[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

Discussion

A few words about model assumptions

Conclusion and take home messages