Several types of dynamical properties of Epidemic Model
|School||Gannan Teachers' College|
|Keywords||Epidemic model Impulsive differential equation Non-autonomous system Extinction Permanence|
Epidemiological dynamics is a very important branch of mathematical biology. Studying the spread of infectious diseases and predicting the trend of theirs development are its vital aims. They offer the theoretical basis for the government institution and the medical and health institutions to make corresponding measures against diseases. In this paper, three kinds of epidemic models are proposed, and the dynamical properties of these models are studied. The main results are summarized as followings:In the first chapter, the purpose and significance of the epidemic dynamics are introduced, and also the research achievements and progress of the epidemic dynamics in these aspects are reviewed. At the same time, the main work and the organization of this paper are given.In the second chapter, a non-autonomous eco-epidemic model with disease in the prey is considered. Some sufficient conditions on the permanence and extinction of the infected prey are obtained. Moreover, by constructing a Lyapunov function, the global attractivity of the model is discussed. Finally, our theoretical results are confirmed by numerical simulation.In the third chapter, a delayed epidemic model with pulse vaccination and nonlinear incidence is formulated. The existence of infection-free period solution is analyzed, and two new threshold values R(?) and R(?) of the impulsive epidemic system are given. By the comparison theory of impulsive differential system, we have: (i) if R(?) < 1, the infection-free period solution is globally attractive, (ii) if R(?) > 1, the system is permanent. Our results indicate that a large vaccination rate or a short pulse of vaccination is benefit for the extinction of the disease.In the forth chapter, we investigate an impulsive epidemic model with time dependent contact rate. The effects of period varying contact rate and mixed vaccination strategy on eradication of infectious disease are studied. By using Floquet theory and comparison theorem, we obtain the conditions of global asymptotical stability of the infection-free periodic solution and permanence of the model. A threshold value R0 for the disease to be extinct or not is established. The results imply that a large vaccination rate is propitious to the extinction of the disease. In the end, the theoretical results are confirmed by numerical simulations.