Mathematical modelling of problems with delay and after-effect

This paper provides a tutorial review of the use of delay differential equations in mathematical models of real problems. We use the COVID-19 pandemic as an example to help explain our conclusions. We present the fundamental delay differential equation as a prototype for modelling problems where there is a delay or after-effect, and we reveal (via the characteristic values) the infinite dimensional nature of the equation and the presence of oscillatory solutions not seen in corresponding equations without delay. We discuss how models were constructed for the COVID-19 pandemic, particularly in view of the relative lack of understanding of the disease and the paucity of available data in the early stages, and we identify both strengths and weaknesses in the modelling predictions and how they were communicated and applied. We consider the question of whether equations with delay could have been or should have been utilised at various stages in order to make more accurate or more useful predictions.