A qualitative mathematical model of the immune response under the effect of stress

In recent decades, many studies have been developed in psychoneuroimmunology that associate stress, arising from multiple different sources and situations, to changes in the immune system, from the medical or immunological point of view as well as from the biochemical one. In this paper, we identify important behaviors of this interplay between the immune system and stress from medical studies and seek to represent them qualitatively in a paradigmatic, yet simple, mathematical model. To that end, we develop an ordinary differential equation model with two equations, for infection level and immune system, respectively, which integrates the effects of stress as an independent parameter. In addition, we perform a geometric analysis of the model for different stress values as well as the corresponding bifurcation analysis. In this context, we are able to reproduce a stable healthy state for little stress, an oscillatory state between healthy and infected states for high stress, and a "burn-out" or stable sick state for extremely high stress. The mechanism between the different dynamical regimes is controlled by two saddle-node in cycle bifurcations. Furthermore, our model is able to capture an induced infection upon dropping from moderate to low stress, and it predicts increasing infection periods upon increasing stress before eventually reaching a burn-out state.