Dynamical Analysis Of Several Discrete Biological Models

Based on the forward Euler discrete method,our paper obtained two kinds of discrete infectious disease models,one is the discrete SIR infectious disease model with constant recovery rate,the other is the discrete SIS infectious disease model with nonlinear incidence rate.Through the theoretical analysis of these two kinds of dynamic models,it is found that when the discretization step length is a fixed constant or a variable constant,the discretization model can produce a wealth of dynamic behaviors.The specific research work of this paper is as follows:Firstly,we discuss a class of discrete SIR epidemic models with constant recovery rate.Firstly,the existence of equilibrium point is discussed,the global asymptotic stability of the disease-free equilibrium and the local stability of the endemic equilibrium are analyzed by using the method of qualitative theories such as the Jury criterion and constructing Lyapunov function.Secondly,the sufficient conditions for saddle node bifurcation,1:1 resonance bifurcation,Flip bifurcation and Neimark-sacker bifurcation are analyzed by central manifold theorem and bifurcation theory.Finally,the correctness of the theoretical derivation is verified by numerical simulation,and the discrete model with more complex dynamic behavior is shown.Secondly,a discrete SIS epidemic dynamics model with nonlinear incidence is studied.Firstly,a continuous SIS epidemic model with nonlinear incidence is given,and the corresponding discrete SIS epidemic model is obtained by using the Euler forward discretization method.In particular,the discretization step is kept to ?t.Then,we discuss the existence of the equilibrium of the discrete SIS model,which is independent of the discretization step.Secondly,we discuss the stability of the equilibrium.The results show that the discretization step length will affect the stability of the equilibrium.When ?t is small,the disease-free equilibrium is stable when R0<1,and at R0>1,the endemic equilibrium is also stable.when ?t is large,the stability of disease-free equilibrium and endemic equilibrium will change.At last,we discuss all kinds of branch problems that may occur when the disease-free equilibrium and endemic equilibrium are unstable.