Our analysis of the global spread of pandemic influenza gives some insight into the spread of genetically drifting interpandemic strains of influenza. The cities in our model include about 620 million individuals, or about one tenth of the world's population. These geographically distributed major population centers should be enough to represent the overall dynamics of a global epidemic, in which influenza strains travel via infected passengers from epidemic regions to those just coming into influenza season. Previous modeling studies have shown that air travel governs the rapid dynamics of epidemic spread around the globe, and that other modes of transport govern the slower local regional diffusion of disease 
. Our more detailed treatment of influenza seasonality in the tropics contributes greatly to the realism of the model without increasing its computational complexity.
Although SE Asia has often been the source of new strains of seasonal influenza, the next pandemic may arise in other parts of the world, as was demonstrated by pandemic H1N1 2009 in early 2009. One of the more alarming scenarios would be a newly reassorted H1N1/H5N1 influenza with high transmissibility and virulence. Therefore, we considered regions with potential person-to-person transmission of H5N1 
The cluster map of global influenza transmission in helps explain several important features of our simulation results. In all of the scenarios we simulated, pandemics peaked in the temperate northern hemisphere during the fall/winter of the first year (), as has been observed historically. The temperate northern hemisphere is highly connected by air travel and may share a common winter influenza season, so influenza prevalence peaks across much of this region appear to be synchronized 
. In the temperate southern hemisphere, epidemics may peak in the fall/winter of either (or both) the first or second year, depending on when the epidemic starts and how transmissible it is. The tropics, which do not have a single unifying influenza season and are less densely connected, has unsynchronized epidemic peaks. Epidemics with a high
are likely to burn out in one season, while those with lower transmissibility may take multiple seasons to reach the more remote parts of the world. In particular, South America is not well-connected to most of the world in our model (). Its strongest links are with the North and Central America cluster and the Europe and North/West Africa cluster, where most cities are out of season during the southern influenza season. This path of transmission from Asia to the temperate northern hemisphere, and much later to South America agrees with phylogenetic analyses of influenza strains around the world 
. How these strains evolve each season after they leave the tropics is open to debate 
To duplicate the observed global dynamics for pandemic H1N1 2009, we set the value of
to 1.8, at the higher end of the estimated range of 1.3–1.7 from early spread in Mexico and the US 
but consistent with another global model of pandemic H1N1 
. Our model predicts that parts of the globe already invaded by pandemic H1N1 2009 will not experience substantial further epidemics (see panel for Mexico City and April in ), unless the virus begins genetic drift under immune pressure. Following pandemic years, increasing levels of population immunity change the age-specific transmission patterns of circulating strains. Further study will be needed to build reliable global simulation models of interpandemic strains. In addition, our models predicted that the temperate northern hemisphere would have had considerable reduction in the influenza illness attack rates had vaccine been distributed in the quantities indicated, i.e., rapid 50% coverage, on October 1. However, that was not the case. In the US, small quantities of vaccine arrived in early October, ramping up to about 20% coverage by December, 2009. We estimate that the effect of vaccination in the US reduced the illness attack rate from about 23% to about 20%. Thus, vaccine would have to be delivered in a more timely fashion and with higher coverage in the US and other countries to have the effectiveness predicted by our model.
The model uses many simplifying assumptions to be tractable, and it may be misspecified in ways that bias our results. Recent models have begun to incorporate more realistic networks of human movement, including ground transportation 
. The addition of commuting patterns do not substantially change the timing of the epidemic peaks 
, but this level of detail may be required to simulate the dynamics of epidemics at finer resolutions 
. The fact that our model is open-source and computationally simple enough to run easily on a laptop makes it more accessible to the public health community than proprietary, computationally intensive models. Our model uses the same next-generation model in all cities. Regional differences in population structure and in the behavior of children and adults, including hygiene, socializing, and propensity to travel, may influence the global spread of influenza. We have performed a simple sensitivity analysis for age structure (see Figure S7
), but this is an area that needs further exploration. We suspect that more accurate next-generation matrices in the cities of our model would increase the relative importance of influenza transmission in the tropics, which includes many countries with very young population. This, in turn, makes accurate modeling of seasonality even more crucial for obtaining realistic simulation results.
The factors that influence the seasonality of influenza are not well understood, so we used the observed influenza activity from past seasons to define periods of high transmissibility. One problem with this approach is that the model predictions do not take into account the conditions of a particular year. A more detailed model would allow seasons to be delayed or truncated by, for example, climate and school calendars 
. Although it would be conceptually easy add such conditions to the model, the amount and availability of required data are significant obstacles. Nonetheless, we believe that the tropical seasonality of influenza in our model is an important improvement on earlier efforts.
The single-strain model that we present here is suitable for pandemics, in which there is little pre-existing immunity in the population. However, the dynamics of seasonal influenza are determined by multiple competing strains, cross-protection, antigenic drift, and waning immunity. None of this is captured in our model, which may limit its use in planning a public health response to inter-pandemic influenza spread.
The transmission cluster map captured several important features of global influenza transmission, and we believe it is a new and useful way to understand the behavior of complex epidemic models. The clustering algorithm could be modified in several ways that might improve the identification of transmission clusters. The current algorithm was not designed specifically to understand infectious disease transmission, so it is insensitive to the effects of seasonality and to the population within each city. Improving the identification of transmission clusters and understanding their use in the design of global vaccination strategies are important extensions of the research presented here.
We investigated a likely global distribution of pandemic influenza vaccine within current possible constraints using an open-source model that we developed to capture the essential features of global influenza transmission while remaining computationally simple enough to be used by any researcher. We show that such strategies are marginally effective for certain regions of the planet depending on the location, timing and transmissibility of the new pandemic strain. The modeling structure and clustering algorithm used for could be used to develop optimal vaccine distribution if global strategies were possible for limited quantities of vaccine. This would be an important next step for the control of both pandemic influenza and interpandemic influenza, and it is a subject of future research and planning.