Stability and Hopf Bifurcation Analysis of Tumors with Immature and Mature Lymphocytes

Abstract

The competition between immune system and tumor is highly intricate. Our aim is to develop a simple authentic mathematical model to understand the crucial factors that influence the outcomes of anti -tumor response. We are focusing on the concept that lymphocytes progress through two developing stages immature and mature lymphocytes. We formulated a new model for anti-tumor immune responses and have examined it characteristics. Steady states are found under specific conditions. Using Routh-Hurwitz criteria, local asymptotical stability of equilibria is investigated. Hopf bifurcation is analysed with a time delay as a bifurcation parameter. The length of the delay is derived to maintain stability. Analytical findings demonstrate the impact of delay on destabilizing the system and generating the periodic oscillations. The system undergoes different phases where the rate of mature lymphocytes influx progresses through many stages. This includes uncontrolled tumor growth initially, then reaching a stable state with considerable tumor presence, followed by periodic oscillations. Further progressing to a stable state with minimal tumor and ultimately attaining a stable state free of tumors. Analytical work is illustrated by numerical simulations.