A perturbation problem arising from the modelling of soluble Fas ligand in tumour immunology

It has been known for many years that immune cells can kill cancer cells by a variety of mechanisms. However, new experimental evidence suggests that cancer cells also express these cell killing mechanisms. This enables the tumour to mount a counterattack against the anticancer immune cells. Based on these observations, we propose an ordinary differential equation model for tumour-immune cell interactions. With initial conditions corresponding to a mixture of cancer and immune cells, numerical solutions of the model show a sharp increase in the level of a chemical regulator associated with the interaction of the two cell types. We investigate this behaviour by constructing an analytical approximation to the solution using singular perturbation analysis. This problem has an unusual asymptotic structure. Instead of the usual solution form, with two outer solutions separated by a single transition layer centred at the point at which the sharp jump in the solution occurs, our solution contains multiple fast time layers, with each layer being necessary to capture the entire dynamics of the sharp transition.