Research On The Stability For The SIQS Epidemic Model With Quarantine And Time Delay

The epidemic exists extensively in moden life.Since 1920s, people had tried to study the rules of the spread of the epidemic diseases, which presented the theoretical proof for making the strategies of this predicting and treating diseases. However, the ability which models predicts and control diseases depend greatly on the assumptions made in the modeling process. In order to gain deeper insights into the mechanism of disease transmission and evaluate therapeutic strategies, much attention has been focused on the design and analysis of mathematical models. Body has immunity ability to diseases and the immunity is not permanent. Therefore, investigating the models including of time delays in such models make them more realistic. To the study of the epidemic models, people are interested in the issue that the parameters determine the disease to die out or prevail in a population. Now people are focused on extinction and permanence of dynamic models. Many models represent the transmission dynamic of differently infectious diseases by developing mathematical models using systems of ordinary differential equations, partial differential equations and function differential equations.In this paper, stability of the disease-free equilibrium and the epidemic equilibrium of two class of delay epidemic model is investigated. And with the condition that is the threshold is more than one, we obtained the permanence of the dynamic model.The main work of this paper is as follows:First, we have studied the SIQS epidemic model with quarantine and discrete time delay, and then obtained the threshold of the dynamic model. The disease-free equilibrium and the epidemic equilibrium are obtained. We mainly use the technique of Liapunov functional to establish the global asymptotic stability of disease-free equilibrium and use the method of linear system and characteristic equation to prove the instability. Then we use the method of limit equation theory, linear system and lowing dimension of the system to establish the global stability of epidemic equilibrium but need conditions which are a certain number of susceptible and no deaths to illness. Lastly we put aside to consider the balance point of view, as a whole we have obtained the permanence of the dynamic model.Second, we also have studied the SIQS epidemic model with quarantine are continuous time delay, then obtained the threshold. We use the method of linear system and characteristic equation to prove the local stability of disease-free equilibrium when the threshold is less then one and the instability when threshold is more then one. We then use the technique of Liapunov functional to establish the global attractive of disease-free equilibrium, hence have obtained global stability of disease-free equilibrium when the threshold is less then one. Then we use linear system and stability theorem to establish the local asymptotic stability but need another sufficient condition. Lastly we also put aside to consider the balance point of view, as a whole we use a similar method in chapter 3 to obtain the permanence of the dynamic model.