Dynamics of delay-differential equations modelling immunology of tumor growth

A phenomenological model of a tumor interacting with the relevant cells of the immune system is proposed and analysed. The model has a simple formulation in terms of delay-differential equations (DDEs). The critical time-delay, for which a destabilising Hopf bifurcation of the relevant fixed point occurs, and the conditions on the parameters for such bifurcation are found. The bifurcation occurs for the values of the parameters estimated from real data. Local linear analyses of the stability is sufficient to qualitatively analyse the dynamics for small time-delays. Qualitative analyses justify the assumptions of the model. Typical dynamics for larger time-delay is studied numerically.