Study On Infectious Disease Models With Incomplete Vaccination

Incompletet vaccination means that not all people will be completely immune to the virus after vaccination,but some people will still be at risk of becoming infected,and this group of people will be at a lower risk than those who are not vaccinated.This kind of situation is very common in our daily life,because of this,the research of infectious disease model with incomplete vaccination can not only describe the actual problem more objectively and truly,but also be more accurate in solving the problem,thus providing an effective reference value for proposing a better and more scientific epidemic prevention plan.In this paper,we mainly study the dynamic properties and numerical analysis of several infectious disease models with incomplete vaccination.The main contents are as follows:Firstly,Considering that it takes some time for human to produce antibodies after vaccination,we study the vaccination models with time delay.In the study,we first solved the disease-free equilibrium point and the basic reproduction number of the model,and found that the model has a unique endemic equilibrium point when the basic reproduction number is greater than 1.We then discuss the stability of the disease-free and endemic equilibrium points under different time delays and the conditions where the model has a Hopf bifurcation at the endemic equilibrium.Further determine the direction of the Hopf branch and the stability of the branch periodic solution by solving the first Lyapunov coefficient of the system based on the canonical form and central manifold theory.The theoretical result accuracy is verified by numerical simulation,and it is also shown that the delay phenomenon will affect the spread of infectious diseases.Secondly,Because human vaccination is discontinuous,multi-dose,and periodic pulse,we studied the infectious disease model with pulse vaccination.For the study of the pulse models,we,at the first,optimized the models,solved the disease-free cycle solution and the reproduction number.Then the conditions for the local asymptotic stability of the disease-free periodic solution and the persistence of the impulsive model system are solved.The bifurcation of the system model is discussed by using the impulsive bifurcation theory,and the numerical simulation analysis is carried out.From the simulation results and images,it can be seen that when drug treatment and non-drug intervention are not implemented,vaccination alone cannot effectively control the spread of the epidemic.The research of infectious disease model with incomplete vaccination can not only describe the actual problem more objectively and truly,but also be more accurate in solving the problem,thus providing an effective reference value for proposing a better and more scientific epidemic prevention plan.Thirdly,we studied the age-structured SVIR infectious disease model with pulse vaccination.The model considers that the parameters in the infectious disease model are affected by the age of the population.For this pulse model with age structure,the model is optimized and the disease-free periodic solution is obtained when the population structure is stable.Then,according to a reduction method,the optimized model is reduced to a six-dimensional impulsive ordinary differential equation system? According to the impulse comparison theorem,the conditions for the model system to be persistent are obtained.Finally,it summarizes the contents and innovations of the full text and puts forward further prospects for the research direction of infectious disease models.