Mathematical modeling, global stability, sensitivity analysis, and optimal control of merchant adoption dynamics in E-Commerce

Document Type : Research Article

Authors

HABIB HASSOUNI ¹

AMINE El Bhih ^{2, 3}

OUMAIMA AARABATE ¹

OMAR BALATIF ¹

¹ Laboratory of Fundamental Mathematics and their Applications, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, Morocco.

² Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sik, Hassan II University, Casablanca, BP 7955, Morocco.

³ Multidisciplinary Research and Innovation Laboratory (LPRI), Moroccan School of Engineering Sciences (EMSI), Casablanca Morocco.

10.22067/ijnao.2026.100313.1935

Abstract

In this paper, we introduce a novel mathematical model to analyse the dynamics of e-commerce adoption among merchants. The model captures the key factors influencing adoption behaviours and examines both the spread of e-commerce and resistance to its uptake. Using the next-generation matrix method, we establish the model's well-posedness and derive the basic reproduction number $R_0$. Two equilibrium points are identified: the rejection equilibrium \( E_0 \), representing a state of non-adoption, and the widespread adoption equilibrium \( E_* \), representing a state of large-scale e-commerce adoption. A global stability analysis, based on LaSalle’s invariance principle and Lyapunov functions, demonstrates that \( E_0 \) is globally asymptotically stable if \( R_0 \leq 1 \), while \( E_* \) is globally stable when \( R_0 > 1 \). Furthermore, the model explores the drivers of e-commerce diffusion and factors contributing to resistance. We formulate the model as an optimal control problem to address these dynamics, introducing control variables aimed at minimising resistance and promoting widespread adoption. Cost-Effectiveness Analysis is also performed to evaluate the efficiency of each intervention strategy. Finally, numerical simulations provide insights into the effectiveness of various policy interventions, illustrating the strategies required to achieve optimal levels of e-commerce adoption.

Keywords

Mathematical modeling

Stability analysis

Pontryagin Maximum principle

Optimal control

Sensitivity Analysis

Subjects

• Modeling and Simulation

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