Stability And Bifurcation Analysis Of Two Kinds Of Infectious Disease Dynamics Models

In this paper,based on the transmission mechanism of tuberculosis,considering the effects of exogenous reinfection and population input rate on the transmission level of tuberculosis,a dynamic model of tuberculosis was established to study the transmission dynamics of tuberculosis when the population presented logistic growth.A general brucellosis model with testing and culling behavior was established based on the characteristics of the brucellosis,and the role of testing and culling in brucellosis prevention and control was studied.The dynamic behavior of the model is studied by stability and bifurcation theory.The details are as follows:In the second chapter,a tuberculosis model with logistic input rate and exogenous reinfection was established.It is found that the model has transcritical bifurcation,backward bifurcation and Hopf bifurcation of codimension 1.Using the normal form theory,it is proved that the model undergos a degenerate Hopf bifurcation and a cusp of codimension 2.At last,the theoretical results are illustrated by numerical simulation.This shows that the changing population has more complex dynamics.In the third chapter,a general brucellosis dynamic model with testing and culling was established based on the transmission characteristics of the brucellosis.The basic reproduction number of the model was calculated,the existence of the equilibria was analyzed.Then,Lyapunov method is used to prove the stability of the equilibria.The results show that when R0<1,the disease-free equilibrium P0 of the model is globally asymptotically stable.When Rc<1<R0,the endemic equilibrium P1*of the model is globally asymptotically stable.When 1<Rc<R0,the positive equilibrium P2*of the model is globally asymptotically stable.Therefore,when R0<1,the disease is gradually died out.When R0>1,the disease is always present and tends to be stable.In the end,the paper summarizes the whole paper,clarifies the practical significance of the study,and provides new ideas for further research.