[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 explain the remaining results presented in the paper by Lipsitch et al. in published in *PLoS Medicine* (the subsections headed, “Effects of resistance on epidemic size” and “Dependence of outcomes o fitness cost and intensity of control” on page 6).

These sections and the accompanying figure 4 require some careful reading. The subsection titled “Effects of resistance on epidemic size” has two paragraphs, but only the first of them relates to epidemic size, the second being on timing. The first paragraph reiterates what we saw previously in figure 3A, that even though use of Tamiflu promotes the spread of resistant virus, the attack rate is reduced at all levels of Tamiflu use. The reason is two fold. The first is that the authors have assumed a modest fitness cost for the resistant virus (10% in the illustration). They relax this in the next section. The second reason is that Tamiflu interferes with spread of the sensitive strain.

Here is the figure, which we will explain in detail:

By slowing the sensitive strain, we also delay the epidemic peak (already noted in our discussion of figure 2). Because the authors assume that almost fully transmissible resistant strains will be rare, resistant spread is slow at first. The effects come from Tamiflu slowing the sensitive strain. This is illustrated in the many panels of figure 4. The dashed curves of different colors in figure 4 represent the average time to getting infected for someone who gets sick during the pandemic. The scale for the dashed curves is the right vertical axis and is measured in days. For each of the dashed colored lines in all the panels (we’ll talk about what the different panels are, shortly), you see that as you increase the intensity of Tamiflu use for prophylaxis and treatment (the horizontal axis) the average time from the beginning of the epidemic to infection for someone who has gotten sick increases rapidly. The different dashed lines represent different effective reproductive numbers of the virus, which can be interpreted here as different intensities of non-drug interventions like social distancing. The least intervention is the black line (R=2.2) while the greatest intervention is the green dashed line (R=1.2). In every instance the curves go up rapidly as you increase Tamiflu use, meaning that the average time to getting sick increases rapidly, i.e., that the epidemic is being slowed. While it is true you have spread resistant virus by Tamiflu use (see here and figure 3), you may have bought valuable time in terms of producing a vaccine and spreading out demand for health services. However the benefits of Tamiflu use during the first season may no longer be available during a second wave if the circulating virus is now resistant. You are certainly better off than if you had not used Tamiflu at all, since lack of use is the same thing as use against a fully resistant virus.

Tamiflu use and no resistance:

If there were no resistance (c_{p}=c_{T}=0) you could slow or stop a pandemic with sufficient use of Tamiflu. Figure 4A shows this. Each of the solid curves represents the attack rate (proportion of the population eventually infected) for a sensitive virus with different effective reproductive numbers. Again, you can consider these varying curves as representing different intensities of non-drug interventions, the lower numbers of R representing *more* intervention. In each case, ramping up the intensity of Tamiflu use reduces the attack rate (the scale for the solid line attack rates is now the left vertical axis). Likewise, the average time for a case to have gotten sick increases rapidly (the dashed lines, right vertical axis).

But we know that resistance *does* occur, although so far the resistant mutants have been less capable than the sensitive ones. Previous sections of the paper showed that if capable resistant mutants arise, even very rarely, they can spread rapidly in the population. The last section of **Results** asks about the influence of fitness cost of the resistant virus. Figure 4A is the comparison panel, the case where there is no resistance at all. Increasing Tamiflu use reduces the attack rate and delays spread. If you use enough you can snuff out the epidemic.

Tamiflu use and rate of emergence:

But once resistance arises (panels 4B to 4G) wiping out the epidemic with Tamiflu use is not possible except when the intensity of non-drug intervention is so strong that it reduces the effective reproductive number of both the sensitive and resistant strains below one when Tamiflu is added (green curves of 4B, 4C, 4D, 4F and 4G,, and blue curves of 4D and 4G where you see the attack rate, left vertical axis, eventually goes to zero with high Tamiflu use). Note that non-drug interventions continue to reduce attack rates and delay spread, the extent depending on the cost of resistance (columns of panels, described next). The second and third rows of panels represent what happens when the appearance of resistance is very rare (row 2, 2 per million for prophylxed people and 2 per 10,000 for treated people) or quite rare (row 3, 2 per 10,000 and 2 per 1000).

Tamiflu use and fitness costs:

The vertical columns represent the fitness costs for being resistant, with the leftmost pair (4B and 4E) being fully capable (no cost to the virus for being resistant) while the rightmost pair (4D and 4G) paying a 40% cost in transmissibility, more like what is seen with current resistant mutants. It is noteworthy that in all cases there is a reduction in attack rate with Tamiflu use (all parts of each of the solid curves is below its starting point at zero Tamiflu use on the horizontal axis) but when there is no cost to resistance and most of the virus is the resistant strain (the solid black lines in 4B and 4E to the right of 0.3 on the horizontal axis) there is very little reduction in attack rate. The authors say (last sentence in the opening partial paragraph on page 7 of the .pdf version of the paper) that for no fitness cost and high transmissibility, the delay is no more than a few months, but this appears to be not exactly what is shown, as panel 4B shows a considerable delay (black dashed line) in the case of very rare mutations (second row of panels).

Panels 4B and 4C also show the reduction in attack rate is not great for widespread use of Tamiflu if the cost to the virus of being resistant is modestly low, although the paradox of intermediate use is evident and combining it with non-drug measures makes a large difference if there is some fitness cost. Selective pressures would likely drive the proportion of resistant viruses to lower fitness costs as the epidemic evolves.

These are a lot of results from the model, and the authors attempt to summarize and interpret them in the final section of the paper, the **Discussion**, which we will examine 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