Optimal scheduling of campaigns on on-line forums

We consider the problem of devising optimal timing of on-line campaigns for diffusion of product or information in a population. In particular, we assume that the firm can influence the diffusion process by rolling out campaigns on online forums (we will refer to these as social media campaigns). The problem is to maximize the penetration of the product or given message in the population within the stipulated time. Taking the competitive Bass model as the model of diffusion dynamics, we pose the problem as continuous-time deterministic optimal control problem. We characterize optimal controls, present some numerical results and provide practical insights. We show that the optimal strategy has a simple and elegant structure. We also obtain strategy map which specify the maximum cost at which the firm must stop investing in social media campaigns. In contrast to earlier papers, we also model competition. This is not only close to reality but interestingly it allows to address some problems in completely different contexts. For example, in limiting the spread of misinformation about an entity or an Internet virus, the objective is to maximize nodes with correct information or security patches by immunizing them and/or incentivizing them to spread the information they have to their neighbors. Our work is, thus, applicable to these problems as well.