Modeling the Dynamics of User Adoption and Abandonment for a Single Product

Abstract

We present a new compartmental differential equation model to explore the dynamics of user adoption and abandonment for a single product. Our model incorporates two distinct types of abandonment: infectious abandonment, driven by interactions among current and former users, and noninfectious abandonment, triggered by factors such as mass media, advertisements, or the introduction of new products. Unlike previous studies, we treat the infectious abandonment coefficient as a variable that changes linearly with the number of previous users, rather than as a constant. This introduces additional complexity to the model while also enriching its dynamical behavior. We investigate the existence of equilibria of the model and derive the threshold quantity

{\mathcal{R}}_0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {R}_0$$\end{document}

. The user-free equilibrium is always present, and its stability is analyzed under the condition

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {R}_0 < 1$$\end{document}

. Furthermore, we show that a user-prevailing equilibrium does not exist when

{\mathcal{R}}_0⩽1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {R}_0 \leqslant 1$$\end{document}

, but at least one user-prevailing equilibrium is guaranteed when

{\mathcal{R}}_0>1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {R}_0 > 1$$\end{document}

. We determine the criteria for the existence of one, two, or three user-prevailing equilibria and establish the conditions under which S-shaped and saddle-node bifurcations can arise. Additionally, we establish criteria for different types of Hopf bifurcations. Our theoretical results are further validated through extensive numerical simulations. To demonstrate the practical applicability of our model, we calibrate it using historical data on LinkedIn registered users. The calibrated model is then employed to provide forecasts for future user adoption trends.

Keywords

Adoption and abandonment dynamics / Nonlinear abandonment rate / S-shaped and Hopf bifurcations / Limit cycle / Numerical simulations / Primary 34D20 / 34D23 / 91D30 / 92D25

References

[1] Alexander ME, Moghadas SM. Bifurcation analysis of an SIRS model with generalized incidence. SIAM J. Appl. Math., 2005, 65: 1794-1816

[2] Backstrom, L., Huttenlocher, D., Kleinberg, J., Lan, X.: Group formation in large social networks: membership, growth, and evolution. In: KDD’06: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 44–54. ACM Press (2006)

[3] Bass FM. A new product growth model for consumer durables. Manag. Sci., 1969, 15: 215-227

[4] Boyer, C.B.: A History of Mathematics, 2nd edn. Wiley, New York (1991)

[5] Cannarella, J., Spechler, J.: Epidemiological modeling of online network dynamics (2014). arXiv:1401.4208

[6] Chang, C., Yeh, M., Chuang, K.: Node reactivation model to intensify influence on network targets. Knowl. Inf. Syst. 24, 567–590 (2018)

[7] Chen, R., Kong, L., Wang, M.: Stability analysis of an online social network model. Rocky Mt. J. Math. 53, 1019–1041 (2023)

[8] Chen R, Kong L, Wang M. Modeling the dynamics of adoption and abandonment of multiple products. Math. Comput. Simulat., 2026, 241: 868-889

[9] Dai G, Ma R, Wang H, Wang F, Xu K. Partial differential equations with Robin boundary conditions in online social networks. Discrete Contin. Dyn. Syst. Ser. B, 2015, 20: 1609-1624

[10] Dasgupta, K., Singh, R., Viswanathan, B., Joshi, A.: Social ties and their relevance to churn in mobile telecom networks. In: Proceedings of the 11th International Conference on Extending Database Technology: Advances in Database Technology, pp. 668–677. EDBT (2008)

[11] Du B, Wang H, Liu M. An information diffusion model in social networks with carrier compartment and delay. Nonlinear Anal. Model. Control, 2018, 23: 568-582

[12] Fernando C, Buttriss G, Yoon H, Soo VK, Compston P, Doolan M. Integration of consumer preferences into dynamic life cycle assessment for the sharing economy: methodology and case study for shared mobility. Int. J. Life Cycle Assess., 2023, 28: 429-461

[13] Fibich G, Levin T, Yakir O. Boundary effects in the discrete bass model. SIAM J. Appl. Math., 2019, 79: 914-937

[14] Graef JR, Kong L, Ledoan A, Wang M. Stability analysis of a fractional online social network model. Math. Comput. Simul., 2020, 178: 625-645

[15] Huo L, Li X. An interplay model for official information and rumor spreading with impulsive effects. Adv. Differ. Equ., 2019, 2018(164): 16

[16] Israwati I. Leveraging influencers to reactivate dormant customers effective tactics and metrics. J. Soc. Commer., 2023, 3: 172-187

[17] Kimmel N, Kong L, Wang M. Modeling the dynamics of user adoption and abandonment in online social networks. Math. Methods Appl., 2025, 48: 1853-1868

[18] Kirkland S, Shuai Z, van den Driessche P, Wang X. Impact of varying community networks on disease invasion. SIAM J. Appl. Math., 2021, 81: 1166-1189

[19] Kong L. Modeling the dynamics of product adoption and abandonment. Proc. R. Soc. A, 2024, 480: 20240034

[20] Kong L, Shi RZ, Wang M. A physics-informed neural network model for social media user growth. Appl. Comput. Intell., 2024, 4: 195-208

[21] Kong L, Wang M. Optimal control for an ordinary differential equation online social network model. Differ. Equ. Appl., 2022, 14: 205-214

[22] Kong L, Wang M. Deterministic and stochastic online social network models with varying population size. Dyn. Contin. Discrete Impuls. Syst. A, 2023, 30: 253-275

[23] Lei C, Lin Z, Wang H. The free boundary problem describing information diffusion in online social networks. J. Differ. Equ., 2013, 254: 1326-1341

[24] Li MY, Wang L. Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment. Nonlinear Anal. Real World Appl., 2014, 17: 147-160

[25] Liu WM, Hethcote HW, Levin SA. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol., 1987, 25: 359-380

[26] Liu X, Li T, Cheng X, Liu W, Xu H. Spreading dynamics of a preferential information model with hesitation psychology on scale-free networks. Adv. Differ. Equ., 2019, 2019(279): 19

[27] Number of LinkedIn registered users worldwide as of 3rd quarter (2024). https://www.businessofapps.com/data/linkedin-statistics/

[28] Palm A. Early adopters and their motives: differences between earlier and later adopters of residential solar photovoltaics. Renew. Sustain. Energy Rev., 2020, 133 110142

[29] Ruan S, Wang W. Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ., 2003, 188: 135-163

[30] Wang A, Li MY. Viral dynamics of HIV-1 with CTL immune response. Discrete Contin. Dyn. Syst. Ser. B, 2021, 26: 2257-2272

[31] Wang, F., Wang, H., Xu, K.: Diffusion logistic model towards predicting information diffusion in online social networks. In: 2012 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW), pp. 133–139 (2012)

[32] Wang T, He J, Wang X. An information spreading model based on online social networks. Phys. A, 2018, 490: 488-496

[33] Wu, S., Sarma, A.D., Fabrikant, A., Lattanzi, S.,Tomkins, A.: Arrival and departure dynamics in social networks. In: Proceedings of the 6th ACM International Conference on Web Search and Data Mining, WSDM, pp. 233–242 (2013)

[34] Zhu, L., Liu, W., Zhang, Z.: Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function. Appl. Math. Comput. 370, 124925 (2020)

Funding

Center of Excellence in Applied Computational Science and Engineering at the University of Chattanooga

Center of Excellence in Applied Computational Science and Engineering) grant at the University of Tennessee at Chattanooga

RIGHTS & PERMISSIONS

Shanghai University