Plenary speakers:

Edward Allen, Texas Tech University

Linda Allen, Texas Tech University

Steve Cantrell, University of Miami

Odo Diekmann, Utrecht University

Simon Levin, Princeton University

Mark Lewis, University of Alberta

Philip Maini, Oxford University

For complete details please visit the conference website*:

http://www.math.mun.ca/~ahurford/aarms/

*please note that there was a service outage for math.mun.ca on Monday Feb 18, but the link should work now.

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**Song. **An original song written by a graduate student about graduate student-supervisor meetings. It’s a catchy tune! Click Here.

**Movie.** This is a movie that I took in 2009 while attending a Mathematical Biology Summer School in Botswana. Click here. (Unfortunately, I encountered some technical difficulties uploading this to YouTube, but it’s still watchable, albeit in micro-mini).

Happy holidays everyone!

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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:

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

What we know already

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

As is shown in the diagram:

- Only
*S**usceptible*citizens can be**I**nfected *I**nfected*citizens either**D**ie*R**ecover.*- Citizens must have been
*I**nfected*before they can*R**ecover*. - Only
*I**nfected*citizens die from the disease. - Once they have
*R**ecovered*, citizens cannot be re-infected. - All
*I**nfected*citizens take the same number of days to*D**ie*or*R**ecover*. - 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

- Fill in the missing values on the DATASHEET (below).
- How many days are citizens infected before they recover?
- What fraction of infected citizens died from the disease?
- Label the axes on the graphs.
- 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?
- 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.

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

- A Biologist’s Guide to Mathematical Modeling by Sally Otto and Troy Day
- Modeling Biological Systems: Principles and Applications by James Haefner
- A Course in Mathematical Biology by Gerda de Vries, Thomas Hillen, Mark Lewis, Birgitt Schonfisch and Johannes Muller
- Dynamic Models in Biology by Stephen P. Ellner and John Guckenheimer

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:

- Mathematical Models in Biology by Leah Edelstein-Keshet
- Mathematical Models in Population Biology and Epidemiology by Fred Brauer and Carlos Castillo-Chavez
- Mathematical Biology by J. D. Murray Part I and Part II

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.

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**Use, misuse and extensions of “ideal gas” models of animal encounter**. JM Hutchinson, PM Waser. 2007. Biological Reviews. 82:335-359.

I haven’t have time to read it yet, but from the title it certainly sounds like it answers some of my questions.

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Yesterday, I came across this paper from PNAS: **Parameter-free model discrimination criterion based on steady-state coplanarity**** **by Heather A. Harrington, Kenneth L. Ho, Thomas Thorne and Michael P.H. Strumpf.

The paper outlines a method for testing the mass-action assumption of a model without non-linear fitting or parameter estimation. Instead, the method constructs a transformation of the model variables so that all the steady-state solutions lie on a common plane irrespective of the parameter values. The method then describes how to test if empirical data satisfies this relationship so as to reject (or fail to reject) the mass-action assumption. *Sounds awesome!*

One of the reasons I like this contribution is that I’ve always found mass-action to be a bit confusing, and consequently, I think developing simple methods to test the validity of this assumption is a step in the right direction. Thinking about how to properly represent interacting types of individuals in a model is hard because there are lots of different factors at play (see below). For me, mass-action has always seemed a bit like a *magic rabbit from out of the hat; just multiply the variables; don’t sweat the details of how the lion stalks its prey; just sit back and enjoy the show.*

Before getting too far along, let’s state the law:

**Defn.** Let be the density of species 1, let be the density of species 2, and let be the number of interactions that occur between individuals of the different species per unit time. Then, the law of mass-action states that .

In understanding models, I find it much more straight forward to explain processes that just involve one type of individual – be it the logistic growth of a species residing on one patch of a metapopulation, or the constant per capita maturation rates of juveniles to adulthood. It’s much harder for me to think about interactions: infectious individuals that contact susceptibles, who then become infected, and predators that catch prey, and then eat them. Because in reality:

Person A walks around, sneezes, then touches the door handle that person B later touches; Person C and D sit next to each other on the train, breathing the same air.

There are lots of different transmission routes, but to make progress on understanding mass-action, you want to think about what happens on average, where the average is taken across all the different transmission routes. In reality, also consider that:

Person A was getting a coffee; Person B was going to a meeting; and Persons C and D were going to work.

You want to think about averaging over all of a person’s daily activities, and as such, all the people in the population might be thought of as being uniformly distributed across the entire domain. Then, the number of susceptibles in the population that find themselves in the same little as an infectious person is probably .

Part of it is, I don’t think I understand how I am supposed to conceptualize the movement of individuals in such a population. Individuals are going to move around, but at every point in time the density of the S’s and the I’s still needs to be uniform. Let’s call this the uniformity requirement. I’ve always heard that a corollary of the assumption of mass-action was an assumption that individuals move randomly. I can believe that this type of movement rule might be sufficient to satisfy the uniformity requirement, however, I can’t really believe that people move randomly, or for that matter, that lions and gazelles do either. I think I’d be more willing to understand the uniformity requirement as being met by any kind of movement where the net result of all the movements of the S’s, and of the I’s, results in no net change in the density of S(t) and I(t) over the domain.

That’s why I find mass-action a bit confusing. With that as a lead in:

**How do you interpret the mass-action assumption? Do you have a simple and satisfying way of thinking about it?**

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**Related reading**

This paper is relevant since the author’s derive a mechanistic movement model and determine the corresponding functional response:

**How linear features alter predator movement and the functional response **by Hannah McKenzie, Evelyn Merrill, Raymond Spiteri and Mark Lewis.

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