Bifurcation analysis of a synthetic drugs model affected by multiple factors*

Synthetic drug addiction represents an escalating worldwide concern. In this paper, we introduce a comprehen-sive synthetic drug addiction model that considers both psychosocial and environmental factors. Additionally, our model encompasses aspects related to relapse and the drug supply chain. Through a dynamic analysis of this model, we uncover a multitude of intricate dynamical phenomena, including saddle-node bifurcation, Hopf bifurcation. Meanwhile, we have also obtained an elliptic-type nilpotent singularity. Furthermore, we provide bifurcation diagrams and corresponding phase diagrams to offer epidemiological insights into these intricate dynamic phenomena.