[This post is frequently revised]
In the previous post, after defining a mathematical model, I then asked how and why do we make models. The theme of how we make models was addressed in the previous post, but I did not yet discuss why we make mathematical models.
To examine why we make mathematical models, firstly, since we are considering mathematical models in biology, our model should address a biological question – or at least link to a biological question that has been posed previously, or will be addressed subsequently. In additional to having biological, and therefore, practical relevance here are some more reasons to make a mathematical model:
- To make a quantitative prediction: Next year there will be seven frogs.
- To make a qualitative prediction: Next year there will be fewer frogs.
- To use information from one scale to understand another (i.e., multiscale modelling): Given our knowledge of how individual frogs reproduce and disperse, we use the mathematical model to predict the future species range of the frog population.
- To derive indirect methods to estimate parameters (or other quantities).
- To clarify the reasoning of a given argument: The frog population in Frog Lake is growing exponentially, however, predator density in Frog Lake has been constant over the last ten years, possibly owing to an extraneous factor. We derive a mathematical model to show that the persistence of the extraneous factor is necessary for the frog population to grow exponentially.
- To investigate hypothetical scenarios (and then make a quantitative or qualitative prediction). Particularly, scenarios on long time scales or large spatial scales for which it may be difficult to collect data or to conduct an experiment.
- To motivate experiments: i.e., the experimental manipulation of extraneous factor in Frog Lake.
- To disentangle multiple causation: Every year frogs die from either (a) old age, or (b) from Undetectable Frog Disease. The goal of the mathematical model is to show that time series frog mortality data would exhibit telltale signs of the relative contributions from (a) or (b).
- To make an idea or hypothesis precise, to integrate thinking and to think about a problem systematically.
- To inform data needs to best answer a given question.
- To identify the processes or parameters that outcomes are most sensitive to (i.e. using a sensitivity analysis).**
- To determine the necessary requirements for a given relationship: Undetectable Frog Disease evolves intermediate virulence only under a convex trade-off between transmission and virulence.
- To characterize all theoretically possible outcomes.***
- To identify common elements from seemingly disparate situations.
- To detect hidden symmetries and patterns.****
What do you think of this list?
As a long range objective of the blog, I would like to come up with an exhaustive list of possible types of motivations for making mathematical models, which I would then like to draw as a Venn diagram (i.e., since 1. and 6. are not mutually exclusive, but 1. and 2. are*).
Note that 1-8 are very related to ‘what is your question?’. It has been said that coming up with a good question is an important component of doing good science, and so understanding the range of possible motivations for wanting to use a mathematical model, therefore, seems central to making smart decisions about how best to direct one’s scientific investigations. It’s a question that’s worth exploring and in the future I will summarize Chapter 1 of A Biologist’s Guide to Mathematical Modelling which also covers this topic.
* In a sense every quantitative prediction is also qualitative, however, from the point of view of characterizing the motivation for a particular modelling project, I think you need to choose one or the other because this will strongly influence the nature of how the project is best carried out.
** Thanks to Prof. Jianhong Wu and Helen Alexander for contributing to the list.
*** 13. is really a more robust version of 6. A bifurcation diagram might be an example of 13. The same is true of 12. and 5. Numbers 9. and 5. seem similar to me, except that 5. is focused on identifying an explanation consistent with a relationship, while 12. just seeks to be precise (without being motivated by a relationship that needs explaining).
**** See this post for the relevant citations.