Stability Analysis of a Rumor-Spreading Model with Two Time Delays and Saturation Effect

Time delay and nonlinear incidence functions have a significant effect on rumor-spreading. In this article, a rumor-spreading model with two unequal time delays and a saturation effect is proposed. The existence, uniqueness, and non-negativity of the solution to this model are shown. The basic reproduction number is determined. A criterion for the existence of a rumor-endemic equilibrium is derived. It is found that there is an interesting conditional forward bifurcation. As a consequence, a complex bifurcation phenomenon is exhibited. A collection of criteria for the asymptotic stability of the rumor-free equilibrium are outlined. In the absence of a time delay, a criterion for the local asymptotic stability of the rumor-endemic equilibrium is presented. In the presence of small time delays, a criterion for the local asymptotic stability of the rumor-endemic equilibrium is established by applying our recently developed technique. Finally, a rumor-spreading control problem is reduced to an optimal control model, which is tackled in the framework of optimal control theory. This work facilitates the understanding of the influence of time delays and the saturation effect on rumor-spreading.