Dynamic Analysis And Control Researches Of Several Epidemic Models

The dynamic behavior of the model has practical significance for grasping the transmission mechanism of infectious diseases and providing reference for prevention and control.In this thesis,we mainly study two kinds of infectious disease models based on vaccination and optimal control problems,and analyze a two-strain infectious disease model for the existence of multi-strain diseases.In the second chapter,an SEIHRVI epidemic model with standard incidence rate is developed,and the effect of vaccination on the control of infectious diseases is studied.The condition for the existence of the endemic equilibrium is given,and the stability of equilibrium is proved by Lyapunov function.Personal protection,vaccination and treatment strategies are introduced into the model.Pontryagin’s maximum principle is applied to analyze the optimal control problem and the optimal control strategy is obtained.The results show that improving the vaccination rate and vaccine efficacy can effectively control the spread of the disease.Attention to self-protection,timely vaccination and treatment can reduce the scale of disease outbreaks to a greater extent.In the third chapter,a model for COVID-19 and malaria co-infection is constructed and investigated the effect of vaccination on the COVID-19-malaria co-infection.First,the sub-models of COVID-19 and malaria are analyzed,and the existence and stability of disease-free equilibrium and endemic equilibrium of sub-models are discussed respectively.Then,the full model of COVID-19 malaria co-infection is discussed,and it is obtained that the disease-free equilibrium is locally asymptotically stable when the basic reproduction numbers R_c0 and R_m0 are both below unity.Additionally,sensitivity analysis of key parameters driving disease dynamics shows that the infection rate of disease has the greatest influence.Then,the prevention and treatment measures of COVID-19 and malaria are introduced,the optimal control problem is proposed,and the optimal control is solved by Pontryagin principle.The simulation results of different control strategies show that vaccination has a positive effect on reducing the number of co-infected individuals.Although applying each strategy singly helps to reduce the burden of co-infection,strategy A increases the number of malaria cases,and strategy B prolongs the cycle of COVID-19 infection.Therefore,measures of control COVID-19 must be combined with efforts to ensure malaria control is maintained.In the fourth chapter,an epidemic model describing two-strain with different nonlinear incidence rates is studied.Four equilibria of the model are found and the basic reproduction numbersR₀¹ andR₀² are obtained.The global stability of equilibrium is analyzed by constructing Lyapunov function.It is proved that whenR₀²≤1,if R₀¹≤1,the both strains die out,and if R₀¹＞1,the strain 1 persists and the strain 2 dies out,while when R₀²＞1,under specific conditions,if R₀¹≤1,the strain 1 dies out and the strain 2 persists,and if R₀¹＞1,the both strains persist.Numerical simulations support the theoretical analysis,and show the influence of the parameters describing the inhibition on the prevention and control of diseases.The results show that the strain with higher basic reproduction number will automatically outperform another strain.