On the dynamics of a viral marketing model with optimal control using indirect and direct methods

The complexity of optimal control problems requires the use of numerical methods to compute control and optimal state trajectories for a dynamical system, aiming to optimize a particular performance index. Considering a real viral advertisement, this article compares the dynamics of a viral marketing epidemic model with optimal control under different cost scenarios and from two perspectives: using numerical methods based on the Pontryagin's Maximum Principle (indirect methods) and methods that treat the optimal control problem as a nonlinear constrained optimization problem (direct methods). Based on the trade-off between the maximization of information spreading and the minimization of the costs associated with it, an optimal control problem is formulated and studied. The existence and uniqueness of the solution are proved. Our results show not only that the cost of implementing control policies is a crucial parameter for the spreading of marketing messages, but also that low investment costs in control strategies fulfill the proposed trade-off without compromising the financial capacity of a company.