Onamy Ramdinpuii
Department of Mathematics, Amity Institute of Applied Science, Amity University Uttar Pradesh, Noida, Uttar Pradesh, India.

Sumit Kaur Bhatia
Department of Mathematics, Amity Institute of Applied Science, Amity University Uttar Pradesh, Noida, Uttar Pradesh, India.

Shivani Bali
Jindal School of Banking & Finance, O. P. Jindal Global University, Sonipat, Haryana, India.

Kuldeep Chaudhary
Department of Mathematics, Amity Institute of Applied Science, Amity University Uttar Pradesh, Noida, Uttar Pradesh, India.

DOI https://doi.org/10.33889/IJMEMS.2026.11.2.029

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Abstract

This research presents a new multi-stage innovation diffusion model that incorporates the delay effect due to the complexity and variability of actual diffusion processes as an evaluation time delay parameter, efficiently using the mathematical framework to provide insight into the progression of awareness of the product, decision-making stage, and finally the adoption of the product. In contrast to traditional diffusion models, this model considers the behavioral delay in the transition from awareness to adoption, therefore, adding realism and applicability to the diffusion process. Initially, a set of differential equations that incorporate the three stages with a single time delay parameter is formulated. The rigorous structure and well-defined nature of the model system is ensured by the positivity and boundedness of the solution to the model system. A unique equilibrium point is established, and the local stability conditions of the equilibrium point are acquired on analyzing the characteristic equation. Existence of Hopf bifurcation has been proven, and the direction and stability of bifurcating periodic solutions have also been obtained using the normal form and center manifold theorem. The global stability conditions for the equilibrium point are obtained on constructing a suitable Lyapunov function. Further, the research contributes by formulating an optimal control problem to maximize the number of adopters while minimizing promotional costs, linking theoretical results to marketing strategy. Numerical simulations, including parameter estimation with real-world data, are carried out to further analyze the model's behavior, support the theoretical results, and estimate the rate of adopters. The findings indicate that accounting for the adoption delay plays a crucial role in shaping the system stability and predicting long-term adoption dynamics.

Keywords- Multi-stage innovation diffusion model, Evaluation time delay, Equilibrium point, Hopf bifurcation, Sensitivity analysis.

Citation

Ramdinpuii, O., Bhatia, S. K. Bali, S., & Chaudhary, K. (2026). Dynamic Analysis and Optimal Control of a Multi-stage Innovation Diffusion Model Incorporating Time Delay in Awareness and Adoption Process. International Journal of Mathematical, Engineering and Management Sciences, 11(2), 706-735. https://doi.org/10.33889/IJMEMS.2026.11.2.029.