Numerical simulations are provided to illustrate the theoretical results.
📌 3 Key Takeaways
🎯
Objective & Hypothesis: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.
📊
Key Findings & Data:The basic reproduction $R_0,$ the user-free equilibrium(UFE) $P_0,$ and the user-prevailing equilibrium(UPE) $P^*$ are identified.
💡
Conclusion & Impact:Numerical simulations are provided to illustrate the theoretical results.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Online social network · stability · Hopf bifurcation
Abstract
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.
WANG, L., WANG, M. (2023). Stability and Bifurcation Analysis For An OSN Model with Delay. Advances in the Theory of Nonlinear Analysis and its Application, Cilt: 7 Sayı: 2, 413–427. https://ajindex.com/articles/956699
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
J. Cannarella and J. Spechler, Epidemiological modeling of online network dynamics, arXiv preprint arXiv:1401.4208 (2014), 1–10
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
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
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
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
L. Kong and M. Wang, Optimal control for an ordinary differential equation online social network model, Differ. Equ. Appl., 14 (2022), 205–214
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
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
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
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
R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361–367
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
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