This paper is concerned with examining the practical aspects of developing mathematical models of immunological phenomena using Marchuk-type delay differential equation models proposed since the early 1970s [cf. G. I. Marchuk, Frontiers in pure and applied mathematics, 209–234 (1991; Zbl 0825.92076)]. The original models prompted much analysis and numerical analysis of delay equations, and this paper discusses how one needs to apply the results of this analysis to mathematical modelling. The authors’ purposes in this paper appear to be to highlight issues that need to be considered, to make some suggestions as to how they can be addressed, and to set the agenda for further investigations needed in the future.
Following a brief review, the authors move on to the issue of formulating a meaningful model (and the estimation of its parameters) and the problem of choosing an appropriate initial function. The authors discuss extensions of the type of model originally employed by Marchuk, so that neutral delay differential equations of differing types can be admitted as possible models. Of course, the idea is that data available from experimentation is to be used both for choosing the model equations and for determining the initial function.
They also present an analysis of, and a computational approach to, the sensitivity of solutions of the models (sensitivity to the choice of model, in particular to the lag parameter \(\tau > 0\), and sensitivity to the initial data). They present the role of sensitivity in modelling and show that the complexity of the sensitivity analysis depends on the form of the model.
It becomes clear that stability is a key requirement because, if small changes in model parameters or initial data might produce large changes in solutions, then the parameter identification could become intractable. The authors show how sensitivity can be estimated, both analytically and computationally.
In conclusion, the authors draw attention to the various gaps in existing analysis and computational methods that will need to be the focus of further work to improve the quality of mathematical modelling methods in this area.