The Dynamical Behavior Analysis Of Two Classes Of Discrete Epidemic Models

This paper study two classes of discrete epidemic models, The discrete models areobtained by using the method of Euler discrete. we give the the positivity and theboundless of solve; A new sufcient condition on the global stability of the equilibriumis established by using Lyapunov function and the comparison principle of diferenceequations and Jury criterion.The first part, we studied a class of discrete SIR epidemic model with time delayand standard incidence rate by using the method of mixture Euler discrete. We givethe positivity and the boundless of solve by the analysis method. The criteria for globalattractivity of disease-free is obtained by applying Lyapunov functional technique asR₀<1and give sufcient condition for permanence of system as R₀>1.The second part, by the backward Euler scheme discretizing the corresponding con-tinuous time SIR epidemic model, we obtain the discrete time SIR epidemic model. Wealso discuss the positivity and the boundless of solve by the analysis method. More-over, we give the sufcient condition for the stability of the free equilibrium and endemicequilibrium by using Lyapunov function method.The third part, we studied a class of discrete cholera epidemic model with timedelay, by using the forward Euler method. we consider the positivity and the boundlessof solve, and the global stability of the equilibrium when τ=0, and the locality stabilityof the equilibrium when τ>0. Moreover, we give the criterion of producing the Hopfbifurcation.