On Problems Of Analysis Of Epidemical Dynamic Systems And Control Of Population Chaotic Systems

Infection has always threatened the health and life of people, and it had brought tremendous mischance to pelope in history. Today, some infection have been eliminated or controlled because of powerful measures that people adopt, but there is a kind of trend for infection to occur again and spread, and various new types of infection emerge endlessly which has made huge loss of life and wealth of people. For studying laws of spread of the interior and interspecific population and correlative social factors, building the mathmaticl models which show the course of disease development is very important. The aims of studying the models are to understand and discover the laws of development and forecast the trend of development, in order to offer the evidence to the correlative department for making policy and resisting the disease better. This work is very important.Biology dynamical systems are a representative kind of nonlinear complex systems. In addition to complexity of system itself and hugeness of structure, there are many factors such as internal population interaction, the interaction between the population and disturbance from outside environment These factors will lead systems to the existence of complex dynamic behavior such as quasi-periodic solutions and bifurcationand chaos. There have been many researches to control biology systems in order to achieve aim of people, but the problem of control complex dynamic characters of biology systems that exist by applying control theory and methods of engineering has begun just now, there are many unsolved problem.In this disertation, to a class of epidemical model, by using the stability theory of dynamical system, the existence and stability problems of equilibrium points and period solutions of are studied when continuous vaccination and pulse vaccination are considered. To some population chaotic models, the problem of chaotic phenomena and chaotic control of population systems are investigated by the control theory and methods such as OGY (Ott, Grebogi, Yorke) method, nonlinear feedback control, feedback linearization control, straight-line stabilization method, and chaotic tracking control. Main results of this distertation are as follows.(1) To two kinds of bilinear incidence SIRS (Susceptible-Infected-Removed-Susceptible) epidemic model with vertical infection and continuous vaccination, different contiounous vaccination metohods are adopted according to different types of new-born, the existence and global stability of infection-free equilibrium and endemic equilibrium (positive equilibrium) is studied by using Lyapunov function method and LaSalle invariable principle respectively. The conditions of the existence and stability of infection-free equilibrium are obtained, as well as the conditions of the existence and stability of endemic equilibrium are obtained.(2) To a class of bilinear incidence epidemic models with continuous vaccination and pulse vaccination and vertical infection, the basic reproduce numbers of SIRS epidemic model are given respectively. Global stability of infection-free equilibrium and endemic equilibrium is proven by using Lyapunov function method and LaSalle invariable principle; the existence and stability of the infection-free period solution is proved by using Floquet multipler theory,comparison theorem and nonlinear analysis method. The conclusion shows that adopting the strategy of pulse vaccination is better than adopting the strategy of continuous vaccination for the same SIRS model with bilinear incidence.(3) It is shown that there exists the chaotic behavior for a class of discrete population difference equation with species flowing in special domain by Lyapunov exponents distinctly. A controller is designed to control the stabilities of equilibria.and periodic orbits by using the essential ideas of the OGY method and eliminate bad chaos that can affect population increase and ecology balance. To a class of single-species model with harvesting coefficient, the existence of chaotic behaviors is validated by Lyapunov exponents. System dynamic behaviors are showed under the different harvesting coefficient range by studying bifurcation diagram detailedly. Bad chaos that can affect population increase and ecology balance is eliminated by using the OGY method, nonlinear feedback control method and chaotic tracking control method, chaotic orbits are stabilized on the anticipant period orbits and chaos is eliminated. The aim of tracking control is more flexible, it is any point in phase space or any given period orbits and chaotic orbits according to needs of actual problem. The result provides a kind of method to maintain population permanent survival.(4) To a class of discrete predator-prey model, it is shown that there exists the chaotic behavior by lyapunov exponents distinctly. For eliminating chaotic phenomena of population, a feedback controller is designed to stabilize the chaotic population to the expected target periodic orbits by using feedback linearization method and establish rational exploiture strategy; at the same time, a controller is designed to stabilize population to any period states by applying chaotic tracking control method. To a discrete coupled logistic model with the symbiotic interaction of two species, firstly existence of chaotic phenomena is validated by applying Lyapunov exponent, System dynamic behaviors of the coupled logistic model is more complex dynamic behaviors than logistic model by analyzing bifurcation diagram detailedly. A controller is designed to stabilize the population to any periodic orbits by using chaotic tracking control method. To a discrete functional response model, a controller is designed to stabilize the population to any reasonable periodic orbits by using chaotic tracking control method and thereby chaos control is realized.(5) To a class of time-delay population model, it is shown that there exists the chaotic behavior by Lyapunov exponents distinctly. A controller is designed to stabilize the chaotic population to fixed point orbits by using the method of straight-line stabilization for system, thereby chaotic phenomena of population are eliminated.