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From Patterns and Models

Our daily lives are steeped in mathematics. According to Gerta Köster, professor at the University of Munich, there is hardly an area that is not influenced by modeling and simulation.

During this pandemic, the world is often explained using simulations.
Gerta Köster: Mathematics has been experiencing an upward trend for some time. Its triumph started about ten years ago, with the advent of AI. Mathematics could do quite well on its own, but would anyone really care about it? Not I! The advancement of mathematics is strongly driven by problems that arise in practice – now, that’s a happy marriage.

Aristotle called mathematics “the art of learning”.

In previous times, it was still possible to earn a Doctor of Philosophy in the field of mathematics. Today, most first-year university programs introduce structures that are ideally suited for learning and practicing logical thinking. One learns to discipline one’s thought processes – but practical application is still best. Today, even the social sciences rely on mathematical models.

This is your area of expertise – you build models and simulate the movement of pedestrians.

The beauty of it is that one never works alone. Mathematics is located somewhere between the discipline of its application and informatics. Together with my students, I look at how people move through urban areas. With “Vadere”, for example, we simulated the evacuation of a Wies‘nzelt (Octoberfest beer tent) and recreated a model of how a subway car was vacated following a bomb attack in London in 2005. For these purposes, we collaborate with psychologists who alerted us to the fact that people do not flee alone but instead identify as victims and help each other.

You take empirical observations and describe patterns in order to derive models?
It is crucial to find out the defining characteristics about the observation, or else you will never finish. It is probably not hair color! In pandemic models, age plays a major role; in pedestrian simulations, it is group membership. The art of “leaving out” is the most challenging task of the modeler who always assembles their model in three steps: description of the real world – for example, capturing observations using a mathematical equation; translation into unequivocal rules of computation, also known as algorithms, and programming. It is imperative to continuously verify that the assumptions are still correct – on every level.

How do I know whether my model is valid?
You never know – that’s the tragic part of modeling! I like citing British statistician George Box who, in the 1970s, said the following: “All models are wrong, but some are useful.” One is bound to make mistakes during simplification and abstraction. The question is: are the assumptions wrong to the extent that their predictions are no longer useful?

Does every question require a new model?

It often helps to see if there are related phenomena; if there are, the borrowing begins. Pedestrian simulations were heavily influenced by models originating in physics; we now know that people do not quite move like grains of sand. For the pandemic, 100-year-old models are being used which were published in 1927 by Kermack and McKendrick.

They still work?
There have always been pandemics, and the principles remain identical. McKendrick was a British doctor; when modeling the Bombay (present-day Mumbai) plague epidemic of 1905/06, he assumed three groups: the non-immune healthy person (susceptible); those who are infected and contagious (infectious); and those people who have either recovered or died (removed). This marked the birth of the SIR model, which places these three groups in relation to the population as a whole and which can therefore simulate different scenarios. It is a macroscopic model that works well for considering the entire population, but it cannot explain how a localized infection from one person to another inside a supermarket will play out. This is the type of model on which we are working right now.

How long will it take?
Probably until the end of the year. We must first conduct a literature search – roughly 200 scientific articles on COVID-19 are published every day. We must filter the information and think about how to simplify, which type of algorithm to create and how to construct the software architecture.

The algorithm is defined as a solution to a mathematical problem which instructs the computer.
An example of a simple model: I will become infected if I breach the distance of 1.5 meters to an infected person. But what do we mean by “breach”? One second? One hour? Verbal communication does not require a strict definition – a computer does. Everything must be closed; there are only zeros and ones – no approximates. An algorithm closes these gaps; it is not in itself a computer program, but it translates facts into numbers.

Which part is the most labor-intensive?
During two of the three phases, you must never allow yourself to be rushed: if there is a mistake in the beginning, during the interdisciplinary collaboration phase, you may pursue the wrong lead for a long time. Actual programming is relatively quick, but a good tool – often tens of thousands of lines of code – will be continually tested. If you change one line of code, everything has to be tested to ensure that this one line did not introduce an error that will wreak havoc in a different area. All of this takes time.

SHORT PROFILE

Gerta Köster, 52, teaches mathematics at the University of Munich. A professor of scientific computing since 2010, she prefers to work in interdisciplinary teams with a focus on application. Her areas of expertise are model building and pedestrian dynamics. Her latest research project (local occurrence of infection) is funded by the BMBF (German Federal Ministry of Education and Research).