[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 now take a look at what happens in the model when we vary the intensity of prophylaxis and treatment. The model treats the fraction of those prophylaxed, f_{p}, and those treated, f_{T}, separately, but for illustrative purposes the paper sets these two figures at the same number. In the previous subsection of the paper (discussed in post XI) they were fixed at 30% and the authors varied whether there was prophylaxis only, treatment only, both, or neither. Now we fix conditions at “Both” and see what happens when the 30% number is varied (page 5 of the paper). Now you can see why they were set equal for illustrative purposes. If you vary them separately, the number of combinations is the product of the number of different levels in each. For example, if you vary prophylaxis and treatment separately into nine levels (say, 0%, 5%, . . . ,40%), the number of combinations is 81, whereas if they are set equal it is only 9. The model can use any combination, but showing them all in the paper is another matter. In this case both prophylaxis and treatment can be shown on one axis, allowing the graphs to be drawn in a clearer fashion. Such are the constraints of presentation, always a compromise between detail and conceptual clarity.

Refresher on the basic reproductive number, R_{0}:

We need to do a quick refresher on the notion of basic reproductive number, R_{0} (also see our explanation over at The Flu Wiki). The easiest way to think of R_{0} is that it represents the average number of new cases a single infectious case would produce if all its contacts were susceptible. It is thus equivalent to what happens on day 2 of this model (after a single infectious case is introduced into the population). After the infection has spread, many contacts of an infectious case are immune (they will have had flu and recovered) or will be other infectious cases, so the *effective* reproductive rate will be less than R_{0}. When the basic reproductive rate (R_{0} or its subsequent effective equivalent) falls below one, the epidemic eventually peters out. So the intent of public health intervention is to lower the effective reproductive rate below one, the lower the better.

Antiviral use is one way to do that, but it requires the viral strain to be sensitive to the antiviral agent (assumed, here, to be Tamiflu). Antiviral use lowers the effective reproductive number for new cases infected with the sensitive strain more than for those infected with the resistant strain. This is what allows the resistant strain to predominate in the population. But the effective reproductive number of the resistant strain can also be lowered if there is a fitness cost to the resistance mutation. In other words, if the cost of becoming resistant is that the virus replicates less well, is more vulnerable to bodily defenses, or is less transmissible for some reason, then its reproductive number will be lowered even if it is resistant to Tamiflu. Whether resistance will spread (and not just predominate) will thus depend on whether the effective reproductive number of the resistant virus is greater than one.

How to account for other interventions:

There are other ways than antiviral use to influence the effective reproductive number. These are the non-pharmaceutical interventions (social distancing, isolation and quarantine, hygiene measures, etc.; we are assuming there is no effective vaccine, but a vaccine would also be a way to reduce R_{0}). So a convenient way to summarize the effects of all the non antiviral interventions is to assume varying effective reproductive numbers (which the authors designate by the upper case letter, R). The results of doing this are a good illustration how models can provide some insight into how some things you wouldn’t anticipate would happen actually *might* happen. Let’s see what these things are.

How to read figure 3:

Here is figure 3 from page 6 of the .pdf version of the paper. It displays the influence on total attack rate (the fraction of the population eventually infected), attack rate by the resistant strain, and the level of dominance by the resistant strain when using Tamiflu to treat and prophylax various fractions of the population. In each panel, one of these three outcomes is shown for four different effective reproductive rates (the four different curves in each panel).

The different effective reproductive rates (different colored curves in the figure, key on the right) are meant to summarize the combined influence of antiviral and non antiviral interventions. The horizontal axis of each panel gives the fraction of the population prophylaxed and treated with Tamiflu. The fractions considered go from 0% to 40%, in other words, no Tamiflu up to enough to treat almost half the population.

In the leftmost panel (3A) we can see that to a certain point increasing the fraction treated and prophylaxed reduces the attack rate, but above that point it increases again. If you assume the effective reproductive rate is low enough, you can snuff out the epidemic (lowest curve in 3A, R=1.2), but even here we see a transient increase in attack rate just before you use enough to do the job completely (that’s the little bump at the end just before the plummet to zero). The authors explain what is going on in the second paragraph of the subsection, “Effects of antiviral use on the size of the epidemic” on page 5. At the start of the epidemic the virus is sensitive and modest antiviral use impedes its spread without producing too much resistance before the effective reproductive number sinks below one. But as you increase the intensity of Tamiflu use, you both produce resistant strains faster and impede sensitive strains faster, so you infect the population with resistant strains preferentially over sensitive strains. If the effective reproductive number for resistant strains is higher than for sensitive strains, the attack rate will rise again. Thus in this rather plausible model, higher levels of Tamiflu use lead to a higher total attack rate than intermediate levels. That’s somewhat of a surprise at first.

The same phenomenon is illustrated in a slightly different way in figure 3C, which shows the increasing fraction of all cases infected with the resistant strain. Except for the lowest effective reproductive rate of R=1.2, by the time you are treating a quarter of the population you have pushed virtually all infections to be from resistant virus. This takes more intense Tamiflu use for the higher reproductive rates. In the model run with the settings indicated, at 25% nearly all the strains are resistant, no matter the reproductive rate. At 10% use, only a small proportion of the strains are resistant, no matter the reproductive rate. These percentage numbers should not be taken literally, because they are dependent on specific assumptions of the model. What they show is that as you ramp up Tamiflu intensity, you can have unexpected effects (attack rate can go up and the proportion of resistant infections can increase suddenly), and that these effects also depend on what else you are doing to lower the effective reproductive rate. The 10% and 25% number might be very different in a real outbreak, but there is good reason to believe the qualitative behavior would still happen at some percentages. So we have learned something.

The authors point out that if you use enough Tamiflu, you can block the pandemic by having so few sensitive strain infections (because you blocked them with prophylaxis) the epidemic fizzles. This usually requires such high Tamiflu use it is unrealistic, and even relatively small numbers of infections with sensitive virus will lead to the spread of resistance.

Some new things we have learned:

In addition to the “intermediate being better” paradox, the authors point out another feature of figure 3 that isn’t evident without the model. Remember the differing reproductive rates can either represent some feature of the virus and population contact rates or the effects of all the other non-drug interventions being applied, such as social distancing. Consider the effects of non-drug intervention, whose objective is to lower the effective reproductive rate. As you expect, under all treatment scenarios attack rates are lower with more intervention (lower R; figure 3A), and higher antiviral use leads to higher attack rates with the resistant strain (figure 3B). But figure 3C also shows that more intervention leads to higher proportions of cases from resistant strains (for 20% Tamiflu use [horizontal axis], compare the proportions from resistant strains for interventions causing an R=1.5 (blue curve), almost 100% [vertical axis], to the proportion with R=2.2 (black curve), only about 30%, keeping in mind that a lower reproductive rate represents more non drug intervention).

Viewed another way, resistant cases are a greater proportion of the total in the presence of nondrug interventions than without them [other things being equal]” (Lipsitch et al.)

Thus one of the things this model produces is a better appreciation for some of the *possible* dynamics of infection and spread of resistant and sensitive virus as Tamiflu prophylaxis and treatment change. Some of these possibilities would not be evident without the model. With the model, they become new possibilities you hadn’t thought of before, and more importantly, they prevent you from some false assumptions, e.g., that if a little Tamiflu is good, more must be better, or that preventing contact will also prevent the spread of resistance. These things may be the case under certain conditions, but are not logically required under all conditions. The model displays certain plausible conditions under which intermediate use is better than high use and shows you in which way. It also suggests that the proportion of resistant cases is inversely related to the nondrug interventions, other things being equal. These are *not* predictions of what *would* happen if Tamiflu were used in a pandemic, because setting the parameters at different values might alter the qualitative behavior. Sometimes more general statements can be made by a careful mathematical analysis of the equations. But they are notice of what *could* happen that you hadn’t thought of, and why. This is one of the important uses of modeling.

More 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