Dynamical Analysis Of The Epidemic Model

Epidemiology dynamics and population dynamics are two great branch inmathematical biology. Epidemiology dynamics is an important method of t theoreticalquantitative study on infectious diseases. By describing the qualitative andquantitative analysis of the dynamic behaviors of epidemic model and the numericstimulations, we show the development of disease, uncover the epidemic law andanticipate the developing trend. It provides theoretical basis for preventing andcontrolling infectious diseases. Population dynamics studies the variation law of thepopulation and structure with time. It also studies how to protect,exploit and utilizethe population by mean of rational manual intervention.In this paper, an epidemic model with nonlinear incidence rate is studied. Weconstruct an SIR model with three dimensional nonlinear incidence rate based onassumptions of logistic population growth. Firstly, we discuss the stability of theequilibrium of the model. Then we draw a conclusion that the system has been ariseda supercritical bifurcation near nontrivial equilibrium according to center manifoldcomputation. Finally, some numerical simulation for justifying the theoreticalresults are also provided.This paper has investigated the dynamics of discrete-time predator-prey systemwith Holling–Ⅱ functional reaction function, and discusses the model in the closedfirst quadrant. Firstly, we discuss the stability of the fixed point of the system. Thenaccording to center manifold theorem and bifurcation theory. we show that the systemproduces a period doubling bifurcation with the parameter changes, then makes anentrance to chaos and Neimark-Sacker bifurcation. numerical simulation is presentedto verify the theoretical results.This dissertation is divided into four chapters:Chapter one introduces the history and the meaning of infectious disease modeland introduces the main works done.Chapter two introduces the main lemmas and theories involved in the paper.Chapter three investigates a SIR model with three dimensional nonlinearincidence rate, based on assumptions of logistic population growth and discusses thestability of the equilibrium of the model and Hopf bifurcation near nontrivialequilibrium. Chapter four studies the dynamics of discrete-time predator-prey system withHolling-Ⅱ functional reaction function, and discusses the model in the closed firstquadrant. We discuss the stability of the fixed point of the system and the2、4、8…bifurcation and Neimark-Sacker bifurcation produced by the system withparameter fluctuations.