Analysis And Control Problem Research For Several Eipdenic Dynamical Systems

The health and life of people are always threatened by the infection. And the infection had brought tremendous mischance to pelope in history. In combination of the research of epidemic and mathematics, the establishing mathematical infectious models based on the methods of dynamics, have become very important research problems which are of the theoretical and practical meaning. 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.In this disertation, based on the theory of stability and bifurcation for dynamical systems, bifurcation for differential-algebraic systems (singular systems), some epidemic models with nonlinear incidence rate are investigated, such as periodic, chaotic, hyperchaotic behaviors, codimension-one bifurcations (transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, fold bifurcation, flip bifurcation, Neimark-Sacker bifurcation) and codimension-two bifurcations (1:1 strong strong resonance,1:2 strong resonance,1:3 strong resonance,1:4 strong resonance, fold-flip bifurcation). For the chaos, hyperchaos, period behaviors of some systems, the controllers are designed, such that the disease gradually disappears. This disertation are summaried as follows.(1) The dynamical behaviors of an SEIR epidemic system governed by differential and algebraic equations with seasonal forcing in transmission rate are studied. The cases of only one varying parameter, two varying parameters and three varying parameters are considered to analyze the dynamical behaviors of the system. For the case of one varying parameter, the periodic, chaotic and hyperchaotic dynamical behaviors are investigated via the bifurcation diagrams, Lyapunov exponents spectrum diagram and Poincare section. For the cases of two and three varying parameters, Lyapunov diagram is applied. A tracking controller is designed to eliminate the hyperchaotic dynamical behavior of the system, such that the disease gradually disappears. In particular, the stability and bifurcation of the system for the case which is the degree of seasonality/β1=0 are considered. Then taking isolation control, the aim of elimination of the disease can be reached. The model has very complicated dynamical behaviors.(2) The dynamic behaviors of an SEIQS epidemic system governed by differential and algebraic systems with a nonlinear incidence rateβIpSq(p>0, q>0) are analyzed. At first, for the different p and q values, the existence and stabilities of the disease-free equilibrium and endemic equilibria for the system are discussed, respectively. Thus, for one parameterβvarying, the conditions of codimension-one bifurcations are obtained by the bifurcation theory of differential and algebraic systems, including transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation. In particular, for two parametersβand p varying codimension-two bifurcation (Bogdanov-Takens bifurcation) for the system is researched via center manifold theory and bifurcation theory.(3) A discrete epidemic model with nonlinear incidence rate obtained by the forward Euler method is investigated. The existence and stabilities of the disease-free fixed point and endemic (positive) fixed point for the system are discussed using stability of discrete dynamics, respectively. For one parameter varying, the conditions of existence for codimension-one bifurcations (fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation) are derived by using bifurcation theory of discrete systems. Flip bifurcation is called period-doubling bifurcation, and it is an important approach to produce chaos. The system has chaos based on flip bifurcation. And the system has period orbit due to Neimark-Sacker bifurcation. In order to eliminate the chaos or Neimark-Sacker bifurcation of the discrete epidemic model, a tracking controller is designed such that the disease disappears gradually. The numerical simulations including bifurcation diagram, Lyapunov exponents spectrum diagram and the dynamic response of the system, not only illustrate the validity of the proposed results, but also display the interesting and complex dynamical behaviors.(4) The dynamic behaviors of the discrete epidemic model with a nonlinear incidence rate are further discussed. For two parameters varying, the dynamical behaviors are researched. Codimension-two bifurcations of the discrete epidemic model, associated with 1:1 strong resonance,1:2 strong resonance,1:3 strong resonance,1:4 strong resonance and fold-flip bifurcation, are analyzed by using bifurcation theorem and the normal form method of maps. The system has the complicated dynamical behaviors via the theoretical analysis. In particular, there is chaos for 1:2 strong resonance and fold-flip bifurcation. Moreover, in order to eliminate the chaotic behavior of the discrete epidemic model, baed on (3) the tracking controller is designed such that the disease disappears gradually. Finally, numerical simulations are obtained by the phase portraits, the maximum Lyapunov exponents diagrams for two different varying parameters in 3-dimension space, the bifurcation diagrams, the computations of Lyapunov exponents and the dynamic response. They not only illustrate the validity of the proposed results, but also display the interesting and complex dynamical behaviors.