Stability and Bifurcation Analysis For An OSN Model with Delay

Yıl 2023, Cilt: 7 Sayı: 2, 413 - 427, 23.07.2023

Liancheng Wang , Min Wang

https://doi.org/10.31197/atnaa.1152602

Cited By: 1

https://izlik.org/JA32RS66SG

Öz

In this research, we propose and study an online social network mathematical model with delay based on two innovative assumptions: (1) newcomers are entering community as either potential online network users or that who are never interested in online network at constant rates, respectively; and (2) it takes a certain time for the active online network users to start abandoning the network. The basic reproduction $R_0,$ the user-free equilibrium(UFE) $P_0,$ and the user-prevailing equilibrium(UPE) $P^*$ are identified. The analysis of local and global stability for those equilibria is carried out. For the UPE $P^*,$ using the delay $\tau$ as the Hopf bifurcation parameter, the occurrence of Hopf bifurcation is investigated. The conditions are established that guarantee the Hopf bifurcation occurs as $\tau$ crosses the critical values. Numerical simulations are provided to illustrate the theoretical results.

Anahtar Kelimeler

Online social network , stability , Hopf bifurcation

Kaynakça

• [1] L. M. Bettencourt, A. Cintrn-Arias, D. I. Kaiser, and C. Castillo-Chvez, The power of a good idea: quatitative modeling of the spread of ideas from epidemiological models, Physica A: Statistical Mechanics and its Applications, 364 (2006), 513–536.
• [2] J. Cannarella and J. Spechler, Epidemiological modeling of online network dynamics, arXiv preprint arXiv:1401.4208 (2014), 1–10.
• [3] R. Chen and L. Kong and M. Wang, Stability analysis of an online social network model with infectious recovery dynamics, Rocky Mt. J. Math., accepted.
• [4] G. Dai, R. Ma, H. Wang, F. Wang, and K. Xu, Partial differential equations with Robin boundary conditions in online social networks, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1609–1624.
• [5] J. R. Graef, L. Kong, A. Ledoan, and M. Wang, Stability analysis of a fractional online social network model, Math. Comput. Simulat., 178 (2020), 625–645.
• [6] L. Kong and M. Wang, Deterministic and stochastic online social network models with varying population size, Dyn. Contin. Discrete Impuls. Syst. A: Math. Anal., accepted.
• [7] L. Kong and M. Wang, Optimal control for an ordinary differential equation online social network model, Differ. Equ. Appl., 14 (2022), 205–214.
• [8] C. Lei, Z. Lin, and H. Wang, The free boundary problem describing information diffusion in online social networks, J. Differential Equations, 254 (2013), 1326–1341.
• [9] X. Liu, T. Li, X. Cheng, W. Liu, and H. Xu, Spreading dynamics of a preferential information model with hesitation psychology on scale-free networks, Adv. Difference Equa., 2019 (2019), No. 279, 19pp.
• [10] X. Liu, T. Li, and M. Tian, Rumor spreading of a SEIR model in complex social networks with hesitating mechanism, Adv. Difference Equa., 2018 (2018), No. 391, 24pp.
• [11] F. Wang, H. Wang, and K. Xu, Diffusion logistic model towards predicting information diffusion in online social networks, 2012 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW) (2012), 133–139.
• [12] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361–367.
• [13] W. Kermack and A. McKendrick, A Contribution to the Mathematical Theory of Epidemics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character., 115 (1927) 700-–721.
• [14] M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, Global dynamics of an SEIR model with vertical transmission, SIAM J. Appl. Math., 160 (1999), 191–213.
• [15] M. Y. Li, H. L. Smith, and L. Wang, Global dynamics of a SEIR model with a varying total population size, Math. Biosci., 62 (2001), 58–69.
• [16] L. Wang, M. Y. Li, and D. Kirschner, Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosci., 179 (2002), 207–217.
• [17] S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems, 10 (2003), 863–874.
• [18] P. ven den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Math. Biosc., 180 (2002), 29–48.
• [19] J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York, 1969.
• [20] N. D. Pavel, Differential Equations, Flow Invariance and Applications, Pitman Publishing Inc., London, 1984.
• [21] J.P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied mathematics. SIAM, Philadelphia, 1976.
• [22] E. A. Barbashin, Introduction to the Theory of Stability, Groningen: WoltersNoordhoff, 1970.
• [23] A. Korobeinikov, Lyaponov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75–83.
• [24] J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method. Academic, New York, 1961.
• [25] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
• [26] L. Wang and X. Wu, Stability and Hopf Bifurcation for an SEIR Epidemic Model with Delay, Advances in the Theory of Nonl. Anal. and its Appl., 2 (2018), 113–127.