Study Of A Discrete SIR Epidemic Model With Delay

Epidemic model is an essential part of mathematical biology model. We usuallyuse continuous mathematical model to characterize epidemic model. The advantage ofcontinuous mathematical model is to do qualitative research, it is appropriate when thepopulation density is large and infectious diseases spread fast. In fact, the change in thenumber of population is discrete,therefore,we also use discrete mathematical model tostudy infectious disease. It can refect the law of the spread of infectious diseases moreaccurately. The discretization of continuous mathematical model is an research thinking.Based on this idea,we study the dynamics behaviors of epidemic model in this paper.In the frst part of this paper, we discuss the dynamics behaviors of a discrete SIRepidemic model with Logistic process and delay. We use the way of discretization of a con-tinuous SIR epidemic model with Logistic process and delay. We study the boundednessand permanence of the model by the method of discrete Lyapunov function and iteration.We calculate the basic reproduction number R0and establish relevant criterions.In the second part of this paper, we study the dynamics behaviors of a discrete SIRepidemic model with density dependent birth rate and delay. We use the method ofdiscretization of a continuous SIR epidemic model with densit dependent birth rate anddelay. We discuss the boundedness and permanence of the model by the comparison prin-ciple of diferential equations and the method of discrete Lyapunov function. We calculatethe basic reproduction number R0and get the sufcient condition for the permance andextinction of disease.In the third part of this paper, we study the dynamics behaviors of a discrete SIRSepidemic model with nonlinear incidence rate and delay. We use the method of discretiza-tion of a continuous SIR epidemic model with nonlinear incidence and delay.We discussthe positivity, boundedness and global stability of the solution of this model.We use themothod induction to prove the positivety of the solution. We study the global stability of disease-free equilibrium and the permanence of disease by the comparison principleof diferential equations and the method of discrete Lyapunov function. On the basis ofpermanence, we get the sufcient condition for the global stability of endemic equilibrium.