Research On The Dynamic Behaviors Of Infectious Disease Model Under Vaccination Strategy

Vaccination is an effective way to prevent infectious diseases from outbreaking and spreading.Considering the vaccination,establishing appropriate dynamic models for various infectious disease and analyzing the dynamic properties of these models are important methods to understand the transmission of infectious disease.This dissertation aims to strategy of vaccination,two kinds of infectious dynamic models with vaccination,SVIS and SVIR models,are established.The effects of vaccination rate and vaccination period on the disease transmission are investigated for continuous vaccination and impulsive vaccination,respectively.In the first chapter,major diseases in history,the current situation of dynamic models for infectious disease,as well as,the required prior knowledge,the innovation and main tasks of this dissertation are introduced.In the second chapter,we assume that the total population remains constant.SVIS models with continuous and impulsive vaccination are established.For the con-tinuous model,the basic reproduction number R₀which determines the extinction of disease is calculated by the stability theory of ordinary differential equation and Lasalle invariant principle.When R₀<1,there exists a disease-free equilibrium point which is globally attractive.When R₀>1,there exist a disease-free equilibrium point and an endemic equilibrium point.The former is unstable.The latter is globally asymptoti-cally stable.For the impulsive model,the Fixed point theorem and bifurcation theory are employed to prove the existence of disease-free periodic solution and endemic peri-odic solution.The asymptotic stability of the disease-free periodic solution is proved by Floquet Theorem.In the third chapter,we assume that the birth rate equals to death rate.SVIR models with continuous and impulsive vaccination are established.For the impulsive model,the Fixed Point Theorem,Floquet Theorem and Comparison Theorem of im-pulsive differential equations are combined to prove the global asymptotical stability of the disease-free periodic solution.The uniform permanence of the disease is proved by the Comparison Theorem of impulsive differential equations.In the fourth chapter,the correctness of the results is verified by numerical simu-lation.