The Research On Backward Bifurcation And Dynamical Behavior Of Some Classes Of Continuous Epidemic Models

This paper focuses on the research of backward bifurcation and dynamicalbehavior of three classes of epidemic models, which includes: backward bifurca-tion and dynamical behavior of a SIS epidemic model with saturated incidencerate and treatment, backward bifurcation and dynamical behavior of SEIR epi-demic model with saturated incidence rate and saturated treatment and back-ward bifurcation and dynamical behavior of SEIRS epidemic model with nonlin-ear incidence rate. The main contents of this paper are summarized as:Section I. First, we describe the background and signifcance of the studyon the model of infectious diseases, and then introduces the present situation tostudy on epidemic models with backward bifurcation. Finally, we summarizesthe main content framework of this paper.Section II. We study the backward bifurcation and dynamical properties ofthe equilibrium point of a SIS epidemic model with saturated incidence rate andtreatment. We assume that the treatment rate is proportional to the numberof the infected in the treatment ability, but when the number of the infectedexceeds the treatment capability of bearing boundary the treatment functionis a constant form. The study fnds that if the treatment capacity is small,the model will appear the backward bifurcation phenomenon. The results ofanalysis show that only by reducing the basic reproduction number below1isnot necessarily able to eradicate the disease.Section III. We study the backward bifurcation and dynamical properties ofthe equilibrium point of SEIR epidemic model with saturated incidence rate and saturated treatment. The study fnd that the model will appear the backwardbifurcation phenomenon under certain conditions when the treatment capacity issmaller or treatment resources are limited and makes demonstration analysis toit. The paper also discusses the dynamic properties of the disease-free equilibriumand the endemic equilibrium point. The results of analysis show that only byreducing the basic reproduction number below1does not necessarily eliminatethe disease.Section IV. We study the backward bifurcation and dynamical properties ofthe equilibrium point of a SEIRS epidemic model with nonlinear incidence rate.The incidence rate is assumed to be a convex function on the infected class. Thestudy fnd that the model will appear the backward bifurcation when it meetsome certain conditions and makes demonstration analysis. We also discusse thedynamic properties of the disease-free equilibrium and the endemic equilibriumpoint in detail. The results of the study show that in order to eradicate thedisease completely, R0has to be reduced below a smaller threshold R0.Section V. We briefy discuss and analyze the results of this paper.