H. Thomas Banks is one of those people I wish I had had a chance to meet. Unfortunately, he died December 31st of last year, so that won’t be happening. Given that I greatly admired his work, and on the assumption that some young scientists read this blog, I thought I would say a few words about some of Banks’ papers that I particularly enjoyed.

H. T. Banks, for those of you who may not have heard of him, was an outstanding applied mathematician. He had wide interests, but most interesting to me was his extensive work on delay-differential equations, given my own interest in the subject.

The first Banks paper I read was a 1978 joint paper with Joseph Mahaffy on the stability analysis of a Goodwin model. Looking for oscillations in gene expression models was a popular pastime in those days. In some ways, it still is. This paper stood out for me as a careful piece of mathematical argument showing that a certain class of models could not oscillate. The paper also contained a solid discussion of the biological relevance of the results. Discovering oscillations in a model may be fun for those of us who enjoy a good bifurcation diagram, but most gene expression networks probably evolved not to oscillate. How much of that lovely discussion was due to Banks, and how much to Mahaffy, I cannot say. But a lot of Banks’ work was just as careful about the relevance of the results to the real world.

Much more recently, Banks was involved in a lovely piece of mathematics laying down the foundations for sensitivity analysis of systems with delays, particularly for sensitivity with respect to the delays. Sensitivity analysis is a key technique in a lot of areas of modelling. The basic idea is to calculate a coefficient that tells us how sensitive the solution of a dynamical system is to a parameter. There are many variations on sensitivity analysis, which you can read about in a nice introductory paper by Brian Ingalls. The Banks paper provided a basis for doing this with respect to delays, and was a key foundation stone for our work work on this topic.

Some years ago, we developed a method for simulating stochastic systems with delays. Our intention was for this method to be used to model gene expression networks. I was therefore pleased and surprised when I discovered that Banks had used our algorithm to study a pork production logistics problem. That just shows what an applied mathematician with broad interests can do with a piece of science developed in another context. Banks and his colleagues went a bit further than just studying one model, looking a models with different treatments of the delays, and finding that these led to different statistical properties, which would of course be of great interest if you were trying to optimize a supply chain.

The few examples above show a real breadth of interests, both mathematically and in terms of applications. You can get an even better idea of how broad his interests were by scanning his list of publications. There are papers there on control theory, on HIV therapeutic strategies, on magnetohydrodynamics, on acoustics, … Something for just about every taste in applied mathematics. There is a place for specialists in science, but often it’s the people who straddle different areas who can make the most important contributions by connecting ideas from different fields. I think that Banks was a great example of a mathematician who cultivated breadth, and was therefore able to have a really broad impact.

So I’m really sorry I never got to meet H.T. Banks. I think I would have enjoyed knowing him.

(If you’re wondering why I’m so late with this blog post: I found out about Banks’ passing from an obituary in the June SIAM News, which because of the pandemic I didn’t get my hands on until about a month ago.)