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The authors have declared that no competing interests exist.

As the number of cases of COVID-19 continues to grow, local health services are at risk of being overwhelmed with patients requiring intensive care. We develop and implement an algorithm to provide optimal re-routing strategies to either transfer patients requiring Intensive Care Units (ICU) or ventilators, constrained by feasibility of transfer. We validate our approach with realistic data from the United Kingdom and Spain. In the UK, we consider the National Health Service at the level of trusts and define a 4-regular geometric graph which indicates the four nearest neighbours of any given trust. In Spain we coarse-grain the healthcare system at the level of autonomous communities, and extract similar contact networks. Through random search optimisation we identify the best load sharing strategy, where the cost function to minimise is based on the total number of ICU units above capacity. Our framework is general and flexible allowing for additional criteria, alternative cost functions, and can be extended to other resources beyond ICU units or ventilators. Assuming a uniform ICU demand, we show that it is possible to enable access to ICU for up to 1000 additional cases in the UK in a single step of the algorithm. Under a more realistic and heterogeneous demand, our method is able to balance about 600 beds per step in the Spanish system only using local sharing, and over 1300 using countrywide sharing, potentially saving a large percentage of these lives that would otherwise not have access to ICU.

The outbreak of COVID-19 [

In the COVID-19 pandemic, demand for intensive care is not uniform across a country. Epidemic outbreaks can take place in different parts of a country and this can lead to substantial variations of demand both through space and time. Some hospitals may receive substantial numbers of patients early in an outbreak, whilst others may be only mildly affected. This demand heterogeneity opens the possibility of balancing the load of patient admissions such that excessive demand is re-routed to the places which have spare capacity. The clinical need for such a system was evidenced by a spontaneous initiative that took place in Madrid (Spain) in early April 2020 [

A natural question is thus, given the available resources of a national health system covering a specific region, whether there exist a principled, adaptive and

The methodology is in principle tailored to address the COVID-19 pandemic situation, but otherwise is general and thus applicable in different countries, at different resolution levels, and for any resource constrained clinical service. The method uses graph-embedded load balancing technology coupled with a simple optimisation kernel, and we showcase its usability by testing it on the UK National Health Service (NHS) and the Spanish health system as examples with different spatial granularity. Note that graph-embedded load balancing [

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).

We coarse-grain data for the UK at the level of trusts, as the main units of NHS organisation. We have _{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:

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 _{ij} defined above. The resulting graph is depicted in panel a of

Spain has a decentralised health system, so we consider that load sharing between hospitals can only take place within each autonomous community (intra-community). Because of that, as a second example here we will consider load sharing at the inter-community level. The network therefore has

Additionally, we will also consider a fully connected network formed of

The basic architecture of the local load sharing model is depicted in _{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.

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

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 |

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

Let us first discuss the parallel mode. In this case, the

In sequential update, the

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).

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:

To assess the global impact of such load sharing configuration, we define the global stress of the whole system

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

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 ^{5} 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

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

The projected number of new infections next week: This quantity can be informed in the first place from an epidemiological model [

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.

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.

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.

As an initial illustration, we first analyse a stress test case where _{max} = ∞. The histogram of

In a second step, we explore how the system behaves when initial demand per trust varies. To do that, we consider a suite of stress tests and assume for each test that all trusts receive the same load –leading to a uniform demand per trust–, and we compute the

Results are shown for both the sequential and parallel mode in panels (c) and (d) of

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

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

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.

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

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

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 [

In both cases we use a sequential multiple-share mode. The

The COVID-19 pandemic is putting the national health systems of several countries under significant pressure. In this scenario, it is important to devise strategies that distribute capacity of hospitals, not only in terms of the number of ICU beds or ventilators, but also overall capacity (critical care, acute capacity, etc). Here, we have detailed such methodology and have implemented and validated it at two different resolutions: at the level of NHS trusts in the UK and at the level of autonomous communities in Spain. All data and code are available

In the context of COVID-19, adopting a load sharing strategy is likely to be beneficial when the whole system is not completely overwhelmed, the projected ICU demand can be accurately estimated, and facilities exist to transfer either patients between ICU departments or ventilators. This is likely early on in the exponential growth phase (of each wave), or in situations where demand is declining either due to interventions or towards the end of the pandemic. When the system is already fully overwhelmed or soon-to-be, this strategy is likely to be inefficient. Furthermore, we also expect this approach to be useful as the epidemic reaches a declining phase, helping to reduce demand and allowing hospitals to return back to normal in a fast an optimised way. Note that we chose to validate the method in two countries (UK and Spain) as we could focus at two different spatial granularities. However, the method is directly applicable to other countries as well, as long as any sort of transfer system can be put in place. From a clinical point of view, an important point to consider is whether the load sharing can be activated at the ICU stage –potentially leading to transferring highly unstable patients who require ambulance with ICU equipment as well as trained personnel– or if, in anticipation to this, transfer needs to be planned at the point of hospitalisation (admission). In the latter scenario, planning needs to further take into account not only baseline ICU capacity, but overall capacity, also factoring in the estimated lag between admission to hospital and the need for ventilators, which for COVID-19 is currently estimated at about 2 to 3 days. The adequate strategy will also depend on the operational capacity of the system and the country where it is applied to. For illustration, this work explicitly considers the transfer of ICU patients, however exactly the same approach can be followed if the load to be shared is not patients but ventilators (the units to be moved are not ICU patients but ventilators, so transfer simply happens in the opposite direction, from receptor to origin). Assuming the receptor has both room and personnel to handle additional ventilators, this alternative would indeed (i) eliminate the burden on transferring highly unstable patients and the associated resources required to make such transfers, and (ii) the risk of transferring infection along with patients. Of course, risk (ii) is removed if one only transfers non-COVID ICU patients. In reality, a combination of these mechanisms (transferring ICU patients and ventilators) for sharing load is possible.

This work is subject to several limitations which we hope will be addressed in future work.

First of all, the baseline ICU demand only takes into account surge capacity in the Spanish case: more realistic analysis of the UK case shall include surge capacity, that is expected to significantly increase the real ICU capacity of each trust.

Second, in the sequential case (where receptors cannot be overwhelmed), overwhelmed nodes can at most share all the excess load, but not more (this latter case would be beneficial if e.g. two-step sharing is needed), therefore multiple-step load sharing strategies have not been explored.

Third, the optimisation process implemented here is based on a stochastic search. This method was chosen for simplicity and computational efficiency, but there is no mathematical guarantee that the suggested configuration is indeed the global optimum. More sophisticated methods such as hill climbing, genetic algorithms or simulated annealing could be used to refine this layer, if at all needed. Other extensions of interest include questions related to dynamic load balancing where the demand varies dynamically.

Fourth, and on relation to having distance between nodes as a limiting factor, note that while we have implemented such restriction (_{max}) in the code, for simplicity in this work we have set _{max} = ∞. The justification is that in two out of three realistic cases considered in this work, distance is already implicitly considered in the topology of the transfer network. For instance, in the UK case (NHS trust network), transfers are already restricted to happen only within the _{max}, or even different values of _{max} for different regions. All these are interesting extensions which would be relevant for a practical application of the model we present.

Finally, we have assumed that the cost of transfer is zero, i.e. the number of ambulances or the human resources are not a constraint, and that there are enough vehicles to transfer ICU patients or ventilators effectively and enough qualified personnel to handle them. All these limitations can be addressed by suitably extending the specifications of the algorithm, leading to multi-criteria optimisation problems.