A friend suggested an interesting solution to the hospital overcrowding situation in New York caused by the first wave of Covid-19 cases. Considering all of the cancelled flights across the country, could the grounded aircraft be used to support a national medevac transportation network? If we could predict when cases would peak in states across the country, then we could chart a transportation plan to ensure no single hospital would exceed its capacity. Setting aside all of the finer details of moving sick patients across the country, I wanted to help quantitatively prove the efficacy of this ambitious idea.
The "SIR model" is a simple compartmental model that can model the spread of infectious diseases. This method divides a population into one of three categories: S - Susceptible, I - Infected, and R - Removed (deceased or recovered and immune). S, I, and R are all non-linear functions of time, and all groups sum to a constant value, of the entire population, across time. The rate of change for each group can be expressed by the following equations (N = total population, γ = recovery rate, β = infection rate, R0 = basic reproduction):
Some assumptions for our scenario:
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Hospital Occupancy = 30% of all infected patients, t-5 days ago. This assumes there is a 5 day window from the time a patient comes into contact with the disease and is admitted to the hospital.
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Death Rate = 4% of all removed / recovered patients. This is a rough estimate, which falls within the wide and varying range of reported percentages.
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Death Rate Without Transportation is increased by 50% when hospital occupancy exceeds beds available.
For each state I created a model to predict when hospital will be exceeded and the difference in deaths with and without an air transportation network. This amounts to anywhere between several hundreds and several thousands of lives, depending on the state. The assumed deaths without air transportation is a very rough number – it is assuming people turned away from the hospital, who would be admitted if there was capacity, have a 50% increase in mortality. Plots for New York and California can be seen to the right.
Each state model individually trains with a root mean squared loss function to minimize the difference between the predicted infections and historical infection data (~January to April 2). The parameters beta and gamma are adjusted to reduce the loss function. Gamma is the recovery rate, which I’ve restricted to around ~1/14 days, since it’s been observed to take patients roughly 2 weeks to recover. Beta is the infection rate and varies a bit more to try and compensate for state dependent factors, like social distancing measures. The video below compares two scenarios based on the predictions of these models, one with an interstate transport network (top) and one without (bottom). The top visual assumes every state has access to all of the hospital beds in the country at every time step (best case scenario, with immediate transportation of patients.) I extended this video to an interactive web-based map, that can be used to toggle participation of individual states in a transportation network and see the effects. The video at the top is a recording of this tool.
Choosing a proper starting value for the susceptible populations posed an issue. I used census data for each state and originally assumed 80% of those values, since about 80% of the US lives in urban / suburban areas. This gave me extremely higher infection populations than what’s being projected in the news. For example, for New York my model predicts ~7M people getting infected and 2M people needing to be hospitalized. To align with the reports from IHME, I reduced the initial susceptible populations to about ~2% of the states’ populations. This seems a little too low, but the resulting projections better align with those of IHME.
The medevac solution was in response to the terrifying projections, in particular Cuomo's claim that ventilator and bed shortages would kill thousands of New Yorkers. The idea and supporting models gained the attention of the New York Times, just days before a new set of IHME predictions were released. Over the two weeks of exploration and modeling the idea became less compelling because:
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Hospital capacity was expanded dramatically
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Hospitalizations stabilized at a much lower level than feared
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Peaks across all states occurred during a much shorter window
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Few days remained to implement a novel transportation network before the peaks
Although this solution was not implemented, its efficacy is a reminder that such high-flying ideas should not be immediately ruled out, but encouraged in times of crisis.
All supporting research is available on Github.