In this paper, a time-delay rumor propagation model with a saturated control
function is established. Regarding time delay as a bifurcation parameter, Hopf
bifurcation is studied. By means of the normal form and the center manifold theorem,
a formula is put forward to determine both Hopf bifurcation direction and bifurcating
periodic solution stability, together with some numerical simulations to illustrate the
relevant theoretical results. Simulation results indicate that an appropriate
government control could make periodic oscillation behaviors of the system become
steady so as to improve the balance of a social system.

• A study on time-delay rumor propagation model with saturated control function