[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.]

We have now gone through the entire paper on modeling the impact of antiviral resistance in an influenza control program, by Lipsitch et al., published in *PLoS Medicine*. Since a number of assumptions were made, we take some time to consider what effects they have on the model’s results. In the Supplementary material available separately online, the authors consider a number of them.

The assumption that some of those whose infections are blocked by prophylactic Tamiflu remain susceptible (i.e., have no effective immune response), has only a modest effect on the extreme contrary assumption, that they *all* become immune. As far as this assumption is concerned, the model slightly over estimates the attack rate, which makes its policy implications conservative. The authors also say in the Supplement (without displaying the results) that assuming Tamiflu treatment only reduces transmission and not length of illness has only a small effect on the attack rate, i.e., extending the effect of Tamiflu to simultaneously reduce transmission and shorten disease duration doesn’t matter much to the results.

More important are the assumptions about randomness inherent in the Law of Mass Action. Mass Action is the theoretical engine that drives spread in the model. The authors discuss a fairly subtle aspect in the Supplement and we will discuss the overall use of the Mass Action principle. The aspect Lipsitch et al. address relates to a possible correlation between receiving prophylaxis and having contact with a treated infective case. The model assumes there is no relationship, i.e., that whether you receive prophylaxis or not is independent of whether a contact with an infected case was treated with Tamiflu or not. But if treated or previously prophylaxed infected contacts are people who, for one reason or another, have better access to resources (and hence get treated with Tamiflu), their susceptible contacts might also be people more likely to be prophylaxed (e.g., if they were family members of a known case or people with the same or similar social circumstances). Taking this kind of correlation into account requires a structural modification of the model. If there are such correlations, treatment is not going to those in most need (contacts of unprophylaxed cases), so the amount of antiviral needed for outcomes equivalent to the original model is increased, but is otherwise qualitatively the same.

The major assumption underlying the model, however, is the Law of Mass Action itself, which we discussed in some of the earliest posts. It is quite clear that although the assumption makes mathematical analysis easier, it is not a realistic description of what happens in real populations. I do not have an equal chance of being a contact of someone from New Zealand as I do of someone who lives in the same city as I do. Our contacts are not random. Is this a fatal flaw? The answer depends to some extent on what you hope the model will do.

Not all members of the susceptible population is equally likely to be infected upon contact with a transmitting case. It may also be true that some people are much more efficient spreaders (“superspreaders”). Either way, in real contact networks it is usually true that a few people have large numbers of efficient contacts but most have few. So even when the average contact rate, R_{0}, is the same in two communities, the pattern and rate of spread might be quite different if we assume the average number of contacts per person is random and equal or if we assume a more realistic and varied contact network. It is an important area of modeling research to isolate the the role of contact patterns to see its effect. The paper by Colizza et al. we mentioned in the preliminary post (part “zero”) examines the effect on global spread of assuming a particular topology for the air transport network. In modeling, as in wet bench experimentation, researchers take a step at a time.

So does the unrealistic assumption of Mass Action invalidate the analysis we have taken such time and effort to explain? It depends on what weight you want the analysis to bear. If you require accurate quantitative estimates, the more conventional Mass Action model will tend to overestimate the spread and attack rate. But if what you are interested in are potential behaviors of a virus spreading in the context of antiviral and other barriers to its transmission, the paper by Lipsitch et al. provides important and valuable information. In this sense it is no different than studying transmission of reassortant H5N1s from one cage of ferrets to another or one cage of mice to another. Both are artificial settings and neither involve humans. But we still can learn about how different genetic endowments of the virus affect transmission, all other things being equal.

In the last post, then, we summarize what lessons to take away from this modeling exercise.

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