Critical bifurcations and chaos in a delayed nonlinear model for the immune response

In this work, we consider the dynamical behaviour associated with the interaction of the immune system with a target population. We consider a model with delayed responses, which consists in a set of two coupled nonlinear differential equations. We show that the stationary solution becomes unstable above a critical delay time of the immune response. In the delay induced oscillatory regime, the minimum density of the target population is smaller than the corresponding stationary value. We obtain the characteristic exponents of this bifurcation and the critical dynamics. We show that, under certain conditions, increasing the delay time induces a series of bifurcations leading to chaos.