[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’ve spent a long time in the previous posts looking inside the black box of a mathematical model for the spread and consequences of antiviral resistance to Tamiflu (described in the paper by Lipsitch et al., published in *PLoS Medicine*). From the last post you will recall the authors assume it is rare to have the emergence of a spontaneous mutation, resistant to Tamiflu but spreading as easily as the sensitive strain. In addition, consistent with data on seasonal flu, they assume prophylactic use is less likely to produce a spontaneous resistant mutant. For illustration they have chosen a rate that is ten times less frequent among the prophylaxed than the infected. In other words, there is a big difference in the emergence of resistance according to whether the person was receiving Tamiflu before or after infection, but in both cases it is very rare. The switch to resistance, when it happens, occurs immediately upon infection. What does “rare” mean? The emergence of resistance is .02% (two per 10,000) among people who were prophylaxed and then become infected anyway (the virus mutated immediately to resistance) and .2% (2 per thousand) among infected people treated with Tamiflu. While this is much lower than emergence of resistance in seasonal flu, the known resistant mutants are genetically impaired. What would happen if they weren’t? Would even rare emergence lead to rapid spread of resistance? The answer seems to be, “yes.” But as always, there are important details to consider.

The **Results** section is divided into five subsections: effects of treatment and prophylxis on the emergence of resistance; effects of antiviral use on the size of the epidemic; effects of non-drug interventions (e.g., social distancing, etc.); effects of resistance on the size of the epidemic; and the effect of resistance fitness costs and other factors. We will go through each section.

The spread of resistant strains with prophylaxis and treatment:

Figure 2 (top of page 5 of the paper in .pdf format) shows what happens when we push the button and run the model described in the paper. It is a bit complicated so it will require explanation. Here is the figure from the paper:

As soon as resistant strains appear, whether in the prophylaxed or treated population, they begin to constitute a larger and larger proportion of the infections because Tamiflu is preferentially impeding the spread of the competing sensitive strain. Initially it is the spontaneous appearance of resistance in a person originally infected with the sensitive strain that contributes most, but as more resistant mutants arise they begin to transmit and eventually most resistant cases are from a resistant virus a person got from another person (i.e., through spread rather than mutation). Here are the details.

There are two parts to the figure, (A) through (D), the top row, and (E), (F) and (G), the bottom row. The top row has time, in days, running along the horizontal axis, so it is telling us what happens as the epidemic unfolds. The entire scale is a bit over a year’s time (400 days). There are also three graphs in each panel, a black one and two colored ones (red and green).

We’ll start out at the right end of the top row. Figure 2(D) shows what happens when you don’t use Tamiflu at all (i.e., neither treatment nor prophylaxis with Tamiflu, labeled above the graph, “Neither”; f_{p}=f_{T}=0 to indicate zero percent of the susceptible population is being treated and zero percent receiving prophylaxis). There are two vertical axes in the graphs in the first row, the vertical axis on the right being the scale for the black graph and the vertical scale on the left the readings for the red and green graphs. Let’s do the black one in Figure 2(D) first . The black graph represents the proportion of the susceptible population that is uninfected. Since we start out with everyone uninfected, you can see that for the first three weeks or so the population accrues new cases very slowly. Since you read off the proportion infected on the right scale, it starts off at 1.0 (i.e., 100% uninfected) and stays close to there for a short span, then drops suddenly as the infection spreads rapidly, eventually reaching about .35 of the population uninfected, or an attack rate of 65% (.35 uninfected means .65 or 65% infected). At about 80 days the epidemic has burnt itself out by having enough people become “Removed” (got well and are immune, or died) and insufficient susceptibles so that the disease has a hard time spreading. In essence, the number of new cases each infected case produces has become less than one and the epidemic peters out. You can see the sudden rise of people infected with the virus in the green hump, which peaks at about 60 days (I am eyeballing the graph for these numbers). You should also be able to see why the hump coincides in time with the sudden drop in the black line. As people become infected the proportion of uninfected must drop and that is exactly what happens. When no new infected people are appearing (end of the hump) the black line levels off, i.e., it doesn’t drop any more.

The other three graphs on the top row show what happens (in the model!) when you either treat with Tamiflu (A), use Tamiflu only prophylactically (B), or do both (C). With the parameter assumptions of this model, prophylaxis blocks transmission of the sensitive strain more effectively, even though resistant mutants arise more rarely. You can see what happens in the top row, panels (A), (B) and (C). With treatment alone, (A), the peak of the epidemic is slightly delayed (the flat beginning of the black curve is prolonged) and the attack rate is lower (now slightly over 50% instead of 65%). With prophylaxis alone, (B), it is delayed still more and the attack rate is now just over 40% (remember you get the attack rate by subtracting the intersection of the black curve with the right vertical axis, from one). While you do better with prophylaxis only on delay of the peak and attack rate, you now have more resistant virus (red hump). Panel (C) shows what happens when you use Tamiflu for both prophylaxis and treatment. The peak of the epidemic is markedly delayed but the attack rate is back up to over 60% and virtually all of the infecting strains are resistant. You have bought a substantial amount of time but at the cost of more infections and predominantly resistant virus.

Resistant infection from spontaneous mutation versus spread from another person:

Panels (E), (F) and (G) in the bottom row show a slightly different view of what is going on (be aware there is a typo in the caption; in describing the curves, the colors black and orange were inadvertently reversed. I have confirmed this with the author). The black lines now represent the proportion of new resistant infections that come from spontaneous mutation (i.e., that start out as sensitive in the infected person but mutate to resistant at the outset of infection). The orange curves are the proportion of all infections that are resistant. The three panels correspond to the three cases above them: treatment only, prophylaxis only, and both (there is no “Neither” panel because these curves are only about resistance). Under all three scenarios, almost all resistant infections arise initially as spontaneous mutations (the black curve is up around 1.0, or 100%), but with treatment only, (A), this drops rapidly and soon most new infections with resistant virus come from spread from another person. The proportion of all viruses that are resistant is relatively small, however (the low orange curve). With prophylaxis only, (F), infection with resistant virus via spread from another person is delayed, but when it happens, a larger proportion of all infections are with resistant virus. (The orange curve effects are also visible in the top row by comparing the heights of the green (sensitive) and red (resistant) curves in panels (A) and (B).) When both treatment and prophylaxis are used, (G), for a long time infections from spontaneous mutations are the predominant means of acquiring the few resistant infections, but again, there is a sudden switch as the epidemic takes off and soon all infections are resistant and they are almost all acquired from another person.

Thus the model shows that when Tamiflu is used and fully or almost fully competent resistant virus arises by rare spontaneous mutation, the peak of the epidemic is markedly delayed but at the cost of almost all the circulating virus being resistant. You do better on overall attack rate if you only use Tamiflu for prophylaxis, but you have less time to prepare. With *all* of the Tamiflu scenarios you do better on attack rate, but you produce resistant virus in the process. Very low rates of spontaneous mutation can spread rapidly through the population in a pandemic.

Remember there are a lot of knobs that can be turned on this model. What we have seen is what would happen if we left all the knobs fixed at certain plausible positions and then used Tamiflu in several different ways. In effect, we have run a controlled experiment, keeping all things constant except what we were interested in, the effect of Tamiflu on the time to the peak of the epidemic; and the proportion of infecting virus that is resistant. This isn’t different, in principle, than what is done at the bench in a lab experiment with mice or ferrets. There, too, you have set knobs at certain plausible positions (e.g., at “ferret”, “intranasal inoculation,” “specific dose,” “specific strain,” etc.) and you want to know what happens when you treat or don’t treat ahead of time with Tamiflu or a vaccine or some other variable you are interested in.

In the next post we’ll move on to the second part of the **Results** section, the effect of different intensities of combined prophylaxis and treatment on the size of the epidemic, as measured by attack rate, the proportion of the population eventually infected).

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