几类流感模型的动力学性质研究

发布时间：2018-03-21 18:51

本文选题：流感模型　切入点：稳定性　出处：《北京建筑大学》2017年硕士论文　论文类型：学位论文

【摘要】：在研究传染病时,研究者通常会采用数学模型来刻画传染病的传播机理和传播规律,从而采取有效的措施预防控制疾病的传播和爆发,减少疾病对人类健康的危害。本文主要针对呼吸道疾病流行性感冒(流感),建立了三类数学模型来探究流感传播的动力学行为。首先,我们研究了一类具有年龄结构和媒体播报的短期SIR流感模型,将人口分为青少年和成人两部分。数值模拟表明:(1)媒体播报在控制流感传播的进程中发挥着重要作用,然而当疫情爆发时,如果媒体播报的持续时间过长,会弱化人们的防范意识。(2)当采取措施控制疫情的爆发时,应考虑人口的异质性,否则,将会低估疾病的规模。其次,我们探讨了一类具有疫苗接种和有限医疗资源的流感模型,对模型进行理论分析,得到了无病平衡点是全局渐近稳定的,并证明了疾病的一致持久性。这意味着疾病将长期存在,危害人们健康。并利用数值模拟验证了分析结果,揭示了提高疫苗的有效性并不能使疾病的规模降低,但可以推迟疾病爆发的高峰期;提高恢复率,可以有效地降低疾病的染病规模。最后,我们建立了两个具有药物敏感和耐药性的双重菌株流感模型,并将药物敏感菌株的感染者分为两类,分别为无症状的和有症状的。有症状的感染者接受治疗之后,一部分形成耐药性,一部分恢复。两个模型的主要区别是:第一个模型的有症状感染者必须经历无症状的染病期,才可以显现症状;而第二个模型在疾病初始就将感染者分为无症状的和有症状的。我们分析了两个模型的无病平衡点、边界平衡点的稳定性,证明了疾病的一致持久性,考虑了治疗所引起的副作用,通过计算得到了获得再生数和整体再生数,并用数值模拟验证了理论结果,评估了治疗所引起的副作用,即:治疗率不是越高越好,治疗率的增加会减小药物敏感菌株的染病规模,使其逐渐灭亡,但会使耐药菌株的染病规模变大,疾病出现反弹现象,对人们的生命健康造成更大的威胁。并对两个模型的染病规模、染病高峰期的到达时间等特征做了比较,得到第二个模型比第一个模型更容易爆发,但持续时间较短,从而为流感的预防和控制提供了理论依据。
[Abstract]:In the study of infectious diseases, researchers usually use mathematical models to describe the transmission mechanism and laws of infectious diseases, so as to take effective measures to prevent and control the spread and outbreak of diseases. To reduce the harm of disease to human health. In this paper, three kinds of mathematical models are established to study the dynamics of influenza transmission. We have studied a class of short-term SIR influenza models with age structure and media coverage. The population is divided into two parts: adolescents and adults. Numerical simulations show that the media play an important role in controlling the spread of influenza. However, when the outbreak occurs, if the media broadcast for too long, it will weaken people's awareness of prevention.) when we take measures to control the outbreak, we should consider the heterogeneity of the population, otherwise, we will underestimate the scale of the disease. Secondly, We discussed a class of influenza models with vaccinations and limited medical resources. By theoretical analysis of the model, we obtained that the disease-free equilibrium is globally asymptotically stable. And proved the consistent persistence of the disease. This means that the disease will persist for a long time and endanger people's health. And the numerical simulation is used to verify the results of the analysis, which shows that increasing the effectiveness of the vaccine does not reduce the scale of the disease. But we can delay the peak of the outbreak, increase the recovery rate, and effectively reduce the scale of the disease. Finally, we have established two double strain influenza models with drug sensitivity and drug resistance. And divide the infected patients of drug-sensitive strains into two categories: asymptomatic and symptomatic. After treatment, some of them develop drug resistance. Partial recovery. The main difference between the two models is: the first model of symptomatic infected persons must go through asymptomatic infection period, can show symptoms; The second model divides the infected person into asymptomatic and symptomatic at the beginning of the disease. We analyze the disease-free equilibrium point, the stability of the boundary equilibrium point, and prove the consistent persistence of the disease. Taking into account the side effects of the treatment, the number of regenerations and the total number of regenerations are calculated. The theoretical results are verified by numerical simulation, and the side effects caused by the treatment are evaluated, that is, the higher the treatment rate is, the better the treatment rate is. The increase in the rate of treatment will reduce the scale of infection of drug-sensitive strains and gradually destroy them, but it will make the scale of infection of drug-resistant strains become larger and the disease will rebound. Compared with the two models, the second model is more likely to erupt than the first model, but the duration is shorter. Therefore, it provides a theoretical basis for the prevention and control of influenza.
【学位授予单位】：北京建筑大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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