Dynamics Of Epidemic Model And Discrete Predator-prey System

Epidemic model and population system are two major models in mathematical biology.The study of epidemic model could provide useful theoretical basis for the preventive and con-trol of propagation of infectious diseases.The survey of population system plays an important role in protecting the endangered plants and animal species and in guiding the developmen-t of ecology.This dissertation mainly dissects the dynamics of epidemiology and population ecology by applying the theory and method of nonlinear dynamical system.It mainly includes three aspects:continuous epidemic model,discrete epidemic model and discrete predator-prey system.Details are as follows:(1)An SIR epidemic model with limited medical resources is investigated.The recovery rate is considered as a function of the number of infective individuals and hospital beds.By applying the center manifold theory and bifurcation theory,we obtained that the model may exist backward bifurcation,saddle-node bifurcation and Hopf bifurcation,which implies that the propagation of infectious diseases not only rely on basic reproduction number but also depend on the number of hospital beds.(2)A multi-groups SIR epidemic model with human movement is explored.Through dynamical system theory and matrix related knowledge,the investigation proves that when Rv<1,the disease-free equilibrium is the only one equilibrium whose coordinate includes zero,meanwhile,it is global asymptotically stable.Moreover,the existence and stability of the non-zero equilibria for a two group epidemic model is explored.(3)We identify the optimal supplementary vaccination strategy for the prevention and con-trol of vaccine-preventable disease(such as,measles)via a three groups epidemic model with circular migration.Based on two reasonable objectives,the best supplementary vaccination is screened in terms of measures:final sizes,peak sizes,the number of vaccine doses needed to achieve prescribed reduction in epidemic sizes,and the likelihood of containing an outbreak.(4)The global dynamics of an epidemic model with vaccination,treatment and isolation is discussed.The work delves into the existence and global asymptotical stability of the equilibria by using dynamical system theory and Lyapunov functional.(5)An discrete SIR epidemic model with constant vaccination strategy is derived through nonstandard finite difference(NSFD)scheme.Based on the discrete dynamical system theory,the existence and stability of the fixed points is analyzed.Finally,we compare the dynamics of discrete model and the corresponding continuous model.(6)A discrete predator-prey system with Holling IV functional response is discussed.The existence and stability of fixed points is obtained through discrete dynamics theory.Sufficient conditions for the Flip bifurcation and Neimark-Sacker bifurcation are provided by bifurcation theory.