Analysis Of Global Dynamics For Coupled SVIR Models With Age Structure

Infectious disease is one of the problems that affect people’s health.AIDS,avian flu and Ebola haemorrhagic fever in recent years are undoubtedly obstacles to people’s healthy life.Scientists are trying to construct appropriate mathematical models for the purpose of describing the development and evolution law of infectious disease accurately,and then to find a scheme to prevent and control the propagation of disease effectively.The first chapter describes the research progress in related fields at home and abroad.We find that there are few studies on the influence of vaccination age and vaccination ratio on the cross population propagation of infectious disease.Therefore,this paper studies a class of coupled SVIR infectious disease model with age structure,vaccination ratio and immune loss effect on vaccination group.The model allows the vaccinated individuals to become susceptible again after vaccine losing protective properties,and the vaccinated individuals satisfy first order partial differential equations structure with regard to vaccination age.In the second chapter,the model that we are going to investigate is expressed into a nondensely defined Cauchy problem by virtue of the linear operator semigroups theory.Subsequently,the existence,uniqueness,point dissipation and asymptotic smoothness of the strong continuous semiflow of the system are studied.Then,the existence theorem of the system global compact attractor is obtained.In the third chapter of the paper,we study the global dynamical behaviors of strongly connected networks with age structure coupled SVIR epidemic models.After some preliminaries,the expressions of equilibrium and the basic reproduction number are given.After analysing the structure of the model,we find that each node involves not only ordinary differential equations but also a first order partial differential equation(PDE).Therefore,the commonly used method for the construction of Lyapunov functional can not be applied immediately.In this paper,we transform the system into a new one making use of Volterra formula,where the equation of the new system on each node is composed of functional differential equations with infinite delay.Utilizing the Lyapunov method and the Graph theory,we reveal that the global stability of endemic equilibrium for the strongly connected network is determined by the basic reproduction number.In the fourth chapter,the global dynamic properties for non-strongly connected model are also investigated.We can find from Graph theory that the non-strongly connected network contains several strongly connected branchs.By analysing the condensed diagraph comprised from each strongly connected branch,we find that the system’s endemic or mixed equilibrium may exist in addition to the disease-free equilibrium.By defining the evaluation function,we prove that all the solutions of the system converge to the maximum value of the evaluation function,and the stability of the endemic equilibrium of the model depends on the basic reproduction numbers corresponding to every strongly connected branch.Finally,we use numerical simulation to verify the theoretical results.