Abstract: Aiming at the shortcoming caused by the lack of prevention measures in the consideration of the existing drug abuse epidemiologic research, an Susceptible-Infected-Treated-Recovered-Susceptible (SITRS) drug abuse epidemiologic model based on prevention and therapy strategy was proposed by introducing prevention mechanism. Firstly, through analyzing the evolution process of the populations correlated with the drug abusers, an autonomous dynamical system was constructed by using ordinary differential equations. Secondly, the existence and local asymptotic stability of the drug-free equilibrium point of the system were proved. Thirdly, the unique existence of endemic equilibrium point was analyzed, and the sufficient conditions for global asymptotic stability of the endemic equilibrium point were obtained. Finally, the necessary conditions of existing backward bifurcation were calculated, the basic reproduction number under comprehensive prevention and therapy strategy and the number under single therapy strategy were compared. The possibility that backward bifurcation may exist and the stability of equilibrium points were verified by the numerical simulations. The study results indicate that, compared with single therapy strategy, comprehensive prevention and therapy strategy can further reduce the basic reproduction number of drug abuse thus more effectively prevent the breeding of drug abuse by increasing publicity coverage rate and education efficiency.

Key words: drug abuse, mathematical epidemiology, equilibrium point, stability, prevention and therapy strategy

摘要： 针对现有毒品滥用流行病学研究未考虑预防措施的不足，通过引入预防机制，提出了基于防治策略的易感-感染-治疗-康复-易感（SITRS）毒品滥用流行病学模型。首先，通过分析毒品滥用相关人群的演化过程，利用常微分方程构建了一个自治的动力系统；其次，证明了系统无毒平衡点的存在性和局部渐进稳定性；然后，分析了地方病平衡点的唯一存在性，并得到了地方病平衡点的全局渐进稳定性的充分条件；最后，计算了出现后向分支现象的必要条件，并比较了综合防治策略和单一治疗策略下的基本再生数。数值模拟验证了存在后向分支的可能性及平衡点的稳定性。研究结果表明相对于单一治疗措施，采用综合防治策略，通过提高宣传覆盖率和教育有效率，能够进一步降低毒品滥用的基本再生数，更有效地防止毒品滥用的滋生。

关键词: 毒品滥用, 数学流行病学, 平衡点, 稳定性, 防治策略