[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.]
Now we are almost ready to run the model described in the paper, “Antiviral resistance and the control of pandemic influenza,” by Lipsitch et al., published in PLoS Medicine. If you have been following up to this point, you will know the model described in the Methods section is a homogeneous population of susceptible, infected and removed people. The people were exchangeable, in an epidemiological sense. They didn’t have genders, ages or pre-existing conditions. Thus they were all equally susceptible and the infected ones in each of the categories had the same degree of infectiousness (although this differed between the categories).
Adding age to the model:
Of the factors omitted, age is probably the one most likely to make a difference on a population basis. So while the methods we have set out in the previous posts described a homogeneous population, the actual model as run on the computer divided the population into six age categories, each with its own population size, its own risk of getting infected and its own risk of passing it to others. Infection can move between age classes, each with specified rates. The numbers used for all of this were not just “made up” but taken from a study of social contact patterns in The Netherlands. The details are not given in the paper itself but in an online Supplement whose links are given with the paper itself at PLoS Medicine. The essence of this refinement is to had a whole lot more boxes to the model, each with a lot of arrows leading in and leading out. So it is not different in principle from the model described in the paper and the preceding posts, just a lot more bookkeeping. That’s a nasty problem for human beings, but trivial for computers.
Lipsitch et al. make the observation that the results using the more complex, age-structured model aren’t very different than the one with a single box for each category, except for a slightly lower attack rate (i.e., fewer people eventually infected in the entire population). This tells us that omitting the age structure wouldn’t have made much difference and whatever difference it would have made would be to slightly overestimate epidemic spread in this model. The results in the paper are from the age structured model, so you needn’t worry abut this, but we observe that this is quite typical in modeling efforts. We often find that adding lots of refinements and complications doesn’t change things very much. The basic logical structure of the model is the prime determinant. Unfortunately it is possible for some kinds of changes to make a big difference, so it is common (and good) practice to test the sensitivity of the model to the various assumptions that have been made. This is usually called a sensitivity analysis. In this particular case the relatively small difference from the reduced model allows the authors to use the equations (2) on page 3 for a mathematical analysis to draw certain qualitative inferences about how the model behaves. This helps them interpret the numerical results produced by the computer runs of the age structured population. They get the best of both worlds. The mathematical analysis is also found in the Supplement, along with the computer code and other details needed if you were going to replicate the simulation.
What is a model “parameter”?
In the course of constructing the model, there were a lot of assumptions made about how easily the virus would be transmitted, the average duration of illness, the fraction of susceptible people who would receive prophylaxis, etc. These assumptions had Greek and Roman letters attached to them and wound up on labels in the arrows in Figure 1 and in equation 2 (see last post). These quantities are referred to as “parameters” of the model, and they can be changed in the computer code, so if you think one or another is unrealistic or you want to see what happens if one is changed, it is easy to do by tweaking one or more parameters and running the model again. This is one of the main ways a sensitivity analysis is done, too.
The emergence of resistance:
Now it’s time to use the model to examine the question at hand, the consequences of the emergence of a strain of virus fully resistant to Tamiflu as a pandemic develops. Lipsitch et al. will assume that a fully or almost fully competent strain (i.e., little or no fitness penalty) is possible, but happens rarely, and that it happens more often in infected people treated with Tamiflu (.2%) than in people who receive Tamiflu prophylactically (.02%). Both of these rates of emergence of resistance are more than ten times lower than the emergence of resistance seen in clinical trials, but those viruses have high fitness costs, i.e., they do not replicate or transmit well. They can make a person very sick or kill them, but they don’t spread as well in the population. The fitness cost of the rare variant is 10% for illustration. A fitness cost of 10% means that the resistant virus is 10% less able to transmit than the sensitive virus, not much of a penalty, but the authors also examine costs that go from none (0%) to 40%.
Because treatment and prophylaxis will stop the sensitive viruses from spreading, they ask if even a very rare rate of emergence of resistance spread quickly through the population, and if so, what would be the consequences. The answer is “yes,” very rare mutations can spread through the population, and the consequences are quite interesting.
We are now ready to push the button on the age structured model and watch what happens to antiviral resistance, its spread, and the consequences for the population. That will start in the next post.
Table of contents for posts in the series: