[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 begin our look at the mathematical part of the model in the paper, “Antiviral resistance and the control of pandemic influenza,” by Lipsitch et al., published in PLoS Medicine. The main model is presented in the first four paragraphs of the Methods section.
The Model: view from 50,000 feet:
The description of the basic model begins in paragraph 2 of Methods. Its essential elements are displayed in Figure 1 (page 3 of the paper and below). It contains all the elements in equation 2 on the lower right of page 3, so if we can get through the figure we are almost done. Since this is the heavy lifting, we’ll spend several posts on this figure:
The model used by Lipsitch et al. is an example of a homogeneous population model, also called a random mixing model (see previous post; it is also frequently called a “compartmental model”). Consider a population of uninfected (“susceptible”) people. The number of susceptible people on some particular day or week is designated by the upper case letter, X. We are starting to do mathematics, using a symbol (“X”) to stand in for something we are interested in, the number of susceptible people at a particular time. From now we will assume the counting is done per day. Thus if you are modeling spread in a town, X is the number of as yet uninfected (susceptible) people in the town on that day.
The virus is in town, too, in the form of infected people (infected people are also considered to be infectious to others, so we could also call this group infectious as Lipsitch et al. do). The number of infected people in the town an a particular day we designate with the upper case letter, Y. After a suitable time, infected people either die or become immune. In this kind of model they are called Removed (or sometimes Recovered, but since they may die as well as recover and become immune, we prefer Removed). Lipsitch et al. designate their number on a particular day with the letter, Z. Since there are now three states a person can be in — Susceptible, Infectious/Infected or Removed — this is often called an SIR model.
If you look at the figure you can see the boxes for X (susceptibles) and Z (removed), but the box for Y is split into three different boxes, each with subscripts, ST, SU and R. Lipsitch et al. assume infected people fall into three sub-categories, those infected with virus sensitive to Tamiflu and treated with the drug (ST), those infected with virus sensitive to Tamiflu but untreated (SU), and those infected with virus resistant to Tamiflu (R). When I say “Tamiflu” I mean “antiviral.” There is nothing specific to Tamiflu about this model. The use of Tamiflu to designate antivirals is my choice, for pedagogical purposes, and doesn’t mean this model only refers to that particular antiviral.
What it means to “run” the model:
So now we have five boxes, one each for the susceptible and removed people on a particular day and three boxes for people infected with sensitive virus, treated and untreated, and resistant virus. The epidemic develops as people move from a susceptible state to the infected state and then to the removed state, peaking when the sum of the Y boxes is at its maximum and ending when the population is mostly in either the X box or the Z box (few or no one in the infected state). As each day passes, new people become infected and treated (or not) and each day some of the infected get better or die (become “removed”). “Running the model” means to watch, day by day, as people move from one box to another.
In order to make the model run we need some rules that tell us how to move people between the boxes each day. If we call the number of susceptible people on day 12, X(12) and similarly for the other categories, the rules typically look like this: on day 12 (for example), if you have X(12) people susceptible people and YST(12), YSU(12) and YR(12) infected people, then on day 13 you will have X(13), YST(13), YSU(13) and YR(13) and Z(13) Removed people.
To get the ball rolling, we’ll start with one person infected with an influenza virus sensitive to Tamiflu on day 1. Everyone else is susceptible. So if you have your rule book and you know how many people are in each category on day 1, it’s just a matter of letting the computer grind it out.
The major modeling task, then, is to write the rule book. The underlying principle of the rules will be the Law of Mass Action (see part IV) and we will get to them straightaway in the next post.
Table of contents for posts in the series: