Neimark–Sacker bifurcation of a chemotherapy treatment of glioblastoma multiform (GBM)

• Main Authors: F. Bozkurt, A. Yousef
• Format: Article
• Language: English
• Published: SpringerOpen 2019-09-01
• Series: Advances in Difference Equations
• Subjects: Differential equation with piecewise constant arguments; Stability; Neimark–Sacker bifurcation; Allee threshold
• Online Access: http://link.springer.com/article/10.1186/s13662-019-2324-9

Abstract In this paper, we propose a system of differential equations with piecewise constant arguments to describe the growth of GBM under chemotherapeutic treatment and the interaction among the glial cells, the cancer cells, and the chemotherapeutic agents. In this work, the cancer cells are considered as two populations: the sensitive cancer cells and the resistant cancer cells. The sensitive tumor cells produce a population that is known as the resistant cell population, where this population has more resistance to the drug treatment than the sensitive tumor cell population. We analyze at first the local and global stability of the positive equilibrium point by considering the Schur–Cohn criteria and constructing a suitable Lyapunov function, respectively. Moreover, we use the center manifold theorem and bifurcation theory to show that the model undergoes Neimark–Sacker bifurcation. To investigate the case for the extinction of the tumor population, we consider the Allee threshold at time t. Simulation results support the theoretical study.