Qualitative Analysis Of Fractional Epidemic Models

Infectious diseases are caused by the pathogen or parasite, which can transmit between humans,between humans and animals and between animals. Infectious diseases have not only been a serious threat to human health, but also bring great disaster to people's livelihood. Mathematical models play a important role in understanding dynamical behavior of the infectious diseases. By the study of dynamical behavior of mathematical models, it analyzes the development of the disease process,reveals its popular law, predicts trends and provides a quantity basis and theoretical foundation for the prevention and treatment decisions for people. Therefore, it is realistic meaning to study dynamics of epidemic models. Based on the analysis and summary of compartmental epidemic models and epidemic models on complex networks, the author researches the dynamic behavior of the above two fractional models respectively through the analysis of the fractional stability theory. This paper takes the following form:In the first Chapter, the author elaborates the significance, current status and progress of the dynamical system of epidemics. The main contents and originalities of this paper are expounded.The second Chapter establishes a delay fractional SIS model, studying stability and bifurcation of the system. Firstly, by the next generation matrix, the basic reproduction number R₀ is obtained.Tnen, selecting delay as the bifurcation parameter, discusses locally asymptotical stability and bifurcation of equilibria through the characteristic root method. The model is shown that the disease free equilibrium is stability for all delays when R₀ <1. When R₀ >1 and ????[0,?₀) the endemic equilibrium is locally asymptotical stability; When R₀ >1 and?>?₀, the endemic equilibrium is not stability. Finally, numerical simulations verify the conclusions.The third Chapter studies a fractional HIV model with immunization, analyzing the effect of vaccines on backward bifurcation of the system. Firstly, selecting the effectivenesss ? and dosage ? of the vaccines respectively as the bifurcation parameter, discuss the backward bifurcation of the system and obtain a new threshold for disease eradication. Then, the disease free equilibrium is found to be globally asymptotically stable when R₀ <1 using Lyapunov function theory and La Salle invariant principle. Finally, numerical simulations are performed to illustrate and verify the conclusions.The fourth Chapter talks about the fractional SIR model with birth and death on complex networks, obtaining the threshold R₀ which determines outcome of the disease. Using Lyapunov function method and LaSalle invariant principle, the disease free equilibrium is proved to be globallyasymptotically stable when R₀ <1 and the disease will be cleared out; the endemic equilibrium is globally asymptotically stable when R₀ >1 and matix M is irreducible and the disease will be persistent. Finally, the simulations are carried out to conform to our analytical results.The fifth Chapter summarizes the rearch work of this dissertation. Furthermore, the future work is made.