Abstract
Tumours evade immune surveillance through a number of different immunosuppressive mechanisms. One such mechanism causes cytotoxic T-cells, a major driving force of the immune system, to differentiate to a state of 'exhaustion', rendering them less effective at killing tumour cells. We present a structured mathematical model that focuses on T-cell exhaustion and its effect on tumour growth. We compartmentalise cytotoxic T-cells into discrete subgroups based on their exhaustion level, which affects their ability to kill tumour cells. We show that the model reduces to a simpler system of ordinary differential equations (ODEs) that describes the time evolution of the total number of T-cells, their mean exhaustion level and the total number of tumour cells. Numerical simulations of the model equations reveal how the exhaustion distribution of T-cells changes over time and how it influences the tumour's growth dynamics. Complementary bifurcation analysis shows how altering key parameters significantly reduces the tumour burden, highlighting exhaustion as a promising target for immunotherapy. Finally, we derive a continuum approximation of the discrete ODE model, which admits analytical solutions that provide complementary insight into T-cell exhaustion dynamics and their effect on tumour growth.