IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v241y2026ipap868-889.html

Modeling the dynamics of adoption and abandonment of multiple products

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475425004549
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2025.10.026?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    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. Fibich, Gadi & Levin, Tomer, 2020. "Percolation of new products," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    4. 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.
    5. 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.
    6. Frank M. Bass, 1969. "A New Product Growth for Model Consumer Durables," Management Science, INFORMS, vol. 15(5), pages 215-227, January.
    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).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Palm, Alvar, 2022. "Innovation systems for technology diffusion: An analytical framework and two case studies," Technological Forecasting and Social Change, Elsevier, vol. 182(C).
    2. Chorowski, Michał & Nowak, Andrzej & Andersen, Jørgen Vitting, 2023. "What makes products trendy: Introducing an innovation adoption model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 616(C).
    3. Liu, Jiawei & Ding, Jie, 2020. "Requesting for retweeting or donating? A research on how the fundraiser seeks help in the social charitable crowdfunding," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    4. Xenikos, D.G. & Constantoudis, V., 2024. "Extended Bass model on the power-law epidemics growth and its implications on spatially heterogeneous systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 656(C).
    5. Hu, Sen & Hu, Bin & Cao, Ya, 2018. "The wider, the better? The interaction between the IoT diffusion and online retailers’ decisions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 196-209.
    6. Gadi Fibich & Amit Golan, 2023. "Diffusion of New Products with Heterogeneous Consumers," Mathematics of Operations Research, INFORMS, vol. 48(1), pages 257-287, February.
    7. Xenikos, D.G. & Constantoudis, V., 2023. "Weibull dynamics and power-law diffusion of epidemics in small world 2D networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 617(C).
    8. Cardol, Hanna & Mignon, Ingrid & Lantz, Björn, 2025. "Rethinking the forecasting of innovation diffusion: A combined actor- and system approach," Technological Forecasting and Social Change, Elsevier, vol. 214(C).
    9. Singhal, Shakshi & Bano, Yasmeen & Gautam, Prerna, 2026. "Changepoint model for energy-efficient technology diffusion: A comparative evaluation of empirical models," Technovation, Elsevier, vol. 149(C).
    10. Shari, Babajide Epe & Dioha, Michael O. & Abraham-Dukuma, Magnus C. & Sobanke, Victor O. & Emodi, Nnaemeka V., 2022. "Clean cooking energy transition in Nigeria: Policy implications for Developing countries," Journal of Policy Modeling, Elsevier, vol. 44(2), pages 319-343.
    11. Cambier, Adrien & Chardy, Matthieu & Figueiredo, Rosa & Ouorou, Adam & Poss, Michael, 2022. "Optimizing subscriber migrations for a telecommunication operator in uncertain context," European Journal of Operational Research, Elsevier, vol. 298(1), pages 308-321.
    12. Tiruwork B. Tibebu & Eric Hittinger & Qing Miao & Eric Williams, 2024. "Adoption Model Choice Affects the Optimal Subsidy for Residential Solar," Energies, MDPI, vol. 17(3), pages 1-19, February.
    13. Simon P. Anderson & André de Palma, 2012. "Competition for attention in the Information (overload) Age," RAND Journal of Economics, RAND Corporation, vol. 43(1), pages 1-25, March.
    14. Van, Tien Linh Cao & Barthelmes, Lukas & Gnann, Till & Speth, Daniel & Kagerbauer, Martin, 2021. "Addressing the gaps in market diffusion modeling of electrical vehicles: A case study from Germany for the integration of environmental policy measures," Working Papers "Sustainability and Innovation" S05/2021, Fraunhofer Institute for Systems and Innovation Research (ISI).
    15. Klingler, Anna-Lena & Luthander, Rasmus, 2018. "Market diffusion of residential PV and battery systems driven by self-consumption: A comparison of Sweden and Germany," Working Papers "Sustainability and Innovation" S18/2018, Fraunhofer Institute for Systems and Innovation Research (ISI).
    16. Edgardo Arturo Ayala Gaytán, 2009. "Social network externalities and price dispersion in online markets," Ensayos Revista de Economia, Universidad Autonoma de Nuevo Leon, Facultad de Economia, vol. 0(2), pages 1-28, November.
    17. Régis Chenavaz & Corina Paraschiv & Gabriel Turinici, 2017. "Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach," Working Papers hal-01592958, HAL.
    18. White, Reilly & Marinakis, Yorgos & Islam, Nazrul & Walsh, Steven, 2020. "Is Bitcoin a currency, a technology-based product, or something else?," Technological Forecasting and Social Change, Elsevier, vol. 151(C).
    19. Shigeno, Hidenori & Matsuzaki, Taisuke & Ueki, Yasushi & Tsuji, Masatsugu, 2023. "The Effect of the Covid-19 Pandemic on the Innovation Process of Small and Medium-sized Regional Firms," 32nd European Regional ITS Conference, Madrid 2023: Realising the digital decade in the European Union – Easier said than done? 278018, International Telecommunications Society (ITS).
    20. Sohn, So Young & Lim, Michael, 2008. "The effect of forecasting and information sharing in SCM for multi-generation products," European Journal of Operational Research, Elsevier, vol. 186(1), pages 276-287, April.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:241:y:2026:i:pa:p:868-889. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.