[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.]

The use of antiviral drugs to prevent or manage a pandemic with influenza/H5N1 is both a mainstay of national and international pandemic plans and a source of controversy. Will there ever be sufficient doses to make a difference? If there were, could they be deployed and administered in time? If so, would the virus continue to be responsive to them or would resistance quickly develop? Along with these big questions, there are many practical ones. What doses are needed and for how long? Should application be on a population basis or targeted to subgroups? Should antivirals be used just for treatment or also for prophylaxis? What about matters of licensing, patenting, intellectual property rights, whether innovation is being suppressed, etc.? — the usual pharmaceutical-related questions, but magnified here by the size of the stakes.

Since there has (as yet) been no pandemic where antiviral drugs have been used, we have no actual experience to guide us. But we do have data from other settings and a kit of tools and techniques from molecular biology, infectious disease epidemiology and public health practice. In the toolkit are mathematical models, ways to ask “what if we did this” questions. We thought it would be useful (and we hope instructive) to discuss the general technique of modeling in the specific context of antiviral drug strategies against H5N1. This is an experiment, using the blog format to walk you through a recent modeling paper, section by section. To do this properly and thoroughly will take many posts, but we have tried to make each sufficiently independent that if that was the only thing you read it, would still contain useful and interesting information. Whether we have succeeded only you can tell us.

We begin with some general observations about modeling. The first obstacle is figuring out what the word “model” means. Before writing this I pulled off my shelf three of the better known books on mathematical modeling and was surprised to discover none spent any time talking about what a model was. They each took the word as understood. Unfortunately the word “model” has so many meanings in English and other languages we have to be careful we don’t unconsciously let connotations from other contexts slop over into the one we are using. For example, there are model citizens (the ideal citizen), fashion models (examples of people selected for certain ideal or useful characteristics), model airplanes (miniature non-working versions of much larger objects), model legislation (a template for similar kinds of laws) and various kinds of models used by scientists such as ball and stick molecular models of crystallographic electron density maps (a translation of one kind of rendering into another), wind tunnel models (miniature working physical versions of larger scale objects), conceptual models (like the solar system model of the atom), animal models of human disease (like using a mouse as an experimental object to stand in for a human being) and mathematical models (using mathematics to strip away the complexities to get to the logical consequences of some explicit assumptions about the world). I’ve probably left out a few uses of the word model here (like statistical model), but you get the idea.

You might think that the different meanings of model would all be so clear from context they would cause no trouble. You would be wrong. Within science itself, confusions over the meanings and uses of models are common and the consequences usually make it more difficult for scientists to talk across disciplines. A good example is the different use of the word model by a biologist (like a virologist) and by a mathematical modeler in epidemiology, a physicist or a computer scientist (this last has still other technical uses of the word model derived from mathematical logic). The purpose of an animal model is to have a common and feasible experimental target for explanation. The use of a mouse, rat, ferret or other agreed upon animal, whose characteristics make it a plausible substitute for human physiology, is both to be able to do experiments you can’t do on humans (for obvious reasons) and to make those experiments comparable so that they can add to and relate to each other. There is some abstraction involved, for a scientist has to decide what features of a mouse are essentially like those of a human (e.g., they are both mammals, mammalian cells have similar features no matter the species they come from, mouse lungs are functionally and in other ways just like human lungs, etc.).

Just as importantly for a biologist, the mouse retains all the complexities of the human biological system. Biologists are very attached to the complexity of their system. They tend to look down on the assumptions of mathematicians and physicists whose standard operating procedure is to strip away all the complexity and operate on the logical skeleton. The classic opening gambit of a physicist — “Consider a spherical cow …” — is the end of the conversation for a veterinarian. For a vet, that real cows aren’t spheres means the physicist has nothing useful to say on the subject. Of course this isn’t true, but it is a reality of cross disciplinary communication. (Evelyn Fox Keller’s book, *Making Sense of Life: Explaining Biological Development with Models, Metaphors, and Machines*, is a fascinating exploration of this interplay in developmental biology and I recommend it highly. Keller is not only a noted science studies scholar and historian but had an earlier career as a mathematical biologist with one of the giants in that field, Lee Segel.)

Obviously the fact that a cow is spherical would be an absurd assumption for many purposes, e.g., explaining locomotion. Cows don’t roll, they walk. But if the issue were to predict that a cow dropped from an airplane would reach a terminal velocity because air resistance would eventually balance the acceleration due to gravity, then adding on the extra complications of ears, legs and tails would be just adding complications. It doesn’t affect the general outcome. A real cow would also reach a terminal velocity, which may or may not be very different from that of a spherical cow, but the poor animal would still not go faster and faster as it fell. On the other hand, dropping a cow over the moon, where there is no atmosphere, would not just be quantitatively different because of the differences in gravitational attraction, but qualitatively different. The cow jumping over (or in this case toward) the moon would keep going faster and faster, eventually making a pretty big crater. On earth this doesn’t happen because of atmospheric resistance (which would nicely roast the cow first). A bovine Armageddon is one less thing you have to worry about.

Here’s another example, this time more mathematical. The common gut bacterium, *E. coli*, inoculated into a fresh broth medium divides once in approximately 20 minutes, i.e., three times an hour. You can see that a single bacterium would give rise to eight at the end of the first hour, 512 at the end of the second hour, etc. This shouldn’t surprise anyone. *E. coli* is present in the gut in the billions or trillions. So this doubling is a pretty good mathematical model for the growth of an organism like *E.coli*, or, for that matter, for the spread of a disease. But it has a very narrow range of validity. If you let the doubling model predict the number of *E. coli* you would have after little more than two days (52 hours), it exceeds the number of atoms in the universe by a wide margin. Obviously this won’t happen. The model has broken down. It’s not just that the model gives the wrong quantitative answer. Like the falling cow that neglects wind resistance, it gives the wrong qualitative answer. *E. coli* populations don’t grow without bound. But the model is essential for understanding what is happening in the first few hours and is the basis for routine food protection regulation.

So the *E. coli* model is wrong but in some circumstances it is useful. This is a general characteristic of all models. Because they involve neglecting some aspect of the thing they are modeling, they are *all* wrong. But many models are useful. A central problem of all modeling is figuring out how to make a model that is useful, to tell when it is, and to figure out what is useful about it.

Modeling is not a Secret Art. Tacit and informal models of bird flu spread are used all the time here, by us and by our commenters. Every time someone speculates about an infected person passing the disease to a relative, co-worker or seatmate on an airplane, they are using an informal model. It isn’t (yet) a mathematical model, but it is an imagining of what would happen and an abstraction of the real world. The person doesn’t have a name or gender or age or state of immunity. There is no airplane flight number. It is a “typical” person and a “typical” airplane trip, generalizations or abstractions. But the details matter. If we start to make underlying assumptions explicit, say by assuming different chances of an airplane passenger passing on the disease if he is using “sneeze hygiene” or not, infectious or not, asymptomatic or not, we can convert this into a mathematical model, even without using mathematical symbols. By making some guesses as to how likely it is to pass on the disease to a contact and how full airplanes are today, you could make some estimates as to how many new cases this person would produce. If we had the patience, time and enough paper and pencil to keep track of this we could make predictions about disease spread in a hypothetical population, but it’s easier if you use a computer to do the grunt work for you. VoilĂˇ. You are a modeler. You also reap the added bonus of making clear to yourself and everyone else what assumptions you have made. Of course mathematical modeling is more complicated and sophisticated, but this is the basic idea.

Maybe you don’t like the fact that you have to make explicit a lot of assumptions. How are you supposed to know how likely it is to pass on the disease? How do you know how many contacts someone will have and how close they are? What if the people he comes in contact with are immune? If having to make these assumptions makes you queasy or uncomfortable, you might ask what your alternatives are. Should you just say, “What if an infected person gets on an airplane. Oh my God!”? You could, and many people just stop there. But experience in all branches of science shows we can do better. Consider a spherical cow.

Mathematical models are now being used in epidemiology to answer questions that can’t be answered any other way. It’s not much different than asking what would happen if you dropped a cow out of an airplane. It’s probably never been done but we can still make some qualitative predictions — for example, that the cow will eventually reach a steady velocity. We can probably even make a pretty good guess what it would be by just assuming the cow is a sphere. Many epidemiologists distrust mathematical models (just as many ordinary people are coming to distrust epidemiological studies), but our view is that both modeling and epidemiology (and statements about Weapons of Mass Destruction) should and can be critically evaluated to see when they are useful and for what.

In subsequent posts we’ll look at one of the current models on antiviral resistance in a bit more detail. When we are done, you may even conclude that it is useful.

Table of contents for posts in the series:

The Introduction. What’s the paper about?

Sidebar: thinking mathematically

The rule book in equation form

Effects of treatment and prophylaxis on resistance

Effects of Tamiflu use and non drug interventions

Effects of fitness costs of resistance