Qualitative Analysis Of The Solution Of A SEIQR Epidemic Model With Saturated Incidence

In recent years,the prevalence and prevention of infectious diseases are the main content of global health and safety concerns.As an important method of studying infectious diseases,infectious disease dynamics has increasingly highlighted the important contributions to the scientific field after the combination of mathematics and ecology.Through the establishment of mathematical models,the spread of infectious diseases and risk prediction are displayed in the form of data,which makes the research more practical.The main contents of the paper are as follows:The first chapter introduces the major impact of infectious diseases on humans in the past and the current research status of related infectious disease mathematical models.In order to carry out the follow-up research smoothly,the second chapter gives relevant preliminary knowledge and theorems.The third chapter mainly considers that pathogens with latent characteristics will cause a wider range of transmission,as well as the need to take appropriate isolation measures when a major disease breaks out,and establish a SEIQR infectious disease model with a saturated incidence.First,the condition R₀ that determines whether the disease is endemic or not is given,and the existence and uniqueness of the solution of the infectious disease model is discussed.The asymptotic stability of the disease-free balance point and the endemic disease balance point was proved by using relevant knowledge such as La Salle's invariance principle?Routh-Hurwitz stability criterion and global asymptotic stability.The rationality of the model was verified by numerical simulation,and the impact of different isolation rates and saturation rates on the patient population was discussed.In Chapter 4,consider that during the patient's isolation period,the fixed isolation intensity may cause additional impacts such as the patient's mental health and isolation costs.Adding a control function to the SEIQR model makes it possible to minimize the medical resources consumed during isolation while controlling the disease.For the new model,the optimal control theory and Pontryagin's principle are used to prove the existence of optimal control,and the mathematical expression of optimal control is given.At the end of the article,considering that infectious diseases are prevalent in a certain area,the distribution area will continue to expand outward over time,and a discussion on the issue of free borders is launched.The local existence and uniqueness of the solution is obtained by using the L^p theory and the principle of compressed mapping.An estimate is given to prove that the solution of the free boundary problem can be extended,and then it is proved that the solution of the free boundary problem is globally unique.