X and the City: Modeling Aspects of Urban Life

"This book has a nice theme, which is using mathematics to understand the city. It’s amazing how many different aspects of city life can be investigated using mathematics. There are aspects of driving, congestion, populations, air pollution, light, gardening — there’s even some description of rainbows. This is quite tricky. As with many cases in this book, mathematical models are constructed to try to understand the problem. A mathematical model is, in essence, a set of equations that you can use to try to understand a process and to predict what’s going to happen in the real world. If you take a problem like congestion, you want to understand how things like the speed or the density of the traffic affects it. So you come up with equations, that you play around with, that will model that situation. With your equations, you can vary the speed of the traffic or the density and use that to predict what will happen. Of course, there’s always the question of whether your model faithfully matches the real world behaviour. How do you check that? How do you choose your parameters? Ultimately, the aim is to gain more understanding. You might, for example, choose speed limits for certain roads in order to try to reduce the congestion. What should those speed limits be? You don’t necessarily want to try different speed limits over a period of months and then see what happens. You want to do some computations and say ‘Ah! 25 miles-per-hour looks like the right speed limit for this particular road’ — thereby saving a lot of time and effort. In the most general usage, modelling and simulation are sometimes used as synonyms, but a distinction is usually made between developing and analysing the model (modelling) and carrying out computational experiments with the model (simulation). I like this book because it is inventive — with a lot of great examples of applied mathematical modelling. The ‘X’ in the title refers to an unknown. I did know of this book before I actually got my hands on it. I had been rather put off by the apparent connection to the TV series, Sex and the City, which I have never watched. But actually there’s not even the slightest connection. There’s no allusion or mention of it even in the chapter entitled “Sex and the City.” He is making use of the logistic differential equation to model the populations of bed bugs and rats. This is a very old topic in mathematical modelling – modelling the growth of populations. One of the examples that we often use in undergraduate courses is the ‘fox and rabbit’ simulation. This is where you’ve got a population of foxes and a population of rabbits. The foxes eat the rabbits and the rabbits reproduce, both of which can happen at different rates. What’s going to happen? Will all the foxes eat the rabbits and there be no rabbits left? Will there be a steady state? Or will the populations oscillate up and down forever? These so-called predator-prey equations go back about a hundred years. There’s a long tradition of mathematical modelling through differential equations to understand how populations vary over time. This is talking about those sorts of ideas. The book also gives an estimate of the answer to the question, ‘How many people have lived in London?’ Estimation is well used — for example, to answer ‘How many squirrels live in Central Park?’ And some practically important questions are also addressed, such as should you walk or run in the rain? This is one of the two most mathematical books out of the five I have chosen. There are quite a lot of equations and mathematical derivations. I think having a first course in linear algebra and in differential equations or calculus is probably what you need to make much headway. But quite a lot of the book is readable anyway. It introduces the problems in a nice gentle way, setting the scene before introducing the technicalities."