Stability Of Two Types Of Discrete Epidemic Models With Distributed Time Delay

Dynamics of infectious diseases is a significant method applied in the theoretical and qualitative research of infectious diseases. There have been many studies by constructing differential equation models of infectious diseases, but little research via discrete models. In terms of parameter estimation and initial value selection, discrete models are more realistic than continuous models, because the data about infectious diseases are generally collected daily, weekly, monthly or yearly. Other advantages of discrete model is that it is much easier to obtain the existence and uniqueness of solutions of the initial-value problem. Compared with continuous models, discrete models are capable of expressing more abundant dynamic behaviors. Many continuous models, which cannot be solved or of theoretical analysis, have to conduct numerical simulation through being converted into discrete model. In this paper, the discrete SIR and SIRS epidemic models with distributed time delay are proposed and studied, respectively.Chapter 1 introduces the background of epidemic models and the research achievements contributed by many scholars in this field, as well as stating the major work of this dissertation.Chapter 2 analyzes the following SIR epidemic model with distributed time delay:The basic reproduction number of the model is R₀=â– . It is concluded that:if and only ifR₀≤1, the only disease-free equilibrium E₀=(b/μ,0,0) of the system is of globally asymptotic stability; and if and only if R₀> 1 the only endemic equilibrium E^*=(S^*,I^*,R^*) of the system is of globally asymptotic stability.Chapter 3 probes into the following SIRS epidemic model with distributed time delay: The basic reproduction number of the model is R₀=â– .The stabilities of the disease-free equilibrium and endemic equilibrium of the system are studied by employing arguments similar to that in Chapter 2. It is concluded that:whenR₀<1, the disease-free equilibrium E₀=(b/μ,0,0) is of globally asymptotic stability and the disease is extinct; whenR₀>1, the system is permanent.