Effect Measure

[A series of posts explaining a paper on the mathematical modeling of the spread of antiviral resistance. Links to other posts in the series by clicking tags, “Math model series” or “Antiviral model series” under Categories, left sidebar. Preliminary post here. Table of contents at end of this post.]

In this post we start to dig into a mathematical model of antiviral resistance in influenza. The modern era of mathematical modeling started early in the last century with attempts to understand malaria spread, almost exactly 100 years ago. For the first 50 of those years scientists used pure mathematics because the problems were too complicated for brute paper and pencil accounting. For example, there is a simple mathematical formula that allows prediction of how big the population of E. coli would be after two days if it doubled every twenty minutes. You don’t have to sit with a paper and pencil and do all the doublings by hand 144 times (144 is 48 hours, or two days, times three, the number of doublings in an hour, one every twenty minutes). In fact this is not even feasible as the numbers get too large. Using the formula you don’t have to do 144 doublings on a piece of paper, even in principle. With ingenuity, formulas like the doubling formula can often be devised for relatively simple situations, but not more complicated ones.

Enter, the computer

The computer changed things dramatically. Now the power of the machine and some new mathematics allow us to explore much more complicated scenarios. We now can instruct the computer to do the paper and pencil work, making it an electronic version of an army of scribes and research assistants keeping track of each step by hand. At the same time we learned how to make inferences about general behavior without doing any calculations at all, because of new theoretical understandings of how the underlying equations work.

Along with this power came a loss of transparency. The mathematical computer model has become like a black box for practitioners and non-specialists and the new theories are beyond their ken. We know what went into the black box and can see what came out of it, but between these two points a lot has to be taken on faith. In this post we want to start the process of looking into the box to see how it works. We are going to do this by parsing a new and excellent modeling paper, “Antiviral resistance and the control of pandemic influenza” by Marc Lipsitch, Ted Cohen, Megan Murray and Bruce Levin, researchers at the Harvard School of Public Health and Emory University in Atlanta. The authors are acknowledged experts in infectious disease and modeling. I chose this paper, from some other, equally excellent ones, for several reasons. First, it is very good work. Second, it deals with a subject, antiviral resistance, of great interest to many of our readers. Third, the model is relatively simple, but not simpler than needed. Fourth, it is published in Public Library of Science, Medicine (PLoS Medicine), an Open Access, online journal. This means that if you want to follow along at home (something I advise), you can easily obtain your own copy of the paper for free. All you need to do is go here. You can either read it in a web version on screen (HTML) or more conveniently print it out by first clicking on .pdf (large or small) on the top right sidebar (Download) section. From now on I will assume you are looking at the .pdf version of the paper, although I hope you will learn something even if you don’t read along. On our journey I’ll be taking my usual liberties to comment on a variety of subjects, including how scientific papers are organized, how scientists read them (or parts of them), and what a non-specialist might use a highly technical paper for besides its technical content.

The paper

The main paper is 11 pages long (in the .pdf version), but there are also two online supplements that contain interesting information. We will refer to them when necessary. Of the 11 pages, the first and last are synopses of the contents, and the second from last is references. The references can be very useful, even if you don’t read the paper. Consider them an old fashioned version of today’s “social bookmarking” services like del.icio.us. They refer to literature the authors found useful. The synopses are very helpful in telling you what the paper is about. They are often the only thing people read, especially if they are not specialists. The first page is a typical scientific abstract, while the last page is something PLoS Medicine uses to make it easier for non-specialists and journalists to understand the significance of the paper. Since we are going to examine the paper in detail we will skip them. You can read them whenever you wish, but if you hang in with us to the end of this series we suggest you read them again as a way to put it all together in your mind. You may find that while each post is understandable on its own, there are so many details that having a picture of the whole journey is more difficult. The synopses are a good way to recapture some of the big picture.

So that leaves 8 pages of paper, per se. It has four sections, set off by bold face headings: Introduction, Methods, Results and Discussion. Another dirty little secret of scientists is that if they read beyond the abstract, they often only read the Introduction and Discussion sections. The Introduction gives context and background to the study while the Discussion highlights the interesting results and explains what the authors think they mean. Admittedly this is not the most critical way to appraise the scientific literature but it is a great time saver and often tells you whether you want to dig deeper. Since we are going to dig deeper, we will go section by section, including the Methods, where most of the mathematics is located, and the Results, where the model output is summarized, illustrated and analyzed. By that time the Discussion will be relatively easy.

We’ll start this in the next post. This will give you time to print out your own copy, take a look at it, and no doubt wonder how you will ever be able to understand the diagram and equations on page 3. We hope we can surprise you by explaining them. In any event, we will try.

Table of contents for posts in the series:

What is a model?

A modeling paper

The Introduction. What’s the paper about?

The essential assumption.

Sidebar: thinking mathematically

The model variables

The rule book

More on the rule book

Finishing the rule book

The rule book in equation form

Ready to run the model

Effects of treatment and prophylaxis on resistance

Effects of Tamiflu use and non drug interventions

Effects of fitness costs of resistance


A few words about model assumptions

Conclusion and take home messages


  1. #1 Tom DVM
    March 22, 2007

    “The mathematical computer model has become like a black box for practitioners and non-specialists and the new theories are beyond their ken. We know what went into the black box and can see what came out of it, but between these two points a lot has to be taken on faith.”

    A technology, in the end, is only as good as the person driving it. There are a lot of drivers out there who should not have a license and epidemiology is a ‘wild west’ profession…no self-regulation or regulation of any kind. There are a few too many ‘guns for hire’ out there who will deliver any result for a price.

    ” With ingenuity, formulas like the doubling formula can often be devised for relatively simple situations, but not more complicated ones.”

    A mathematical model, like an experiment, is only as good as the accuracy and control of variables…the point being that it is difficult to understand nature when you live an work in complete isolation from the object of your study.

    We can’t model what we don’t understand. Modelling may work for something like the replication of bacteria but the emergence and conditions of a pandemic (from an animal and not a human disease source) is multi-factoral.

    The nature of natural phenomena including the emergence of exotic disease is beyond our capabilities and therefore, results in this particular area of study should be viewed with our eyes completely open.

  2. #2 gilmore
    March 22, 2007

    PDF of paper downloaded, but diagram and equations on page 3 are going to be a bitch to figure out.

  3. #3 revere
    March 22, 2007

    Tom: You’ve made up your mind, so why bother to read this? You seem to have no experience in modeling and aren’t even aware of the many models that are successfully used everyday. Your general statements about modeling are not correct. Why don’t you just sit back and read the series (or not) and comment when it’s over?

    gilmore: I plan to lead you through the figure and equations step by step. They are not as bad as they look, but require a little bit of concentration. Lots of details. It’s probably more important to get the general idea, which you can do if you understand a couple of the many arrows in the figure (although I will do them all). Once you’ve got the figure you’ve got the equations, because they are essentially the same. That’s one of the lessons, here.

  4. #4 frostieb
    March 22, 2007

    I’m with you Revere. Thanks for doing this. I’m as interested in the general approach to reading the paper as the specifics. Many times in my life I’ve ventured into Medical School libraries in search of information. I’m often stymied by the specifics due to a lack of precise understanding of the way the language is used, but am often able to glean alot at a higher level.

    As regards the figure on page 3 – it looks like a state diagram to this old computer scientist. Any relation?

  5. #5 revere
    March 22, 2007

    frostieb: You could convert it to one, although it’s not quite the same as the boxes aren’t in different states. They have contents. But if you discretize the model you essentially have a finite state automaton. There is a theorem that says you can put ODEs into a canonical form so that they are also Turing capable.

  6. #6 J.P.
    March 22, 2007

    Tom DVM seems to be saying: “Don’t confusing me with the facts, my mind is already made up!”

    Great work revere, most of us appreciate your efforts.

  7. #7 traumatized
    March 22, 2007

    to Tom DVM:
    I find that a much more widespread problem is lack of intellectual curiosity and petrification from fear of anything more complicated than counts and averages.

  8. #8 anon
    March 22, 2007

    I’ll decide whether to download and print it
    after your last post in the series and then maybe
    (re)read it all 😉

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