The Implementation of Optimally Controlled Social Media Advertisements for Diffusion of an Innovation

Abstract

Nowadays, social media advertisements have gained a considerable wide reach in the marketplace. They have become a significant medium for disseminating necessary information for the diffusion of a new product or an innovation. In this paper, a non-linear mathematical model is proposed and analyzed to study the impact of social media advertisements on the diffusion of an innovation in a marketplace. The impact of word-of-mouth and the crowding factor of social media advertisements are also considered in the modelling process. The long-term behaviour of the model is analyzed by using the stability theory of the system of differential equations. Since, in order to run advertisements through social media, a substantial budget is required to be allocated, which may lead to a net deficit of the organization. Therefore, we have extended the proposed model to control the non-adopter population as well as to minimize the cost associated with social media advertisements. To enhance the efficiency of our model, we have incorporated a time delay factor due to the fading of memory of adopters to visualize the challenge of memory decay in efficacy of social media advertisements. Further, numerical simulations are also conducted to validate and illustrate the analytical results.