Mission drift and black fever

Ever since clicking ‘publish’ on my first post in the Great Models series, I didn’t feel right about it. I like to be original and interesting, however, I chose to write about the ‘mosquito theorem’ because it was a safe choice – because the work was foundational and because Ross’ achievements are highly and widely respected. I had named this blog post series Great Models and given the name, I felt compelled to chose work that was undeniably significant. In keeping with my blog mission, in the future I want to make the choices of Great Models more with respect to elegant model derivation for a given question, and not with respect to the gravity of the conclusions. The next model that I plan to discuss, I chose just because it’s fun. This will shatter my expectation that I have to choose only work that is already widely acclaimed.

Since that post, I started reading The Malaria Capers by Robert S. Desowitz. The dust jacket of the book states that despite the progress of 21st century science, malaria is more prevalent in tropical regions now than 50 years ago – and that truly the situation is worse because now a large number of mosquitoes are resistant to insecticides. The first part of the book was a story about how scientist’s determined the transmission route of kala azar (visceral leishmaniasis). An interesting aside is that kala azar appears to be a relatively new infectious disease, one that did not cause it’s first epidemic until 1824 (pp. 34).

To understand the spread of kala azar, the first problem for scientists was to identify the infectious agent. Initially, this was thought to be a hookworm because hookworm infections were common and sometimes found in patients that had died from kala azar. The infectious agent, in fact, was a protozoan first observed by Dr. William Boog Leishman. However, at the time it was quite some work to determine what the specks that Leishman had seen inside macrophages actually were. Initially, it was thought that these bodies could be trypanosomes.

The so-called Leishman-Donovan bodies were then put in a saline solution. This revealed a flagellated elongated life stage and this transformation implied an insect vector in the transmission of the black fever. Bed bugs were very common and unpopular at the time and so these arthropods were scientist’s first guess. To incriminate the bed bugs it was necessary to show that the protozoan could survive in these insect’s intestines and then moved to the salivary glands (where they could be transferred during biting) or that they were defacated and rubbed into the bite wound. This could not be demonstrated for bed bugs and instead it was determined that the silvery sandfly was the guilty party. The silvery sandfly first became a candidate vector when the range maps of this sandfly and kala azar epidemics were overlayed and found to be suspiciously related.

None of this story has much to do with modelling, but it highlights the challenge of making inferences in science when there are multiple plausible hypotheses and only incomplete knowledge. Would solving the kala azar problem be any simpler today?

With a more advanced taxonomic key for microbes and with modern day molecular techniques, it would be much easier to determine that the Leishman-Donovan bodies were not an organism that had ever been seen before, and to slot them into the tree-of-life as an animal-like protist. However, determining the insect vector might still be roughly as challenging now as it was in the 1900’s because the knowledge gains made in the areas of vector ecology and epidemiology over this period have been less dramatic.

Problems with multiple plausible hypotheses and incomplete information are a type of problem where modelling can make a great contribution. Here, the mathematical model is used to build a bridge between each hypothesis and the available evidence, to understand if any of the hypotheses are consistent with the limited information on hand, and even more so, to determine what additional characteristics and additional pieces of evidence must exist if each of these hypotheses are to be consistent. For example, modelling might be used to determine if the seasonal variation in kala azar is consistent with the seasonally driven vector population biology combined with the current best explanation of vector-human epidemiology. In fact, it was mentioned in the book that major kala azar epidemics occur on a 15-year cycle and, to me, that sounds like a great modelling problem for someone!

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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!