Generalized Hopf Bifurcation in a Cancer Model with Antigenicity under Weak and Strong Allee Effects

TL;DR

This paper analyzes a cancer-immune interaction model incorporating Allee effects, identifying various bifurcations and their implications for tumor control and immune response under different antigenicity levels.

Contribution

It provides an analytical and numerical bifurcation analysis of a cancer model with Allee effects, revealing how antigenicity influences tumor dynamics and control.

Findings

Higher antigenicity enhances tumor control under strong Allee effects.

Multiple bifurcations, including Hopf and generalized Hopf, are identified and interpreted biologically.

Numerical continuation confirms analytical bifurcation curves and limit cycle behaviors.

Abstract

This article deals with an autonomous differential equation model that studies the interaction between the immune system and the growth of tumor cells with strong and weak Allee effects. The Allee effect refers to interspecific competition, and when the population is small, it can retard population growth. The work focuses on describing analytically, using a set of parameters, the conditions in the phases of the immunoediting theory, particularly in the equilibrium phase, where a latent tumor would exist. Saddle-Node, Saddle-symmetric, Hopf, generalized Hopf, and Takens-Bogdanov bifurcations get presented for both Allee effects, and their biological interpretation regarding cancer dynamics gets discussed. The Hopf and generalized Hopf bifurcation curves get analyzed through hyper-parameter projections of the model, where it gets observed that with a strong Allee effect, more tumor control persists as it has higher antigenicity, in contrast to the weak Allee effect, where lower antigenicity gets observed. Also, we observe that the equilibrium phase persists as antigenicity increases with a strong Allee effect. Finally, the numerical continuation gets performed to replicate the analytical curves' bifurcations and draw the limit and double limit cycles.