[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 need to finish the **Methods** section of the mathematical model in the paper, “Antiviral resistance and the control of pandemic influenza,” by Lipsitch et al., published in *PLoS Medicine*. Then we can move on to **Results**. In this post we will deal mainly with paragraphs 3 and 4 of **Methods** on page 3 of the paper. We hope some of you are still following along at home. This is a long post because we have spelled everything out in excruciating detail and we wanted to finish the whole figure. It isn’t as bad as it looks, but if you have trouble with it or don’t want to take the time, just get the general idea. In the next post we’ll summarize the rule book (and take care of equation (2)) and then we can talk about **Results**. You won’t need the details of the model to understand them.

Here’s the figure again:

We turn first to the fraction of the susceptible population lucky enough to have received Tamiflu prophylaxis in the hope they can either avoid becoming infected at all or even better, become infected enough to become immune but not become symptomatic or infectious. The fraction who receive prophylactic Tamiflu is denoted by f_{p}, and is set at 30% for illustrative purposes. Let’s talk about what prophylactic Tamiflu does to whether the person will transmit the virus to others. If you are exposed to a resistant strain, prophylaxis is assumed to have no effect on your probability of infection. Thus the number of new cases infected with resistant virus daily is the bottom arrow in the diagram labeled β_{R} Y_{R}. It really represents β_{R} X Y_{R} new cases infected with resistant strain, i.e., a per person probability of β Y_{R} times the number of people in X. Again, this is the Law of Mass Action at work, as detailed in previous posts. So for this arrow, prophylaxis is irrelevant. What about people infected with virus sensitive to Tamiflu? They are represented by the star like set of arrows in the middle of the diagram emanating from the left of the two points between the X box and the three Y boxes. There is another star-like figuration to the right of it with one arrow leading in from the first star and three leading out to the Y boxes. We’ll deal with that one after we do the one on the left.

Leading *into* the center of the left star is one arrow coming *out* of the X compartment, labeled β_{ST} Y_{ST} + β_{SU} Y_{SU}. That label represents the rate of new cases daily coming from exposure of susceptibles (X) to people infected with virus sensitive to Tamiflu, some of whom have been treated with the drug (Y_{ST}) and some of whom have not (Y_{SU}). The model allows the probability of being infected to be different in those two cases, with rates β_{ST} and β_{SU}. Since these are per capita rates, the actual number of new cases infected with the sensitive strain each day would be (β_{ST} Y_{ST} + β_{SU} Y_{SU}) times X. All the arrows in this star are multiplied by X to get the daily number of cases, so we will omit mentioning it each time, instead just dealing with the arrow label itself (which also omits it).

Now that we have done the single arrow leading *into* the center of the star (connecting the right hand side of the X box), we have to do the five arrows coming *out* of the center of the star. All of these new daily cases have either received prophylactic Tamiflu or not. If f_{p} of them have been prophylaxed, then 1-f_{p} have *not* been prophylaxed (for example, if 30% have received prophylaxis, then 70% have not; 30% is the same as a proportion of 0.3, 1-.3 =.7, or 70%). That’s the label on the arrow coming out of the center at the 3 o’clock position. It really represents a factor of (1-f_{p}) times what went in, which was (β_{ST} Y_{ST} + β_{SU} Y_{SU}). So now we have unprophylaxed susceptible people getting sick every day at the rate of (β_{ST} Y_{ST} + β_{SU} Y_{SU}) (1 – f_{p}). Only the extra factor of (1-f_{p}) is written on the arrow to make the figure easier to read.

So that takes care of the in-arrow and one out-arrow (the one at 3 o’clock). Now we need three other assumptions to take care of the other three out-arrows. We assume that a susceptible person receiving prophylaxis will have a lowered risk of infection with the sensitive strain. That’s the point of prophylaxis, after all. Call the fraction by which risk of infection is lowered, e_{p}, i.e., the probability of infection of those prophylaxed is reduced by a fraction e_{p}. Since the fraction of those prophylaxed is f_{p}, the probability of infection of the prophylaxed is now f_{p} e_{p}. Lipsitch et al. consider two ways that the risk of infection can be changed by prophylaxis. A proportion a_{p} are partially blocked in the sense that they don’t get sick or infectious but become immune. That moves them immediately to the Z compartment, and you can see them in the 2 o’clock arrow arcing over the diagram and going to Z. Its label is f_{p}e_{p}a_{p} because it is the fraction partially blocked, a_{p} times the fraction of lowering of infectivity, e_{p} times the fraction of the total susceptibles prophylaxed, f_{p}. Now if a fraction a_{p} of the blocked have partial blockage of infection than the remaining fraction, that is those completely blocked, is 1-a_{p}. In these people the blockage is so complete that they don’t even get sufficiently infected to mount an immune response, i.e., they stay susceptible. That’s the 11 o’clock arrow arcing back to the X compartment, now labeled f_{p}e_{p}(1-a_{p}). Two more arrows to go in the left star. Remember that prophylaxis has no effect on risk of infection from a resistant strain (the bottom arrow leading from X to Y_{R}).

To get the other two arrows from the left hand star we need one more assumption, one that is at the center of the model’s purpose. The authors assume that a fraction c_{p} of people receiving prophylaxis will have a virus that undergoes an immediate spontaneous mutation to resistance. That’s the arrow labeled f_{p}c_{p} coming out of the star at about 4 o’clock and going directly to the Y_{R} box because those people now are infected with virus resistant to Tamiflu. In f_{p}(1-e_{p}-c_{p}) of the cases prophylaxis fails. The f_{p} is the fraction prophylaxed. The fraction by which transmission is blocked (e_{p}) plus the fraction that have a spontaneous mutation among the prophylaxed (c_{p}) represent people that have either blockage or mutation (you cannot have both at the same time, since blockage only refers to sensitive strains), so (1-(e_{p} + c_{p})) = (1 – e_{p} – c_{p}) is the fraction of the prophylaxed where there is neither blockage nor resistance. Thus that arrow points into the Y_{ST} box, because the use of prophylaxis means these people are also receiving Tamiflu and are in the same category as infected people receiving Tamiflu.

Don’t be discouraged if you are having trouble following this. I had to really concentrate and make notations on the figure to be sure I was keeping track of everything. If you bother to do this you will find it all makes sense. If you don’t bother, at least you know in some general terms the kind of thinking that went into it.

Finally we have to do the right hand star, the one smack in the middle of the figure. That requires yet another assumption, the proportion of infected people treated with Tamiflu. The authors call this proportion, f_{T} (we already noted in the last post the minor error in the first sentence of paragraph 4 of the paper where this is given as f_{p} instead of f_{T}). We have previously dealt with the incoming arrow (at 9 o’clock of the right hand star, coming out at 3 o’clock of the left hand star) which represents all the people infected with a virus sensitive to Tamiflu that have received no prophylaxis (a fraction 1-f_{p} of those infected with a sensitive virus and so labeled). Since f_{T} of these will be treated, a fraction (1-f_{T}) will *not* be treated and they go directly to the Y_{SU} box, non-prophylaxed people infected with a sensitive virus who aren’t treated. So that takes care of the arrow at 3 o’clock. Two more arrows to go (1 o’clock and 5 o’clock).

The last assumption we need is the fraction of those previously unprophylaxed (that fraction was (1 -f_{p}) who got treatment (a fraction f_{T}) and upon being treated their previously sensitive virus spontaneously changes to a resistant virus. That fraction is denoted c_{T} and accounts for the two arrows labeled f_{T}c_{T} and f_{T}(1-c_{T}), the latter being those not previously prophylaxed but treated whose virus remains sensitive.

We have now accounted for all the arrows in Figure 1 on page 3 of the paper. Next post we’ll put it all together and then move on to what happens when we let this model run.

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