Stability And Branching Analysis Of Cholera Models With Time Delays

With the continuous updating of infectious disease dynamics research and theory,delay differential equation is more widely applied to the study of infectious disease model.Delay differential equations are used to describe dynamic systems whose state depends not only on the current state,but also on the historical state.This lag phenomenon is common in the spread of infectious diseases.Firstly,we briefly introduced the background and status quo of the study on the dynamics model of infectious diseases.Using mathematical model to explore the mechanism of the infectious diseases can play a role that experimental disciplines cannot:it can help people better understand the law of the outbreak and spread of infectious diseases,so as to take measures to prevent and control.We further introduced the research status of cholera,a water-borne infectious disease.Because cholera has a variety of transmission routes,its model has a certain high dimension and complex strong coupling,so it is very difficult to study and analyze such model.Secondly,we establish a cholera epidemic model with time delay and a general infection generation function.The existence and uniqueness of the solution of the model is proved by the qualitative theory of the abstract ordinary differential equation.Using comparison principle,we obtain the positive and uniform boundedness of the solution and the environmental reproduction number.By calculating the basic reproduction number of infectious diseases,the existence of equilibria is obtained.Furthermore,the global stability of the disease-free equilibrium is proved by applying Lyapunov stability theory and Lyapunov-LaSalle invariance principle when the basic reproduction number is less than 1.Finally,we study the Hopf bifurcation of the system,and we show that the time delay can cause the loss of stability of the positive equilibria,resulting in the Hopf bifurcation and periodic oscillation of the solution of the system.The sufficient conditions for the existence of local Hopf bifurcation at the positive equilibrium point are discussed by studying the distribution of characteristic roots of the transcendental equations whose coefficients depend on the time delay.On this basis,we derive the explicit formulas determining stability,direction and other properties of bifurcating periodic solutions by using the center manifold argument and the normal form theory of functional differential equations.Then,we give global Hopf bifurcation analysis of equilibria by the global Hopf bifurcation theorem,and the global existence of periodic solutions are given.