Dynamical Analysis Of Two Types Of Infectious Disease Models With Age Structure

Based on the pathological background and transmission mechanism of hand-foot-and-mouth disease and tuberculosis,this thesis establish two types models of age structure with time delay.By analyzing the dynamic behaviors of models,displaying the effects of age structure and time delay on the spread and control of the disease,providing scientific theoretical basis for the prevention and control of these two types of infectious diseases.The specific research work in this thesis is as follows.In the first chapter,the pathological background of hand-foot-and-mouth disease,tuberculosis and the state current of research on the age structure are summarized briefly.At the same time,the pre-knowledge of this article is also introduced.In the second chapter,a hand-foot-and-mouth disease model with immunity age is investigated,where the constant treatment rate and the loss of acquired immunity are incorporated.Firstly,the well-posedness of the model is verified by changing it into an abstract non-densely defined Cauchy problem,and the conditions for the existence of disease-free equilibrium and the endemic equilibria are found.Secondly,the global stability of the disease-free equilibrium is proved by using the Lyapunov function,and the instability and local stability of different endemic equilibria are verified.The existence of saddle-node bifurcation and Hopf bifurcation are analyzed,and the Bogdanov-Takens bifurcation will occur for the model under some conditions.Finally,the correctness of the theoretical analysis is verified by numerical simulations.The results show that both non-periodic and periodic behaviors are appeared when the disease persists in the population,where the duration that the recovered individual stays in the recovery class plays an important role in the spread of the disease.In the third chapter,the dynamics of an age-structured tuberculosis model with relapse are studied.The time delay in the progression from the latent individuals to becoming the infectious individuals is considered in this model.Firstly,the dynamic behaviors are performed for the model,including presenting a formula for the basic reproduction number of the model,proving the global stability of the disease-free equilibrium and the local stability of the endemic equilibrium.Secondly,addressing the persistence of the solution semi-flow and the existence of a global attractor and establishing some results about stability and instability of the solutions for the model.Finally,the model is applied to describe tuberculosis transmission in China.The figures show that the number of the total population and the number of the annual newly reported TB cases both match the statistical data well.Moreover,the number of the total population,the latent individuals,the infectious individuals,the PPD positive rate and the prevalence rate from 2020 to 2035 all are predicted.In the fourth chapter,the current work is summarized briefly,at the same time,the deficiencies of this thesis and the issues that need further research are discussed.