What is Poincare-Bendixson Property

By clinical data, drug treatment sometimes is ineffective to eradicate the infection completely from the host in some human pathogens such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), and human T cell lymphotropic virus type I. Therefore, mathematical modeling can play a significant role to understand the interactions between viral replication and immune response. In this chapter, the author investigates the global dynamics of antiviral immune response in an immunosuppressive infection model which was studied by Dadi and Alizade (2016). In this model, the global asymptotic stability of an immune control equilibrium point is proved by using the Poincare–Bendixson property, Volterra–Lyapunov stable matrices, properties of monotone dynamical systems and geometric approach. The analysis and results which are presented in this chapter make building blocks towards a comprehensive study and deeper understanding of the dynamics of immunosuppressive infection model.