Modeling the dynamics of user adoption and abandonment for a single product

We introduce a compartmental differential equation model to study the dynamics of user adoption and abandonment for a single product. The model integrates two forms of abandonment: infectious, driven by user interactions, and non-infectious, prompted by external influences. Notably, the infectious abandonment coefficient varies linearly with the number of previous users. We investigate the existence of equilibria of the model and derive the threshold quantity ℛ0. The user-free equilibrium is always present, and its stability is analyzed under the condition ℛ0 < 1. Moreover, a user-prevailing equilibrium does not exist when ℛ0 ≤ 1, but at least one user-prevailing equilibrium is guaranteed when ℛ0 > 1. We further characterize conditions for multiple equilibria and various bifurcations, including saddle-node, 𝑆-shaped, and Hopf bifurcations, and formulate an optimal control problem. Numerical simulations validate our theoretical findings, and the historical LinkedIn and YouTube data calibrate the model to forecast future user adoption trends.