Some Mathematical Models Of Infectious Diseases And Their Analysis

In this thesis, we study the dynamics of some mathematical models on infectious diseases, make the mathematical analysis and discuss the effect in the prevention and con-trol of infectious diseases. For different models, we discuss the critical thresholds, namely, the basic reproduction numbers, which determine whether the diseases die out or not. Applying the theory on the asymptotical autonomous system and the limit system, the Poincare-Bendixson property, the theory of the second compound matrix and Lyapunov function, etc., we study the global dynamics of corresponding models. Moreover, by means of numerical simulations, we discuss the trend of the epidemiology under different values of parameters.The arrangement of the thesis is as follows.In Chapter 1, a brief introduction is given for the background of mathematical mod-eling on infectious diseases, including some basic compartmental models. Moreover, a survey of the results in this thesis is also presented.In Chapter 2, we discuss a model of rabies with the interaction between human and dogs. Some suggestions are provided to the prevention and control of rabies according to the corresponding analysis and simulations.In Chapter 3, we discuss an SIRS model with vaccination and isolation. A critical vaccination rate and a critical isolation rate for the disease are obtained.In Chapter 4, we discuss an SIR model with vaccination and varying total population. We study the global dynamics of both the proportional model and the original model, and the relations between these two models.In Chapter 5, taking into account the social distance induced by media coverage, we discuss an SIS epidemic model with two patches.In Chapter 6, we revisit a two-patch SIS model with infection during transport, and study the influence of travel in the transmission of infectious diseases.