Stockpiling Ventilators for Influenza Pandemics

In preparing for influenza pandemics, public health agencies stockpile critical medical resources. Determining appropriate quantities and locations for such resources can be challenging, given the considerable uncertainty in the timing and severity of future pandemics. We introduce a method for optimizing stockpiles of mechanical ventilators, which are critical for treating hospitalized influenza patients in respiratory failure. As a case study, we consider the US state of Texas during mild, moderate, and severe pandemics. Optimal allocations prioritize local over central storage, even though the latter can be deployed adaptively, on the basis of real-time needs. This prioritization stems from high geographic correlations and the slightly lower treatment success assumed for centrally stockpiled ventilators. We developed our model and analysis in collaboration with academic researchers and a state public health agency and incorporated it into a Web-based decision-support tool for pandemic preparedness and response.

We could formulate a static multiple linear regression model to study ILI hospitalizations in the following manner: . (1) However, to incorporate the evolution of the predictors over time, which has significant importance in forecasting of ILI hospitalizations, we instead posit a dynamic linear regression model: . ( The critical difference between equations (1) and (2) is that the regression parameters are no longer static, evidenced by introducing the time subscript in equation (2). The estimation of the random parameters in equation (2) can be performed recursively using the Kalman filter (1). .
Next, the 1-step-ahead predictive distribution of given is also Gaussian, i.e., , with and as follows: .
After obtaining the observation , the filtering distribution of is, again, Gaussian, i.e., . The parameters and can be computed as follows: , where is the forecast error. Our discussion here follows the book (1), which we refer to for further details.

Multiple Steps Ahead Forecasting
For the purpose of producing demand scenarios for our optimization model, we must forecast hospitalizations and in turn, ventilator demand, many weeks into the future. .
The distribution of given is also Gaussian, i.e., , with and as follows: .
We use 1 year of historical seasonal influenza data to construct the prior for and , and we use 2009 pandemic data to fit the model and forecast = 40 weeks into the future.

From Hospitalizations to Peak-Week Demand for Ventilators
We index the health service regions (HSRs) in Texas by . The DLM predicts hospitalizations on a weekly basis for each of the 8 HSRs in the form of a multivariate Gaussian distribution, providing the means ( ) and variances ( ) for each region. We estimate the region-to-region correlations ( ) using historical data, and we assume this correlation to be identical for each pair of regions. To estimate the peak-week demand for ventilators from the forecasted hospitalizations, we employ 4 additional parameters: 1) , the proportion of hospitalized ILI patients requiring ICU care; 2) , the proportion of ICU patients requiring ventilation; 3) , the proportion of ventilated patients requiring 2 weeks of ventilation, under the assumption that at most 2 weeks is needed; and, 4) , 1-week lagged temporal correlation in ILI hospital admission in region at time generated by the DLM.
We calculate the mean weekly demand for ventilators in region at time as follows: . ( We obtain the corresponding variance of weekly demand for ventilators, involving temporal correlation ( ), as follows: .
We choose the peak-week demand in a region as the week with the largest mean according to equation (3). With the estimated region-to-region correlation ( ), we employ a standard Monte Carlo sampling algorithm (2) to generate independent and identically distributed (i.i.d.) samples of peak-week demand for ventilators as input to the optimization model, which we describe next.

Two-Stage Optimization Model
To optimize stockpiling decisions, we construct a 2-stage stochastic program. We index the regional sites by . 1. Given existing stockpiles at the regional sites, optimize the number of centrally held ventilators.
2. Given an existing central stockpile, optimize the number of ventilators at each site.
3. Jointly optimize the central and regional stockpiles, allowing us to assess the advantages of stockpiling ventilators centrally versus at the sites.
Model (4) is stated in the form of the third variation above, but the first 2 variations can also be handled by fixing decision variables or , respectively, to prespecified values.
We cannot solve model (4)  The objective function in (5a) is identical to that in (4a). Constraint (5b) is analogous to constraint (4b), where we add index to variable because shipments from the central stockpile to the sites occur after observing the demand realization. In constraint (5c), , are the samples of ventilator demands, and in constraint (5d), is the proportion of centrally held ventilators dispatched to the site that can be used. These 2 constraints take care of the 2 positive-part operators in constraint (4c) by using 2 new decision variables, and . Given that these variables capture the positive parts, constraint (5e) is analogous to constraint (4c), and constraint (5f) again captures non-negativity of all decision variables. While we state models (4) and (5) for a fixed value of , we view this as a bi-criteria model in which we can explore the tradeoff between the cost of the total stockpile (which we assume is proportional to the number of ventilators) and the limit on expected shortfall ( ).