Temporal and spatiotemporal variations in a mathematical model of macrophage–tumor interaction

In the present paper we study a three-component mathematical model of tumor–immune system interaction. A number of solid tumors contain a high proportion of macrophages and these immune cells are known to have a remarkable impact on the progression and dormancy of such tumors. We assume these macrophages as the main immune system component facilitating tumor destruction. Stability criteria of the basic model around the steady state of coexistence are derived. Next, we consider the process of macrophage activation as non-instantaneous by using a distributed delay with a weak kernel and obtain a range for the macrophage death rate that ensures system stability. Finally, we incorporate the spatial irregularity of solid tumors by making the delay nonlocal. Analysis of the resulting spatiotemporal model gives a number of thresholds in terms of different system parameters that guarantee tumor stability. Numerical simulations are performed to justify analytical findings.