Dynamical behaviors of a heroin population model with standard incidence rates between distinct patches

Abstract The dynamical behaviors of a stochastic heroin population model with standard incidence rates between distinct patches are investigated in this paper. We firstly derive the fitness of a global positive solution. Then, under some moderate conditions, we show that the heroin population model always admits a stationary Markov process in the long run. More precisely, the coexistence of the drug users and the susceptible is studied by constructing multiple proper functions when the intensities of the white noises are under control, and the spectral radius of the irreducible matrix is greater than one ( R 0 > 1 ). For the elimination of the drug users, if the intensities of the white noises are large, then the elimination of drug users in a heroin population would exponentially decrease with probability one. Especially, when the spectral radius of the irreducible matrix is less than one ( R 0 1 ), the elimination of drug users who are not in treatment also follows the exponential decline with probability one. As a consequence, two examples and their corresponding numerical simulations are demonstrated to present the validity and feasibility of the theoretical results.