Stability Analysis Of The Infectious Disease Model

The prevention and treatment of the epidemic disease is a big problem which is related to the human healthy and the economy of the country. It is an important evidence of the prevention and treatment work to do the quantitative research on the spreading rules of the disease. Then what is the epidemic dynamics. It is a science, which formats a mathematics model to reflect the varying rules according to the occurrence development and environment changes of the disease. By probing into the characteristics of the model. We can show the developing procedure of the disease, predict the spreading rules and the developing tendency, analysis the causes and key elements of the spreading of the disease, seek the best strategy of preventing and controlling it, provide the theoretic base and quantity evidence for the prevention and treatment.In chapter II. This article establishes two dynamic models for various infectious diseases, SEIR model and SIRS model. For SEIR model, we derived approximate equations for the epidemic threshold as well as the spreading dynamic, and analyzed the stability of this model, show the necessary condition for the stability of epidemic. We then discuss a particular condition of this model. For SIRS model, we derived approximate equations for the epidemic, established a system model for the spreading processes, then we made local and global stability analysis to the equilibrium points of the system equations, obtained the necessary conditions for the equilibrium points. The global stability of the equilibrium points of the system is proved by constructing Lyapunov function. Finally, we discuss a particular condition of this model.In chapter III ,This paper focuses on nonlinear transmission rateβIS3 of SIR model, by using Hurwitz Criterion, V function, Bendixson-Dulac Criterion study its global stability and gradual global stability, solving the system reached the point of balance.In chapter IV, In this paper, we establishes and analyzes the ecology - epidemic disease SIS model which come from preying. We acquired local stability by eigenvalue analysis at the equilibrium and obtained global stability of the solution by constructing appropriate Lyapunov function, discussed uniform persistence of the system.In chapter V , The paper researches on the epidemic disease through the establishment of a cellular automata based on the dynamics model. Bases on the model, emulated some epidemic diseases that with different propagating properties and also the processes to control the propagating. Meanwhile researches on the factors that could influence the disease's propagation, especially, the infectors migration. Finally, we make some suggestions on how to control these probleme.