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Modeling the dynamics of adoption and abandonment of multiple products

Author

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  • Chen, Roger
  • Kong, Lingju
  • Wang, Min

Abstract

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.

Suggested Citation

  • Chen, Roger & Kong, Lingju & Wang, Min, 2026. "Modeling the dynamics of adoption and abandonment of multiple products," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PA), pages 868-889.
    • Handle: RePEc:eee:matcom:v:241:y:2026:i:pa:p:868-889
    DOI: 10.1016/j.matcom.2025.10.026
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    References listed on IDEAS

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    1. Liu, Liang & Qu, Bo & Chen, Bin & Hanjalic, Alan & Wang, Huijuan, 2018. "Modelling of information diffusion on social networks with applications to WeChat," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 318-329.
    2. Palm, A., 2020. "Early adopters and their motives: Differences between earlier and later adopters of residential solar photovoltaics," Renewable and Sustainable Energy Reviews, Elsevier, vol. 133(C).
    3. Wang, Tao & He, Juanjuan & Wang, Xiaoxia, 2018. "An information spreading model based on online social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 488-496.
    4. Frank M. Bass, 1969. "A New Product Growth for Model Consumer Durables," Management Science, INFORMS, vol. 15(5), pages 215-227, January.
    5. Graef, John R. & Kong, Lingju & Ledoan, Andrew & Wang, Min, 2020. "Stability analysis of a fractional online social network model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 625-645.
    6. Fibich, Gadi & Levin, Tomer, 2020. "Percolation of new products," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    7. Zhu, Linhe & Liu, Wenshan & Zhang, Zhengdi, 2020. "Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function," Applied Mathematics and Computation, Elsevier, vol. 370(C).
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