Stability and Bifurcation Analysis For An OSN Model with Delay

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.