Chapter 1 An Introduction to Modelling and Differential Equations

“The profound study of nature is the most fertile source of mathematical discoveries.” - Joseph Fourier (1768-1830)

1.1 Modelling

Mathematical Modelling is a general procedure in the Mathematical Sciences wherein ideas about observed processes are presented and interrogated using mathematical principles and techniques. There may exist many different conceptual formulations of observed processes and so there may be many equivalent models for those processes. In the sections that follow, we shall discuss some of the conceptual aspects of building models that describe physical phenomena.

The Modelling Process

To gain an understanding of the processes involved in mathematical modelling, consider two worlds which can be depicted with the properties below:

  • Real-world system
    • Observed behaviour
  • Mathematical World
    • Models
    • Mathematical operations and rules
    • Mathematical conclusions

Suppose we want to understand some behaviour or phenomenon in the real world. We may wish to make predictions about that behaviour in the future and analyze the effects that various situations have on it.

For example, when studying the populations of two interacting species, we may wish to know if the species can coexist within their environment or if one species will eventually dominate and drive the other to extinction. In the case of the administration of a drug to a person, it is important to know the correct dosage and the time between doses to maintain a safe and effective level of the drug in the bloodstream.

In order to construct and use models in the mathematical world to help us better understand real-world systems, it is important to gain an understanding of how we link the two worlds together. To begin, we would first have to define what we mean by a real-world system.

Definition 1.1 (System) A system is an assemblage of objects with some interaction or interdependence.

Definition 1.2 (Model) A model is a conceptual representation of a given object or process that captures some collection of aspects of that object or process.


The modeller is interested in understanding how a particular system works, which features cause changes in the system, and how sensitive the system is to certain changes. They are also interested in predicting what changes might occur and when they occur.

Remark. Physical systems can be very complicated, with subtle intricacies that are either not well understood, or difficult to describe mathematically. Therefore, the impetus of mathematical modelling is to produce an approximation that is sufficient to describe the important parts of the system. This means that the domain of applicability of a model can be very narrow compared to the conceptually perfect mathematical representation of such a system.


Suppose the goal is to draw conclusions about an observed phenomenon in the real-world. One approach would be to conduct some experiments or real-world behaviour trials to observe the effects thereof. This would be the process of drawing conclusions from the real-world behaviour. However, this is not always a viable method. In many cases there may be prohibitive financial or social costs which inhibit our ability to run such experiments. For example, determining the concentration level at which a drug would be fatal, or studying the effects of a failure in a nuclear power plant. It may even be the case where it is not possible to carry out such experiments, for example, investigating a specific change in composition of the ionosphere and its corresponding effect on the polar ice caps. We may also be interested in expanding our conclusions beyond that of the very specific trial we have run. For example, we may have run a trial in Johannesburg with a temperature of \(32^{\circ}C\), humidity \(42\%\), and other very specific conditions. Even if we do succeed in drawing a conclusion from the real-world behaviour under these very specific conditions, we may not have necessarily explained why the particular behaviour has occurred. This gives us motivation to derive indirect methods of studying real-world behaviour.

The alternative approach to draw conclusions about an observed phenomenon in the real-world can be outlined as follows. The first step would be to make observations of the system we are attempting to model. When making observations about real-world behaviour we will usually not be able to identify or consider all the factors associated with the phenomenon. As such we would make simplifying assumptions that eliminate some of the factors. Take for example a situation such as the effects of a failure in a nuclear power plant near a city such as Cape Town. Initially it might be found that we do not need information about the humidity levels and we would neglect this factor in our model. Our next step would then be to conjecture tentative relationships among the factors to create a model. After this, we would construct a model using these relationships. After having constructed a model we would move to the next step of applying a comprehensive mathematical analysis to the resultant model. We must be careful at this step to note that any conclusions we obtain from this model are strictly to do with the model and not the real-world behaviour, since we have made simplifications to the model, and our observations on which we have based our model could contain errors. In summary, we have a rough outline of the procedure:

  1. Through observation, identify the primary factors involved in the real-world behaviour.
  2. Conjecture tentative relationships among the factors.
  3. Apply mathematical analysis to the resultant model.
  4. Interpret mathematical conclusions in terms of the real-world problem.

We shall consider several scenarios in detail using this approach in Chapters 3 and ??.

Mathematical Models

“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?” - Albert Einstein (1879-1955)

Let us begin with a useful definition.

Definition 1.3 (Mathematical Model) A mathematical model is a mathematical construct designed to study a particular real-world system or phenomenon.


There are two approaches we can take when modelling a phenomenon mathematically. We can either take pre-existing models of well studied phenomenon and build on these models, or we can construct a completely new model for our special phenomenon.

When constructing a model it can often be the case that we reach a solution so complex and intractable that there is little hope of analyzing or even solving the model. This can be caused by a number of conditions, and can often cause our model to have very little utility in understanding the phenomenon. These complexities may arise when attempting to use a model given by a system of equations. The problem may be so large that it is impossible to capture all the information in a single mathematical model. An example of such a problem is the global effects of population interaction, resource use and pollution. In such cases we can attempt to replicate the behaviour directly by conducting various experiments in order to collect data and analyze the phenomenon using statistical techniques or curve-fitting.

It is important to note that there is a trade-off among factors in selecting a specific type of model. Three of these factors are listed below.

  • Fidelity: The preciseness of a model’s representation of reality.
  • Cost: The total cost of the modelling process.
  • Flexibility: The ability to change and control conditions affecting the model as required data are gathered.

First let us consider fidelity. Observations made from real-world behaviour would be a direct representation of reality and thus we would have the greatest fidelity from these types of models. After this we would expect that experiments are the next best since they are scaled down versions of the real phenomenon, followed by simulations which may not account for all the intricacies of the system. Constructed models would be tailored for the specific problem and would probably outperform pre-existing models.

The cost of real-world observations would generally be high since gathering data is an expensive process. The cost of experiments and simulations would also generally be high since the equipment required is usually expensive. Constructed models will generally have some cost associated to the tailoring to the specific problem and pre-existing models would generally be the cheapest.

The flexibility of constructed models would generally be the greatest since we can vary the models easily. Pre-existing models can be used as a base and built upon and are generally available. Experiments and simulations would only be as flexible as the technology, which is available at the time, allows for. Finally, real-world observations would generally be the least flexible since we cannot change the environment in which we make these.

In the beginning of Chapter 3, we will use these elementary discussions around the modelling process and the concept of a mathematical model to discuss the technical aspects of model building by unpacking the steps to follow in order to construct a mathematical model. Before we can do that though, we need a deeper understanding on the ways in which we can mathematically model change.

1.2 Differential Equations

“It is well known that the central problem of the whole of modern mathematics is the study of transcendental functions defined by differential equations.” - Felix Klein (1849-1925)

Before we do anything, it is important to understand the key differences between an algebraic equation and a differential equation.

Now that we know this key difference, the question remains: what actually is a differential equation?

Why are we interested in studying differential equations? Well, it is due to the fact that differential equations arise naturally in models in science, engineering, and economics. The question is why did they arise naturally in these physical processes and phenomena. The answer is that these physical, biological, and economic systems are all marked by the same thing: change. And so, differential equations model these problems by describing how they change. So we are interested in describing a systems dynamics (compare to statics).

What do I mean by this?

Exercise:

- Complete the task at the end of the above video.
- Summarize the concepts and contents of this Chapter.

Summary

At the end of this Chapter, you should:

  • understand the difference between a system and a model;
  • be able to explain the concept of a mathematical model;
  • know the difference between an algebraic and differential equation;
  • understand what a differential equation is, and why it is useful; and
  • be able to conceptualize differential equations as dynamic models.