A mathematical tumor growth model for exploring saturated response of M2 macrophages

Abstract

This study addresses a tumor–macrophage interaction model to examine the role of the saturated response of M2 macrophages. We find the equilibrium point of the model and analyze local stability at each equilibrium. We show that tumor-free equilibrium is always stable, whereas, under certain conditions, the tumor-dominant and interior equilibrium are asymptotically stable. Moreover, stable and unstable limit cycles and period-doubling bifurcation have been observed at the interior equilibrium point. A remarkable result has been observed: in the presence of a saturated response of M2 macrophages, with a relatively higher activation rate of M2 macrophages due to tumor cells, the disease spreads more quickly in the body. Hence, M1 macrophages cannot stabilize the system, and aperiodic oscillations are observed. Furthermore, we show that a better immune response can reverse that system’s unstable nature. Numerical simulations verify the analytical results.

Keywords

• Tumor
• Macrophages
• Limit cycles
• Hopf bifurcation
• Equilibrium
• Time series