Stability Analysis Of Two Types Of Delayed Epidemic Models

Epidemic models is an important part of biomathematics,and have received extensive attention from many scholars at home and abroad in recent years.Use the law of the spread and development of epidemics to establish appropriate mathematical models,and then use the dynamic properties of mathematical models to reveal the internal transmission mechanism of epidemic model,predict their development and change trends,analyze the causes of epidemics,and provide its quantitative basis and theoretical basis for people with timely preventive treatment.In this paper,two kinds of epidemic models with time delay are considered,and theoretical analysis and summary are done mathematically.The main contents are as follows:1.In a type of SIR epidemic model with time delay,the existence of each equilibrium point is analyzed firstly,and then the system is linearized to obtain the corresponding characteristic equation and characteristic value of the system at each equilibrium point.Using the stability theory of delay differential equations,the local asymptotic stability of the system at each equilibrium point is analyzed.When the infected person and the recovering person coexist,the critical point and conditions for the Hopf bifurcation of the system are obtained.Finally,the conclusion is verified by numerical simulation.2.In an epidemic model with time delay in which the virus itself mutates,the dynamic properties of the Hopf bifurcation are studied.First,an important threshold for the prevalence of infectious epidemic-the basic reproductive number is obtained,and then according to the corresponding characteristic equation of the system at the equilibrium point,the conditions of Hopf bifurcation of the system could be determined.By using the central manifold theory and the normal form method,we could be obtained that the normal form of the Hopf bifurcation of the system on the central manifold.According to the formula for calculating the stability of periodic solution of Hopf bifurcation,the stability of periodic solution are determined,and the rationality of the theory was proved by numerical simulation.