Online AppendixLimiting Spread of Resistant OrganismsPage 1

Online Appendix

Contents:

  • Construction of the Critical Care Transfer Network
  • Simulations on the Network: Simulating the Spread of Disease
  • Simulations on the Network: Comparing Alternative Infection Control Strategies
  • Running the Models
  • Approach to Sensitivity Analyses
  • Result of the Sensitivity Analyses

Construction of the Critical Care Transfer Network

We examined the potential for interhospital transfers of critically ill patients to spread highly resistant organisms. In order to do so, we used Medicare claims to map these interhospital transfers, and aggregate them. In the language of network science, we represent the transfers as a weighted, directed graph. We have examined this network in some detail in previous publications, but provide a focused discussion here to allow readers to more easily interpret the simulation data.[1]

We define a critical care hospitalization as any hospitalization during which the patient used an ICU or a CCU. As suggested by Halpern and colleagues, we excluded step down units and psychiatric intensive care units. [2, 3] We define a critical care transfer as occurring when a patient has two critical care hospitalizations at different hospitals, and when the admission date for the second hospitalization is the same or one day greater than the discharge date for the first hospitalization.

The critical care transfer network is the aggregation of all such transfers in one year. For our primary analysis, we used the MedPAR files from Medicare for 2005. The nodes in the transfer network are the hospital. The edges are the transfers, which go from a sending (first) hospital to the receiving (second) hospital. The weights on the edges are the number of transfers that occurred in a year. In general, these transfer relationships are asymmetric – most hospitals that exchange critical care patients do so in only one direction. Since this directionality is a key feature of actual hospital transfers, it is represented in the network.

For a sensitivity analysis, we examined similar data from the state of Pennsylvania for a 2 year period during 2004-2006. Data definitions were the same. These all-payer data were provided by the Pennsylvania Health Care Cost Containment Council.

Simulations on the Network: Simulating the Spread of Disease

We simulated the potential spread of an organism on this weighted, directed network. We will first describe the base case—that is, how the simulation works in the absence of any infection control resources for the maximal infectivity condition. Then we describe the modifications for the moderate infectivity condition. The next section considers the impact of infection control resources. The model is stochastic and run in discrete time. Each “tick” of the model is one day.

For one run of the simulation, we chose a hospital at random as the initiating site for the evolution of the highly resistant organism. Each hospitals probability of being selected is proportional to its number of critical care beds. (But another way, the start site is selected uniformly at random from the distribution of critical care beds.) On each sequential day, all transfer partners for the hospital are checked for possible transmission. Transmission occurs probabilistically, proportionate to the average number of transfers per day from the infected hospital to each receiving hospital. More formally, in the absence of infection control, the daily probability of transmission from hospital i to hospital j is pij = Ii * tij /365, where Ii = 1 if hospital i has been colonized by a highly resistant organism and 0 otherwise. tij is the total number of transfers observed in the network during 2005 -- the weights of the network edges, obtained from network construction. If hospital i never transfers patients to hospital j, then pij = 0. (In sensitivity analyses using the Pennsylvania data, the denominator was 730 as that data was for 2 years.)

The preceding paragraph described the basic mechanic by which interhospital transmission was operationalized. This process is then repeated on each subsequent day for each hospital to which the highly resistant organism has spread. Once a hospital acquires a highly resistant organism, it never clears the organism in this simulation; repeated transmissions of the organism to a hospital do not have any incremental effect on that hospitals’ transmission rate.

The moderate infectivity simulations build on this same approach, with two additional modifications. First, hospitals cannot transmit the organism to another hospital for 7 days after receiving transmission itself. Thus pij = 0 for the first 7 days after a hospital acquires the highly resistant micro-organism in transfer. Thereafter, the basic probability of transmission is one-tenth lower: pij = 0.1 * tij /365. All other mechanics of the simulation are the same.

Simulations on the Network: Comparing Alternative Infection Control Strategies

We considered 4 approaches to allocating infection control resources. In our base case we distributed 500 infection control resources.

In the random approach, hospitals were selected uniformly at random to receive an infection control resource. Because the number of hospitals was substantially greater than the number of infection control resources, only 16% of hospitals received at least one resource, less than 2% received at least two, and fewer than one hospital was expected to receive three or more resources.

In the degree centrality approach, the degree centrality was calculated for each hospital from the directed, weighted network. Infection control resources were allocated to each hospital proportional to the geometric mean of two network measures: in-degree (the number of transfers the hospital receives) and out-degree (the number of transfers it sends out). If the geometric mean kof the in-degree and out-degree of one hospital was twice as high as that of another, it would receive twice as many resources. Following this constraint, resources were allocated to the top hospitals, such that the last hospital to receive a resource would receive a fraction of what the top hospital received, in proportion to the ratio of their degrees k.

In the betweenness centrality approach, betweenness centrality was calculated for each hospital from the directed, weighted network. Again, the allocation was done proportionally, in this case in proportion to the betweenness of each hospital.

In the greedy algorithm, a set of simulations are run to iterativelyselect hospitals to receive units of resource, until the total amount of resource has been exhausted. At each iteration, a hospital is identified to receive a unit of resourceif it has the highest average number of beds that become exposed to the micro-organism at the hospital and all downstream hospitals where the infection subsequently spreads. The average number of exposed beds is calculated by simulating spread, with resource allocated thus far in place,over a year from all possible initial hospitals where the infection could originate. Once a given infection control resource is allocated, the iterative greedy algorithm then repeats itself, identifying the hospital at which the next infection control resource would most reduce the beds with the highly resistant organism.

Running the Models

Having established the basic mechanics of the simulations, each model was run repeatedly. The key outcome variable was the total number of critical care beds which had been exposed to the micro-organism. For the maximal infectivity condition, this was at the end of 1 “year”, or 365 simulation ticks. For the moderate infectivity condition, this was at the end of 5 years, or 1825 simulation ticks. We also examined the moderate infectivity condition at “1 year”, and it displayed similar results. However, the initial spread of under this condition was predictably slower, with its 7-day built in delay and lower transmission probabilities. Therefore, we present 5-year results for greater interpretability. Each simulation condition was run 33,060 times (starting 10 times from each hospital), with graphical examination of the results to insure that the outcome of interest were stable at the completion of the runs.In fact, the results are nearly indistinguishable if one simulates infections starting once from each hospital, or if one increases the number of runs ten-fold by starting infections ten times from each hospital.

Approach to Sensitivity Analyses

This model required a number of simplifying assumptions in order to examine the core dynamic. Thus, we modeled interhospital transmission of a generic highly resistant micro-organism, rather than of a particular organism. We tested the sensitivity of our conclusions to several policy-relevant choices: what happens if the organism spreads more slowly? What happens if you look only within a single state, rather than for the nation as a whole? What happens if you use all-payer data, rather than Medicare data? What happens if you allocate quite a bit more infection control resources? How rapidly do the allocations become outdated as the hospital transfer patterns evolve?

In theory, an infinite series of models could be used to test for some particular combination of parameters that might lead to idiosyncratically divergent results. Further, one might consider alternative probability distributions, or alternative approaches to distributing the infection control resources. We cannot prove that some areas of widely divergent behavior do not exist in the parameter space. A full search of the parameter space is beyond the scope of the current paper, and indeed is not technically feasible using existing equipment in a reasonable period of time. Our approach has been to combine network and clinical expertise to search the domains of the parameter space that we believed to be most likely to be of interest.

Result of the Sensitivity Analyses

We replicated our results in the Pennsylvania critical care data for 2004-2006, examining the spread of infection only within the single state, with all-payer data. In this Pennsylvania analysis we allocated 25 units of infection control (a number proportional to the relative number of hospitals in Pennsylvania). The greedy algorithm was clinically and statistically significantly better than the other algorithms, with similar patterns to the national data.The best performing allocation system was, again, highly unequal, with 14 of 25 resources being allocated to one hospital, and only 5 hospitals receiving any resources.

In order to test the stability of resource allocations over time, we simulated the situation in which resource allocations are based on transfer data for hospitals in one year (1998), but highly resistant micro-organisms spread over the network of transfers between hospitals for a later year. Had the greedy algorithm been used with 1998 transfer network data – rather than 2005 – there were would have been 1,092 critical care beds (s.d. 978) exposed at the end of the first year of spread on the 2005 network. This is a 15.6% increase versus allocations using the 2005 network (which had a mean of 944 exposed beds), but still statistically and substantively better than the performance of the other tested algorithms that used 2005 data.

We reran our nationwide analyses for 1,000 infection control (rather than 500) units for nationwide data, and for 84 rather than 25 units for Pennsylvania, and found substantively identical results.

We further reran our analyses with a sigmoidal relationship between the number of infection control resources and the transmission probability and found substantively identical results.

In order to test for bias caused by limiting our data to Medicare only, rather than all-payer data, we replicated the greedy resource allocation between the Medicare-only and the all-payer network in Pennsylvania. The results were substantively quite similar at a hospital-by-hospital basis and statistically, with a Pearson rank correlation of 0.76.

Appendix References

1.Iwashyna TJ, Christie JD, Moody J, Kahn JM, Asch DA, (2009) The Structure of Critical Care Transfer Networks. Medical Care 47: 787-793

2.Halpern NA, Pastores SM, Greenstein RJ, (2004) Critical care medicine in the United States 1985–2000: An analysis of bed numbers, use, and costs. Critical Care Medicine 32: 1254-1259

3.Halpern NA, Pastores SM, Thaler HT, Greenstein RJ, (2007) Critical care medicine use and cost among Medicare beneficiaries 1995-2000: Major discrepancies between two United States federal Medicare databases. Critical Care Medicine 35: 692-699