Geometric ergodicity and stochastically ultimate boundedness of a stochastic heroin epidemic model

Ergodicity analysis plays an essential role in exploring the longterm dynamic behaviors of stochastic biological epidemic models. This paper studies a three-compartment stochastic heroin transmission model that describes the spread of heroin abuse among susceptible populations and two classes of drug users. We establish the geometric ergodicity of the stochastic system by verifying the minorization and Lyapunov conditions, indicating that the Markov process converges to its limiting distribution at a geometric rate. Additionally, we demonstrate that the solution of the stochastic system is stochastically ultimately bounded. To intuitively support our theoretical deductions, we carry out complete numerical simulations. Beyond pure mathematical analysis, all our theoretical findings carry clear practical value for real-world heroin prevention and control. Based on our analytical and numerical results, we further put forward targeted operable suggestions for public security and anti-drug departments to formulate scientific long-term drug control policies.