Stability and bifurcation analysis in a basic model of the immune response with delays

The dynamics of the Mayer model with delays has been studied. Firstly, the linear stabilities with one delay and two delays are investigated, respectively. It is found that there are stability switches for time delays, and Hopf bifurcations when time delays cross through some critical values. Then, using the normal form method and the center manifold theorem, the direction and stability of the Hopf bifurcations are determined. Finally, some numerical simulations are carried out to illustrate the results.