本文選題：流感模型　切入點：穩(wěn)定性　出處：《北京建筑大學》2017年碩士論文　論文類型：學位論文

【摘要】：在研究傳染病時,研究者通常會采用數(shù)學模型來刻畫傳染病的傳播機理和傳播規(guī)律,從而采取有效的措施預防控制疾病的傳播和爆發(fā),減少疾病對人類健康的危害。本文主要針對呼吸道疾病流行性感冒(流感),建立了三類數(shù)學模型來探究流感傳播的動力學行為。首先,我們研究了一類具有年齡結構和媒體播報的短期SIR流感模型,將人口分為青少年和成人兩部分。數(shù)值模擬表明:(1)媒體播報在控制流感傳播的進程中發(fā)揮著重要作用,然而當疫情爆發(fā)時,如果媒體播報的持續(xù)時間過長,會弱化人們的防范意識。(2)當采取措施控制疫情的爆發(fā)時,應考慮人口的異質性,否則,將會低估疾病的規(guī)模。其次,我們探討了一類具有疫苗接種和有限醫(yī)療資源的流感模型,對模型進行理論分析,得到了無病平衡點是全局漸近穩(wěn)定的,并證明了疾病的一致持久性。這意味著疾病將長期存在,危害人們健康。并利用數(shù)值模擬驗證了分析結果,揭示了提高疫苗的有效性并不能使疾病的規(guī)模降低,但可以推遲疾病爆發(fā)的高峰期;提高恢復率,可以有效地降低疾病的染病規(guī)模。最后,我們建立了兩個具有藥物敏感和耐藥性的雙重菌株流感模型,并將藥物敏感菌株的感染者分為兩類,分別為無癥狀的和有癥狀的。有癥狀的感染者接受治療之后,一部分形成耐藥性,一部分恢復。兩個模型的主要區(qū)別是:第一個模型的有癥狀感染者必須經歷無癥狀的染病期,才可以顯現(xiàn)癥狀;而第二個模型在疾病初始就將感染者分為無癥狀的和有癥狀的。我們分析了兩個模型的無病平衡點、邊界平衡點的穩(wěn)定性,證明了疾病的一致持久性,考慮了治療所引起的副作用,通過計算得到了獲得再生數(shù)和整體再生數(shù),并用數(shù)值模擬驗證了理論結果,評估了治療所引起的副作用,即:治療率不是越高越好,治療率的增加會減小藥物敏感菌株的染病規(guī)模,使其逐漸滅亡,但會使耐藥菌株的染病規(guī)模變大,疾病出現(xiàn)反彈現(xiàn)象,對人們的生命健康造成更大的威脅。并對兩個模型的染病規(guī)模、染病高峰期的到達時間等特征做了比較,得到第二個模型比第一個模型更容易爆發(fā),但持續(xù)時間較短,從而為流感的預防和控制提供了理論依據。
[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.
【學位授予單位】：北京建筑大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175

【參考文獻】

相關期刊論文 前1條

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本文編號：1645111