Mathematical modeling and bifurcation analysis of pro- and anti-tumor macrophages.

• A novel mathematical model of tumor immune interactions system consisting of tumor cells, M1 and M2 macrophages is proposed. • The model captures two major functions of macrophages with both pro- tumor and anti-tumor effects. • Bifurcation diagrams of steady states are drawn in different parameters planes. • The sensitivity analysis is performed to illustrate the role of parameters with respect to tumor size. • The decay of tumors is not only the result of activation of M1 and M2, but also the effect of transition between M1 and M2. More and more evidences show that macrophages can induce both pro-tumor and anti-tumor effects. The study on genetically engineered macrophages to kill tumor cells has become a new direction for anti-cancer therapy in recent years. In this paper, we propose a mathematical model of interactions between tumor cells, M1 and M2 macrophages, and study the stability and bifurcations around the interior equilibrium. It is shown that the model exhibits stable periodic oscillations bifurcating from Hopf bifurcation, which indicates the long-term coexistence of tumor cells, M1 and M2 macrophages as well as long-term tumor relapse. Furthermore, sensitivity analysis is performed to illustrate the effect of parameters on tumor size. Our results show that an increase in ξ (the ratio of activation rates by tumor cells of M2/M1 macrophages) leads to a decrease of tumor cells, which can be explained that ξ has significant effects on γ 1 (the transition rate from M1 to M2 macrophages) and γ 2 (the transition rate from M2 to M1 macrophages). Hence, the decay of tumor cells is not only the result of direct activation of M1 and M2 macrophages by tumor cells, but also the combined effect of the transition between M1 and M2 macrophages. [ABSTRACT FROM AUTHOR]