【作者】 徐敏；

【导师】 丁永生；

【作者基本信息】 东华大学 ， 管理科学与工程， 2007， 博士

【摘要】 随着艾滋病这种严重威胁人类健康和生存的传染病在全球范围内迅速传播与蔓延,对其传播动力学过程进行研究,进而提出如何降低其传播风险的解决方案在其风险管理与控制工作中占有十分重要的地位。目前绝大部分对艾滋病传播过程进行建模都建立在常微分方程的基础上,而没考虑随机干扰对其传播过程的影响,这样就造成对其传播与发展趋势无法进行较准确预测,从而也就难以提出有效措施降低其传播的风险。本论文的目的是运用随机模型对艾滋病传播动力学过程进行研究,对未来感染者人数进行更加准确地预测;并分析影响其传播的关键因素,探讨可能降低其传播风险的策略,为其社会行为风险管理与控制提供一定的决策支持。论文主要分为两大部分,当考虑疾病传播率受到环境因素的干扰时,其由一固定常数变为一随机过程,论文第一部分研究了环境因素的干扰对艾滋病传播过程的影响,主要成果包括:考虑实际情况中环境因素的干扰,结合已有模型,将随机模型扩展到艾滋病传播过程的研究,建立了艾滋病传播过程的随机微分方程模型,将确定型情形与随机干扰的情形进行了对比,然后结合具体数据,利用数值解方法,对中国未来若干年艾滋病病毒感染者人数进行预测,得到中国社会未来若干年艾滋病病毒感染者人数占总人口数比例的一个分布趋势,分析了疾病传播率对传播过程的影响,提出了通过调节传染率控制指标来达到一定的控制目标的方案,并结合模型进行了相关的风险分析。在此基础上,证明了艾滋病传播过程的随机微分方程模型解的存在唯一性定理,并得出它是有界的结论,然后利用伊藤公式分析了其渐进稳定性,得出稳定条件,分析了p阶矩稳定的条件,根据已经证明的定理,结合具体数据,通过仿真验证了疾病传播率的一个临界值,得出能使艾滋病传播速度降低直至消亡的一个条件,并深入探讨了艾滋病爆发所可能产生的经济后果。现实中,政府一定会对艾滋病的传播与蔓延进行干预,会进行疫苗研制,新药品的开发,也会收治病人,本文第二部分考虑收治率的介入对模型的影响,分析了存在环境因素干扰下更为复杂的模型的相关性质。主要成果包括:将收治率引入原模型,建立了带有收治率的艾滋病传播的随机微分方程模型,利用伊藤公式得出能使解渐进稳定的一个条件,证明了解的渐进稳定性、几乎必然指数稳定性及几乎必然p-阶矩指数稳定,通过仿真进一步验证和说明了在其它参数不变的情况下,能够使艾滋病感染者人数占总人口的比例降低直至最终消失的收治率的一个临界值,更进一步地,从经济效益的角度,建立了收治病人的收益—支出模型;接着,对艾滋病的社会行为风险进行了分析,针对带有收治率的模型,通过仿真,计算了感染者比例超过一定目标的风险概率,在此基础上,分析了感染者的知识、态度、行为对模型的影响,得出感染者比例的概率分布所满足的福克尔-普朗克前向方程,再通过仿真验证了相应结果,比较了在感染者的知识、态度、行为不同的情况下,未来感染者比例的不同变化趋势,并得出结论:只有当感染者积极面对艾滋病、积极接受治疗,才能在最大程度上降低艾滋病的传播速度,最后结合模型结果,从不同角度对艾滋病的社会行为风险进行了进一步地分析。更多还原

【Abstract】 The AIDS epidemic belongs to a kind of disease which threats the life and health of people seriously. As AIDS spreads all over the world, to research its transmission dynamics and then render some strategies to decrease its transmission risk plays a very important role in risk management and control for AIDS. Until now, most of the work focuses on ordinary differential equation (ODE) on modeling AIDS transmission dynamics without consideration of the effect of stochastic disturbance on its transmission. Consequently, accurate prediction for its spread can not be obtained. And in such a way, effective way to decrease its transmission risk can not be offered. This thesis aims to research AIDS transmission with application of stochastic model and predict the number of people infected with HIV in the future more accurately, then analyze the critical attributes affecting its course of transmission and study some strategy to decrease its transmission risk in order to render some decision support for the risk management and control of its social risk behavior.This thesis can be separated into two main parts. The result in the first part covers the effect of the environmental disturbance on AIDS transmission. When the disturbance of the environmental noise is taken into account, the disease transmission is changed from a constant to a stochastic process. In such case, we obtain the following results: Consdering the effect of the enviorment on AIDS transmission, the stochastic differential equation (SDE) model for AIDS transmission is extended and formulated, based on its orginal model. The deterministic and stochastic situation is compared. With application of numerical solution, the future proportion of the population infected with HIV against total population is forecasted in next few years in China with actual data, and then the distribution trend for the proportion of the population infected with HIV against total population is obtained. The effect of the disease transmission on AIDS transmission is analyzed and some strategies are presented to reach a certain control by means of adjusting the transmission rate control index, and the corresponding risk analysis is given meanwhile. Based on this, the existence and uniqueness for the solution is proved, and it is concluded that the solution is bounded and belongs to the interval [0, 1]. And then the asymptotic behavior is analyzed and the condition for the stability is obtained, and the p-th moment stability is also attained. Next, by simulation, a critical value for disease transmission rate is demonstrated with the help of actual data and some condition for the speed for AIDS spread to fall and then finally for AIDS to die out is obtained. And, the effect of AIDS’s broken out on economics is also analysed.Actually, the government takes action to prevent AIDS from spreading. They are dedicated to develop the vaccine and new medicine to treat AIDS. They also pay attention to assemble the patients and treat them. When the treatment rate is considered into the model, the property of the more complex model is analyzed and the main contributes in the second part includes: With consideration of treatment rate, a stochastic model with treatment rate is formulated to describe AIDS transmission. A condition for the asymptotic behavior of the solution is obtained as to the extended model with application of Ito’s formula, and the asymptotic behavior is analyzed and the condition for the stability is obtained, and the p-th moment stability is also attained. Then a critical value is illustrated for the treatment rate to make the proportion of the population infected with HIV against total population fall and then for AIDS to die out by simulation when other parameters is unchanged. Furthermore, the input-output model for treatment of patients is formulated based on the model. Then the risk of social behavior leading to AIDS is analysed. And, by simulation, the risk probability is computed of the proportion of the population infected with HIV against total population exceeding the given control value based on the stochastic model with treatment rate. Based on this, the effect is analyzed of the knowledge, attitudes and action of people infected with HIV on the model, and the Fokker-Plank forward equation is obtained for the probability distribution of the proportion of the population infected with HIV against total population and this result is illustrated by simulation. The changing trend of the proportion of the population infected with HIV against total population in the future is compared among different situations as far as the knowledge, attitudes, behavior of the infected person are concerned. And a conclusion can be made that the speed for AIDS transmission can be decresed to the largest degree only if the infected person can be confronted with HIV/AIDS very bravely and is willing to get treatment positively. Finally, further analysis is given for the risk of AIDS’ social behavior based on the analysis of the model.更多还原