Mathematical models in different potential crisis situations

Abstract

Mathematics today is an unavoidable tool in almost all areas of science and technology of everyday life. Not only that it is unavoidable, but it has become essential in modelling dynamical phenomena of real-life situations. Today new interdisciplinary fields of science are rapidly evolving like mathematical biology, mathematical chemistry, mathematical economy, mathematical physics, mathematics for computer science, etc. The fields also help mathematics to evolve - many mathematical approaches got developed precisely because of challenges in its applications. Motivated by working at our University of Applied Sciences Velika Gorica, where we have both undergraduate and graduate modules that educate students about crisis management, we started investigating how mathematics is involved to crises at some deeper levels. We started exploring how mathematics models potential crisis situations.

The relationship between mathematics and crisis was investigated thoroughly by the Danish mathematician Ole Skovsmose. According to him, there are at least three connections between mathematics and crises: mathematics picturing a crisis, mathematics constituting a crisis and mathematics formatting a crisis. The first mentioned relationship is due to mathematics picturing different parts of our reality, and hence crises, using its models. A mathematical model is an abstract representation of a real-world system expressed through mathematical concepts and mathematical language. The purpose of mathematical modelling is to simplify complex processes and systems to better understand them, predict their future behaviour or optimize decisions. Mathematical models use mathematical formulas, equations, graphs or other mathematical structures. Mathematics constituting a crisis means that some reality processes that can end in critical situations do not exist without mathematical algorithms, like making transactions and decisions on the stock markets today. The crisis can happens due to misreadings and misusings of mathematics or maybe due to not understanding the process enough and hence not applying the appropriate models and approaches in specific situations. The third role where we say that mathematics formats a crisis is about how mathematics forms the view on crisis situations and hence forms how we approach and take action in these situations.

We reviewed books and papers from different areas where crisis might emerge, like epidemiology (disease spread), climate changes and economy (and finances) and cyber security to see which methods are used here. Among many others, we find that the common method used in mathematical models in these three areas are differential equations. Differential equations are often used to describe everyday dynamical processes, i.e. processes that change in time, and interactions of factors that are a part of these processes. Differential equations are mathematical equations that connect (unknown) mathematical functions and their derivatives, i.e. their rates of change. They are used to predict future states and hence prevent potential crisis situations. They belong to quantitative methods where one is interested more why and how some phenomenon works.

Today, more and more mathematical methods and tool are combined, both qualitative and quantitative, to create a unique appropriate approach to a specific interdisciplinary problem for which a solution is being sought. Differential equations are just one of them. Some of the other tools are machine learning, Markov Chains, linear programming, Monte Carlo simulations, graph theory, cryptography, game theory, …

In this paper we explain and further explore the three relationships of mathematics and crisis in disease spread and cyber security. We explain in more detail how differential equations are connected to our reality. We use concrete examples of differential equation models from the first three areas mentioned in the previous paragraph. Altogether, the aim of this paper is to show and convey to the wider audience how mathematics and crisis are connected and how mathematics, with its models, embodies relationships, dynamics and changes of the objects from different areas of the real world, with an emphasis on crises situations.