We use a mathematical model to find the optimal vaccine allocation at different time points of an epidemic. For both developed and less developed countries, when faced with low supplies of vaccines, it is always optimal to concentrate vaccine in high-risk children to provide them with direct protection, as they are part of the high-transmission chain and they are among the most vulnerable.
For a developed country, as vaccine supplies increases, it becomes optimal to allocate the resources in the high-transmission group, i.e., children at low-risk. This makes sense, since by protecting the high-transmission group, we stop the chain of transmission and indirectly protect the high risk groups. However, this policy is optimal only up to a certain time during or after the exponential rise phase of the epidemic, when too many high-transmission people have already been infected and have acquired natural immunity. After this point in time, it is optimal to concentrate vaccine in high-risk groups protecting them directly. Minimizing mortality, as opposed to hospitalizations, tends to push this threshold to the left in time so that the protection of high-risk adults starts earlier.
This is also true for less developed countries when minimizing hospitalizations. In contrast, when averting deaths, it is better to allocate vaccine in the high-risk groups first and then cover high-transmission groups. Once vaccine supplies reach a certain coverage level, then it becomes important to vaccinate the high-transmission groups in the earlier stages of the epidemic, but this policy becomes suboptimal once the peak of the epidemic has passed. This is because by allocating this much vaccine in children earlier on in the epidemic, we would be able to block transmission and mitigate the disease, but if vaccination took place later on in the epidemic, there are too many people already infected and this strategy is no longer optimal (see ).
These results highlight several important components of influenza epidemic control with vaccines. First, the proportion of population that is children is extremely important. For a less developed country, where the high-transmission group accounts for the majority of the population, one needs large amounts of vaccine to indirectly protect the high-risk groups by vaccinating the high-transmission ones. However, in a developed country, where high-risk groups represent a smaller fraction of the population, it is possible to reduce and even mitigate transmission by vaccinating the high-transmission groups, if this is done early in the epidemic. The second important point is that timing of the vaccination is extremely important and greatly determines where the efforts should be concentrated. Finally, while using the optimal policy greatly reduces the size of the epidemic if done early on, all vaccination policies perform poorly after the peak of the epidemic. It is very difficult to identify in real time where one is in an epidemic, but it would be even more difficult to switch the vaccination target groups during an epidemic, both politically and logistically. Given that timing is crucial, this suggests the necessity of better surveillance and preparedness. In this context, our results could be used to set general a priori guidelines for vaccine distribution on a given population. For example, pandemic H1N1 peaked in the US in early October 2009, and during the same period vaccination with limited supplies of vaccine began. As a result, most vaccine was delivered and administered well after the peak. The mass vaccination of children that occurred had a minimal effect on protecting others and reducing general morbidity in the population. Despite the early accurate prediction of when the epidemic would peak in the US 
, the vaccine arrived during the peak, for logistical reasons, and had a limited effect.
As a first approach, we used excess in infant mortality and excess in female adult mortality as proxies for excess in influenza-related mortality and hospitalizations in a LDC compared with a DC. The results were very sensitive to these parameters. This suggests that we need studies to more accurately determine these numbers. Hospitalizations and mortality rates in a LDC are very different from those in a DC. People tend to not seek medical attention unless strictly necessary due to the lack of health insurance and the economic cost. In a pandemic situation, the health system in a LDC is likely to quickly run out of essential medications, to lack essential health personnel or to reach full capacity. This will in turn increase mortality. None of these factors were considered in our model. Furthermore, our results are extremely sensitive to the population structure, both in the percentages of people at high risk and in the contact pattern among them. Here, we assumed that the contact patterns were identical in a DC and a LDC. This is an important limitation since the model depends strongly in this assumption. Given the uncertainty for the parameters for pandemic H1N1 2009, we agree with Dushoff et al. 
that one should be cautious in interpreting the results offered by simple models.
The model presented here is extremely simple. While we are able to draw general conclusions, our results may not be appropriate for specific countries. Adding more structure to the model (for example adding more age groups, changing the probabilities of transmission in each group, and adding other details) will make a more realistic model for a specific situation, and hence more realistic predictions. We assumed that the vaccine efficacy was the same in all groups. This is a limitation since we know that the efficacy is reduced in the elderly and takes more time to develop in children. Finally one could expect somewhat different results if the objective function were replaced by other functions, such as final illness attack rates, remaining years of life lost, economic burden or a combination of these.
Previous work 
has suggested that in presence of low vaccine supplies, high-risk groups should be prioritized but high-transmission groups should be vaccinated with larger quantities of vaccine. Our results agree with this strategy for a population with a structure similar to the one in the US as long as vaccination starts before the peak of the epidemic. However, we suggest that there is a threshold in the time when a switch in the optimal strategy occurs, after which, vaccine would be more effective if allocated directly to the high-risk groups. This is in agreement with the results found by others 
. The particular time for this threshold is strongly dependent on the values of the model parameters, in particular on the vaccination coverage and population structure, but in general, occurs some time during the exponential phase of the epidemic or right at the peak. Our results suggest that if vaccination occurs too close to the peak of the epidemic, then all the strategies considered performed poorly, in agreement with recent work 
. Our results are novel in that we compared optimal strategies for both a developed country and a developing country, taking into account differences in the population structure and excess in influenza-related mortality and hospitalizations.