Stability and optimal control of two products innovation diffusion system

Abstract

The current study is aimed at analysing a four-compartmental system to examine the interaction and market dissemination of two product categories. In the first category, the product is low-priced and preferred by economically backward people. In the second category, the product is high-priced and liked by people in high-income groups. The non-user population is the potential buyer of products in the market, and frustrated people are fed up with both products. The nonnegativity and boundedness of the solution to the system help in ensuring the well-defined nature of the model. The viability of the model with UFE (user-free equilibrium), an equilibrium point in the absence of users of low-priced products, and interior equilibria is analysed. The analysis of the local and global stability of the model’s equilibria is conducted with reference to ℛ0. We examine the system’s transcritical bifurcation using the Centre Manifold Theory at ℛ0=1. The Hopf bifurcation’s occurrence condition has been identified. The frustration class population gets reduced using the optimal control theory. The extended optimum control model is used to build the Hamiltonian function, which is then solved using Pontryagin’s maximum principle to obtain the lowest cost. To realize which parameters impact the basic influence numbers most, we also run a sensitivity analysis on them. Finally, simulations are conducted to support the theoretical model.