Qualitative Analysis Of A Diffusion Model Of Vaccination With Vaccination

Infectious diseases have been harmful to human health and also have a strong impact on the stability of society and the economical development. For a long time, human beings have been struggling with infectious diseases. Controlling infection sources, cutting off infection routes and protecting susceptible persons are all the methods to solve increasingly rampant infections diseases. However, vaccination is one of the most effective means to prevent and control the diseases.In the dynamics of infectious diseases, the most influential is "compartment model" known as the SIR model proposed by Kermack and McKendrick in 1927. To study measles, chicken pox and other infectious diseases, Kermack and McKendrick proposed an SIS model in 1932. In addition, SIRS model and SEIR model are studied by lots of scholars. Considering that the vaccination has a very obvious effect on the control of infectious diseases, in 2003,Gumel and Moghada proposed a SVI model, and the term V means vaccination. In reality, when a contagious disease is prevalent in one region, it usually has sustainable immunization part due to the disease that is isolated or cured. In this paper we will study a SVI model by introducing remove item R, which means recovering healthy people. SVIR epidemic model is first established, and then the SVIR model is analyzed and simulated. Specifically, this paper contains five chapters.In chapter 1,the background and research status about the development of infectious disease model and the main contents of this paper are introduced.Chapter 2 deals with the ordinary differential SVIR system. The local stability of disease-free equilibrium is discussed and the global stability of disease-free and endemic equilibrium is investigated by constructing Lyapunov functional.Further, we show that the model has transcritical bifurcation at the situation of disease-free equilibrium while R0=1. The vaccination and treatment of diseases by using maximum principle is also considered.Chapter 3 is devoted to the PDE (Partial Differential Equation) model. The uniform boundness of the solutions to the model is given, the local stability of disease-free equilibrium, the global stability of disease-free and endemic equilibrium by using Lyapunov functional are studied.In chapter 4, Matlab is used to simulate the theoretical results and the graph are provided to verify the correctness of the theoretical conclusions.In chapter 5,we first summarize our results, then present some work for future consideration.