2.1 Transfer networks

We first define the network over which load and resources can be shared. Demand and capacity data –and thus, load sharing– can be coarse-grained at different resolutions: hospitals, postcodes, trusts, and broader regions. In this paper, we consider two levels of resolution: NHS trusts (UK) and autonomous communities (Spain).

2.1.1 NHS trust network.

We coarse-grain data for the UK at the level of trusts, as the main units of NHS organisation. We have N = 141 trusts across the UK, where each trust corresponds to a conglomerate of m hospitals. For each trust, we define a single central position by finding the centroid of the polygon whose vertices are the hospitals belonging to that trust. While spatial coordinates are given in terms of latitude and longitude, we make a small angle approximation and interpret latitude and longitude as cartesian coordinates Since sin α ≈ α when α ≪ 1, it is easy to see that an increase of a small angle α leads to a linear increase h ≈ Rα where R is a constant. In particular, under this approximation the centroid coordinates of trust reduces to the arithmetic mean of the coordinates of each hospital in the trust (1) In the event that the net capacity c_i of each hospital is also available, then instead of computing the centroid, one can compute the center of mass by appropriately weighting the contribution of each hospital: (2) where is the normalised capacity of hospital j we normalise it such that x and y still have dimensions of length, and m is the number of hospitals in trust i. The distance between two trusts corresponds to the Euclidean distance between the centroids or the centers of mass if more precise coordinates need to be used, instead of these we can use Haversine formula. (3) In our case we do not have information on the actual ICU capacity of each specific hospital within a given trust, so we choose to use centroids instead of centers of mass.

Once we have defined the location each of the 141 NHS trusts, we assign a vertex to this spatial location and proceed to tessellate this set. We build a regular geometric graph with degree k = 4, where each vertex i is connected to the four closest vertices according to the distance d_ij defined above. The resulting graph is depicted in panel a of Fig 1. Each trust will only be allowed to transfer patients or resources to the trusts in their topological neighborhood, modelling the fact that transfers only take place between close trusts.

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Fig 1.

(a) NHS trust network, where nodes are NHS trusts and the network is a 4-regular geometric graph tesselating the set of nodes, i.e. any node is connected to its four closest nodes. (b) Spanish contact network, where nodes are autonomous communities and two nodes are linked if the communities share a common border. In this network we have discarded both archipelagos (Canary islands and Balearic islands) as transfer between these and mailand is not realistic. We have not plotted Canary islands as it is off-scale. Background images have been generated using Natural Earth [23] and GADM [24].

https://doi.org/10.1371/journal.pone.0241027.g001

2.2 Local load sharing model

The basic architecture of the local load sharing model is depicted in Fig 2. For each node, the algorithm takes projected-ICU-demand data (aggregated at the NHS trust level or the autonomous community level, depending on the example), matches with its baseline-ICU-capacity (aggregated number of ICU beds or ventilators which are available to be used at that specific time by the trust itself or others), and generates a local-stress value for each node accordingly: For those nodes where the local stress is positive (meaning that demand surpasses the available capacity and thus there is a need to load-share), the algorithm explores which neighboring nodes (extracted from the topological neighborhood of the node under analysis) could accept a transfer. A transfer is possible if two conditions are met: (i) there is at least a node in the neighborhood of the origin node whose local-stress is negative (i.e. the receptor node has freely available space after its own demand is met), and (ii) the physical distance between the origin and the receptor node is smaller than a certain upper bound d_max. This maximum distance models at the same time several possible constraints, e.g. the fact that ICU patients can only be outside an hospital for a limited amount of time or that effective transfers require the distance between origin and receptor to be small.

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Fig 2. Scheme of the local load sharing model.

Red and orange denotes an overwhelmed unit with varying levels of stress, green denotes a unit with capacity.

https://doi.org/10.1371/journal.pone.0241027.g002

Once the receptor is chosen, a ‘solidary’ load is shared to the receptor. As a rule of thumb, we choose this load to be either 50% of the excess capacity of the receptor (that is, of |local-stress|, i.e. capacity after having met internal demand), or the total excess demand of the origin trust, whichever is smaller. The rationale for this definition is based on the fact that receptor nodes would probably be willing to accept to release only a percentage of their capacity while they keep another percentage in anticipation for future internal demand. The exact number (50% in this paper) is flexible and different practical implementations can assume different percentages.

2.3 Sequential vs parallel update

In this work we have systematically considered two alternative algorithmic updates, mirroring the fact that the local decision of a given node to transfer or not can be carried out either sequentially or in parallel for the rest of the nodes.

Let us first discuss the parallel mode. In this case, the projected-ICU-demand of all N nodes are updated in the new step independently from each other, and transfer between nodes take place independently. Note that this update can be problematic in practice, as e.g. two nodes might decide independently to transfer part of their excess load to a third node, what might mean overwhelming the third node. On the other hand, as we will show below, configurations which are globally more optimal are available using this mode.

In sequential update, the projected-ICU-demand of each node is sequentially updated after each local load share is performed. In other words, we consider the update of all N nodes in order. That means, for instance, that in a given step of the algorithm all N nodes are updated in order, such that the node p is updated taking into account the old status of nodes p + 1, p + 2, …, n (that haven’t been updated yet) and the new status of nodes 1, 2, …, p − 1. Sequential updates have the positive implication that no receptor will be overwhelmed from the simultaneous load sharing of different nodes.

Incidentally, the algorithmic difference between the sequential and the parallel update mode is similar to the difference between Jacobi and Gauss-Seidel numerical schemes when solving systems of linear equations. In practice, the code we have implemented asks the user to choose which processing mode is used (sequential or parallel).

2.4 Random search optimisation

The basic local load sharing model is run for all nodes (NHS trusts or autonomous communities), and as a result a possible load sharing configuration is extracted, consisting of the specified origin and destination of all the packets of ICU patients shared:

Trust i shared x loads to trust j

To assess the global impact of such load sharing configuration, we define the global stress of the whole system (4) where the sum runs over all trusts j, and Θ(x) is a rectified linear unit (ReLu), defined by Θ(x) = x if x > 0 and zero otherwise. So essentially global-stress counts the total demand of ICU units in excess of capacity, in all those trusts which are projected to be overwhelmed.

Now, in the event there is a node with positive local stress (i.e. with an excess of demand and a need to transfer load) and more than one candidate receptor, how to choose the adequate node where the load is shared to? A natural choice would be to follow a majority heuristic, i.e. transfer the share to the receptor with largest availability, i.e. with lowest local stress. However this choice does not always yield solutions which are globally optimal. This situation is illustrated in Fig 3, where the majority heuristic would suggest that node ‘2’ should transfer its excess load to node ‘3’ as its local stress is lower than the one at node ‘1’. However doing this precludes node ‘4’, also in need of load-sharing, to transfer loads to node ‘3’. In this case the optimal solution (in the sense of minimising global-stress) would be that node ‘2’ load-shares to node ‘1’, thus enabling node ‘4’ to load-share to node ‘3’.

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Fig 3. Cartoon of a simple chain graph where some nodes require to load share and a trivial majority heuristic rule provides a suboptimal solution.

https://doi.org/10.1371/journal.pone.0241027.g003

The example above is just a cartoon in an extremely simple graph. In more practical applications where transfer networks are more complex, the number of possible configurations is much larger and thus the issue is even more acute. To address this issue, here we implement a so-called random search optimisation approach, which consists in two steps. First, if more than one receptor is available for transfer in the topological neighborhood of a given node, then the algorithm selects the receptor at random. Second, once the algorithm chooses the configuration for all N nodes, it is then re-run 10⁵ times, such that in each realisation a different configuration is stochastically chosen. In this way the algorithm stochastically samples the search space. The quasi-optimal run with the lowest global-stress is finally retained. While this approach is computationally scalable (and can be implemented to be run in real-time in practical cases), the solution is guaranteed to be at least a local optimum, but one cannot discard that other configurations not sampled might have lower global-stress since search is not exhaustive.

2.5 Input variables

Now we briefly discuss the main input data required to run the local load sharing model:

• projected-ICU-demand: This is an input data to the algorithm. In practice this demand is either informed by some surveillance protocol or otherwise modelled, as it could be estimated following a complex multi-step flow [12], which can be summarised as follows:
  1. The projected number of new infections next week: This quantity can be informed in the first place from an epidemiological model [8, 26] which provides predicted numbers of contagion at different spatial resolutions. Alternatively, or in the absence of such a model, it could be estimated from various sources of data [27] including prescription data [28] or through direct questionnaires Data can be retrieved and processed from apps and other surveillance systems such as centralised webpages where citizens submit their symptoms. These questionnaires, coupled with a classification algorithm, can estimate the number of latent infected people in a certain region or postcode. A post-processing of these numbers is then carried out, taking into account (i) age demographics and (ii) associated infection-to-ICU rates.
  2. The projected number of patients already in the hospital which progress to ICU by next week: this number is estimated from real data of hospital admissions and average admission-to ICU likelihood.
  3. The projected number of patients already in ICU this week which will still require ICU next week: this number takes into account both the fatality ratio and the estimated discharge time.
  
  As a proof of concept, in this work we assume different types of artificial ICU demands (uniform and heterogeneous distributions) in the UK case, whereas in the spanish case we consider realistic demand as of 30th March 2020, i.e. during the first epidemic wave. We test how the load sharing algorithm performs under different demands.
• baseline-ICU-capacity: This list is extracted from public available databases [29, 30]. In the case of autonomous communities these quantities already have into account some enhancement provided by surge capacity [30], whereas in the case of NHS trusts we only use baseline data, so we expect such capacity to be significantly increased in practice.

3.1 Single-share in the UK NHS trust network

In this first section we assume that each trust can only submit a unique load to a unique receptor trust, to be selected randomly from the trust’s topological neighborhood.

3.1.1 Stress test with fixed, uniform-load ICU demand.

As an initial illustration, we first analyse a stress test case where projected-ICU-demand is artificially set to a uniform value of 20 ICU beds per trust (i.e. all trusts receive a demand of 20 beds) whereas we set all baseline-ICU-capacity to its real value, and d_max = ∞. The histogram of baseline-ICU-capacity is reported in panel (a) of Fig 4, whereas the histogram of local-stress, before and after the load sharing procedure is performed, is depicted in panel (b) of the same figure (we are only showing the parallel mode here). The procedure is capable of reducing the global stress of the system from an initial value of global-stress = 611 ICU beds in excess in overwhelmed trusts, to a final value of global-stress = 101 after the optimal load sharing is performed, i.e. a transfer and subsequent treatment of 510 ICU patients.

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Fig 4.

(a) Histogram of the baseline-ICU-capacity (number of beds) per trust for the NHS trust network. (b) Illustration of the histogram of local-stress (expected demand of number of beds above capacity) per trust, before and after applying the load sharing procedure. In the synthetic example, all trusts have a uniform projected-ICU-demand = 20, whereas the baseline-ICU-capacity is informed by data and shown in the left panel. Before the load sharing procedure, global-stress = 611, and after the procedure, the new global-stress = 101, i.e. a reduction of a total of 510 ICU patients (83%). (c and d) Response of the UK healthcare system in terms of global-stress (c) and net reduction in number of ICU beds (d) after the load sharing procedure is applied, as a function of the initial demand per trust (uniform demand across trusts). Different lines correspond to different modes: absence of load sharing (red); single-share, parallel mode (green); single-share, sequential mode (blue); multiple-share, sequential mode (black). Results are similar across modes. We can see three regimes: an initial regime where the load sharing procedure easily removes all signs of overwhelming, a second regime where although the procedure cannot remove all signs of overwhelming, the net reduction is maximised, and a third regime where the load sharing procedure is less and less efficient due to the fact that the whole system is overwhelmed.

https://doi.org/10.1371/journal.pone.0241027.g004

3.2 Multiple-share in the UK NHS trust network

In this second section we relax the single-share assumption and allow each trust to share multiple loads to multiple receiving trusts, selected from the trust’s topological neighborhood at random. For this analysis we drop the parallel mode and only consider the sequential processing mode, where real values of local-stress are updated in a sequential way as load sharing is performed.

In the uniform-load stress test, enabling a multiple-share option in the sequential mode provides an improvement in the net reduction of cases when compared to the single-share case. However, the improvement is not large (see panel (d) of Fig 4), and puts the multiple-share sequential mode on a similar footing to the single-share parallel mode, while guaranteeing that no receptor trust is overwhelmed. This result is easy to interpret: there is not much gain in being able to share loads to several receptor nodes at once (as opposed to only one), because on average this possibility will only be useful in a handful of cases. In other words, this result is a byproduct of imposing a uniform-load.

A different result is expected if the initial demand on each node is not uniform. Suppose, for instance, that we have a few trusts that are extremely overwhelmed, and could in principle share loads with several receptors (more than one available receptor in its topological neighborhood), but suppose that those receptors are small trusts with only a small number of available ICU beds. In that case, a single-share approach is clearly deficient, but a multiple-share approach could indeed provide a notable improvement. We illustrate this case in what follows.

3.2.1 Heterogeneous-load stress test.

Instead of loading a uniform demand in each trust, we now test the scenario where demand is heterogeneous, and we only overwhelm ‘large’ trusts. To model such demand, we assume that if the trust originally has a baseline-ICU-capacity larger than a certain pre-defined threshold τ, then we set an initial value for projected-ICU-demand for this trust equivalent to 120% its baseline-ICU-capacity (i.e. we set put that node in a situation with positive stress, 20% above capacity). Similarly, for those trusts whose baseline-ICU-capacity is smaller than the threshold τ, we set an initial projected-ICU-demand equivalent to 80% of their corresponding baseline-ICU-capacity (i.e. 20% below capacity).

We then apply the load sharing procedure sequentially and compare the net reduction of the global level of stress (number of ICU patients that can be efficiently transferred) for the single-share and the multiple-share options. In Fig 5 we plot these results as a function of the threshold τ. First, note that for very small or very large values of τ both methods are similar. This is expected because in those limiting cases, either all nodes are overwhelmed (τ very small) or virtually no nodes are overwhelmed (τ very large). Thus in both cases there is no gain in performing a multiple-share over a single-share, as the net reduction of stress is small (either because we cannot transfer any load as all nodes are overwhelmed, or because no nodes are overwhelmed and there is no loads to share). Second, for the large range of intermediate values of τ where load is heterogeneous, we indeed find that the multiple-share option is much more efficient than the single-share one for a large range of values of τ, as expected.

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Fig 5.

Global stress (panel a) and Net reduction (panel b) offered by the sequential load sharing procedure vs the threshold τ (see the text), for a single-share and a multiple-share option, in the UK system undergoing a synthetic heterogeneous-load stress test. The multiple-share option clearly outperforms the single-share one in this case.

https://doi.org/10.1371/journal.pone.0241027.g005

3.3 Multiple-share in the Spanish autonomous communities contact networks

We now consider the second case: the Spanish healthcare system at the level of Spanish autonomous communities. Recall that there are 17 autonomous communities in Spain, and healthcare is decentralised so that each autonomous community runs its own system in a semi-independent way. To explore load sharing effects at the inter-community level, instead of adapting the 4-regular network to this context we have constructed two transfer networks: (i) a local contact network of 15 nodes (all autonomous communities in mainland Spain), where two nodes are linked if they share a border, and (ii) a fully connected network of all 15 autonomous communities in mainland Spain. The former allows for faster transfers, whereas the latter requires using national rail resources [23].

In both cases we use a sequential multiple-share mode. The ICU-baseline-capacity for each node is extracted from public data and considers both baseline and surge capacity on 30 March 2020 [28], and the projected-ICU-demand is initially set in terms of the ICU occupation number on 30 March 2020 [28]. The average is 63% of the national health system capacity, i.e. all autonomous communities are below capacity showing 63% load. We then increase the demand in each autonomous community, and explore how the load sharing procedure alleviates overwhelming. In the top panels of Fig 6 we illustrate a scenario, where the Spanish health system is globally overwhelmed (about 200% of the initial demand recorded on the 30th March 2020, or 130% above surge capacity). After load sharing using the contact network, some autonomous communities substantially alleviate such excess and for some others such excess is completely removed. In the ideal scenario where a fully connected network can be used, the load sharing is greatly enhanced. In the bottom panels of Fig 6 we plot the global stress and net reduction (total number of ICU beds or ventilators which are effectively transferred) as a function of the national health system saturation (in %), for either using the local contact network or the fully connected network. Both cases enable substantial transfers (about 600 for the local contact network and up to 1300 if transfer is done countrywide, for a single step of the algorithm). Both cases are indeed able to delay global overwhelming, and in the case of the fully connected network the algorithm can maintain the local stress of every autonomous community below capacity even when the true global saturation is around 100%.

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Fig 6.

(Top panels) Color-coded local stress of each autonomous community in Spain, before (left panel) and after load sharing (middle and right panels). Background images have been generated using Natural Earth [23] and GADM [24]. Canary islands is absent because it is an isolated node and is off-scale. The initial demand is 200% above the real demand as of 30th March 2020, i.e. an approximate demand 130% above surge capacity. In the middle panel a local contact network is used, whereas in the right panel a fully connected network is used. The local contact network is able to transfer and alleviate about 467 ICU beds or ventilators, whereas the fully connected network can transfer 933 units. (Bottom panels) Global stress (left panel) and Net reduction (total number of ICU beds or ventilators efficiently transferred, right panel) offered by the sequential, multiple-share load sharing procedure performed on the Spanish health system, coarse-grained at the level of autonomous communities, as a function of the global saturation percentage of the system (note that capacity has already been enhanced thanks to surge capacity). Orange dots correspond to the contact network, whereas purple squares correspond to the fully connected network.

https://doi.org/10.1371/journal.pone.0241027.g006

For the local contact network, we can distinguish a first phase of steep increase, where only a few communities are overwhelmed and the algorithm is maximally efficient, until the saturation reaches about 120% of the capacity. Then in a second phase, the procedure is still able to transfer many beds or ventilators –even if some autonomous communities will still be overwhelmed–), peaking at a maximum of about 600 beds or ventilators when the system is globally at 170% capacity. As the system gets more and more overwhelmed globally, the load sharing algorithm loses efficiency and the amount of loads that can be shared starts to decrease.