Bifurcation Analysis Of A Nonlinear State-dependent Feedback Control SIR Model

So far,infectious diseases are still one of the major causes of human death world-wide.Human beings have a hard struggle with infectious diseases.How to prevent and control infectious diseases remains one of the focus of human work.Through the long-term research of scholars from all over the world,although the outbreak of tra-ditional infectious diseases has been prevented and controlled,the outbreak of new infectious diseases such as SARS,H1N1,dengue fever and Ebola virus has brought new threats and challenges to mankind.Integrated prevention strategies,includ-ing quarantine,isolation,vaccination and treatment,are widely used to reduce the spread of such infectious diseases.Based on the scale of susceptible population,how to reasonably and effectively implement the mitigation prevention and control of infectious diseases and evaluate its effectiveness has important practical significance and research value.Thus,the scale of susceptible population is selected as the basis for implement-ing state-dependent feedback control.According to the limitation of resources,a classic SIR model with nonlinear state-dependent feedback control is proposed and investigated in which integrated control measures,including vaccination,treatment and isolation,are applied once the number in the susceptible population reaches a threshold level.First,the existence and global stability of the disease free period-ic solution(DFPS)are addressed,and the threshold condition is provided which can be used to define the control reproduction number Rc for the model with state-dependent feedback control.The DFPS may also be globally stable when the control reproduction number Rc is less than one.Secondly,to show that the threshold dy-namics are determined by the Rc,we employ bifurcation theories of the discrete one-parameter family of maps,which are determined by the Poincare map of the proposed model,to reveal transcritical and pitchfork bifurcations related to all of the interesting parameters which are the vaccination rate,the threshold vaccination,the birth rate.The main results indicate that under certain conditions a stable or unstable interior periodic solution could be generated through transcritical and pitchfork bifurcations.The stable DFPS and the interior equilibrium of the SIR model can coexist once an unstable interior periodic solution is bifurcated(Rc<1<R0 here),i.e.backward bifurcation occurs.Moreover,the function of Rc with respect to the threshold level of susceptible population is U-shape(or U-shape)curve which reveals some important issues related to disease control,i.e.there exists an optimal threshold level such that Rc reaches the minimum.Therefore,to achieve the best prevention and control effect,the scale of susceptible population should be monitored timely during the outbreak of infectious diseases.