Modeling the dynamics of adoption and abandonment of multiple products

We propose a compartmental differential equation model aimed at analyzing the dynamics of user adoption and abandonment across two products. This model captures two distinct abandonment processes: infectious abandonment, influenced by interactions among current and past users, and noninfectious abandonment, triggered by factors like mass media, advertisements, or the introduction of new products. We define critical thresholds, represented as R1 for product one and R2 for product two, and explore the existence and local stability of various equilibrium states, including user-free scenarios, dominance of product one, and dominance of product two, based on these thresholds. We prove the global stability of these equilibria under specific conditions and identify a necessary criterion for the coexistence equilibrium. We find conditions under which one product persists while the other fades away. Additionally, we conduct a comprehensive analysis of an associated optimal control problem. We first prove the existence of an optimal control pair and then determine the system conditions necessary for this optimal control pair. Extensive numerical simulations are conducted to validate our theoretical findings. Finally, we demonstrate the effectiveness of the model by fitting it to historical data on the daily active users of Facebook and Instagram. By calibrating the model with derived parameter values, we make predictions regarding the future user counts for both platforms.