考虑控制策略及医疗资源的传染病模型的建立与分析

发布时间：2018-05-27 06:07

本文选题：干预措施 + 医院病床数　；参考：《中北大学》2016年硕士论文

【摘要】：传染病一直以来都是涉及面最广、影响最深、危害最大的社会问题之一。为了有效地控制疾病的流行与爆发,政府部门一方面会采取诸如:隔离,对新兴疾病的报道,带口罩,注射疫苗等干预策略;另一方面,政府会在短时间内提供相关的医疗资源如:医护工作者,病床,设备等用于疾病的控制。利用数学建模的方法可以有效地分析疾病的流行趋势从而为疾病的控制提供一定的理论依据。为了探索医疗资源以及政府干预措施对疾病控制的影响,本文做了以下研究:1.建立了考虑政府干预策略及医疗资源限制的SIR模型。通过分析发现系统会出现一系列复杂的动力学性态:后向分支,Hopf分支,Bogdanov-Takens分支。我们发现在疾病爆发初期,政府相关部门若能及时采取干预措施则只需要提供较少的医疗资源就可以有效地控制疾病的爆发;否则需要投入大量的医疗资源对疾病进行控制。2.建立并研究了考虑医院资源的SEIS模型。通过分析发现基本再生数R0不再是疾病流行的阈值即当R0小于1时,系统会出现两个正平衡点。进一步,我们得到系统发生后向分支的参数条件。结果证明:医院的病床数目小于某个临界值时,疾病就会流行。为此,可以通过增加医院病床数的方式来对疾病进行控制。3.在现实生活中,接种疫苗是一种较为有效的控制策略。为了探索此种控制策略与医疗资源对疾病控制的影响。在第五章,我们研究了一类具有接种项且考虑医院病床数的SVIS模型。通过分析,我们发现基本再生数R0是疫苗接种率φ的函数且不再是疾病流行的阈值,当R0大于1时,系统会有一个正平衡点;当R0小于1时,系统会出现两个正平衡点或者无正平衡点。当系统存在两个正平衡点时,其中染病者数量较小的是鞍点,染病者数量较大的为非鞍点。通过对模型的进一步分析,我们发现当传染率较大或者病床数目较小时系统都会出现后向分支。因此,当R0小于1时,通过增加病床数和减少疾病的传染率是可以消灭疾病的。
[Abstract]:Infectious diseases have always been one of the most widespread, most influential and harmful social problems. In order to effectively control the epidemic and outbreak of the disease, government departments will, on the one hand, adopt intervention strategies such as isolation, reporting of emerging diseases, wearing masks, vaccinating; on the other hand, The government will provide medical resources such as health care workers, hospital beds and equipment for disease control in a short period of time. The method of mathematical modeling can be used to analyze the epidemic trend of disease and provide some theoretical basis for disease control. In order to explore the impact of medical resources and government interventions on disease control, this paper did the following research: 1. 1. A SIR model considering government intervention strategy and medical resource restriction was established. It is found that there are a series of complex dynamical states in the system: backward bifurcation Hopf bifurcation Bogdanov-Takens bifurcation. We found that in the early stage of the disease outbreak, if the relevant government departments can take intervention measures in time, they can effectively control the disease outbreak by providing less medical resources; otherwise, they need to invest a large amount of medical resources to control the disease. 2. A SEIS model considering hospital resources was established and studied. It is found that the basic reproduction number R0 is no longer the threshold of disease prevalence, that is, when R0 is less than 1, two positive equilibrium points will appear in the system. Furthermore, we obtain the parameter conditions for the backward bifurcation of the system. The results show that when the number of hospital beds is less than a certain threshold, the disease will be prevalent. To this end, you can increase the number of hospital beds to control the disease. 3. Vaccination is an effective control strategy in real life. In order to explore the impact of this control strategy and medical resources on disease control. In chapter 5, we study a class of SVIS models with inoculation term and hospital bed count. Through analysis, we find that the basic reproduction number R0 is a function of vaccine coverage rate 蠁 and is no longer the threshold of disease prevalence. When R0 is greater than 1, the system will have a positive equilibrium point, and when R0 is less than 1, the system will have a positive equilibrium point. The system will have two positive equilibrium points or no positive equilibrium points. When there are two positive equilibrium points in the system, the number of infected persons is the saddle point, and the number of infected people is non-saddle point. Through further analysis of the model, we find that the backward branching occurs when the infection rate is high or the number of beds is small. Therefore, when R 0 is less than 1, the disease can be eliminated by increasing the number of beds and reducing the infection rate.
【学位授予单位】：中北大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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