Dynamic analysis and optimal control of a novel fractional-order 2I2SR rumor spreading model. (English) Zbl 1529.34049
Summary: In this paper, a novel fractional-order 2I2SR rumor spreading model is investigated. Firstly, the boundedness and uniqueness of solutions are proved. Then the next-generation matrix method is used to calculate the threshold. Furthermore, the stability of rumor-free/spreading equilibrium is discussed based on fractional-order Routh-Hurwitz stability criterion, Lyapunov function method, and invariance principle. Next, the necessary conditions for fractional optimal control are obtained. Finally, some numerical simulations are given to verify the results.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 91D30 | Social networks; opinion dynamics |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
References:
| [1] | K. Bansal, S. Arora, K. Pritam, T. Mathur, S. Agarwal, Dynamics of crime transmission using fractional-order differential equations, Fractals, 30(01):2250012, 2022, https://doi. org/10.1016/j.nonrwa.2015.05.014. · Zbl 1371.92080 · doi:10.1016/j.nonrwa.2015.05.014 |
| [2] | D. Daley, D. Kendall, Stochastic rumours, IMA J. Appl. Math., 1(1):42-55, 1965, https: //doi.org/10.1093/imamat/1.1.42. · doi:10.1093/imamat/1.1.42 |
| [3] | D. J. Daley, D. G. Kendall, Epidemics and rumours, Nature, 204:1118-1118, 1964, https: //doi.org/10.1038/2041118a0. · doi:10.1038/2041118a0 |
| [4] | W. Deng, C. Li, J. Lv, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 48:409-416, 2007, https://doi.org/10.1007/ s11071-006-9094-0. · Zbl 1185.34115 · doi:10.1007/s11071-006-9094-0 |
| [5] | F. Liu et al., Fractional differential equations, Academic Press, New York, 1999. · Zbl 0924.34008 |
| [6] | S. Galam, Modeling rumors: The no plane Pentagon French hoax case, Physica A, 320:571-580, 2003, https://doi.org/10.1016/S0378-4371(02)01582-0. · Zbl 1010.91088 · doi:10.1016/S0378-4371(02)01582-0 |
| [7] | J. Graef, L. Kong, A. Ledoan, M. Wang, Stability analysis of a fractional online social network model, Math. Comput. Simul., 178(82):625-645, 2020, https://doi.org/10.1016/ j.matcom.2020.07.012. · Zbl 1523.91025 · doi:10.1016/j.matcom.2020.07.012 |
| [8] | T. Grein, K. Kamara, G. Rodier, A. Plant, D. Heymann, Rumors of disease in the global village: Outbreak verification, Emerg. Infect. Dis., 6(2):97-102, 2000, https://doi.org/10. 3201/eid0602.000201. · doi:10.3201/eid0602.000201 |
| [9] | C. Huang, H. Liu, X. Shi, X. Chen, M. Xiao, Z. Wang, J. Cao, Bifurcations in a fractional-order neural network with multiple leakage delays, Neural Netw., 131:115-126, 2020, https: //doi.org/10.1016/j.neunet.2020.07.015. · Zbl 1483.34098 · doi:10.1016/j.neunet.2020.07.015 |
| [10] | J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Anal., Real World Appl., 26:289-305, 2015, https://doi.org/ 10.1016/j.nonrwa.2015.05.014. · Zbl 1371.92080 · doi:10.1016/j.nonrwa.2015.05.014 |
| [11] | H. Kheiri, M. Jafari, Optimal control of a fractional-order model for the HIV/AIDS epidemic, Int. J. Biomath., 11(6):1850086, 2018, https://doi.org/10.1142/ S1793524518500869. · Zbl 1400.92321 · doi:10.1142/S1793524518500869 |
| [12] | H.-L. Li, L. Zhang, C. Hu, Y.-L. Jiang, Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54:435-449, 2017, https://doi.org/10.1007/s12190-016-1017-8. · Zbl 1377.34062 · doi:10.1007/s12190-016-1017-8 |
| [13] | J. Li, H. Jiang, X. Mei, C. Hu, G. Zhang, Dynamical analysis of rumor spreading model in multi-lingual environment and heterogeneous complex networks, Inf. Sci., 536:391-408, 2020, https://doi.org/10.1016/j.ins.2020.05.037. · Zbl 1475.91256 · doi:10.1016/j.ins.2020.05.037 |
| [14] | D. Maki, M. Thomson, Mathematical models and applications, Prentice Hall College Press, New Jersey, 1973. |
| [15] | P. Naik, J. Zu, K. Owolabi, Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control, IEEE Trans. Syst. Man Cybern., 137:1-30, 2020, https: //doi.org/10.1016/j.chaos.2020.109826. · Zbl 1490.37112 · doi:10.1016/j.chaos.2020.109826 |
| [16] | G. Ren, Y. Yu, Z. Lu, W. Chen, A fractional order model for rumor spreading in mobile social networks from a stochastic process, in 2021 9th International Conference on Systems and Control (ICSC), Caen, France, IEEE, 2021, pp. 312-318, https://doi.org/10.1109/ ICSC50472.2021.9666673. · doi:10.1109/ICSC50472.2021.9666673 |
| [17] | P. Shu, Effects of memory on information spreading in complex networks, in 2014 IEEE 17th International Conference on Computational Science and Engineering, Chengdu, China, IEEE, 2015, pp. 554-556, https://doi.org/10.1109/CSE.2014.126. · doi:10.1109/CSE.2014.126 |
| [18] | J. Singh, A new analysis for fractional rumor spreading dynamical model in a social network with Mittag-Leffler law, Chaos, 21(1):013137, 2019, https://doi.org/10.1063/1. 5080691. · Zbl 1406.91326 · doi:10.1063/1.5080691 |
| [19] | P. van den Dreessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(1-2):29-48, 2022, https://doi.org/10.1016/s0025-5564(02)00108-6. · Zbl 1015.92036 · doi:10.1016/s0025-5564(02)00108-6 |
| [20] | C. Wang, Z. Tan, Y. Ye, H. Kang L. Wang, N. Xie, A rumor spreading model based on information entropy, Sci. Rep., 7:9615, 2017, https://doi.org/10.1038/s41598-017-09171-8. · doi:10.1038/s41598-017-09171-8 |
| [21] | H. Wang, H. Jahanshahi, M.-K. Wang, S. Bekiros, J. Liu, A.A. Aly, A Caputo-Fabrizio fractional-order model of HIV/AIDS with a treatment compartment: Sensitivity analysis and optimal control strategies, Entropy, 23(5):610, 2021, https://doi.org/10.3390/ e23050610. · doi:10.3390/e23050610 |
| [22] | J. Wang, H. Jiang, T. Ma, C. Hu, Global dynamics of the multi-lingual SIR rumor spreading model with cross-transmitted mechanism, Chaos Solitons Fractals, 126:148-157, 2019, https://doi.org/10.1016/j.chaos.2019.05.027. · Zbl 1448.91220 · doi:10.1016/j.chaos.2019.05.027 |
| [23] | X. Wang, Z. Wang, X. Huang, Y. Li, Dynamic analysis of a delayed fractional order SIR model with saturated incidence and treatment functions, Int. J. Bifurcation Chaos Appl. Sci. Eng., 28(14):1850180, 2018, https://doi.org/10.1142/S0218127418501808. · Zbl 1412.34231 · doi:10.1142/S0218127418501808 |
| [24] | X. Wang, Z. Wang, J. Xia, Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders, J. Franklin Inst., 356(15):8278-8295, 2019, https://doi.org/10.1016/j.jfranklin.2019.07.028. · Zbl 1423.92240 · doi:10.1016/j.jfranklin.2019.07.028 |
| [25] | Y. Wang, J. Cao, C. Huang, Hopf bifurcation of a fractional tri-neuron network with different orders and leakage delay, Fractals, 30(03):2250045, 2022, https://doi.org/10. 1142/S0218348X22500451. · Zbl 1500.34060 · doi:10.1142/S0218348X22500451 |
| [26] | R. Yang, D. Han, Multi-message topic dissemination probabilistic model with memory attenuation based on Social-Messages Network, Int. J. Mod. Phys. C, 30(07):1940011, 2019, https://doi.org/10.1142/S0129183119400114. · doi:10.1142/S0129183119400114 |
| [27] | S. Yang, C. Hu, J. Yu, H. Jiang, Exponential stability of fractional-order impulsive control systems with applications in synchronization, IEEE T. Cybern., 50(7):3157-3168, 2020, https://doi.org/10.1109/tcyb.2019.2906497. · doi:10.1109/tcyb.2019.2906497 |
| [28] | S. Yu, Z. Yu, H. Jiang, S. Yang, The dynamics and control of 2I2SR rumor spreading models in multilingual online social networks, Infm. Sci., 581(1):18-41, 2021, https://doi.org/ 10.1016/j.ins.2021.08.096. · Zbl 1539.91104 · doi:10.1016/j.ins.2021.08.096 |
| [29] | Y. Zhang, Y. Yu, X. Cui, Dynamical behaviors analysis of memristor-based fractional-order complex-valued neural networks with time delay, Appl. Math. Comput., 339:242-258, 2018, https://doi.org/10.1016/j.amc.2018.06.042. · Zbl 1428.93082 · doi:10.1016/j.amc.2018.06.042 |
| [30] | L. Zhao, C. Huang, J. Cao, Dynamics of fractional-order predator-prey model incorporating two delays, Fractals, 29(01):2150014, 2012, https://doi.org/10.1142/ · Zbl 1487.34162 · doi:10.1142/S0218348X21500146 |
| [31] | L. Zhu, W. Liu, Z. Zhang, Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function, Appl. Math. Comput., 370:124925, 2020, https://doi.org/10.1016/j.amc.2019.124925. https://www.journals.vu.lt/nonlinear-analysis · Zbl 1433.91122 · doi:10.1016/j.amc.2019.124925 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
