Epidemic dynamics on a delayed multi-group heroin epidemic model with nonlinear incidence rate

For a multi-group Heroin epidemic model with nonlinear incidence rate and distributed delays, we study some aspects of its global dynamics. By a rigorous analysis of the model, we establish that the model demonstrates a sharp threshold property, completely determined by the values of R 0 : if R 0 ≤ 1 , then the drug-free equilibrium is globally asymptotically stable; if R 0 1 , then there exists a unique endemic equilibrium and it is globally asymptotically stable. A matrix-theoretic method based on the Perron eigenvector is used to prove the global asymptotic stability of the drug-free equilibrium and a graph- theoretic method based on Kirchhoff s matrix tree theorem was used to guide the construction of Lyapunov functionals for the global asymptotic stability of the endemic equilibrium.