| 摘要: | 如今，隨著社會與科學技術的發展，通過採用高科技和創新性的醫療方法，許多種類的傳染性疾病已經能夠被有效的控制。除了傳統的思想和方法，基於動態數學模型的研究方向方法正成為研究傳染性疾病的一項可選用的重要工具。一個恰當的愛滋病或B型肝炎數學模型可以幫助我們找到一種新的治療方法，並可用於估計或預測耐藥性、副作用或藥物效用的進展。在本論文中，我們的貢獻被提出在三個主要章節。在第3章中，其愛滋病數學模型考慮了一些未知參數和不可測的CD8+T細胞計數，然後提出一種藥物治療的切換控制策略，可以有效地抑制並清除感染細胞和血漿病毒顆粒，重建免疫穩態，增加健康細胞。在本章節中使用了李亞普諾夫函數來設計兩種分別針對不同情況下的切換控制，使愛滋病數學模型系統的狀態在不受未知參數和不可測細胞數影響的情況下，漸近接近健康均衡狀態。第4章討論了一個更為複雜的愛滋病數學模型,並設計了三種控制策略。本章提出的三個定理分別合成了視為三種藥物治療控制策u1(t)或/和u2(t)。這樣使所有狀態收斂到愛滋病數學模型的無感染平衡點。在這三個定理中，藥物治療控制策略可以是狀態函數或常數。另外，抗病毒治療通常可以有效地減輕B型肝炎病毒感染者的病毒負擔，切斷感染源，在疾病控制方面取得了若干貢獻。因此在第5章中考慮了B型肝炎數學模型。在三個主要定理中，分別設計了三種不同的藥物治療控制策略u1(t)或/和u2(t)以漸進地完成最終收斂到無感染平衡點的任務。請注意到本章節中所設計的藥物治療不是一個固定值，而可以是時變的並依賴於病人的狀態。
綜上所述，本文在李雅普諾夫函數，愛滋病數學模型和B型肝炎數學模型的基礎上，提出了幾種針對這些模型的控制設計方法。最後，給出了幾個算例和模擬結果來說明了本文提出的不同參數控制的有效性、有效性和準確性。
;Nowadays, with social and technological development, many kinds of infectious diseases have been under control in the mass by means of the high-tech and novel method in the use of medical research. Usually, besides the traditional ideas and approaches, the research based on dynamic mathematical modeling is becoming a significant tool to study infectious diseases. In addition, a feasible HIV or HBV dynamic mathematical model can facilitate us to find out a novel therapy and estimate/forecast the progress of drug resistance, side-effect or drug effect. In Chapter 3, an HIV mathematical model has considered some unknown parameters and unmeasurable CD8+T cell count. Then, a switching control strategy of drug treatment is proposed to suppress and sweep out the infected cells and the virus particles of blood plasma, and rebuild the immunologic homeostasis to increase the healthy cells. In terms of the Lyapunov function, the switching control corresponding to two different cases is designed to make the state variables of the HIV system approach the health equilibrium asymptotically without the influence of unknown parameters and unmeasurable cell counts. In Chapter 4, a more complicated HIV model is discussed and three types of control strategy for this model are designed. According to the Lyapunov function, this chapter synthesizes three theorems to as three types of control strategy for drug treatments u1(t) or/and u2(t), respectively, are designed to make all states converge to the infection-free equilibrium point of HIV model. In these three theorems, u1(t) or/and u2(t) can be a function of states or a fixed constant within a certain range. Similarly, in Chapter 5, antiviral therapy can usefully reduce the viral burden and switch off certain infectious sources for HBV-infected patients, and it has achieved several contributions and successes in disease control. In this chapter, an HBV mathematical model is considered. Three main theorems in which three different control strategies for drug treatments u1(t) or/and u2(t) are designed, respectively, to complete the final task of converging to the infection-free equilibrium point of HBV model asymptotically. Note that the designed drug treatments u1(t) or/and u2(t) are not a fixed value, but it is time-varying and dependent on states.
In conclusion, on the basis of Lyapunov function and nonlinear dynamic mathematical models of HIV and HBV, this dissertation proposes several methods to design controls for HIV and HBV models. Finally, several examples and simulation results are given to illustrate the availability, effectivity and accuracy of the proposed controls with different parameters in this dissertation. |
