Modeling And Analyzing Of The Dynamics Of HIV Infections Based On Fractional Differential Equations

Since the first case of AIDS was reported during the early 1980s, large amounts of work have been done on modeling the dynamics of HIV infection. Mathematical models of HIV infection (or HIV-immune system) dynamics were proposed not later than 1986, and the disease has become the subject of intense modeling efforts. However, all of these models have been restricted to integer-order differential equations.With the rapid development of fractional-order differential equations (FODEs), many mathematicians and applied researchers have tried to model real processes using the fractional-order differential equations. FODEs are closely related to fractals which are abundant in biological systems. They allow for the possibility of modeling phenomena which traditional differential modeling cannot accomplish. Particular emphasis is that a major difference between fractional-order models and integer-order models is that fractional-order models possess memory, while the main features of immune response involve memory. Hence, we attempt to model HIV infection with immune response using a fractional-order system.Firstly, this thesis presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation. We obtain that small changes in order and initial condition cause only small changes of the solution under some conditions. It provides the bases of theory to the numerical simulations for FODEs.Then, for fractional-order HIV infection models, only when non-negative solutions or positive solutions exist, the models is physical. Thus we discuss the problem of existence of positive solution for a class of ordinary (delay) differential equation with fractional order. Some sufficient conditions for their existence and unique existence of positive solution are gained.Next, we introduce fractional-order into a model of HIV infection of CD4~+ T-cells. We show that the model established possesses non-negative solutions as desired in any population dynamics. We obtain a restriction on the number of viral particles released per infectious cell in order for infection to be sustained. Under this restriction, the fractional-order system has a unique positive equilibrium - the infected steady state. By using stability analysis on fractional-order system, we obtain sufficient condition on the parameters for the stability of the infected steady state. Numerical simulations are presented to illustrate the results.Moreover, we propose a fractional HIV infection model with immune response and frequency dependence. We reveal, mathematically, how the order of a fractional differential system affects the dynamics of system. In the case of one-virus model, the fractional-order system has an interior equilibrium under some restriction. By using stability analysis on fractional-order system, we obtain sufficient condition on the parameters for the stability of the interior equilibrium. Our analysis shows that the interior equilibrium which is unstable in the classical integer-order model can become asymptotically stable in our fractional-order model. For the model of viral diversity, we find that strange chaotic attractors can be obtained under fractional-order model with frequency dependence. That is, the effect of viral diversity and the frequency dependence results in collapse of the immune system and makes the behavior of the system dynamics complex. However the chaotic motion may disappear and the fractional-order system stabilizes to a fixed point if the value of the order decreases.Also, a fractional-order model for the immunological and therapeutic control of HIV is studied qualitatively. We show the fractional-order model possesses a unique non-negative solution. The equilibria are found and their local stability are investigated. The global stability of the infection-free equilibrium is established. The optimal efficacy level of anti-retroviral therapy needed to eradicate HIV from the body of an HIV-infected individual is obtained.Furthermore, the optimal control theory is applied to a model related to HIV infection dynamics. Using an objective function that it minimizes the infectious viral load and count of infected helper T cells, and uses minimal dosage of anti-HIV drugs, we solve for the optimal control in the fractional-order system. The effects of mathematically optimal therapy is demonstrated. Finally, a summary of the thesis is made, and the deficiency in the project and the further development are narrated respectively.