An innovation diffusion model with two innovations under two simultaneous effects in a competitive market: co-existence through optimal control

Abstract

Since mass media plays an important role in influencing the non-adopter population, the adoption rate changes with the media awareness rate. Motivated by this concept, in this paper, a model of three non-intersecting classes of non-adopters, adopters for product-I and adopters for product-II is proposed. Under the influence of media coverage and word-of-mouth, the dynamic behaviour of the system is investigated. The basic influence numbers R 0 1 and R 0 2 associated with the first and second innovations help in performing stability analysis. It is observed from stability analysis that adopter-free equilibrium is conditionally stable. Also, the system has no stable interior equilibrium point. The basic influence numbers determine the sustainability of a particular product in the market. The optimal control theory is used to reduce the frustration rate in both adopter classes. The Hamiltonian function is constructed using the extended optimum control model and is then solved according to Pontryagin's maximum principle to get the cost. Also, coexistence is possible with the implementation of optimal control. Sensitivity analysis has been performed for both the basic influence numbers R 0 1 and R 0 2 . Lastly, numerical experimentations have been executed to assist analytical findings with distinct sets of parameters.