Dynamics Analysis And Application Of Fractional-order HIV Infection Model

HIV infection is a serious disease of impacting the global health and perplexing the medical experts in recent hundred years, and its research is always a hot issue of concerns. Fractional calculus theory is a new theoretical branch of many mathematicians and applied scientists concerned in recent years, because fractional calculus has memory function relative to the integer order calculus, and it is very important to study dynamics of all kinds of cells in our body. So for decades biologists and mathematicians are interested in introducing the fractional-order to mathematical model for HIV infection. Research of fractional order HIV infection model is a kind of dynamic analysis based on the theory of fractional differential equations. Mathematics have been committed to use the mathematical model to analyze the dynamic changes of HIV infection in various types of cells in the body in order to use the medicine at the best moment so as to achieve the best therapeutic effect.Based on the previous research on fractional HIV model, this paper give its further stud-ied. The thesis is divided into seven chapters:In chapter 1, we briefly introduced the research background of related problems, the main work and the states of study at home and abroad.In chapter 2, we introduce the necessary preliminaries such as definitions, properties of fractional calculus and some related lemmas and so on.In chapter 3, we deal with existence of fractional-order HIV infection model’s solution with CTL immune response by Banach fixed point theorem, and then study its stability by Lyapunov stability theorem. In studying existence and stability of HIV infection model, we first use Banach fixed point theorem prove existence of model’s solution, then we use gener-alized Routh-Hurwitz theorem to study its stability by analyzing its characteristic equation.In chapter 4, we increase time delay to the model and investigate its local stability by analyzing the characteristic equations and global stability by Lyapunov stability theorem, we also carry out detailed analysis on its threshold dynamics and find its threshold delay. In studying delay’s effect on HIV model’s stability, we first analysis the influence of different time delay poses to the stability of the HIV model in theory, then we find the model’s critical delay of stable and unstable in the numerical simulation by using the simulation method, and then verify the correctness of the conclusion.In chapter 5, we introduce Mittag-Leffler stability, and then study Mittag-Leffler stability of a fractional-order model for HIV dynamics in HIV-specific helper cells. In studying Mittag-Leffler stability of HIV infection model with two HIV-specific helper cells, we first choose a Lyapunov candidate function to verify model’s local stability, then we analyse its global stability in the sense of Mittag-Leffler stability, at last we simulate the dynamics of several kinds of cells in the process of HIV infection by using the simulation method.In chapter 6, we investigate optimal control for a HIV multitheraphy model with drug-resistant mutants, and then apply numerical simulation to analyze impact of optimal control for HIV infection model.In chapter 7, we make a brief summary for this paper and introduce our future work.