【作者】 张玲；

【导师】 夏米西努尔；

【作者基本信息】 新疆大学 ， 应用数学， 2014， 硕士

【摘要】 最近十年,非法使用毒品已经成为全世界的一种社会和公共健康问题.实质上非法使用毒品增加了血液传播病毒的危险,相关毒品犯罪活动,家庭问题以及发达和发展中国家的公共健康医疗花费.据估计中国是全世界最多的注射吸毒者的国家.在全国范围内已注册了133万吸毒者,然而实际上吸毒者的数量更高.吸毒者有很多高风险行为,例如共用注射器具和销售性药物,这些大大加剧了感染艾滋病毒的传播.人类免疫缺陷病毒(HIV)和丙型肝炎病毒(HCV)传染的一个主要途径是共用注射器具.而毒品滥用者大部分是静脉注射,所以为了减少艾滋病的流行就必须控制毒品使用.海洛因是毒品滥用中最主要也是最严重的毒品之一.海洛因的使用一旦上瘾不仅会伤害身体,更会影响家庭及周围的人群.因此研究海洛因的一些数学模型对于预防吸毒,挽救吸毒者、控制吸毒问题都是有非常重要的价值.本文研究的具体内容概括如下:第一部分讨论了具有时滞的离散海洛因传染病模型的全局行为.通过使用非标准有限差分技术,我们得到了具有时滞的离散海洛因传染病模型.证明了解的正性和地方平衡点的存在,并说明了若基本再生数小于1,无毒平衡点全局稳定；若基本再生数大于1,使用海洛因是持久的.第二部分研究了一类具有不同意识阶段和分布时滞的海洛因传染病模型.模型涉及到有意识吸毒者和无意识吸毒者.建立合适的阈值R0,证明当R0<1时,无毒平衡点是全局渐近稳定的;当R0>1时,系统的毒品扩散平衡点是全局渐近稳定的.第三部分构建了具有饱和发生率和两个时滞的海洛因传染病模型.由平衡点的稳定得到海洛因扩散的基本再生数.利用合适的Lyapunov函数证明当基本再生数小于1时,无毒平衡点是全局渐近稳定的;当基本再生数大于1时,唯一的毒品扩散平衡点是全局渐近稳定的.更多还原

【Abstract】 In recent decade, illicit drug use has become a social and public health issue aroundthe word and virtually increases the risk of transmission of blood-borne viruses, drug-related criminal activities, family issues and public health medical costs in both developedand developing countries. It is estimated that China is the most injecting drug users ofthe world. It has registered1.33million drug users across the country, however the actualnumber of drug users is much higher. Drug users have a lot of high-risk behaviors, suchas sharing injection equipment and selling sexual medicine, which greatly increase thespread of HIV infection. The main reasons of Human Immunodefciency Virus (HIV)and Hepatitis C Virus (HCV) transmission is sharing injection equipment. While mostdrug abusers use intravenous injection. So, we must control drug use to reduce the AIDSepidemic. Heroin is one of the most important and main drugs in the drug use. Onceusing heroin, it will not only hurt the body, but also afect the family and people around.Therefore, it is very important to study heroin models of drug use to prevent the drugabuse, save drug addicts and control the drug problems. The main contents in this papercan be summarized as follows:In the frst part of this paper, we discus global dynamics of a discretized heroinepidemic model with time delay. We derive a discretized heroin epidemic model withdelay by applying a nonstandard fnite diference scheme. We obtain positivity of thesolution and existence of the unique endemic equilibrium. We show that heroin-using freeequilibrium is globally asymptotically stable when the basic reproduction number R0≤1,and the heroin-using is permanent when the basic reproduction number R0>1.In the second part of this paper, we study a distributed delayed heroin epidemicmodel with diferent conscious stages. The model allows for conscious drug users andunconscious drug users. The threshold property of R0is established. It is shown thatdrug-free equilibrium is globally asymptotically stable when R0<1; When R0>1, it is proved that the drug spread equilibrium of the system is globally asymptotically stable.In the third part of this paper, we construct a heroin epidemic model with saturationincidence rate and two distributed delays. We obtain the basic reproduction number ofthe heroin spread for the stability of equilibria. We show that drug-free equilibrium andthe unique drug spread equilibrium are globally asymptotically stable when the basicreproduction number is less than one and basic reproduction number is greater than onerespectively by using the proper Lyapunov functionals.更多还原