# 臺灣博碩士論文加值系統

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 運用數學模型來研究流行病在人群中傳播所引起的疫情，已有百年歷史。自從愛滋病（英文簡寫為AIDS）被醫學發現公布，就引起了生物、醫藥與數學界的注意，並試圖找出處理這類可怕傳染病的方法；至今已有許多相關研究被發表。然而這些研究大都著眼於人群中AIDS的傳播，而鮮少探討病患個人病情的發展。 本論文根據AIDS的傳染機制，並針對目前較常用的病情判定標準：健康T細胞密度、染病T細胞密度及HIV（俗稱愛滋病毒）密度，建立了一個AIDS患者病情演化的數學模型以及兩個在藥物控制下的AIDS病情演化數學模型。簡化起見，這些模型不考慮潛伏期之類複雜問題。 由於這些數學模型都是非線性的微分方程組，其分析解不易求得；因此，我們藉由數值計算方式求得模型的近似解，並用其狀態軌跡圖來顯現數學模型之平衡點的穩定性，及模型中各參數在病情演化上所扮演的角色。狀態軌跡圖顯示，若藥物能降低感染係數 或染病T細胞所釋出的HIV數量 ，則AIDS患者的病情就能獲得改善。
 It has been hundreds of years, since mathematical models were applied to the study of epidemic disease. From the time when AIDS was inspected, a lot of researchers in the related fields including biology, medicine and mathematics, were interested in and tried to deal with this kind of epidemic. Up to now, there have been many reports involving in AIDS; however, most of these reports concerned only the spread of the epidemic disease in the population, and rarely concerned patients’ condition. This thesis was based on mechanism for AIDS. By figuring out the cell concentration of healthy T-cell density, infected T-cell density and HIV density, patients could be identified whether they were infected with AIDS. Without taking the medical issue such as the incubation period into deep consideration, the study constructed one AIDS mathematical model without treatment and two mathematical models with treatment. Because these mathematical models are written in nonlinear differential equations, it is usually not easy to find the analytical solution. Therefore, we adopted numerical methods to find out the approximating solutions. From the graphs of these approximating solutions, we found that the equilibrium states of mathematical models were asymptotically stable and also discovered some crucial approaches to control AIDS symptoms. For instantce, if the infection coefficient or the amount of HIV released from an infected T cell could be decreased, then AIDS symptoms would be under control.
 致謝i中文摘要ii英文摘要iii章節 第一章 緒論1 第二章 愛滋病簡介3 第三章 傳染病的數學模型8 第四章 愛滋病的數學模型20 第五章 數值實驗31 第六章 結論39參考文獻41
 [1]衛生署疾病管制局，http://nidss.cdc.dov.tw。[2]姜啓源等人編著，高等教育出版社出版，數學建模案例選集，第107-123頁。[3]馬知恩等人編著，科學出版社出版，傳染病動力學的數學建模與研究，第259-270頁。[4]E Beretta, Y Kuang. Modeling and Analysis of a Marine Bacteriophage Infection. Math. Biosci., 1998, 149: 57-76.[5]E Beretta, Y Kuang. Modeling and Analysis of a Marine BacteriophageInfection with Latency period. Nonlinear Anal. Real World Appl., 2001, 2: 35-74.[6]H Hethcote. The Mathematics of Infectious Diseases. SIAM Review, 2000, 42: 599-653.[7]Isihara P A. Immunological and Epidemiological HIV/AIDS Modeling. UMAP Journal 2005, 26(1): 51-89.[8]Perelson A S, Nelson P W. Mathematical Analysis of HIV-1 Dynamics in Vivo, SIAM Review, 1999, 41(1): 3-44.[9]R Culshaw, S Ruan. A Delay Differential Equation Model of HIV Infection of CD4+ T-cells. Math. Biosci., 2000, 165: 27-39.[10]R M Anderson, R M May. Population Biology of Infectious Diseases I, Nature, 1979, 180: 361-363.
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