Ronald Ross’ mosquito theorem

I had promised to write about models that I think succeed on a very high level and so let me start that series of posts by discussing Ronald Ross’ so-called mosquito theorem.

If you go running in Kingston, Ontario, along the west bank of the start of the Rideau Canal, past the dock and towards the Kingston-Whig Standard headquarters, you will come across a Celtic cross that reads:

In memory of an estimated one thousand Irish labourers and their coworkers who died of malaria and by accidents in terrible working conditions while building the Rideau Canal, 1826-1832.

I didn’t think too much of that at the time, but later I was trying to recall the disease that was mentioned. Malaria? In Kingston? In Canada? Could that be right?

Ronald Ross. This picture was taken from Wikipedia and is in the public domain.

And so I read up on the details (see for example here and here) and, sure enough, malaria had been endemic in North America and was eradicated through the use of insecticides and mosquito control.

Ronald Ross identified the anophelene mosquitoes as the vector for malaria transmission and developed a mathematical model that showed that malaria could be eradicated as long as the number of mosquitoes per human was brought below a threshold value, which was significant because malaria could then be eliminated without needing to kill every mosquito. Given that introduction, it’s not hard to see why I might consider this a great mathematical model: (1) malaria was eradicated in North America – an extremely impressive achievement, and (2) the ‘mosquito theorem’ turned out to be correct.

Ross sounds like an amazing guy and I’m not sure if there is a lesson here or if it’s just that he was flat out brilliant. Having said that, I think one of the reasons Ross’ work was so successful was because mathematical modelling was just one facet of his arsenal – he likely derived the model because he needed the result to help design and lobby for prevention strategies. If there is any lesson here for today’s theoretician it might be that Ross’ success underscores the importance of collaboration with empiricists, ecologists, clinicians and other experts.

In 1902, Ross was awarded the Nobel Prize in medicine and physiology, however, this was after Ross’ experimental work on malaria transmission and before Ross derived his first epidemiological model in 1908. Yet, despite Ross having made numerous and varied contributions, the Nobel Laureates webpage singles out mathematical modelling as Ross’ greatest contribution:

He made many contributions to the epidemiology of malaria and to methods of its survey and assessment, but perhaps his greatest was the development of mathematical models for the study of its epidemiology, initiated in his report on Mauritius in 1908, elaborated in his Prevention of Malaria in 1911 and further elaborated in a more generalized form in scientific papers published by the Royal Society in 1915 and 1916. These papers represented a profound mathematical interest which was not confined to epidemiology, but led him to make material contributions to both pure and applied mathematics.

A quote on parsimony

This morning’s reading has lead me to a nice quote:

Model building is the art of selecting those aspects of a process that are relevant to the question being asked – J.H Holland*

What I like about the quote, is that it not only highlights the principle of parsimony (as the Einstein quote did), but it highlights that the question being asked is the element of the scientific problem that should be referenced to determine if an aspect of the model should be kept in or kicked out.

In a world where we might identify ourselves as a landscape ecologist, a toxicologist, or even an expert in neural networks – consider this: there are unlikely to be any discipline specific guides to parsimonious model building. And my reason for wanting to catalog the different types of questions was that this could, possibly, serve as a useful framework; where the same types of questions share the same types of guiding principles regarding how best to achieve parsimony.

Holland, JH (1995) Hidden Order. Addison-Wesley, New York, USA.

Deriving models that are simple, but not too simple

This picture of Albert Einstein is from Wikipedia and is in the US public domain

In the first few pages of A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution by Sally Otto and Troy Day they paraphrase Albert Einstein (pg 7) who said:

“Everything should be made as simple as possible, but no simpler”

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After I give a talk, I am often asked questions such as: “you assumed that space is homogeneous, but isn’t there a mountain range to the west?” or “could you expand your model to consider the influence of hunting on adult wolf survivorship?” And for a split second this thought races through my head: They’re right. I am wrong. My work is wrong! This is terrible, I must add in hunting to fix it.

It is tempting to think that a more complex model is better. Will other scientists assume that I aren’t skilled enough to include hunting in the model? Will they not understand that this was my deliberate choice – the choice not to include it?

As a final comment, if some asks “could you expand your model to consider the effect of climate change?” at the end of one of my talks, I will return the question by asking what they think would change if I had explicitly included this. The question above, without further elaboration, could imply that I didn’t include climate change because I overlooked it. Returning the question helps to draw attention to the challenges that modellers face and to highlight the types of careful considerations that go into model construction.

Overview of mathematical modelling in biology II

[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:

  1. To make a quantitative prediction: Next year there will be seven frogs.
  2. To make a qualitative prediction: Next year there will be fewer frogs.
  3. 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.
  4. To derive indirect methods to estimate parameters (or other quantities).
  5. 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.
  6. 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.
  7. To motivate experiments: i.e., the experimental manipulation of extraneous factor in Frog Lake.
  8. 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).
  9. To make an idea or hypothesis precise, to integrate thinking and to think about a problem systematically.
  10. To inform data needs to best answer a given question.
  11. To identify the processes or parameters that outcomes are most sensitive to (i.e. using a sensitivity analysis).**
  12. 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.
  13. To characterize all theoretically possible outcomes.***
  14. To identify common elements from seemingly disparate situations.
  15. 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.

Overview of mathematical modelling in biology

A mathematical model is a set of mathematical equations system of mathematical constructs that represent the nature of biological, physical, and/or chemical process, or even processes that occur in other science or non-science disciplines. But, how and why do we make mathematical models?

Below, I have provided my view on how we make mathematical models (Figure 1). The steps involved are model derivation, model analysis, and interpretation. The middle ‘analysis’ step is the one that mathematicians and physicists excel at. This step may draw on skills that are developed in probability, statistics, dynamical systems and other mathematics classes. In fact, considering just the two bottom boxes alone (from Model to Model results) would constitute a problem in pure mathematics. Applied mathematics involves two extra steps, which are the transformation of the biology into the mathematics, and the inverse transformation back into the biology after the mathematical results are obtained.

Figure 1. Mathematical modelling involves the steps of model derivation, analysis and interpretation.

The skill of model derivation is taught, to some degree, in mathematical modelling classes: we learn that stochastic models are appropriate for questions involving small population sizes and that models should be just simple enough. The skill of model derivation is also what I refer to as the art of mathematical modelling. Compared to the abundance of classes, books, and theorems that are available to help us with model analysis, there are relatively few directives on how to go about model derivation, and the directives that exist are subjective and vague.

Having said that, I am not criticizing the just simple enough advice: actually, I think this is fantastic advice; quite possibly, the-very-best-holy-grail of all advice for aspiring modellers. And yet, this advice is distinctly hard to pin down. This brings me to my main point. Just as painters and sculptures take courses in art appreciation, perhaps us modellers could take some lessons from a (blog) journey on model appreciation. See you next time? I hope so, because I know that I, for one, would like to get to the bottom of this!