Dynamic Behavior Analysis Of An Age-Structured Hand-Foot-Mouth Disease Model With Time Delay

The dynamics model of infectious diseases is an important tool for interpreting the laws of in-fectious disease transmission.In this dissertation,using Hille-Yosida operator,Perron-Frobenius theory,and Hopf bifurcation theory,several types of hand,foot,and mouth disease（HFMD）epi-demic model with age structure and time delay are studied.The dynamic behavior of the model solution is studied to understand the epidemic law of HFMD and provide strategies for predicting the onset of HFMD.The main research contents include well-posedness,regularity,asymptotic stability,uniform persistence,and the existence of periodic solutions.The first two chapters introduce the epidemic status of hand,foot,and mouth disease,the research status of dynamic models,and summarize the main work,research results,and theoretical basis of this dissertation.In Chapter 3,the infection age and time delay are introduced into the HFMD epidemic model,and the asymptotic properties of the solution of the model are investigated.First,define the basic regeneration number （?）,based on the distribution of characteristic roots of the characteristic equa-tion,analyzed the stability of disease-free equilibrium E₀and epidemic equilibrium E_*of system,when （?）>1and （?）<1.At the same time,using a method similar to that in the literature,it is proved that when （?）>1,there is a global attractor in the system,resulting in a consistent persistence of the infection.In Chapter 4,considering the saturated incidence（?）,a HFMD model with age structure,time delay and saturated incidence rate is established.Define the basic regeneration number （?）based on the principle of C₀-semigroup is used to study the local and global asymptotic stability of disease-free equilibrium E₀.Based on the distribution of the roots of the relevant characteristic equations,the epidemic equilibrium E_*is discussed and proved global attractiveness by construct-ing a Lyapunov function.Chapter 5 divides the infected person I（t）into asymptomatic infected person I（t）and symp-tomatic infected person A（t）to explore the stability of the hand foot mouth disease model and conduct Hopf bifurcation analysis.Firstly,define the basic regeneration number （?）and certify that （?）is the threshold value that determines the extinction or survival of the disease.That is,the research of the local and global asymptotic stability of disease-free equilibrium E₀.At the same time,the root distribution of the corresponding characteristic equation is investigated,and the local stability of epidemic equilibrium E_*is proved.The existence of Hopf bifurcation near the epidemic equilibrium is explored through bifurcation theory.Through the analysis of the above three models,relevant conclusions are obtained.As the isolation rate increases,the number of latent individuals,infected individuals,and cured individ-uals will decrease.Introducing isolation mechanisms is an effective way to control the number of people infected with HFMD.When the number of infected persons reaches saturation,the param-eterσof the saturation incidence rate will affect the transmission rate of the disease.The higher theσ,the more beneficial the control of HFMD.When the basic regeneration number is greater than 1,an increase in the delay ofτwill lead to an increase in the number of symptomatic infected individuals and a decrease in the number of asymptomatic infected individuals.In addition,an increase in the proportion of symptomatic infections is beneficial for the control of HFMD.