[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 continue our examination of the paper, “Antiviral resistance and the control of pandemic influenza,” by Lipsitch et al., published in *PLoS Medicine*. We have gotten to the point where our population is divided into five categories. We follow the course of the epidemic on each of its days by examining how many people are still susceptible (denoted by X), how many are infected with sensitive virus and are being treated with Tamiflu (Y_{ST}), how many are infected with sensitive virus but are untreated (Y_{SU}), how many are infected with resistant virus (Y_{R}); and how many have either died or recovered and are immune (Z). These last are called “Removed” because they are no longer part of the process of viral spread. I think most people would agree that if we had an accounting of these five categories, day-by-day, we’d also have a pretty good idea of the progress of the epidemic, including the spread of antiviral resistance. Here’s the strategy from now on.

How we run the model:

We agree that if we stipulate how many people are in each of these categories on some particular day, this will determine how many there are on the next day. We can do this with a “rule book” that allows us to look up the in each of the five boxes on a specific day and gives us the numbers in each box for the next day, etc. If we start off with everyone uninfected on day zero and introduce a single, untreated person infected with a virus sensitive to Tamiflu on day one, we can use the rule book, day-by-day, to follow the epidemic. To spell this out, we look up the value for day 2 when day 1 has X-1 susceptibles, Y_{ST}=1 untreated people infected with a sensitive virus and the other Y boxes and the Removed boxes are initially empty. We do the same for day 3, using the value we got for day 2, etc. That’s all there is to running the model. All we need is a rule book. The rule book is the key. It “is” the model.

The Rule Book:

The rule book is based on the Law of Mass Action (see earlier posts). We’ll spell this out, as it isn’t intuitive. Suppose there is a number, Y, of infected people in town. If your chance of encountering any of them is roughly the same, then the more of them there are, the greater your chance of an encounter. How does that chance increase as the number of infected people increases? Maybe the change is related in some way to the square of the number of infected people plus the cube of the number all of this divided by some other number or even something much more complicated. There are infinite number possibilities for this. The Law of Mass Action picks one, just about the simplest possible. It says that for every new infected person we add, we also add some fixed chance of being infected for each susceptible in the population. Said another way, the per person chance of infection each day is proportional to the number of infected people on that day. Note that if there are no infectious (infected) people (they all got well or died), the risk for the susceptibles becomes zero and no new infected people will be added. Likewise, if there are no susceptible people, left the risk will be zero. In both cases the epidemic is over.

Is picking the simplest such relationship risky? It depends. In this case, no. That’s because the amount of added risk for a susceptible person when there is one more infected person around (called the transmission rate constant and designated by the Greek letter, β) is very small in this model. The transmission constants, β, are all on the order of a ten ~~thousandth~~ millionth [hat tip bar] or less (you can find values in Supplementary Table S1, online; they were not picked out of the air but obtained from a study of social contacts and influenza spread in The Netherlands). Under such circumstances, even very complicated relationships “look” proportional. The surface of the earth, which we know is curved, still looks flat to us. For getting around in ordinary life we don’t go very wrong by assuming it is flat. Similarly, here. Tiny β’s confine us to a local area that “looks flat” no matter how complicated the big picture is. This is a very common technique in applied mathematics. It works well enough to be able to put a space ship precisely at a certain point over Mars despite a journey of many million miles.

Using the Rule Book:

So the rule is that the average per person risk of infection of a susceptible on a particular day is β times Y (the number of infecteds), and since there are X susceptible people on that day, the total number of new infected people is β times Y, X times (written βXY). That means there will also be βXY fewer susceptibles the next day, since we assume the total number of people in town isn’t changing appreciably over the course of the epidemic. The numbers change day to day as the susceptible number decreases and the infected numbers increase by this amount and decrease by the number that get well or die. In essence, people are flowing through the boxes (or compartments) from X to Y to Z. When the flow stops, the spread is over. At that point there will be none or few in the Y boxes or compartments. All the people will be in the X or Z compartments.

Moving on, using the Mass Action principle, the number of people infected on a particular day is the number susceptible on that day (X) times the number of infecteds in each Y category times the transmission rate constant, β. But we wouldn’t expect this transmission rate to be the same if the contact was between a susceptible person receiving prophylactic Tamiflu versus one receiving nothing (one of the purposes of prophylaxis is to decrease transmission, after all); or between either kind of a susceptible person and someone infected with the resistant strain (one of the ideas is that resistant strains might be less fit and hence less transmissible); or someone who is infected with a sensitive strain but is not being treated and a susceptible person with or without prophylaxis. I’m sure you are now confused by all the possibilities, but we need to keep track of them, and the figure shows us how.

Deciphering Figure 1:

Since there are three flavors of infected people (Y_{ST}, Y_{SU}, Y_{R}), we need three transmission rates, (β_{ST}, β_{SU} and β_{R}). So the number of new people infected with a sensitive strain produced by a susceptible person encountering one of the Y_{ST} or Y_{SU} people will be β_{ST} X Y_{ST} + β_{SU} X Y_{SU}; and the total number of new people infected with a resistant strain will be β_{R} X Y_{R}. These people will leave the X compartment and head to one of the three Y compartments.

You can see this in the figure where two arrows lead out of the X box and are labeled with the terms we just set out (not exactly; the X term is left out to simplify the figure because it is common to all of the arrows out of the X box). One of the arrows comes out of the middle of the right side of the X box, the other from the middle of the bottom of the X box. The sensitive virus encounters have been combined into one arrow because the same thing happens to each of the terms in the next step. Lipsitch et al. could have made the two terms into separate arrows, but at the risk of complicating the figure.

If you followed it so far, you shouldn’t have any trouble with the rest of it because we will be doing the same thing. This will also take care of equation (2), which is just a repeat of the figure without the boxes and the arrows. We’ll be doing it for each of the arrows to give you lots of practice in seeing what’s going on.

On day 1 there aren’t any resistant viruses, so we will have to account for how they arise, and we will have to make some assumptions on the proportion of the population that receives prophylactic Tamiflu and the proportion of sick people who get Tamiflu for treatment. We’ll also need to make some simple assumptions about how long someone stays sick before moving to the Z box and being “Removed” from the process. This sounds complicated and it is, but only in the bookkeeping sense. Even if you don’t follow every single step (because your attention wanders or your patience or interest gives out), I hope you will see that the details aren’t that mysterious. We are just following the rule book and counting up people in each compartment from day to day.

We’ll pursue the details in the next post.

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