To fix notation, we assume here that we seek to optimize the number of lives saved (i.e. minimize the number of lives lost due to H1N1pdm); however, with suitable modifications to the definitions of all terms, we could equally attempt to prevent hospitalizations or intensive care admissions. There are at least two possible benefits, and three harms, to predispensing. The first benefit of predispensing is that a patient possessing a predispensed course of antivirals will likely begin therapy earlier in the course of H1N1pdm infection than one must acquire a prescription (possibly including a visit to a physician) and fill it, thereby most likely delaying treatment. This benefit accrues whether or not there is a shortage of antivirals, since it simply reflects the time required to acquire and fill the prescription. A second possible benefit occurs if the demand for antivirals exceeds supply, so that not all patients who attempt to acquire antivirals can do so. In this situation, predispensing a course assures that it is in the hands of someone who, if untreated, would be likely to develop severe disease, rather than (potentially) going to someone who will not likely develop severe disease, and whose benefit from taking the antiviral would therefore be less.
The first harm of predispensing occurs only if total demand for antivirals exceeds supply, namely, a predispensed course is unavailable to anyone who may need it other than the person who has received it and who may not get infected. If demand exceeds supply, this means one more person, possibly someone who would benefit greatly from having the antiviralis unable to obtain it. A second harm of predispensing is the possibility that a predispensed course will be used by a high-risk individual who would not otherwise be able to obtain an antiviral drug, and this usage will lead to a severe side effect. We have no way of assessing the likelihood of such an outcome; in particular, the major severe side effects reported to date are neuropsychiatric events in oseltamivir recipients, and the manufacturer’s data review concludes that there is no evidence of a causal link between oseltamivir and these adverse events [25
]. We will not model this second harm in our analysis. Yet another possible harm to predispensing is preliminary evidence that people treated with antivirals have lower levels of antibodies compared to people who weren’t [33
]. We are not aware of data quantifying additional risk for re-infection, if any, after antiviral treatment compared to no antiviral treatment. Therefore we will not model this potential harm either.
To assess the net benefit or harm from predispensing, we define some notation, summarized in . We divide the whole (total) population into two groups: a high-risk group for which we are considering predispensing, and the general population, which includes all individuals not in the high-risk group. These constitute respectively a proportion q (high-risk) and 1 − q (general population) of the total population. Within the general population, risks may vary, so some individuals may be at higher risk than others. All quantities in our notation are defined as proportion of the total population, hence lie between 0 and 1. Let T be the total supply of antivirals (as a proportion of the population); thus, if enough antivirals are available for treatment of 20% of the population, then T = 0.2. Let D be the demand for antivirals in the general population, that is the number of individuals who would receive antiviral treatment in the general population if supply were not limiting; in practice, some of this demand may be unmet as the supply is constrained. Let pd be the death rate (per capita) from the infection over the whole course of the epidemic in the whole population (if no antivirals are used); the death rate in a high-risk group is R·pd. Note that these death rates are not conditional on infection (i.e., are not case-fatality proportions) but are unconditional, reflecting the risk of infection times the risk of dying from infection. Here the number R is the relative risk of dying in a high-risk group compared to the whole population. R can be estimated from the existing epidemic data while pd may be hard to estimate a priori; however we shall see that pd is factored out of our equations and need not be known. Let ps be the probability that a course of antivirals obtained during the epidemic would save a life of a person who would die otherwise. We assume that this probability is the same between high-risk individuals and members of the general population. This key assumption may be incorrect, and is discussed later. Let ps·B be the probability that a predispensed course of antivirals would save a life of a high-risk person who would die otherwise. B can be thought of as the “relative benefit” in preventing mortality of receiving a predispensed course of antivirals compared to receiving a non-predispensed course by a high-risk individual who would die otherwise. B captures the benefit of early vs. delayed treatment; however, B also may be reduced to the extent that a predispensed course is taken for non-influenza illness, in which case it cannot subsequently save a life. Thus B exceeds one to the extent that early treatment is better than delayed treatment, but it is decremented in proportion to the probability that the course is wasted before it is needed. If a pandemic is imminent or ongoing, the risk of wastage is small, especially outside of the season (winter in temperate climates) when other respiratory infections are most common.
Parameters of the model, as a proportion of the total population (including high-risk and general populations)
Finally let L
be the total number of people in the general population who would die during the epidemic and whose lives would be saved if they received antivirals upon demand. Since the total fraction of the whole population(general + high-risk)who would die during the epidemic and whose lives would be saved if they received antivirals during the course of the epidemic is pd
, clearly L
. We make additional assumptions about how the probability of receiving antivirals behaves if demand exceeds supply; these are made explicit in Appendix A
With this notation in place, we can define the conditions under which it is advantageous to predispense 1 course each to a proportion q of the population. Our main result is the following:
MAIN RESULT: Predispensing saves more lives than not predispensing when any of the following conditions hold:
- Supply of antivirals exceeds demand even after predispensing (D ≤ T − q), and B > 1.
- Demand D in the general population exceeds supply T even without predispensing, and the following conditions are met:
- Supply T exceeds demand D in the general population without predispensing, but after predispensing, demand exceeds supply (T − q ≤ D ≤ T), and the following conditions are met:
- Sufficient condition: The above results show that no matter what the demand for antivirals, predispensing is advantageous when
The justification of this result and the assumptions underlying it are presented in Appendix A
. The sufficient condition at the end tells us when a group is at high enough risk that it is worth predispensing to them even if we have no idea of the expected antiviral demand.
This result can be seen graphically in –, which consider a hypothetical case in which there is a supply adequate for T = 0.2 of the population, and predispensing to q = 3% of the total population is under consideration. The parameter that varies between the figures is the relative benefit B of saving a life by a predispensed course vs. a course obtained by prescription. We are not aware of data allowing an assessment of this parameter; we consider 3 scenarios: B = 2 (), B = 1.3 () and B = 1 (). We consider it likely that B = 1 is an extreme case and that in reality B > 1.
FIGURE 1 Predispensing is beneficial (shaded area) when R exceeds a certain threshold that depends on the projected demand. Here a supply adequate for T = 20% of the population is assumed, and predispensing to q = 3% of the total population is under consideration. (more ...)
FIGURE 3 If predispensed courses are no more likely to be lifesaving than non-predispensed courses taken by the same person, the conditions for benefits of predispensing are even more restrictive. Parameters are as in , except that treatment of a high-risk (more ...)
FIGURE 2 If predispensed courses are 1.3 times more likely to be lifesaving than non-predispensed courses taken by the same person, the conditions for benefits of predispensing are somewhat more restrictive. Parameters are as in , except that treatment (more ...)
The far left side of each of the figures shows low levels of demand, in which there is no harm to predispensing, so predispensing to any group for which B > 1 may be beneficial. At the far right, competition for antivirals is strong, so all antivirals are used, mostly by the general population, and it is beneficial to predispense even to a group that gets modest benefit from antivirals in order to capture the benefit of early treatment. In the middle, when demand is similar to supply, it is beneficial to predispense only to groups that benefit disproportionately from antivirals (R > 5 for B = 2, R > 16.7 for B = 1.3 and never for B = 1).
Finally, we note that it is not always true that the benefit increases with the quantity of antivirals predispensed. Under certain conditions it would be better to predispense to a subset of the high-risk population, defined as those with a risk above a certain threshold. This is discussed further in Appendix C
, where a simple criterion is given ensuring that each successive predispensed dose increases the net benefit, provided that the quantity of antivirals predispensed is not too large.