Multi-Stability and Multi-Instability Phenomena in a Mathematical Model of Tumor-Immune-Virus Interactions

Abstract

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.

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

“…Throughout this study, we consider a doubling time of about 18 hours, resulting in a proliferation rate r = 0.927 (see also [27]). -The death rate δ of cells infected with VSV varies between studies: from δ = 0.69/day in [51] to δ ∈ (0.665, 6.654)/day in [7].…”

“…However, it was shown in [10] that the Vesicular Stomatitis Virus (VSV) eliminated tumours only in some mice. This immunisation protocol was first modelled and studied mathematically in [27], where the authors considered two compartments (lymphoid tissue -where the immune memory cells are located, and the peripheral tissue -where the tumour is located) to model the detailed interactions among tumour cells, viruses, and immune cells. Despite the complexity of that model (described by seven equations), the authors were able to detect and investigate a multi-instability phenomena where all steady states were unstable.…”

“…Despite the complexity of that model (described by seven equations), the authors were able to detect and investigate a multi-instability phenomena where all steady states were unstable. However, because of the large parameter space, the authors did not engage in a systematics search for chaos, and chaos was not detected for the parameter values investigated in [27].…”

“…In regard to the fourth steady state TVI, for the parameter values considered in this article, there is only one type of TVI state, which is stable for ω ≥ 1.32/day and unstable for ω < 1.32/day. The stability of the TF, TO and TV states can be established analytically, while the stability of the TVI state can be established numerically, as discussed in [27] and here in Appendix B.…”

“…This model (which is a simplification of the model introduced in [27]) is based on a particular experimental protocol derived in [10] to investigate the combined anti-tumour effect of vaccination and oncolytic virotherapy. However, our model is sufficiently general to be applied to other combinations of cancer therapies.…”