Mathematical analysis of a reaction-diffusion gambling information diffusion model

This study develops and analyses a non-linear reaction–diffusion model for information spread, motivated by gambling-related information diffusion. The model is formulated as a susceptible–exposed–excessive–skeptical (SEIZ) system, in which individuals transition between information-related states through exposure, interaction, engagement, and resistance, while spatial diffusion captures heterogeneity across the population. The mathematical analysis establishes the existence, uniqueness, positivity, and boundedness of solutions, and derive both gambling-free and endemic equilibria. Two threshold quantities, \(\mathcal {R}_0^{ext} = \beta /\varepsilon\) and \(\mathcal {R}_0 = \alpha /\sigma\) , are obtained for the non-excessive and excessive states, respectively, yielding conditions for the persistence or elimination of these states. Diffusion-driven stability further assess how spatial effects modify the dynamics of the corresponding homogeneous system. For numerical analysis, the reaction–diffusion system is discretized using a semi-implicit Euler finite-difference method and the resulting non-linear system is solved by Newton’s method. The numerical experiments support the theoretical consistency, stability, and convergence of the method. Sensitivity analysis demonstrates that variations in the parameters entering the threshold quantities directly change the predicted qualitative dynamics. Although framed around gambling-related information, the main contribution of this work is the formulation and analysis of a general non-linear reaction–diffusion model that can be adapted to a broad class of spatial information and behavioral diffusion processes.