刘风

摘 要：现有毒品滥用流行病模型假设吸毒者康复后对毒品拥有永久“免疫”力，而忽视了其再次成为毒品易感者的可能性。针对这一问题，通过考虑社区治疗和隔离治疗两种措施，分析了毒品滥用人群的演化过程，提出了基于暂时“免疫”力的毒品滥用流行病模型，并计算了模型的基本再生数，讨论了模型平衡点的存在性和稳定性。当基本再生数小于1时，模型存在一个局部渐进稳定的无毒平衡点;当基本再生数大于1时，模型存在唯一的地方病平衡点，并利用几何方法证明了地方病平衡点的全局稳定性;当基本再生数等于1时，如果满足一定条件，模型出现后向分支现象。数值模拟验证了上述所有结果。研究结果表明提高隔离治疗率、改善社区治疗效果和降低接触传染率可以有效抑制毒品滥用的流行。

关键词：毒品滥用;数学流行病学;无病平衡点;地方病平衡点;全局稳定性

中图分类号：TP391.9

文献标志码：A

Abstract： The recovered drug users maybe become susceptible to drug again， but this possibility is neglected in the existing drug abuse epidemic model in which the drug users are assumed to be permanently immune to drugs after recovery. Aiming at the problem， the evolution process of drug abuse population was analyzed with considering both community treatment and isolation therapy， and a drug abuse epidemic model based on temporary immunity was proposed. Furthermore， the basic reproduction number of the proposed model was calculated and the existence and stability of the proposed model equilibrium were discussed. It is shown that the proposed model has a drug free equilibrium which is locally asymptotically stable and a unique endemic equilibrium when the basic reproduction number is less and more than unity respectively. And the global stability of the endemic equilibrium was proved by using a geometric approach. Otherwise， the proposed model has the phenomenon of backward bifurcation under certain conditions when the basic reproduction number is equal to unity. The above results were verified by the numerical simulations. The results indicate that the prevalence of drug abuse can be effectively inhibited by increasing the rate of isolation therapy， improving the effect of community treatment and reducing the infection rate.

英文關键词Key words： drug abuse; mathematical epidemiology; Drug Free Equilibrium （DFE）; endemic equilibrium; global stability

0 引言

毒品滥用又称为物质滥用，是指蓄意过度使用鸦片类、大麻类等精神活性物质以达到使人兴奋的效果，从而导致产生依赖性症状的现象。近年来，新型合成毒品的出现更是加剧了毒品滥用的蔓延。2008 — 2017年《中国禁毒报告》数据显示，我国登记在册的吸毒人员由112.67万人上升到255.3万人，增长了1.27倍，年均增长率为9.5%。可见，十年间我国毒品滥用问题整体呈现出不断上升的趋势。毒品滥用不仅严重损害吸毒者个人的身心健康，同时还会加快艾滋病等疾病的传播速度，导致各类刑事案件的大量发生，社会危害性极大。……