Multi-stability and multi-instability phenomena in a mathematical model of tumor-immune-virus interactions

Recent advances in virology, gene therapy, and molecular and cell biology have provided insight into the mechanisms through which viruses can boost the anti-tumor immune response, or can infect and directly kill tumor cells. A recent experimental report (Bridle et al. in Molec. Ther. 18(8):1430–1439, 2010) showed that a sequential treatment approach that involves two viruses that carry the same tumor antigen leads to an improved anti-tumor response compared to the effect of each virus alone. In this article, we derive a mathematical model to investigate the anti-tumor effect of two viruses, and their interactions with the immune cells. We discuss the conditions necessary for permanent tumor elimination and, in this context, we stress the importance of investigating the long-term effect of non-linear interactions. In particular, we discuss multi-stability and multi-instability, two complex phenomena that can cause abrupt transitions between different states in biological and physical systems. In the context of cancer immunotherapies, the transitions between a tumor-free and a tumor-present state have so far been associated with the multi-stability phenomenon. Here, we show that multi-instability can also cause the system to switch from one state to the other. In addition, we show that the multi-stability is driven by the immune response, while the multi-instability is driven by the presence of the virus.