tao, On the fractional Adams method, Comput. Math. with Appl., 58(8) (2009), 1573-1588.
[37] H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, and Z. Teng, Dynamical analysis of a fractional-order predator prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435-449.
[38] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1(2) (2015), 87-92.
[39] K. S. Miller and B. Ross,An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
[40] G. D. Moody and D. F. Galletta, Lost in cyberspace: The impact of information scent and time constraints on stress, performance, and attitudes online, Journal of Management Information Systems, 32(1) (2015), 192-224.
[41] J. D. Murray, Mathematical Modelling in Epidemiology, Springer, Berlin, 1980.
[42] P. A. Naik, M. Yavuz, S. Qureshi, J. Zu, and S. Townley, Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, The Eur. Phys. J. Plus, (2020), 1-42.
[43] M. Nekovee, Y. Moreno, G. Bianconi, and M. Marsili, Theory of rumour spreading in complex social networks, Phys. A: Stat. Mech. Appl., 374(1) (2007), 457-470.
[44] O. Oh, M. Agrawal, and H. R. Rao, Community Intelligence and Social Media Services: A Rumor Theoretic Analysis of Tweets During Social Crises, MIS quarterly, 37(2) (2013), 407-426.
[45] K. M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 115 (2018), 127-134.
[46] F. Ozkose, et al., A fractional modeling of tumor-immune system interaction related to Lung cancer with real data, The Eur. Phys. J. Plus, 137 (2022), 1-28.
[47] A. Pimenov, et al., Memory effects in population dynamics : Spread of infectious disease as a case study, Math. Model. Nat. Phenom, 7(3) (2012), 204-226.
[48] I. Podlubny, Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, 1998.
[49] A. R. Putri, M. A. Saida, and M. Syafwan, Dynamics of the Rumor Spreading model of Indonesia Twitter Case, BAREKENG: Jurnal Ilmu Matematika dan Terapan, 16(2) (2022), 625-634.
[50] S. Sathe, Rumor spreading in LiveJournal, Mini-Project Report, Dynamical Networks, 2008.
[51] A. Taghvaei, T. T. Georgiou, L. Norton, and A. Tannenbaum, Fractional SIR epidemiological models, Scientific Reports, 10(1) (2020).
[52] P. Veeresha, D. G. Prakasha, and H. M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos, 29(1) (2019).
[53] B. Wang and L. Q. Chen, Asymptotic stability analysis with numerical confirmation of an axially accelerating beam constituted by the standard linear solid model, J. Sound Vib., 328(4-5) (2009), 456-466.
[54] T. K. Yamana, X. Qiu, and E. A. Eltahir, Hysteresis in simulations of malaria transmission, Adv. Water Resour., 108 (2017), 416-422.
[55] M. Yavuz, F. Ozkose, M. Susam, and M. Kalidass, A new modeling of fractional-order and sensitivity analysis for hepatitis-b disease with real data, Fractal fract., 7(2) (2023).
[56] M. Yavuz and W. Y. A. Haydar, A new mathematical modelling and parameter estimation of COVID-19: a case study in Iraq, AIMS Bioengineering, 9(4) (2022).
[57] Z. L. Zhang and Z. Q. Zhang, An interplay model for rumour spreading and emergency development, Phys. A: Stat. Mech. Appl., 388(19) (2009), 4159-4166.
[58] L. Zhao, et al., Rumor spreading model with consideration of forgetting mechanism: A case of online blogging LiveJournal, Phys. A: Stat. Mech. Appl., 390(13) (2011), 26192625.
[59] X. Zhao and J. Wang, Dynamical model about rumor spreading with medium, Discrete Dyn. Nat. Soc., (2013).
[60] J. Zhou, Z. Liu, and B. Li, Influence of network structure on rumor
Baishya, C., Naik, M., & R. N., P. (2024). Rumor spread dynamics and its sensitivity analysis under the influence of the Caputo fractional derivatives. Computational Methods for Differential Equations, 12(2), 236-265. doi: 10.22034/cmde.2023.55650.2316
MLA
Chandrali Baishya; Manisha Krishna Naik; Premakumari R. N.. "Rumor spread dynamics and its sensitivity analysis under the influence of the Caputo fractional derivatives". Computational Methods for Differential Equations, 12, 2, 2024, 236-265. doi: 10.22034/cmde.2023.55650.2316
HARVARD
Baishya, C., Naik, M., R. N., P. (2024). 'Rumor spread dynamics and its sensitivity analysis under the influence of the Caputo fractional derivatives', Computational Methods for Differential Equations, 12(2), pp. 236-265. doi: 10.22034/cmde.2023.55650.2316
VANCOUVER
Baishya, C., Naik, M., R. N., P. Rumor spread dynamics and its sensitivity analysis under the influence of the Caputo fractional derivatives. Computational Methods for Differential Equations, 2024; 12(2): 236-265. doi: 10.22034/cmde.2023.55650.2316