On The Dynamical Behaviors In Eco-epidemic Models

The study of interact populations with in?uence of epidemiological parameters istermed as eco-epidemiology. It is a combination of epidemiology dynamics and ecologyand plays a very important role for controlling disease spread between populations. Inrecent years, some authors have paid more attention to predator-prey and competitivemodel with disease and acquired some glorious results on the research. In this paper,we perform researches of three models of eco-epidemiology in predator-prey model, whichinclude: a nonautonomous epidemic model with disease in the prey, an eco-epidemicmodel with distributed delay and a pulse vaccination eco-epidemic model with saturationincidence.The main contents in this paper can be summarized as follows:1. Firstly, in the first section, we introduce the background, developmental processand significance of eco-epidemiology. Then we introduce the research results of eco-epidemiology. Finally, the organization of the paper is given.2. In the second section, a nonautonomous eco-epidemic model with disease in theprey is studied. Some su?cient and necessary conditions on the permanence and ex-tinction of the infective prey are established and the global attractivity of the model areobtained by constructing a Lyapunov function. Lastly, an example is given to show thatthe periodic model is global attractive if the infective prey is permanent.3. In the third section, an autonomous eco-epidemic model with distributed delay isanalyzed. We obtain the system's permanence and the basic reproduction number R1 isidentified which determines the disease is extinct or not. And some su?cient conditions forthe stability of disease-free equilibrium and the endemic equilibrium have been obtainedby constructing a Lyapunov function and linearizing method.4. In the forth section, we investigate an autonomous eco-epidemic model withpulse vaccination and saturation incidence rate. Firstly, we show the existence and globalattractivity of the disease-free periodic solution. Further, we obtain the basic reproductive number R? which determines whether the disease is extinct or not. Our results indicatethat a proper pulse vaccination rate and pulse period will lead to eradication of the disease.