Change in criticality of Hopf bifurcation in a time-delayed cancer model

• Main Authors: Israel Ncube, Kiara Martin
• Format: Article
• Language: English
• Published: University of Szeged 2019-11-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: delay differential equations; cancer; tumour; equilibria; hopf bifurcation; criticality
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=7764

The main goal of this work is to conduct a rigorous study of a mathematical model that was first proposed by Gałach (2003). The model itself is an adaptation of an earlier model proposed by Kuznetsov et al. (1994), and attempts to describe the interaction that exists between immunogenic tumour cells and the immune system. The particular adaptation due to Gałach (2003) consists of replacing the Michaelis–Menten function of Kuznetsov et al. (1994) by a Lotka–Volterra form instead, and incorporating a single discrete time delay in the latter to account for the biophysical fact that the immune system takes finite, non-zero time to mount a response to the presence of immunogenic tumour cells in the body. In this work, we perform a linear stability analysis of the model's three equilibria, and formulate a local Hopf bifurcation theorem for one of the two endemic equilibria. Furthermore, using centre manifold reduction and normal form theory, we characterise the criticality of the Hopf bifurcation. Our theoretical results are supported by some sample numerical plots of the Poincaré–Lyapunov constant in an appropriate parameter space. In a sense, our work in this article complements and significantly extends the work initiated by Gałach (2003).