Ageing and disease can be understood in terms of a loss in biological homeostasis. This will often manifest as a constitutive elevation in the basal levels of biological entities. Examples include chronic inflammation, hormonal imbalances and oxidative stress. The ability of reactive oxygen species (ROS) to cause molecular damage has meant that chronic oxidative stress has been mostly studied from the point of view of being a source of toxicity to the cell. However, the known duality of ROS molecules as both damaging agents and cellular redox signals implies another perspective in the study of sustained oxidative stress. This is a perspective of studying oxidative stress as a constitutive signal within the cell. In this work a computational modelling approach is undertaken to examine how chronic oxidative stress can interfere with signal processing by redox signalling pathways in the cell. A primary outcome of this study is that constitutive signals can give rise to a 'molecular habituation' effect that can prime for a gradual loss of biological function. Experimental results obtained highlight the difficulties in testing for this effect in cell lines exposed to oxidative stress. However, further analysis suggests this phenomenon is likely to occur in different signalling pathways exposed to persistent signals and potentially at different levels of biological organisation.