Bifurcations and Chaotic Dynamics in a Tumour-Immune-Virus System

¹ Division of Mathematics, University of Dundee, Dundee, United Kingdom, DD1 4HN
² School of Mathematics and Statistics, University of St. Andrews St. Andrews, United Kingdom, KY16 9AJ
³ Department of Biology, McMaster University, Hamilton, ON, L8S 4L8, Canada
⁴ Michael G. DeGroote Institute for Infectious Disease Research, McMaster University Hamilton, ON, Canada, L8N 4L8
⁵ McMaster Immunology Research Centre, McMaster University, Hamilton, ON, Canada, L8S 4K1
⁶ Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada, L8S 4K1

^* Corresponding author. E-mail: r.a.eftimie@dundee.ac.uk

Abstract

Despite mounting evidence that oncolytic viruses can be effective in treating cancer, understanding the details of the interactions between tumour cells, oncolytic viruses and immune cells that could lead to tumour control or tumour escape is still an open problem. Mathematical modelling of cancer oncolytic therapies has been used to investigate the biological mechanisms behind the observed temporal patterns of tumour growth. However, many models exhibit very complex dynamics, which renders them difficult to investigate. In this case, bifurcation diagrams could enable the visualisation of model dynamics by identifying (in the parameter space) the particular transition points between different behaviours. Here, we describe and investigate two simple mathematical models for oncolytic virus cancer therapy, with constant and immunity-dependent carrying capacity. While both models can exhibit complex dynamics, namely fixed points, periodic orbits and chaotic behaviours, only the model with immunity-dependent carrying capacity can exhibit them for biologically realistic situations, i.e., before the tumour grows too large and the experiment is terminated. Moreover, with the help of the bifurcation diagrams we uncover two unexpected behaviours in virus-tumour dynamics: (i) for short virus half-life, the tumour size seems to be too small to be detected, while for long virus half-life the tumour grows to larger sizes that can be detected; (ii) some model parameters have opposite effects on the transient and asymptotic dynamics of the tumour.