Exploring the Dynamics of Social Media Addiction and Depression Models With Discrete and Distributed Delays

Web-based networking significantly influences daily interactions and user well-being. This study analyzes social media addiction and depression models that incorporate distribution delays to enhance control strategies. Unlike previous deterministic models, our approach integrates both discrete (Dirac-delta) and distributed (gamma) delay distributions to assess the linear stability of disease-free and endemic equilibria and the occurrence of Hopf bifurcation. We find that while both equilibria maintain stability under short delays, increased mean delays lead to instability through Hopf bifurcation across both delay types. Notably, the gamma distribution demonstrates a stability switch; the endemic equilibrium initially remains stable, destabilizes as delays lengthen, and restabilizes with further delay increases. Analytical results confirm the direction and stability of these bifurcations, supported by numerical validations. This research fills a significant gap by combining discrete and distributed delays, providing insights crucial for developing effective interventions and shaping public health policies to mitigate the adverse effects of social media addiction on mental health