The U.S. Government holds millions of treatment courses of antiviral medications its Strategic National Stockpile (SNS), but there are no published criteria for sequential release of these medications to States and Territories during the course of a worldwide influenza pandemic, such as the one currently caused by the pandemic influenza A (H1N1) virus.. We employed advanced optimization methods to evaluate millions of antiviral distribution schedules that vary in both the amount of each release and whether allocations to states are proportional to overall population or recent flu activity. We find that, for the case of 2009 (H1N1) pandemic flu, a simple distribution schedule of periodic small releases proportional to state population densities is a near-optimal policy. For this antiviral distribution scheme to have an impact on transmission of the current H1N1 virus, however, a higher proportion of cases must be identified and treated promptly with antivirals than is currently the case. This paper provides both practical guidance for future flu intervention and new methods that are readily adaptable to optimizing a broad range of infectious disease control policies.

In March/April 2009, a new swine-origin strain of influenza A/H1N1 virus emerged into human populations in California and Mexico and has since escalated into a pandemic. The U.S. experienced low levels of sustained H1N1 transmission throughout the summer months of 2009, and in the Southern hemisphere the novel virus largely displaced seasonal influenza, causing more severe morbidity in a younger adults and children than is typical with "the flu." Starting in September, the U.S. began to experience large geo-temporally distinct outbreaks of influenza like illness and confirmed H1N1 disease in States and Territories. As of mid-October, total cases reported by surveillance systems as well as pediatric deaths had surpassed levels typically seen for the entire flu season ending in mid-Spring 2010.

During the early weeks of the epidemic, the U.S. Centers for Disease Control and Prevention (CDC) distributed 11 million of the 50 million antiviral treatment courses from the federally held portion of the national antiviral stockpile. This distribution went to States and Territories (as well as to four large cities) on a pro rata basis (that is, in proportion to population size). Since the recipients had local stockpiles as well, this allowed the CDC to exceeded the pre-determined target of distribution of 31 million treatment courses of oseltamivir and zanamivir prior to the acceleration phase of the pandemic

Key policy statements have called for the use of mathematical models to support the development of an evidence-based policy for eﬀectively deploying the remaining antiviral stockpile and other limited or costly measures to limit morbidity and mortality from H1N1

Here, we use a new algorithm that efficiently searches large strategy spaces to analyze the optimal use of the U.S. antiviral stockpile against pandemic influenza prior to widespread and effective vaccination. We assume, in line with CDC guidance, that antivirals will be used exclusively for treatment of symptomatic individuals rather than wide scale pre-exposure prophylaxis. We apply our algorithm to a U.S. national-scale network model of H1N1 transmission that is based on demographic and travel data from the U.S. Census Bureau and the Bureau of Transportation Statistics and assumes disease parameters estimated for H1N1 during the initial March-April 2009 outbreak in Mexico City.

We couple a fast, scalable, and adaptable optimization algorithm to a detailed simulation model of influenza transmission within and among the 100 largest cities in the United States. In brief, the method involves running many stochastic simulations of H1N1 transmission, each requiring an intervention policy as an argument. The optimization algorithm dictates the choice of intervention policy for each simulation, with the goal of identifying the optimal intervention policy given the highly stochastic simulation output.

A time-based intervention policy is a series of actions

The naive approach to finding the optimal path through the tree is to simulate multiple disease outbreaks for each intervention policy (path) and record the expected morbidity or mortality (or other public health outcome measure). However, such exhaustive searches are computationally intractable for large trees. We can more efficiently search for the optimal policy by prudently sampling paths from the tree.

To strategically search the tree, we use an optimization algorithm called Upper Confidence Bounds Applied to Trees (UCT)

Specifically, suppose we are descending through the tree and have arrived at node

Initially,

Our model includes the 100 largest metropolitan areas in the United States, which we identified by aggregating Census Bureau Statistical Areas (CBSA) that share a common airport

Within each city, disease transmission is modeled using a compartmental model with five compartments: susceptible, exposed, asymptomatic infectious, symptomatic infectious, and recovered (Fig. 2b). Progression from one compartment to another is governed by published estimates for H1N1 transmission and disease progression rates, as given in Table 1. Epidemics are initialized assuming that there are 100,000 cases of H1N1 in the United States (corresponding to the late June CDC estimate of over one million H1N1 cases

The model considers 11 possible antiviral stockpile actions every month over a twelve month period: distribution of 0, 1, 5, 10, 25 or 50 million courses apportioned either pro rata or proportional to current prevalence. We assume that, post distribution, unused courses decay through misuse or loss at a rate

Each optimization is based on 48 hours of computation on the Linux Lonestar Cluster at the Texas Advanced Computing Center, which offers a peak performance of 10.7 GFLOPS per optimization process. Roughly 1,000,000 simulations can be done in this time; however, for this relatively small action space, the optimal policy is typically identified within six hours of computation.

Without antivirals, the expected cumulative number of cases after 12 months is almost 128 million. We found several optimized time-based intervention policies that decrease the spread of disease. Figure 3a shows the comparison of these strategies with a range of naive distribution policies in which fixed fractions of the antiviral stockpile are distributed monthly. Notably, one of these naive policies - monthly releases of 5 million regimens pro rata for 10 months - consistently performs as well as the optimized policies and better than other naive policies, particularly at uptake rates between 20 % and 60 %. Naive monthly distributions of less than 5 million regimens fail to meet the demand while larger monthly distributions result in greater wastage and rapid exhaustion of the stockpile (Fig. 3b).

The optimal policy returned by the algorithm is sensitive to the levels of uptake (

A mixed pro rata and prevalence-based distribution policy favored pro rata distributions, particularly for low to intermediate levels of uptake (Fig. 4a). This combined with the comparable performance of exclusively pro rata policies (Fig. 3a) suggests that prevalence-based distributions are probably unnecessary.

The optimal pro rata policies (Fig. 4b) vary considerably with uptake. At realistically low levels of uptake (between 10 % and 25 %) the optimal strategies are either a single distribution of the entire stockpile or a single distribution of half of the stockpile followed by a slow steady stream of smaller releases. For example, at 15 % uptake, the optimal policy is to distribute a twelve-month sequence of 25M, 1M, 10M, 10M, 1M, 0, 1M, 0, 1M, 0, 1M, and 0 courses. At higher levels of uptake, which are probably unlikely without a national campaign to increase rates of antiviral treatment for mild cases, the optimal strategies are a bit more complex, involving longer series of variable distributions. Between 35 % and 65 % uptake, the optimal strategies involve initial delays of a few months during which only small quantities are distributed. This delay prevents the wastage of antivirals that will go unused during the initial period when prevalence is relatively low. This delay strategy is even more pronounced when optimizing distributions for the entire epidemic period (Supplementary Information).

The optimization provides information about the relative performance of different actions. At uptake levels less than 25 %, the best strategies are expected to reduce the number of cases by at most 15 %. At even lower levels of uptake (less than 15 %), all initial (August) pro rata distributions perform equally well because only a very small proportion of the entire stockpile will ever be used to treat H1N1 (Fig. 4c). As uptake increases, there is a trade-off between meeting the increasing demand for treatment and risking wastage of courses that would otherwise be used to treat future cases of H1N1. Between 20 % and 30 % uptake, the benefits of increased coverage outweigh the consequent wastage, and thus the optimal action is a large release in August. Between 35 % and 60 %, wasted courses are more likely to have been used for future treatment, suggesting a steady distribution throughout the 12 month period (Fig. 4b). At uptake greater than 60 %, an overwhelming fraction of infected individuals seek treatment, and any distribution schedule effectively mitigates the epidemic for 12 months.

Since avian influenza H5N1 became a potential public health threat in 2003, public health agencies around the globe have been diligently planning for the next influenza pandemic. While the concerted response to H1N1 reflects this careful preparation, several expected and unexpeted events, including its apparent North American origin, the rapid overburdening of U.S. laboratory capacity, non-uniform testing and treatment policies among U.S. states, and delays in production of a viable vaccine, all reinforce the need for a dynamic and quantitative playbook for pandemic mitigation using pharmaceutical countermeasures.

By adapting an established algorithm to optimize disease mitigation policies, we advance from the traditional candidate strategy approach to rapid and systematic analysis of numerous policy options. This is just one of many possible optimization methods suitable for this purpose

Unexpectedly, our analysis suggests that the optimal distribution schedule for the U.S. national antiviral stockpile may be quite simple. A monthly distribution of 5 million regimens to states consistently matches or outperforms other policy options, regardless of the level of uptake or rate of misuse (Fig. 3a). Slight variations on this policy, for example, regular distributions of 1M or 10M courses are predicted to perform significantly worse at intermediate levels of uptake. Since there can be many optimal policies, the results of the optimization algorithm do not necessarily have the same simple structure as the 5M strategy. Our optimization allowed for the possibility of distributions proportional to prevalence, although such actions are not consistent with the current strategic national stockpile policy and would likely be both politically and logistically difficult. Notably, the results suggest that prevalence-based distributions are not expected to enhance the impact of antivirals.

The expected impact of this simple 5 million course monthly distribution schedule is highly sensitive to the rate of antiviral treatment (

Our analysis did not consider the threat of antiviral resistance. Currently circulating strains of seasonal influenza have acquired resistance to oseltamivir

From rapid genetic sequence analysis to automated syndromic surveillance systems, public health emergency response is rapidly improving in technical capabilities both in the U.S. and worldwide; the rapid response to and characterization of the novel pandemic influenza A (H1N1) virus is a testament to this. However, planning the policies of public health response to such identified and emergent threats remains a highly non-quantitative endeavor. We present here a policy optimization approach that is highly modular and can be easily adapted to address multiple additional issues. Our hope is that the quantitative methods will assist clinical experts in developing effective policies to mitigate H1N1 using a combined arsenal of vaccines, antivirals and NPI’s. Specifically, a very similar analysis can be used at the international level to optimize global allocation of the WHO's limited antiviral stockpile to resource-poor countries. One can substitute any stochastic model of disease transmission, at any scale, for our national-scale, U.S. H1N1 model. In addition, while the optimization algorithm is particularly well suited for time-based interventions, any well-behaved policy space can be used

The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the Centers for Disease Control and Prevention. The authors thank John Tegeris at BARDA for providing up-to-date information about the U.S. federal and state Strategic National Stockpiles of antivirals, the BARDA flu vaccine group for providing up-to-date estimates for the availability of H1N1 vaccines, the City of Milwaukee Health Department and the Harvard Center for Communicable Disease Dynamics for sharing unpublished data on the receipt of oseltamivir in Milwaukee. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin (http://www.tacc.utexas.edu) for providing HPC resources that have contributed to the research results reported within this paper.

This work was supported by grants to LM from NIH Models of Infectious Disease Agent Study (MIDAS) (U01-GM087719-01), the James S. McDonnell Foundation, and NSF (DEB-0749097) and grants to BP from CIHR (PTL-97125 and PAP-93425) and the Michael Smith Foundation for Health Research.

The authors have declared that no competing interests exist.