Dynamical Analysis Of Pertussis Models

Infectious disease dynamics is a way to study the spread of infectious diseases through the establishment of mathematical models,including the spread speed,spatial scope.transmission route and dynamics mechanism of infectious diseases,which can effectively reflect the spread law of infectious diseases and give suggestions for the effective prevention and control of infectious diseases.According to the transmission mechanism and immune mechanism of pertussis,this paper establishes the infectious disease model of pertussis from two different perspectives.One is the pertussis model with household structure,which is based on the time of family members together;the second is the pertussis model with age structure from the perspective of covert infection.The dynamic behavior and biological meaning of the two models are discussed respectivelyIn chapter 1,we first introduce the background knowledge of pertussis,including the characteristics of patients,the modes of transmission,the ways of transmission,the preventive measures and the impact on people's life and economy;Secondly,we summarize the research progress of pertussis infectious disease model at home and abroad and some preparatory knowledge needed in this paperIn chapter 2,according to the transmission mechanism of pertussis and the existing prevention and control measures,a pertussis model with household structure is estab-lished.Firstly,the expression of the basic reproduction number R0 is obtained by the method of the next generation matrix,and the Lyapunov function is constructed to prove the global asymptotic stability of the disease-free equilibrium point E0.Secondly,we prove the existence and uniqueness of the endemic disease equilibrium point,the local stability of the endemic disease equilibrium point is proved by using Gersgorin theorem,and the global asymptotic stability of the endemic disease equilibrium point is proved by Lyapunov-LaSalle's invariant principle.Finally,the results of the model are verified by numerical simulation.In chapter 3,from the perspective of covert infection of pertussis,we set up an age-structured pertussis model with covert infection.To begin with,we prove the exis-tence and uniqueness of the positive solution of the system.Then,the expression of the effective reproduction number R(?)is obtained by analyzing the system,and the global asymptotic stability of the disease-free equilibrium point is proved by using the method of characteristic line and Fatou's lemma.Next,the existence of the endemic equilibrium state is analyzed,and the existence of backward bifurcation is proved by using the implicit function theorem.In the end,the main results of this chapter are simulated numerically with Matlab.In chapter 4,we briefly summarize the work of this thesis,and put forward some problems and directions that are worth further research and exploration.