Prophylactic interventions such as vaccine allocation are some of the most effective public health policy planning tools. The supply of vaccines, however, is limited and an important challenge is to optimally allocate the vaccines to minimize epidemic impact. This resource allocation question (which we refer to as VaccIntDesign) has multiple dimensions: when, where, to whom, etc. Most of the existing literature in this topic deals with the latter (to whom), proposing policies that prioritize individuals by age and disease risk. However, since seasonal influenza spread has a typical spatial trend, and due to the temporal constraints enforced by the availability schedule, the when and where problems become equally, if not more, relevant. In this paper, we study the VaccIntDesign problem in the context of seasonal influenza spread in the United States. We develop a national scale metapopulation model for influenza that integrates both short and long distance human mobility, along with realistic data on vaccine uptake. We also design GreedyAlloc, a greedy algorithm for allocating the vaccine supply at the state level under temporal constraints and show that such a strategy improves over the current baseline of pro-rata allocation, and the improvement is more pronounced for higher vaccine efficacy and moderate flu season intensity. Further, the resulting strategy resembles a ring vaccination applied spatiallyacross the US.
There is an abundance of literature on the modeling, analysis, and control of epidemics. We briefly mention three areas that are closely related to our paper, namely, mobility modeling, disease modeling, and designing interventions to control the spread of epidemics. We refer to [11] [12] for surveys on these topics. We develop a framework for national seasonal/pandemic influenza planning using realistic datasets, a mechanistic model of disease spread, and a greedy optimization algorithm for vaccine allocation. Our specific contributions are discussed below.
Our approach in building the national scale model involves two broad steps: Model calibration is the process of estimating parameters of the computational model that can reproduce observed characteristics in the ground truth. In the context of epidemiology, beyond forecasting, calibrated models allow us to perform counterfactual (i. e. , what-if) studies, and address resource allocation questions like VaccIntDesign. In this section, we will briefly describe our approach and the ground truth used for the two-stage calibration of the national-scale influenza model. We begin with the assumption that the ground truth of interest y is a noisy version of the simulation model η (⋅) at some unknown input parameter configuration θ ^. We use a gaussian error model, which are simple and adopted widely for many applications, including epidemics [30,31]. We adopt importance sampling [32] scheme to produce posterior realizations of the calibration parameters. We begin with sampling from an easy-to-sample importance distribution Im (θ) (say, uniform), and run the simulation model η at each of those samples. The importance weights are computed as the ratio of the posterior distribution (proportional to the product of likelihood and prior distribuion) and importance distribution evaluated for each of the samples. The samples along with the normalized weights then constitute an estimate of the posterior distribution. Further, it is often useful to factorize the likelihood function, if possible, when the simulation model is required to be calibrated to several different criteria [31]. One possible way is to sequentially calibrate the model to different criteria. In addition to simplifying the computation of importance weights, the approach allows user to introduce more samples as needed using the intermediate calibrated parameter space. More details on the statistical framework and the two-stage posterior exploration is provided in S1 Appendix. Additional methods. Finally, in our case, since we are interested in using a single calibrated model for the optimization study (as against a weighted ensemble provided by the posterior distribution), we consider the Maximum a Posteriori (MAP) estimate i. e. , model configuration with highest frequency to be the calibrated model. We now consider the problem of determining the spatial allocation of vaccines across the US to minimize a chosen objective function. In addition to the complexity introduced by non-linear dynamics of the disease model, we also need to account for the temporal constraints imposed by vaccine production and delivery logistics. Formally, the VaccIntDesign problem involves determining the vaccine allocation vector X that minimizes the total attack size given by f (X). This can be expressed as: minimize X f (X) subject to ∑ i X i, t ≤ B t, for all t, where Bt is the total number of vaccines available at time t. Our goal in the VaccIntDesign problem is then determine the amount of vaccine allocated to each patch i at time t, denoted by Xi, t. The VaccIntDesign problem is very challenging, and its exact complexity remains open. A strategy that has been useful in many kinds of intervention design problems is to design a greedy allocation, which selects each decision variable based on the marginal improvement to the objective function. If the problem involves submodular maximization, such a strategy is guaranteed to give a constant factor approximation; see, e. g. , [22] [23]. In contrast, VaccIntDesign involves a minimization, and the objective functions are neither submodular or supermodular, in general. Nevertheless, the greedy strategy is a reasonable approach for designing vaccine allocation strategies, and we study it here with an allocation step size of L. The algorithm begins with an initial zero allocation. For each week w, the algorithm allocates the next set of L vaccines to the state s which leads to the maximum reduction in the objective value f (X). The algorithm is repeated for week w, until we exhaust Bw, and then proceed to the next week’s supply of vaccines. Note that the computation of marginal benefit of allocating L additional vaccines to state s subject to population constraints, can be computed in parallel. As a generalization, we have also included the lookahead duration d (in weeks) as an additional parameter. This means that the potential allocations at a greedy stage of week w are evaluated by their reduction of attack size at week min (w + d, T) where T is the total duration of the epidemic. While this includes the total attack size (full lookahead, when d ≥ T) as a special case, it also allows us to explore the resulting trade-off due to varying forecast horizons. The detailed algorithm is provided in S1 Appendix. Additional methods.
For the current study, we begin with the disease model calibrated to the 2014-15 influenza season. Given the best fit model Mθ⋆, we define the optimization study scenarios as follows: A scenario is defined by a (v, E) tuple and is derived by setting the vaccination efficacy to v in model Mθ⋆ and calibrating the transmissibility β to achieve national attack size of E under pro rata vaccine allocation. We do this to simulate multiple seasons that spread spatially like the 2014-15 season, but vary in their severity (captured by the national attack size E) and the efficacy of seasonal vaccine (captured by v). In our study setting, we construct 12 scenarios, where v takes values in {0. 2,0. 35,0. 5} and E takes values in {40,61,73,86} where the values are in millions of cases, corresponding to different severity levels based on past seasons of seasonal influenza. Thus for each target attack size E, we have three scenarios, in which E is achieved by assigning vaccines at v efficacy. In our study (restricted to contiguous US, including DC), the number of states S = 49. Also, we set the number of weeks W = 40, roughly the period from September to May corresponding to the influenza season. Therefore, the allocation profile X has 1960 spatio-temporal dimensions. The temporal constraint B is based on historical vaccine uptake schedue available from CDC FluVaxView [36]. CDC FluVaxView provides monthly coverage estimates nationally for the past influenza seasons. We scaled it by the national population to get a vaccine uptake schedule and converted it to the temporal constraint B. Note that CDC also provides the vaccine supply and distribution schedule [37], however, we noticed a considerable delay between the supply and uptake schedules, so we chose to use the uptake schedule to reflect ground reality.
Current policies for vaccine interventions are designed based on a host of social and political issues, and tend to be fairly simplistic. For instance, Department of Health and Human Services (HHS) directives for targeting pandemic vaccines are based on age group [5], and the allocation of the national vaccine supply and other resources is typically done proportional to the state population. There has been a lot of interest in developing more effective interventions, e. g. , [2] [4] [25] [26]. For instance, Medlock et al. [2] developed an optimal vaccination strategy for the H1N1 outbreak; their model showed a prioritization for a different age group than the ones recommended by CDC directives. All prior methods are restricted to simple models, and only focus on non-temporal interventions in which the allocation is done ahead of time. In reality, vaccine supply varies over time, and the real problem involves finding an allocation that respects the supply constraints and optimizes the epidemic outcomes. Our current model can be extended in several ways. Firstly, the model calibration process can be refined to match more detailed trajectories of influenza spread, like the ILI % time series, or the in-season burden estimates being produced by CDC starting 2018-19 season [33]. Such approaches can then be used to do real-time forecasting and provide vaccine allocation recommendations for an ongoing influenza season. Further, instead of selecting the MAP model for vaccination study, one could use an ensemble of calibrated models based on the posterior distribution, thus being able to quantify uncertainty in the vaccine allocation policy’s effectiveness. Another aspect of the real-world dynamics currently not being captured in our model is that of residual immunity. The national influenza model can be improved by taking into account the co-circulating and dominant influenza strains, as well as the strains present in the recommended vaccine for the season. Note that while improving over pro-rata allocation, greedy algorithm, even with the lookahead duration, may lead to sub-optimal policies. One can develop algorithms that earmark resources for regions with high spreading capacity, thus potentially improving the effectiveness of vaccine allocation. Finally, the logistics of the supply of medical resources, such as medicines, medical equipment (e. g. , ventilators), and medical staff is also very complex. The health infrastructure is generally optimized for typical demand for such resources, and any surge, as would happen during a pandemic outbreak, would place a severe strain on hospitals. Ajao et al. [27] show that over 50,000 ventilators might be needed in the event of a national influenza pandemic outbreak. Since local and state health systems are usually unprepared for such a surge in demand, the Office of the Assistance Secretary for Preparedness and Response (ASPR) maintains a stockpile of mechanical ventilators in strategic locations [38], which can be deployed during an emergency. While existing efforts partially address the question of optimizing stockpile redistribution [28], a mechanistic model like the one developed in this paper will help design better national-scale studies for pandemic preparedness exercises, and develop strategies for allocation of vaccines and other resources during such emergencies. In conclusion, we have presented a national level seasonal influenza model, based on short-range and long-range mobility datasets, and used it to optimize the spatio-temporal allocation of vaccines. For the scenario under consideration, we find that the national attack size can be reduced by up to 17% by allocating the early vaccines to regions around the origin of the epidemic. Most states still end up with close to their overall pro-rata quota of vaccines, however, these findings demonstrate that shifting when and where these vaccines are administered has a sizable impact on the national attack size. Achieving these optimal outcomes would require better surveillance and the ability to accelerate vaccine uptake at will, which presents multiple challenges. However, the study shows there is ample room for improvement and this framework provides means for developing a play-book for epidemic containment.
Annual vaccination campaigns continue to be one of the prime measures which help alleviate the burden of seasonal influenza. Due to production and logistic constraints, there is a need for prioritization policies associated with vaccine deployment. While there is general consensus on age-based or risk-based prioritization, spatial optimization of vaccine allocation has not yet been explored in sufficient detail. In order to do this, we develop a mechanistic model of influenza spread across the United States, and propose a greedy mechanism for spatial optimization. We test the methodology on different realistic scenarios with temporal constraints on vaccine production.
medicine and health sciences
influenza
applied mathematics
immunology
social sciences
human mobility
simulation and modeling
vaccines
preventive medicine
algorithms
mathematics
infectious disease control
vaccination and immunization
human geography
research and analysis methods
public and occupational health
infectious diseases
geography
earth sciences
biology and life sciences
vaccine development
viral diseases
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Optimizing spatial allocation of seasonal influenza vaccine under temporal constraints