[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 the previous posts we have walked through the **Introduction** and **Methods** sections of the paper, “Antiviral resistance and the control of pandemic influenza,” by Lipsitch et al., published in *PLoS Medicine*. The **Methods** section sets out the detailed model, which is summarized in the Figure in the upper left and the equations on the lower right of page 3 of the .pdf version of the freely downloadable paper. In this penultimate post on the **Methods** section, we’ll clean up a few details and give an overview. Then, on to **Results**.

Reading the figure:

To summarize, here is how to read figure 1. Start on day zero with a population of susceptible people, all in box X. On day one seed the population with a single person, infected with a strain of the virus sensitive to Tamiflu. That person is initially untreated. Thus we start the model running with X-1 people in box X and one infected person in the box Y_{SU}, an untreated index case. On each succeeding day the rule is that you move a fraction of people from box X to each of the Y boxes and the Z box and a fraction from each of the Y boxes to the Z box by the cumulated factors on each of the arrows. Thus to get from X to Y_{ST} you traverse two pathways, one along the first arrow out of the X box at 3 o’clock, β_{ST}Y_{ST} + β_{SU}Y_{SU}, then either along the top branch f_{p}(1-e_{p}c_{p}) or the middle brach, which has two legs, giving (1-f_{p}) times f_{T}(1-c_{T}). Each of these is a fraction of the number of people in the X box on that day, so the total number of people who move is X times β_{ST}Y_{ST} + β_{SU}Y_{SU} times f_{p}(1-e_{p}-c_{p}) + X times β_{ST}Y_{ST} + β_{SU}Y_{SU} times f_{T}(1-c_{T}). Looking at the Y_{ST} box you see that it also loses people to the Z box at the rate of v_{T} per day (just the rate is on the arrow label) for a total of v_{T} times Y_{ST} people per day.

The figure and the equation are the same:

The quantity (β_{ST} Y_{ST} + β_{SU} Y_{SU}) is clumsy to write over and over again, so the authors abbreviate it in equation 2 with another Greek letter, λ_{S}. You can see that abbreviation as the second last line of the equation at the bottom right of page 3. Instead of an equal sign with two bars there is one with three bars. This indicates it is a definition. A lot of mathematics just *looks* hard and the use of Greek letters often makes it look even harder. But what we have done so far is just bookkeeping, using the Law of Mass Action to move things around. Even equation (2) is just bookkeeping. Using the abbreviation, the total number of people added to and lost from compartment Y_{ST} on the next day is:

λ_{S}[f_{p}(1-e_{p}-c_{p}) + (1-f_{p})f_{T}(1-c_{T})]X – v_{ST}Y_{ST}.

I wrote all this out so you can see that these are exactly the terms in the third line of equation 2 to the right of the equal sign.

The left of the equal sign, dY_{ST}/dt, you can consider shorthand for the amount of change in the Y_{SU} box each day, the arrow going in, minus the arrow going out. The number of people each day riding the arrow going in is just λ_{S}[f_{p}(1-e_{p}-c_{p}) + (1-f_{p})f_{T}(1-c_{T})]X and the number of people per day riding the arrow going out is v_{ST}Y_{ST} (which has a minus sign n front of it because they are leaving the box Y_{ST}).

Thus the figure and equation 2 say the same thing. It is a good exercise to follow the arrows and check that each line of equation 2 (except for the last two) is just another way to write the pathways between the boxes shown in figure 1, each of which was explained in detail in the last two posts.

A little detail: equation (1):

We need to make a special comment about equation 1 and the last line of equation 2. Here the authors introduce the Greek character ξ, used to prevent tiny fractions of a resistant case from affecting the findings initially.

The authors want to model what happens if the emergence of fit and resistant viruses does occur, but only very rarely. The way the model is set up, extremely small fractions of a case would still “transmit” infection, but in real life you need at least one full case for that to happen. .001 cases (a thousandth of a case) doesn’t transmit infection. Equation 1, with its Greek letter ξ, is a standard way to prevent any transmission until the cumulated fractions of resistant cases amounts to at least one full case. That is also what the last line of equation 2 says: don’t let resistant viruses be transmitted until there is at least one full case of resistance in the population.

How to account for everything else – R_{0}:

The last detail to take care of is how to handle all the other things that are going on to manage a pandemic, like social distancing, handwashing hygiene, isolation, etc. We also don’t know how good an emerging pandemic strain will be in reproducing itself in the population. Since the object of the model is to examine antiviral resistance, all these other factors are lumped into a single component, the basic reproductive rate of the virus, R_{0}. R_{0} expresses (on average) how many new cases a newly introduced infectious case would produce in a totally susceptible population, that is, how many new cases you would expect on day 2. Many people think of this as some kind of intrinsic property of the virus, but it is really a property of the virus, the host it infects and the environment in which they live (see our entry at The Flu Wiki on this). Lipsitch et al. consider a range of possible R_{0}s (1.2, 1.5, 2.0, 2.2) and point out that you can consider a virus with an R_{0}=1.5 to be either one with an R_{0}=1.5 without intervention, or one with an R_{0}=2.2 with an intervention like social distancing. In other words, they are lumping all the non drug interventions into the size of R_{0}. More on this later.

We are now ready to move on to **Results** 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