Hopf bifurcation in Marchukʹs model of immune reactions

This paper deals with stability and bifurcation analysis of Marchukʹs model of an immune reaction. There are two possible stationary states in the model. One of them describes the healthy state of an organism and the second describes the chronic form of disease. Local stability of the first state does not depend on time delay, but in the case of the chronic state, it may be important. This paper investigates the stability of the chronic state and possibility of the Hopf bifurcation occurrence depending on time delay. chronic state is stable in the model without delay, then it is stable for small delays, but always is unstable for large delay. In this case, the Hopf bifurcation is possible. Periodic solution may occur for values of delay larger or smaller then critical value, i.e., the bifurcation may be supercritical or subcritical. If the chronic state is unstable for the model without delay, then it cannot be stable for any delay.