Analysis Of Low-grade Glioma Treated With Radiotherapy Or Chemotherapy And Optimal Control

At present, cancer has become the main threat to human life. In this paper, firstly, we estab-lished and analyzed the mathematical model of the low-grade glioma treated with radiotherapy or chemotherapy, then, we analyzed the Optimal control.In the first chapter, we briefly introduce the background of the tumor and the research progress of mathematical model of the tumour. We also list the main mathematical theory knowl-edge including Hurwitz criterion, Lasalle invariant set theory, Pontryagini s minimum principle.In the second chapter, we analyzed the mathematical model of the low-grade glioma treat-ed with radiotherapy or chemotherapy. In the first section of this chapter, we proved the non-negativity and boundedness of solutions as well as existence condition of the positive equilibri-um. There exists positive equilibrium if and only if 0< c< c0. In the second section of this chapter, we analyzed the local stability of equilibrium. Through the analysis of characteristics roots of jacobian matrix, we recognized that:if 0< c< c0, the zero equilibrium is not stable but the positive equilibrium is locally asymptotic stable; if c> c0, the zero equilibrium is locally asymptotic stable. In the third section of this chapter, we analyzed the global stability of the equilibrium. From Lasalle invariant theory and Lyapunov function, we found that:if c≥c0, the zero equilibrium is globally asymptotic stable.In the third chapter, we analyzed the Optimal control of the low-grade glioma treated with radiotherapy or chemotherapy. The optimal controls of the tumor should be:after maximal control, singular optimal control would be taken. That’s to say, the control should be bang-bang at first, then the control should be singular optimal control. By taking these controls, the tumor can be eradicated.