Stability and bifurcation for Marchuk’s model of an immune system

Abstract

The investigation of an immune system has long been and will continue to be one of dominant themes in both ecology and biology due to its importance. In this paper, we consider Marchuk’s model of an immune system where this model is governed by a system of three differential equations with time. This model has two equilibrium states which are healthy state and chronic state. It is healthy state when the antigen reproduction is small while chronic state is when antigen reproduction rate is large. The objectives of this paper are to analyse the stability of this model, to summarize this stability using bifurcation diagram and to discuss interaction between the healthy and chronic states at stationary solution. The methods involved are stability theory and bifurcation theory. Our results show that healthy states are saddle and only one chronic state is asymptotically stable for a region of parameter considered. For the bifurcation’s case, as we increase the value of a parameter in this model, the chronic state shows that there are increment in the number of antigen, plasma cell and the antibody production.