DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.715 DYNAMICAL SYSTEMS SERIES B Volume 19, Number 3, May 2014 pp. 715–733 GLOBAL STABILITY FOR A HEROIN MODEL WITH TWO DISTRIBUTED DELAYS Bin Fang Beijing Institute of Information and Control Beijing 100037, China and Department of Mathematics, Xinyang Normal University Xinyang 464000, China Xuezhi Li Department of Mathematics, Xinyang Normal University Xinyang 464000, China Maia Martcheva Department of Mathematics, University of Florida 358 Little Hall, PO Box 118105, Gainesville, USA Liming Cai Department of Mathematics, Xinyang Normal University Xinyang 464000, China (Communicated by Pierre Magal) Abstract. In this paper, we consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determines the stability of the equilibria. Using the direct Lyapunov method with Volterra type Lyapunov function, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the unique drug spread equilibrium is globally asymptotically stable if the basic reproduction number is greater than one. 1. Introduction. It is well-known that heroin is an opiate drug that is synthesized from morphine [1]. It is more soluble in the fat cells so it crosses the blood-brain barrier within 15-20 seconds, rapidly achieving a high level syndrome in the brain and the central nervous system. Its rapid action causes both the ‘rush’ experience by users and the toxicity [2]. Heroin users are at high risk for addiction. It is estimated that about 23 percent of individuals who use heroin become dependent on it. Over the past two decades, China has faced a dramatic increase in illicit drug abuse accompanying rapid economic reform and development [3]. In addition to their deleterious somatic and psychological effects, heroin abuse and dependence 2010 Mathematics Subject Classification. Primary: 92D25; Secondary: 93D20. Key words and phrases. Heroin model, distributed delay, basic reproduction number, Lyapunov function, global stability. This work is Supported partially by the National Natural Science Foundation of China (11271314, 11371305, 61203228). M. M. is supported in part by NSF DMS–1220342. 715