Dynamic modelling and optimal control of herd behaviour with time delay and media

A dynamic model of herd behaviour with delay time and media is established and analysed to discover the latent mechanism that represents how capital flows in the stock market. We prove that solutions of the model are uniformly bounded, and the contagion threshold [Formula: see text] is obtained. The stability of positive equilibrium points of the model with zero and nonzero delay is discussed. An optimal control problem with media is formulated, and Pontryagin's maximum principle is applied to find an optimal strategy to control herding. Several numerical simulations show the effect of media and delay on herd behaviour. Finally, the practical meaning of the presented model is briefly discussed.