The spreading of misinformation in Online Social Networks (OSNs) is quite similar to the spreading of infection in biological diseases. As the biological virus spreads and makes one infected as well as those who come into contact with the infected person, the nature and behavior of the spread of misinformation in an online social network is similar. So in order to understand the functioning of misinformation and to control over epidemic outbreak of misinformation in OSNs, epidemic models can be quite handy. The introduction of Bifurcation theory in the epidemic model explains the qualitative behavior of the system with changes in parameters. In this paper, we have introduced the Susceptible, Infectious, and Protected (SIP) model with Bifurcation for the propagation of misinformation in OSNs. Here Bifurcation is due to the limited number of users who are ready to adopt the Security and Privacy Policies of Social Network Sites. We define the threshold number for the system of equations and explain the stability of an Infection Free Equilibrium (IFE), representing the absence of misinformation. We have discussed about the endemic equilibrium point and bifurcation conditions along with its nature and stability at those points. Also, we have shown global stability at the endemic Equilibrium of the system. Finally, numerical simulation has been used to show the existence of bifurcation in the system with a change in the value of parameters.

Developing mathematical models for reliable approximation of epidemic spread on a network is a challenging task, which becomes even more difficult when a wireless network is considered, because there are a number of inherent physical properties and processes which are apparently invisible. The aim of this paper is to explore the impact of several abstract features including trust, selfishness and collaborative behavior on the course of a network epidemic, especially when considered in the context of a wireless network. A five-component differential epidemic model has been proposed in this work. The model also includes a latency period, with a possibility of switching epidemic behavior. Bilinear incidence has been considered for the epidemic contacts. An analysis of the long term behavior of the system reveals the possibility of an endemic equilibrium point, in addition to an infection-free equilibrium. The paper characterizes the endemic equilibrium in terms of its existence conditions. The system is also seen to have an epidemic threshold which marks a well-defined boundary between the two long-term epidemic states. An expression for this threshold is derived and stability conditions for the equilibrium points are also established in terms of this threshold. Numerical simulations have further been used to show the behavior of the system using four different experimental set-ups. The paper concludes with some interesting results which can help in establishing an interface between epidemic spread and collaborative behavior in wireless networks.