Study On Asymptotically Dynamical Behaviors Of Kinds Of Parasitic Disease Propagation Models

In nature, many diseases are caused by parasites invaded the hosts, such as echinococcosis, distomatosis, termatodiasis, paragonimiasis, trypanosomosis, etc. Those diseases may regulate host density or result in host extinction in a manner. Therefore, the prevention and controlling of the parasitic disease are major problems which related with the human health and people’s living. The quantitative research on the regularity of epidemic may provide an important reference basis for the works of prevention and controlling. In recent years, considerable works have been carried out from different angles to describe all kinds of phenomena in the interactive and transmission processes of the parasitic disease. Given this, in this paper, we will aim to the interactive process of host-parasite, consider the influence of the factors（such as Allee effect, delay effect and environmental fluctuation, etc） on the transmission of parasitic disease, and discuss the effect mechanism of these factors from microcosmic（host-parasite interactive process） and macrocosmic（the transmission process of disease） viewpoints. More precisely, the main contents are as follows:In the first chapter, we introduce the research significance and research situation of domestic and overseas of the parasitic disease, and briefly introduce the main results obtained in the paper.In the second chapter, we study a discrete-time host-parasitoid model with Holling II functional response and Allee effect. We first obtain the local stability conditions of the fixed points by Jury criterion. Then, we show that the system undergo the flip bifurcation by using center manifold and bifurcation theory. Numerical simulations are carried out to verify stabilizing effect of Allee effect, and to exhibit the complex dynamic behaviors. Particularly it is shown that the addition of Allee effect has a positive effect impact on the local stability and bifurcation diagram.In the third chapter, we consider a dynamic model of host-parasite with a intracellular delay and treatment. Our research shows that the dynamical behaviors of model is determined by the threshold: R₀. If R₀ <1, the infection-free equilibrium E₁ is globally asymptotically stable. If 1 <R₀ <3, the infection equilibrium E* is locally asymptotically stable for τ≥0. By using Hopf bifurcation theorem, we obtained that if R₀ >3, the infection equilibrium E* losses stability and lead to a Hopf bifurcation with a family of periodic solutions at Ï„₀=Ï„. On the basis of this, by use of the normal form theorem and center manifold argument, we obtain the formulae which determine the direction, stability and period of bifurcation periodic solution. Finally, numerical simulations are carried out to verify our theoretical results.In the fourth chapter, we consider a nonlinear SIRS epidemic model, and analyze the influence of Levy noise on the asymptotic behaviors of the disease-free equilibrium and endemic equilibrium, respectively. First, we prove the existence and uniqueness of global positive solution. Next, we investigate the effect of Levy noise on the asymptotic behaviors of disease-free equilibrium P₀ and the endemic equilibrium P*, respectively. Finally, the theoretical results are validated by the numerical simulation. Our results show that the sample trajectories of the stochastic model is far away the corresponding deterministic solution trajectories under the influence of the Levy noise. Meanwhile, the Levy noise is beneficial to the controlling of the disease.In the last chapter, we give a summary of this thesis and provide some future direction of research on this field.