| Using differential equation to establish and study epidemic models has become an impor-tant research topic in applied mathematics,This paper uses stability theory and bifurcation theory to study four different types of epidemic models,and focuses on the existence and stability of the equilibria and various possible bifurcations,such as backward bifurcation,Hopf bifurcation and Bogdanov-Takens bifurcation.The whole paper is composed of six chapters:In Chapter 1,the development history and some preliminaries are given.In Chapter 2,an SIRS model with climate change rate function is studied,discovered that when the basic reproduction number R0 ≤ 1,the system has only one disease-free equilibriunm E0 which is global asymptotic stability.If R0>1,there exist two equilibria:the disease-free equilibrium E0 and an endemic equilibrium E*which is global asymptotic stability when v>0 or p1>q.When v = 0,the form of Hopf bifurcation can be determined by calculating the first Lyapunov coefficient.Numerical analysis also shows that the impact of climate factor λ on disease.The bigger λ is,the more unstable of the condition,and the more infected people appear.When A tends to infinity,the entire host,will gradually adapt to the harsh climate and the value I*tends to a stable state.In Chapter 3,an SIS model with saturated incidence rate and half-saturated treatment function is studied.It is demonstrated that when some parameters meet the values of specific regions,the backward bifurcation will be produced,which means R0 = 1 will no lornger be the threshold which determines the elimination of the disease.Moreover the stability of the disease-free equilibrium E0 and cndemic disease equilibria E1,E2 arc discussed.The form of Hopf bifurcation is determined by calculating the first Lyapunov coefficient.It is also proved that under some conditions,the system will produce Bogdanov-Takens bifurcation.Finally,three bifurcation curves are given by the Bogdanov-Takens criterion,that is,saddle-node bifurcation,Hopf bifurcation and homoclinic bifurcation.In Chapter 4,a class of SIR epidemic model with bilinear incidence function and linear saturated treatment function is researched.The existence and stability of the local equilibria and the occurrence of a backward bifurcation are studied.A sufficient condition for the occurrence of a backward bifurcation is given when the basic reproduction number is less than 1 and the saturation treatment.rate is small.The existence of at most four equilibria is proved and a sufficient condition for their existences is given.In Chapter 5,a class of SIR.epidemic model with saturated incidence rate and selected treatment function is worked on,in which p and q represent different treatment rates cor-responding to different levels of the disease respectively.When p ≤ q,there exists only one endemic equilibrium E2,which is globally asymptotically stable.Meanwhile,the disease-free equilibrium point E0 is globally asymptotically stable when R0<1.When p>q,there exists multiple equilibria when R0<R0<1,which means that the backward bifurcation may occur.The occurrence of Hopf bifurcation is proved and the form is also analyzed.It is also proved that under some conditions,the system will produce Bogdanov-Ta.kens bifurca-tion.Finally,three bifurcation curves are given by the Bogdanov-Takens criterion,that is,saddle-node bifurcation,Hopf bifurcation and homoclinic bifurcation.Chapter 6 is the summary of the paper and the prospect of the future research. |
