Bifurcation Analysis Of A Class Of Tumour-immune Models With Nonlinear State-dependent Feedback Control

The state feedback control strategy is a control technique that implements control only when a state of the system reaches a pre-set threshold level.It is widely used in the field of cancer treatment and infectious diseases.Therefore,it is important to develop mathematical models to characterize and investigate the kind of threshold strategy.In this Thesis,a tumour-immune model with nonlinear state feedback control is developed and a pulsed SIR model under media coverage is investigated.The bifurcation analyses of the models are carried out by tools such as impulsive semi-dynamic system,bifurcation theory,and Poincare map.The Thesis is divided into the following parts.Firstly,a tumour-immune model with nonlinear feedback control was proposed.The threshold conditions for tumour eradication and tumor persistence for the pulsefree model are investigated.Based on the global properties of the pulse-free model,the definition domains of the impulsive set and phase set in different cases are determined,and the Poincare map of the model is defined in the impulsive set.Then,the existence and orbital asymptotic stability of the tumor-free periodic solution of the system is proved by applying the Analogue of Poincare criterion.In addition,by using the bifurcation theory of a family of discrete one-parameter mappings determined by the Poincare map,we investigated the transcritical and subcritical pitchfork bifurcations of the model with respect to key parameters.Secondly,a nonlinear impulse SIR model with media coverage was proposed to describe the effect of vaccination and isolation measures determined by the number of susceptible individuals.The basic reproduction number is defined and the existence and stability of the solution of the pulse-free model are proved.Then,the Poincare map of the model in different cases is defined.The existence and orbital asymptotic stability of the disease-free periodic solution of the model is proved by the Analogue of Poincare criterion.The results show that the disease-free periodic solution is globally asymptotically stable when R0<1.Even if R0>1,the disease-free periodic solution remains stable when SH<1/R0,which indicates that the statedependent pulse strategy still works in preventing infectious disease outbreaks by choosing a suitable threshold SH.In addition,the bifurcations near the disease-free periodic solution concerning some key parameters are investigated by utilizing the Poincare map and bifurcation theory.