Recent experimental studies have suggested the ratio between T-helper and T-suppressor lymphocytes as an index of immunosuppression in HIV, cancer, immunosenescence and inflammatory and auto-immune diseases. However, a quantitative understanding of the impact of this ratio on the immune response has lagged behind data and its validity as a tool for prognostic monitoring or therapeutic target remains an open question. In this work, we use statistical physics and dynamical systems approaches to analyze the time-dependent response to an antigen, of a simplified model of the adaptive immune system, which comprises B, T-helper and T-suppressor lymphocytes. The model is remarkably robust against changes in the noise level and kinetic parameters, but it is very sensitive to changes in the ratio between T-helper and T-suppressor lymphocytes, exhibiting, in particular, a transition from a responsive to an immuno-suppressed phase, as the ratio is lowered below a critical value, which is in line with experiments. This result supports the validity of the T-helper/T-suppressor ratio as an index of immunosuppression and may provide a useful theoretical benchmark to interpret and compare experiments.