Dynamic Analysis Of SVIR Infectious Disease Model With Prophylactic Vaccination And Therapeutic Effect

The SVIR model is a useful tool for describing the transmission process of infectious diseases and evaluating the effectiveness of vaccination and therapy in controlling them.Vaccination can greatly reduce the risk of susceptible individuals being infected,thus slowing down the spread of the virus.Therapy can also play a role in reducing the mortality rate among patients and speeding up the recovery process for those who have recovered.This dissertation investigates three progressive SVIR infectious disease models that incorporate preventive vaccination and therapy.The main research contents consist of the following:(1)Developing Model Ⅰ,a comprehensive SVIR infectious disease model that takes into account an infectious rate function with psychological factors and a therapy function with medical limitations.The infectious rate function is a non-monotonic morbidity function,and the therapy function is a continuously differentiable saturation function.The existence of equilibrium points in Model Ⅰ is discussed,the basic reproduction number is calculated,and the local stability of both the disease-free and endemic equilibrium points is proven using the Hurwitz criterion.The global stability of the disease-free and endemic equilibrium points is demonstrated using the Lyapunov function and the second additive complex matrix,respectively.(2)Building Model Ⅱ on the basis of Model Ⅰ by increasing the number of vaccinations to three.The existence of equilibrium points in Model Ⅱ is discussed,and the basic reproduction number is calculated.The paper proves the local stability of both the disease-free and endemic equilibrium points by using the Hurwitz criterion and the global stability of the disease-free equilibrium points by constructing the Lyapunov function.The stability condition of Model Ⅱ under three vaccinations is obtained.(3)Obtaining Model Ⅲ by modifying the time delay in Model Ⅱ.The paper investigates the existence of positive equilibrium points in ModelⅢ,calculates the basic reproduction number,and analyzes the branching situation of Model Ⅲat the positive equilibrium point using Hopf bifurcation theory.Through the analysis of the above three models,this study confirms the importance of appropriate psychological intervention,vaccination,and therapy in controlling infectious diseases.The developed SVIR models are valuable in predicting the epidemic development trend and providing theoretical guidance for slowing down disease transmission.