Mathematical biology – by way of example

Mathematical biology takes many different forms depending on the practitioner. I take mine with one math and two biologys (the so-called “little m, big B”), but others like it stronger (“big M, little b”). Under my worldview, mechanistic models are a tool to analyze biological data; a tool that infuses our knowledge of the relevant biological processes into the analytical framework. That might sound very pie-in-the-sky, and so I’ve made up an example to illustrate what I mean. This example has been constructed so that it doesn’t require any advanced knowledge: if you know how to add and multiply – that’s all you’ll need to answer these questions.

In the example below, the relevant biological processes are described in the section what we know already. You will need to use logical thinking to relate the what we know already section to the data reported in the DATASHEET so that you can answer the questions.

If you have ever wondered ‘what is Theoretical Biology?’ this example helps to answer that question too. Specifically, the required steps to do modelling, as inspired by this example, would be: 1) to write down the information that goes in the what we know already section (you’d refer to these as the model assumptions); 2) to devise a scheme to relate what we know already with the biological quantities of interest (this is the model derivation step); and 3) to report the results of your analysis (model analysis and interpretation).

As you work through this example, think about the types of questions that you are able to answer and how fulfilling it is that careful thinking has enabled us to draw some valuable conclusions. Understand too, that a criticism of mathematical modelling is that, in reality, everything might not happen quite as perfectly as we describe it to happen in the what we know already section. These sentiments capture the good and the bad of mathematical modelling. Mathematical models enable new and exciting insights, but our excitement is temped because these insights are only possible owing to the assumptions that have been made, and while we do our best to make sure these assumptions are good, we know that these assumptions can never be prefect.

If this sounds like fun, then have a go at the example below. If you want to email me your answers, I can email you back to let you know how you did  (see here for my email address).

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INFLUENZA X

A new and unknown disease, Influenza X, has swept through a small town (popn. 100). Your task is to describe the characteristics of the disease. Health officials want to know:

  1. How many days are citizens infected before they recover?
  2. What fraction of infected citizens died from the disease?
  3. What is the rate of becoming infected?

What we know already

During the epidemic, citizens can be classified into one of these four groups:

  • Susceptible
  • Infected
  • Recovered, or
  • Dead

As is shown in the diagram:

  • Only Susceptible citizens can be Infected.
  • Infected citizens either Die or Recover.
  • Citizens must have been Infected before they can Recover.
  • Only Infected citizens die from the disease.
  • Once they have Recovered, citizens cannot be re-infected.
  • All Infected citizens take the same number of days to Die or Recover.
  • During the epidemic no one enters or leaves the city. No babies are born; no one dies of anything other than Influenza X.

During the epidemic all that was recorded was the number of citizens who were Susceptible, Infected or Recoverd on each day and the number of people who had Died up until that point. This information is summarized in the DATASHEET provided at the end of this post. This information is also presented graphically below and you’ll get a better understanding of the data by considering how the graphs and the DATASHEET are related (Question 4).

 QUESTIONS

  1. Fill in the missing values on the DATASHEET (below).
  2. How many days are citizens infected before they recover?
  3. What fraction of infected citizens died from the disease?
  4. Label the axes on the graphs.
  5. The transmission rate of Influenza X is 0.008 (the units have deliberately been omitted). Consider the graphs above and describe how this rate was estimated?
  6. How is the unknown quantity from the DATASHEET calculated?

DATASHEET

Some definitions
  • If a patient is infected on Day 1 and recovers on Day 4 that patient is infected for 3 days (i.e., Day 1-3 inclusive).
  • Infected (cumulative) on Day T means the total number of citizens who have been infected any time from Day 1 to Day T (inclusive). Citizens who subsequently Died or Recovered are included in this number.

Related reading
For some of my older posts describing Mathematical Biology, you can start here.

How to make mathematical models (even at home)

As a WordPress blogger, I get a handy list of search terms that have led people to my blog. A particularly memorable search term that showed up on my feed was ‘how to make mathematical models at home’. What I liked about this query was that it suggests mathematical modelling as a recreational hobby: at home, in one’s spare time; just for fun. This speaks to an under-appreciated quality of mathematical modelling – that it’s really quite accessible once the core principles have been mastered.

To get started, I would suggest any of the following textbooks*:

Now, I know, you want to make your own mathematical model, not just read about other people’s mathematical models in a textbook. To start down this road, I think you should pay attention to two things:

  • How to make a diagram that represents your understanding of how the quantities you want to model change and interact, and;
  • Developing a basic knowledge of the classic models in the ecology, evolution and epidemiology including developing an understanding of what these models assume.

This would correspond to reading Chapters 2 and 3 of A Biologist’s Guide to Mathematical Modeling.

A good way to start towards developing your own model would be to identify the ‘classic model’ which is closest to the particular problem you want to look at. If you’re interested in predator-prey interactions, this would be the Lotka-Volterra model, or if you’re asking a question about disease spread, then you need to read about Kermack and McKendrick and the SIR model. Whatever your question, it should fall within one of the basic types of biological interactions, and the corresponding classic model is then the starting point for developing your mathematical model. From there, the next step is to think about how the classic model you’ve chosen should be made more complicated (but not too complicated!) so that your extended model best captures the nuances of your particular question.

Remember that the classic model usually represents the most simple model that will be appropriate, and only in rare circumstances, might you be able to justify using a more simple model. For example, if the level of predation or disease spread for your population of interest is very low, then you might be able to use a model for single species population growth (exponential/logistic/Ricker) instead of the Lotka-Volterra or SIR models, however, if predation and disease spread are negligible, then it arguably wasn’t appropriate to call your problem ‘predator-prey’ or ‘disease spread’ in the first place. Almost by definition, it’s usually not possible to go much simpler than the dynamics represented by the appropriate classic model.

That should get you started. You can do this at the university library. You can do this for a project for a class. And, yes, you can even do this at home!

Footnotes:

*For someone with a background in mathematics some excellent textbooks are:

but while the above textbooks will give you a better understanding of how to perform model analysis, the ‘For Biologist’s’ textbooks listed in this post are still the recommended reading to learn about model derivation and interpretation.

Report from the 4th Annual AARMS Mathematical Biology Workshop

On the weekend, I attended the 4th Annual AARMS Mathematical Biology Workshop hosted at Dalhousie University in Halifax. Workshop participants were a mix of graduate students, postdocs and professors originating from Memorial University of Newfoundland, Dalhousie University, the University of New Brunswick, York University, Wilfred Laurier University and the College of William and Mary. The research presented covered a wide range of model types including partial differential equations, ordinary differential equations, time delays, stochastic processes, algorithms to build maximum agreement forests, and power laws. The applications of these models covered a variety of biological situations including cellular processes, the immune system, phylogenetics, epidemiology, ecology, and ecosystem models for marine systems.

For me, this meeting was a great way to meet other local researchers who have similar interests in mathematical biology. In identifying common interests, I think this helps in understanding good candidate topics for future workshops and summer schools, and to know where we can go for advice on different aspects of our research. For graduate students, I hope that these types of initiatives help to provide a breadth of exposure to better understand what type of research you like, and to get ideas for where you might like to take your research in the future.

… and Halifax was beautiful! Thanks to David Iron, the other workshop organizers, and to AARMS for providing funds to help support student travel.