On a discrete model of tumour–immune system interactions with blockade of immune checkpoints

Abstract

A discrete-time model of tumour–immune system interactions based on a published model is derived. The tumour is assumed to grow exponentially while the immune system can either slow down or shrink tumour growth. However, cancerous tumour cells have many mechanisms to impair the immune system. The mathematical model therefore incorporates immunotherapy of immune checkpoint inhibitors to study tumour dynamics. It is proven that the tumour can grow to unboundedly large if its intrinsic growth rate exceeds a critical value or if its initial size is beyond a threshold when first detected. In addition, a region of initial conditions for which the tumour will eventually reach an unbounded size is derived when the tumour growth rate is smaller than the critical value. As a result, the immunotherapy fails to control the aggressive tumour. There are two positive equilibrium points when the tumour growth rate is smaller than the critical value. We verify that a saddle-node bifurcation occurs at the critical tumour growth rate whereas the equilibrium with the larger tumour size is always unstable. The equilibrium with the smaller tumour size may undergo a subcritical Neimark–Sacker bifurcation in certain parameter regimes. Parameter values derived from various clinical studies are simulated to illustrate our analytical findings.