Dynamics Of Age-structured HIV-1 Virus Stochastic Model

Since the stochastic differential equation theory were put forward by the Japanese mathematician Ito Qing in the 20th century, the stochastic differential equation has developed rapidly in the past hundred years, the theory is widely used in marine engineering, financial statistics, infectious diseases, ecology and other issues, which is of great significance to biology mathematics, financial statis-tics and other subjects. In the thesis, by using the probability theory, stochastic analysis and random dynamical system, ect, we study the dynamics properties of age structured HIV-1 virus stochastic model. The frame structure of this thesis is as follows:Chapter 1 introduces the development history of probability theory and math-ematical biology, related background information of HIV-1 virus and some of the preparatory knowledge which is used in this thesis, it also introduces the main contents and frame structure of this thesis.Chapter 2 mainly studies the asymptotic behavior of constructing age struc-tured HIV-1 virus stochastic model from ordinary differential equations and par-tial differential equations. By the Lyapunov function, Lyapunov stability theo-rem, martingale inequalities and stochastic analysis techniques, ect, we prove the existence and uniqueness, stochastic ultimate boundness and global asymptotic stability of solutions for age structured HIV-1 virus stochastic model.Chapter 3 mainly studies the existence of periodic solutions of age structured HIV-1 virus stochastic model. Firstly, by using the periodicity of correlation function, B-D-G inequality, Fubin theorem, local martingale theory and stochas-tic analysis techniques and other related knowledge, we prove the p-order uniform boundedness, the existence of unique local maximum solution and moment esti-mation. Secondly, we study the existence and uniqueness of periodic solutions of age structured stochastic HIV-1 virus stochastic model.Chapter 4 mainly studies the asymptotic stability behavior of age structured HIV-1 virus stochastic model driven by Levy process. Firstly, by using the non Gauss theory, generalized Ito formula, Kunita inequality, probability properties,probability measure and stochastic analysis techniques and other related knowl-edge, we prove the existence of unique, p-order boundedness, uniform Holder continuity and cauchy property of transition probability of solution for age struc-tured HIV-1 virus stochastic model driven by Levy process. Secondly, we study the asymptotic stability of age structured HIV-1 virus stochastic model driven by Levy process.