Stability and Bifurcation Analysis For An OSN Model with Delay

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.