Long Time Behavior for a Stochastic Heroin Epidemic Model Under Regime Switching

ABSTRACT In this article, we expand a model of the heroin epidemic from a deterministic model to a stochastic model by incorporating Brownian motion and regime switching. The objective is to examine the comprehensive impact of Brownian motion and regime switching on system dynamics. We establish a critical value that fully characterizes its long time behavior. It is found that if this critical value is less than one, the number of heroin drug users converges to zero or extinction occurs. Conversely, if this critical value exceeds one, the system persists in mean and has a unique stationary distribution. Finally, three compelling examples are provided to demonstrate the effectiveness of our findings.