Hospital management in the COVID-19 emergency: Abelian Sandpile paradigm and beyond
Roberta Martucci, Corrado Mascia, Chiara Simeoni, Filippo Tassi
HHOSPITAL MANAGEMENT IN THE COVID-19EMERGENCY: ABELIAN SANDPILE PARADIGMAND BEYOND
ROBERTA MARTUCCI, CORRADO MASCIA, CHIARA SIMEONI,AND FILIPPO TASSI
Abstract.
In this article, we propose a mathematical model –based ona cellular automaton– for the redistribution of patients within a networkof hospitals with limited available resources, in order to reduce the risksof a local/global collapse of the healthcare system. We attempt at de-veloping a conceptual tool to support making rational decisions relevantto the optimisation of the allocation of patients into accessible medicalfacilities. The strategy is based on a version of the Abelian Sandpilemodel for the Self-Organised Criticality, with the idea of testing theparadigm for the management of patients among the COVID-19 hospi-tals in Italian regions. In particular, we compare the novel proposal tothe standard management of connections between hospitals, showing anumber of advantages at a local and global level, by means of a reli-able indicator function introduced for measuring the effectiveness of theallocation strategies. Introduction
Since the beginning of 2020, people from all parts of the world are strug-gling with the COVID-19 coronavirus pandemic, which has led to a hundredmillion of infections and over 2 million deaths worldwide [20, 49]. The healthemergency is not over yet and it will take months before being under control,possibly becoming endemic among most of the world population [32].The current data suggest the paramount importance of acting promptly, andespecially for trying to reorganize the medical facilities in order to avoid satu-ration of the critical care capacity in the various territories (refer to Figure 1for the evolution of employment of the intensive care units in Italy). Indeed,several countries have been experiencing recurring epidemic waves, with aconsequent worsening of the social and economic conditions, thus increasingdramatically the pressure on the healthcare systems.The emergence of infectious diseases has to be considered as an inher-ent property of human and animal populations, which cannot be generallyavoided and/or foreseen: before the present COVID-19 emergency, SARS(2002-2004) and influenza A/H1N1 (2009-2010) are other recent notable in-cidents [50, 51].
Mathematics Subject Classification.
Key words and phrases.
Healthcare system management; Predictive logistics; Cellularautomata; Abelian Sandpile Model; Allocation processes. a r X i v : . [ m a t h . O C ] F e b R. MARTUCCI, C. MASCIA, C. SIMEONI, AND F. TASSI data intensive care unit
March
April
May
June
July Agost September
October
November Figure 1.
Occupancy level (aggregate data) of intensive careunits in the Italian healthcare system.
Most epidemics are characterised by outbreaks in localized geographical ar-eas, and there may be no pharmaceutical solutions already available to safe-guard the people’s health when the disease begins to spread largely around.Moreover, the development of effective vaccines typically takes a ratherlong time, and therefore temporary alternative strategies must be imple-mented (quarantine, travel restrictions, activities closing, social distancing,face masks obligation, ...) However, the demand for specific healthcare facil-ities such as hospitals consistently increases, reaching and sometimes exceed-ing critical levels. For example, in the case of COVID-19, the saturation ofintensive care units depends on the amount of patients requiring mechanicalsupport for ventilation and proper equipment, so that any healthcare systemcan be overwhelmed if the number of severe cases becomes very high (referto Figure 2 for the present-day situation of intensive care units in Italianregions).In that context, the rapid filling of the so-called COVID-19 hospitals inthe areas most affected by the disease represents a serious danger whiledealing with sudden emergencies at different scales (villages, cities, regions,countries). Hence, it is compelling to develop new arrangements of thehealthcare system, for optimizing the overall structure especially in termsof an efficient distribution of patients among the most suitable facilities.In this article, we present innovative intervention options, by comparingthe standard organisation of the Italian healthcare system with a novel pro-posal based on the Abelian Sandpile paradigm. We provide a systematicapproach for an improved planning of the healthcare system by means ofa mathematical model for the self-organisation of critical issues, which iscapable to achieve some specific optimization inside the hospital manage-ment. In particular, we aim at reducing the risks of a local/global collapseof hospitals in times of crisis, while improving their functioning in normal
OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 3
Saturation of intensive care units (data updated to the 23/01/2021)
30 100PiemonteValle d'AostaLombardiaBolzanoTrentoVenetoFriuli Venezia GiuliaLiguriaEmilia RomagnaToscanaMarcheUmbriaLazioAbruzzoMoliseCampaniaPugliaBasilicataCalabriaSiciliaSardegna
La saturazione delle terapie intensive
La saturazione delle terapie intensive..
UmbriaLazio Lombardia
Italia
Figure 2.
Saturation of intensive care units in the Italian regions(updated to January 23, 2021). Source: Il Sole 24 Ore – elaborationon data from the Italian Civil Protection Department times. More specifically, the basic model is grounded in the framework ofcellular automata and postulates a large network of links between hospitalsin a cooperative style of communication. Unfortunately, such connectionsare currently very limited, which drastically restricts the possibilities of real-location for the supernumerary patients, and the strong urgency to enhancethe hospital network is an important conclusion of the present analysis.The content of this article is organized as follows.In Section 2, we introduce the Abelian Sandpile paradigm, with a shortdescription of its basic concepts and properties. Section 3 focuses on theadaptation of such paradigm to build up a novel proposal for the organisationof the healthcare system. In Section 4, we provide a gallery of examplesaiming at illustrating how the proposed model works in the present contextof COVID-19 epidemic. More specifically, we analyze two cases relevant torealistic applications: the central and peripheral outbreaks. The article isconcluded by Section 5 which includes some generalisations of the modeland (provisional) conclusions.
Abbreviations
ASM – Abelian Sandpile Model
SOC – Self-Organized Criticality
CAM – Cellular Automaton Model
SRH – Sandpile with Redistribution to the Hub
SID – Sandpiles with Internal Dissipation2.
Abelian Sandpile paradigm
The notion of Self-Organized Criticality (SOC) was originally introducedby Bak, Tang and Wiesenfeld [4, 5] starting from a basic example proposed asa model for sandpiles (we refer to [3, 22, 37] for a general introduction to thisbroad subject). Since then, the concept has been expanded in many different
R. MARTUCCI, C. MASCIA, C. SIMEONI, AND F. TASSI directions, spanning from classical topics of physics (sandpile avalanches [27],distribution of earthquakes [35, 47], amplitude of solar flares [39]) to lessstandard economic and socio-political contexts [2, 7, 9, 13, 24, 38, 42, 44, 52],passing through computer networks and biological applications [1, 40, 53,15]. At the same time, a huge effort to extend the mathematical tools to dealwith theoretical questions has been made, thus contributing to drive SOCinto an extraordinary crossroads of probabilistic approaches, graph theory,algebraic geometry, mathematical analysis and optimisation [6, 8, 10, 11,12, 23, 28, 41].2.1.
SOC and Sandpiles.
The original application concerns with model-ing of sandpiles, which are regarded as a manageable prototype of SOC, andfundamental contributions have been made by Dhar [17, 16, 18] notably inshowing the crucial property of commutativity. On this account, the ad-jective Abelian has since been added to the technical terminology, leadingto the actual meaning of Abelian Sandpile Model (ASM). Nevertheless, asusual in the most active fields of science, the terminology is not unique,and indeed similar topics are explored under different names (avalanches,chip-firing games, forest-fire models, parking functions, probabilistic aba-cus, Rotor-Router or Eulerian-walker models, ...) with more or less the samemeaning. We refer to [14, 19, 29, 30, 36] for introductory presentations ofthe ASM and its various applications.We are particularly interested to the load balancing property, which de-notes the method of distributing a set of tasks across a group of resourceswith the purpose of making their overall processing more efficient by balanc-ing the workload of each operating unit. In the abstract formulation, unitsare represented by vertexes/nodes of a graph/network, with the correspond-ing connections represented by edges/links. The objective is to balance theloading process by allowing nodes to exchange particles with their neighborsthrough the incident edges.For the problem covered in this article, we build a specific ASM with criticalheight provided by the capacity of the medical facilities, and we propose toapply the Abelian Sandpile paradigm to achieve a methodical and efficientdistribution of patients across the hospitals, in order to maintain the oc-cupancy below the saturation level for dealing with eventual sudden andunexpected emergencies.A sandpile is also a type of Cellular Automaton Model (CAM), namely adiscrete mathematical system fulfilling the following essential conditions:1. the evolution takes place without external interventions;2. the overall structural development depends only on local rules.To a large extent, the CAM is capable of simulating the dynamics of complexsystems which are disposed to organize themselves through unstable criticalstates until reaching a stable configuration. In addition, the CAM imple-mentation makes it possible to generate global coherent patterns startingfrom suitable local instructions, without any external supervisor in chargefor understanding the process in its entirety [31, 43, 48].In practical applications, each restricted portion of space contains a finitenumber of cells, which assume a finite set of (time-dependent) states. The
OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 5 initial configuration ¯ z is the combination of a ground state configuration z with some perturbation w , and for instance¯ z = z + w . After a given time interval ∆ t >
0, the system typically comes to a newstate z , designated final configuration, which is determined by the changesof state of the single cell together with all the others, according to thepreliminarily established series of (local) rules. Therefore, the evolution isdescribed by means of a function Φ mapping some input ¯ z defined (from z and w ) over the graph/network Γ to an output z , as specified by thesimple formula z = Φ(¯ z ).It is worth stressing that, in order to properly manage the patients be-tween hospitals of the healthcare system, an optimal underlying ethical-cooperative paradigm has to be stipulated [34].2.2. Sandpiles on a general network.
We start by recollecting somebasic definitions in graph theory [45, 25].
Definition 1. A graph (or network ) is a pair Γ = (
X, E ) where X is aset whose elements are called vertexes (or nodes ) and E is a set of pairedvertexes called edges (or links ). A rooted graph (or pointed graph ) isa graph in which one vertex, denoted by x ∗ , is distinguished as the root ofthe graph. In what follows, the root x ∗ of the graph is also referred to as the hub ,coherently with the subsequent meaning of the specific application to thehealthcare system. Definition 2. An adjacent vertex y ∈ X of x ∈ X is a vertex which isconnected to x by an edge, so that ( x, y ) ∈ E . The neighbourhood I ( x ) of a vertex x is the subgraph composed of the vertexes adjacent to x and allthe edges connecting vertexes adjacent to x . The degree (or valency ) ofa vertex x , denoted by deg( x ) , is the number of edges which are incident tothe vertex x . Let the graph Γ be connected, finite (with a finite number of vertexesdenoted by x , x , . . . , x p ), simple (i.e. there are no loops connecting anyvertex with itself) and undirected (i.e. the edges are bidirectional). Weassume that a stock of identical particles is initially allocated at any vertex x i of the graph Γ. The configurations z , ¯ z and z , collecting the number ofelements located at x i for any i , are natural-valued functions defined on thegraph. The rules of the evolution are established to guarantee stability ofthe sandpile dynamics, starting from the definition of the height function given by z : X → N p (refer to Figure 3 for the graphical representation ofa sandpile on a two-dimensional Cartesian grid). Definition 3.
The height z i = z ( x i ) is unstable (at x i ∈ X ) if z i ≥ deg( x i ) , where deg( x i ) is the degree of the vertex x i . Otherwise, the vertexis stable .A stable configuration is a configuration of vertexes which are stable forany index i . A maximum stable configuration (or minimally stablestate ) is a configuration in which all the vertexes have a height z i of one unit R. MARTUCCI, C. MASCIA, C. SIMEONI, AND F. TASSI
Figure 3.
A simple sandpile with variable height (on a two-dimensional Cartesian grid) lower than the corresponding threshold value deg( x i ) . An almost stableconfiguration is a configuration of vertexes which are stable for any index i different from the root, and a maximum almost stable configuration is defined accordingly. In that framework, an unstable vertex is forced to topple over its neigh-bouring vertexes, thus causing a change of state in the entire configuration.
Definition 4. A toppling (or firing ) from the vertex x i is the mapping Ξ from the graph Γ to itself determined by the following (local) rules: Ξ( z ) j = z j − ∆ i → j with ∆ i → j := deg( x i ) if x j = x i − if x j ∈ I ( x i )0 otherwiseand ∆ i → j is called the toppling matrix [17] . In principle, different stability criteria could be proposed involving valuesother than the degree deg( x i ) of the vertex x i , and modified redistributioncriteria could also be considered, but we focus on the above definitions forthe sake of simplicity in presentation.The addition of a particle to the sandpile is schematized by summingto the ground state configuration z = ( z , z , . . . , z p ) the vector δ i with 1at some fixed index i and zeros elsewhere. By iterating this procedure m times for possibly different choices of the index, the initial configuration ¯ z is finally given by z + w with w = δ i + δ i + · · · + δ i m . It is a straight-forward consequence that the operation of adding particles to a sandpile isassociative. Moreover, the final configuration z resulting from the evolu-tion is independent of the order in which the topplings are performed, and OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 7 therefore the operation of toppling is commutative [17].As an illustrative case, we assume that deg( x i ) = deg( x j ) for two adjacentvertexes x i and x j . Then, we consider a sandpile in which both x i and x j are at their critical height, that is z i ≥ deg( x i ) and z j ≥ deg( x j ). Accord-ing to the dynamics described above, a toppling from the vertex x i causesthe vertex x j to become unstable, and viceversa, and the sandpile comes toa configuration in which the height z k for some k ∈ { , , . . . , p } increasesby a value (cid:52) i → k + (cid:52) j → k . Such procedure being symmetrical, by applyingthis argument repeatedly, we conclude that the same stable configurationis reached irrespective of whether x i or x j is toppled first, and in generalregardless of the sequence in which unstable vertexes are toppled.2.3. Two-dimensional Cartesian grids.
We focus on the simple case ofa two-dimensional Cartesian grid, with two standard schemes for adjacentvertexes given by the von Neumann neighbourhood, consisting of the fourcells obtained moving one step towards North/East/South/West (refer toFigure 4(left)), and the Moore neighbourhood, which includes also the fourdiagonal cells (refer to Figure 4(right)). Of course, selecting a more generalnetwork provides a higher degree of flexibility and allows to closely representthe connections between various medical facilities of the healthcare system(refer to Figure 5).
Figure 4.
The von Neumann neighbourhood (left) of the (blue)central cell (an extended neighbourhood includes also the pinkcells) and the Moore neighbourhood (right)
We consider a special grid of p = n cells organized over a squared matrixof size n × n , where the particles are dropped randomly being allowed tostack on top of each other, so that a configuration of the sandpile is describedby an element of the space of natural-valued matrices M n ( N ) := (cid:8) A ∈ R n × n : a ij ∈ N for any i, j = 1 , , . . . , n (cid:9) . The position of each vertex x i (with its corresponding height z i ∈ N ) isdetermined by the index i ∈ { , , . . . , p = n } according to the following R. MARTUCCI, C. MASCIA, C. SIMEONI, AND F. TASSI common practise: . . . nn + 1 n + 2 n + 3 . . . n n + 1 2 n + 2 2 n + 3 . . . n ... ... ... . . . ... n ( n −
1) + 1 n ( n −
1) + 2 . . . . . . n For the von Neumann neighborhood, we have deg( x i ) = 4 for any index i associated with an element of the bulk of the matrix, and for instance I ( x n +2 ) = (cid:8) x (North) , x n +3 (East) , x n +2 (South) , x n +1 (West) (cid:9) . Similarly, for the Moore neighborhood, we have deg( x i ) = 8 for any index i associated with an element of the bulk of the matrix, and for instance I ( x n +2 ) = (cid:8) x (North) , x (North-East) , x n +3 (East) , x n +3 (South-East) ,x n +2 (South) , x n +1 (South-West) , x n +1 (West) , x (North-West) (cid:9) . Example 1.
We analyze the simple case n = 3 ( p = n = 9) with the vonNeumann neighbourhood. By combining the perturbation w = 4 δ withthe ground state configuration z = , we obtain the following initial andfinal configurations¯ z = + 4 δ =
00 0 0 −→ z = Ξ(¯ z ) = As expected, the threshold value deg( x ) = 4 is reached at the centralcell, thus inducing a destabilisation of the corresponding vertex, with theconsequence that four particles are poured into the adjacent vertexes –theelements of the von Neumann neighbourhood– to compose a stable finalconfiguration.As a matter of fact, the toppling from an unstable vertex changes the stateof the adjacent vertexes, sometimes causing the appearance of instabilitiesat some other vertex. Then, subsequent topplings over the adjacent vertexesare generated, triggering a sequence of events which are evocative of sandpileavalanches and stopping only when all the cells return strictly below theirthreshold capacity.If the graph is infinite and connected, with a finite number of particles,all vertexes become stable after a finite number of topplings; moreover, itcan be proven that the final (stable) configuration z depends solely on theinitial configuration ¯ z , independently of the order in which the topplingsare performed [17].On the other hand, if the graph is finite, appropriate boundary conditionsmust be implemented: assuming that the particles are evacuated from theboundaries, an analogous result of stability holds [14], but other types ofboundary conditions may not guarantee the same property. During theoperation of toppling, no particles are created as they are redistributed toneighbouring cells. For dissipative boundary conditions, the toppling canactually lead to the loss of particles if it occurs on a boundary cell, and werefer to this case as open boundary conditions. OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 9 Application to healthcare system management
In this article, we pursue the idea of applying the Abelian Sandpile par-adigm to the management of patients among the COVID-19 hospitals inItaly.The current organisation assigns responsibility for the healthcare system tothe regions, which are restricted territorial bodies with their own statutes,powers and functions established by the Italian Constitution. We refer tothe Lazio region when selecting (the order of magnitude of) the number ofmedical facilities relevant to the numerical simulations. More specifically,the healthcare system of the Lazio region is composed of about 100 hospi-tals, mostly located in the metropolitan area of Rome, which are structuredwithin a network of reciprocal connections [33].
Figure 5.
Arrangement of hospitals on a general unstructuredand undirected network.
In that framework, the height function of the nodes (hospitals) indicatesthe number of hospitalised patients, and the links between different nodesrepresent the possibility of a direct exchange of patients (that is subjectto constraints of geographical proximity, together with inherent organisa-tional requirements of the connecting medical facilities). The node of thenetwork exhibiting the largest number of links is the hub , which usuallytakes on a strategic function for the whole system, whilst the nodes withfewer links identify the hospitals with unfavorable geographical location andreduced capacity/functionality (each hospital having different resources, inthe general case the threshold value changes from node to node).3.1.
Reassignment of outgoing particles.
We have already discussedabout the crucial role played by the boundary conditions associated withthe ASM settled on finite networks, and indeed the existence of stable con-figurations is strongly influenced by the presence of a dissipation mechanismsuch as the open boundary conditions.Because the loss of particles through the edge of the network and the hy-pothesis of an infinite network are unrealistic assumptions for practical ap-plications, we have to implement a suitable reassignment law for particlesgenerating a critical height at some boundary cells. Hence, we are induced toredistributing the outgoing particles to the root of the network, namely the hub, which is typically a collecting node of the healthcare system. There-fore, we call the resulting process a Sandpile with Redistribution to the Hub(SRH), and we notice that actually the SRH is equivalent to the ASM whenthere are no topplings occurring at the edge of the network.In particular, for two-dimensional Cartesian grids with odd size, we in-corporate the reassignment of outgoing particles from cells toppling at theborder of the matrix by proposing that they are reallocated to the centralcell. Furthermore, in case the threshold capacity is reached in several cells atthe same time, we impose that the first cell to topple is always the hub, andthen additional toppling is performed from the other nodes. In particular,the hub is allowed to topple only once, at the very beginning of the topplingprocedure.We summarize the essential steps of the CAM workflow –which turns outto be also the implementation scheme of the numerical codes– as follows.1.
Initialisation . We choose a (stable) ground state z = ( z , z , . . . , z p )which satisfies the condition that z i < deg( x i ) for any index i ;2. Inflow . We combine the ground state z with a perturbation w =( w , w , . . . , w p ) which indicates the number of new patients requir-ing hospitalization, and we obtain the initial configuration ¯ z = z + w ;3. Hub toppling . If z ∗ ≥ deg( x ∗ ), the hub performs a toppling towardsthe adjacent nodes of its neighbourhood I ( x ∗ ) ;4. Additional toppling . If z i ≥ deg( x i ) for some x i (cid:54) = x ∗ , these nodesperform (a sequence of) topplings until reaching an almost stableconfiguration z = Φ(¯ z ) ;5. Iteration . We reinitialize the ground state with z equal to z , andwe restart from 2.3.2. Comparison with the standard healthcare system manage-ment.
In order to to compare the efficiency of the novel proposal withthe standard organisation of the Italian healthcare system, we provide a re-formulation of the current management of connections between hospitals interms of the CAM paradigm.Perhaps surprisingly, the connection between medical facilities is presentlymainly determined by their geographical proximity, corresponding to thebasic adjacency of nodes. If a patient comes to a hospital where there areno places available, because the threshold capacity has been reached, thenthe single patient is reallocated to the nearest hospital with available places,and the reassignment is limited to the patients exceeding the threshold value.That being the case, the redistribution of patients is handled manually ateach hospitalization, without foreseeing the possibility of repeated similarevents which are instead highly probable during sudden and unexpectedemergencies.We summarize the essential steps of the standard workflow as follows, byassuming the reassignement law from the edge of the network at the hub asin Section 3.1.1-2.
Initialisation/Inflow . We proceed as in Section 3.1 ;
OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 11 Hub redistribution . If z ∗ ≥ deg( x ∗ ), the hub reallocates only theexceeding patients to the adjacent medical facilities, starting fromthe less crowded ones (in case of equal crowding, a random choice isoperated) ;4. Additional redistribution . If z i ≥ deg( x i ) for some x i (cid:54) = x ∗ , theredistribution procedure is repeated for these nodes, until reachingan almost stable configuration z = Ψ(¯ z ) ;5. Iteration . We reinitialize the ground state with z equal to z , andwe restart from 2.We illustrate the comparison by considering the following simple example. Example 2.
Let the healthcare system be represented by a two-dimensionalCartesian grid of size 3 ×
3, with the von Neumann neighbourhood (so thatdeg( x i ) = 4 for any index i ) and reassignment of outgoing particles to thehub located at the central cell. We choose z = and w = , so that ¯ z = z + w =
11 0 2 and the initial configuration is unstable. The standard organization managesthis critical situation by moving a single particle (patient) from the centralcell towards one in the von Neumann neighbourhood, and preferably theSouth cell, to obtain Ψ(¯ z ) = , where Ψ denotes the evolution mapping of the grid according to the stan-dard approach. Then, we reinitialize the ground state with Ψ(¯ z ) and theperturbation w is chosen as above, so that the configuration Ψ(¯ z ) + w isalso unstable with respect to the central cell. By selecting randomly one ofthe neighbouring cells, another single particle is transferred, and after twoadditional iterations, the system reaches the final configurationΨ(¯ z ) = . On the other hand, the ASP paradigm suggests to transfer at the same timeall the four patients initially allocated to the hub towards the von Neumannneighbourhood, to obtain Φ( z ) = . Finally, under the assumption that new patients are always collected at thehub, the system can accept exactly 3 more particles from the additionaliterations, before reaching a critical situation as before. Of course, it could be argued that, being the final configurations the same, there would be noobvious evidence to prefer the novel approach to the standard organization.However, the deciding factor is that we have unified within a single time-step the efforts for reallocating patients among the medical facilities, whichis usually regarded as a source of stress for the overall healthcare system.Hence, we have left the structure time to reorganize without suffering aconstant state of emergency, which is precisely the issue of the so-calledpredictive logistics.4.
Examples and numerical simulations
We collect numerical results from a selection of preliminary cases, whichare nevertheless useful to understand the alternative methods of healthcaresystem management provided by the two approaches described in Section 3,and their inherent dynamics. We consider two-dimensional Cartesian grids,which are arranged into n × n (squared) matrices with the Moore neigh-bourhood, so that deg( x i ) = 8 for any index i ∈ { , , . . . , p = n } , and weassume the reassignment to the hub of outgoing particles from the edge ofthe network as in Section 3.1.We recall that each cell/node of the grid represents a hospital, and theheight function reproduces the number of patients in the medical facilities,expressed as the percentage of capacity already achieved.We start by focusing on the possible outputs of a basic case consisting ina single hospital originally attaining its threshold capacity, which is locatedat the hub (central cell), with a (randomly chosen) stable ground stateconfiguration. To highlight possibly critical situations, we boldmark allvalues above the threshold capacity, and we encircle the values 5, 6 and 7dangerously close to the saturation level. Example 3.
For n = 3, the hospitals are p = n = 9, and we choose aninitial configuration with all new patients coming to the hub x , given bythe perturbation w = 4 δ , so that z = and ¯ z =
24 3 3 . Because the hub has to manage a number of hospitalisations greater than itsthreshold capacity, some patients must be transferred to adjacent facilities.According to the standard strategy, only four patients need a different al-location to be found among the neighbourhing cells, ending at the (stable)configuration Ψ(¯ z ) = . As it is clearly seen, an additional iteration with w = 4 δ determines a newunstable/critical configuration at the central node (with again four patientsin excess), demanding for a further reorganisation of the hub by meansof another transfer operation. On the other hand, by employing the SRH OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 13 approach, we obtain a different outcome given byΦ(¯ z ) = , such situation being more favorable in terms of load-balancing since an ad-ditional iteration with perturbation w = 4 δ ∗ does not produce any unstableconfiguration. Moreover, there is still a certain amount of available placesat the hub, and therefore the novel proposal solves the criticalities possiblygenerated by an emergency.Despite its simplicity, the above example suggests to attribute to eachconfiguration the value of a given functional, aiming to provide an easilyreadable indicator of the effectiveness of the allocation strategy. This is nec-essarily a very delicate issue.As a first attempt, a possible choice is to count the total number of med-ical facilities attaining a given fraction of their threshold capacity, whichare denoted by critical points. Such choice is actually quite questionable,since it does not take keep memory of the value of incoming patients at thebeginning of the iteration.A different (rough) quantitative measure of the system management effi-ciency is F ( w , z ) := w (cid:80) j w j · z , which intends to evaluate the risk that a new patient comes to a givenlocation, taking into account the previous history of the system (in thiscase, the effect of the perturbation w ).For the case illustrated in the Example 3, the function F equals the stateof the hub x , so that F ( w , ¯ z ) = ¯ z = 11 , F (cid:0) w , Ψ(¯ z ) (cid:1) = Ψ(¯ z ) = 7 , F (cid:0) w , Φ(¯ z ) (cid:1) = Φ(¯ z ) ∗ = 3 . The minimum value is achieved for the configuration Φ(¯ z ) correspondingto the SRH approach (or, equivalently, to ASM). Example 4.
For n = 5, the hospitals are p = n = 25, and we choose aninitial configuration with all new patients coming to the hub x , given bythe perturbation w = δ + δ + 4 δ δ + δ + δ , so that z = and ¯ z = . According to the standard strategy, an admissible solution could beΨ(¯ z ) = , and the same observation previously done holds, that is such configurationlikely determines an instability located at the hub after a subsequent processiteration, since the number of hospitalised patients is very close to the criticalthreshold.On the other hand, by employing the SRH approach, we obtainΦ(¯ z ) = , where the number of patients hospitalised at the hub is far away from thecritical level. Indeed, by computing the indicator function F at the differentoutcomes, we deduce that F (cid:0) w , ¯ z (cid:1) = 6 . , F (cid:0) w , Ψ(¯ z ) (cid:1) = 5 . , F (cid:0) w , Φ(¯ z ) (cid:1) = 4 . , since (cid:80) j w j = 10, and the smallest value is achieved by the configurationresulting from the SRH approach; thus, such strategy must be preferred tothe standard one.Next, we explore a case where two subsequent topplings occur. Example 5.
For n = 5, we choose the ground state and the initial config-uration as follows, z = and ¯ z = , where w = 4 δ . Then, according to the standard strategy, a final configu-ration is given by Ψ(¯ z ) = . For applying the SRH approach, we pass through an intermediate configu-ration − +
45 5+1 2 + 1 4+1 53 5 4 5 3 =
45 6 3 5 53 5 4 5 3 , OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 15 in which the cell x is unstable, and finally we arrive at the final configu-rationΦ(¯ z ) = − = . The matrices Ψ(¯ z ) and Φ(¯ z ) are represented in Figure 6 with differentcolors assigned to cells, corresponding to the degree of saturation achieved. Simulation of the current model Proposal
Number of full hospitals = 5
Proposal emptymore than half of available bedsless than half of available bedsalmost fullfull (a) (b)
Critical points: 2
Critical points: 4 nz = 2
Simulazione modello attuale nz = 4
Proposta
Figure 6.
Model comparison from the Example 5: final configu-rations Ψ(¯ z ) (left) and Φ(¯ z ) (right). The colors attached to eachcell/node represent its corresponding relative capacity (0=green,1/2=yellow, 3/4=magenta, 4/5=red, 6/7=black) At the bottom of each subfigure, we report the counting of critical points,given by the number of cells with values 6 or 7. Although a quick glancecan make us suspect that the left configuration is preferable, the situationis indeed different. In fact, first of all we distinguish the presence of twoalmost critical values 7 inside the configuration plotted on the left, againstthe presence of only one value 7 from the second approach. Furthermore,the hub –the facility most at risk taking into account recent history– hasa value 7 in the first case, whilst a value 4 occurs in the second case, thusmaking the latter situation preferable.Finally, we compute the function F corresponding to the different config-urations: since w / (cid:80) j w j is everywhere zero except at the hub, where it isequal to 1, the values of F coincide with the values at the hub, which aregiven by F ( w , ¯ z ) = 11 , F (cid:0) w , Ψ(¯ z ) (cid:1) = 7 , F (cid:0) w , Φ(¯ z ) (cid:1) = 4 , suggesting again the advantages of configurations proposed by the SRH ap-proach.4.1. Simulation of multiple central outbreaks.
We analyze the casewith more than one hospital in the same restricted area (including the hub)which is concerned with new incoming patients. One might think of thisevent as the emergence of multiple outbreaks within some smaller districtof a larger area. We consider a larger network, with n = 9, whose size iscomparable to the capacity of the healthcare system in the Lazio region inItaly [33].We consider the perturbation w = 2 δ + δ +5 δ +2 δ and the groundstate z = , so that (cid:80) j z j = 250 and (cid:80) j w j = 10, corresponding to the initial configu-ration ¯ z = = . OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 17
Then, according to the standard strategy, a final configuration is given byΨ(¯ z ) = . Hence, the central subgraph around the hub is composed by the elements . In the subsequent steps, any additional patients coming to the hub desta-bilises the configuration. On the other hand, by employing the SRH ap-proach, we obtainΦ(¯ z ) = − , and the corresponding central subgraph around the hub is . The difference between Ψ(¯ z ) and Φ(¯ z ) is transparent regarding, specifi-cally, the number of patients hospitalised in the hub. A further iterationwith the same perturbation w would lead to a new critical situation for thehub in the first case, but it does not in the second.Figure 7 provides a representation of the two configurations Ψ(¯ z ) andΦ(¯ z (cid:1) .Counting the critical points in Figure 7 could give the (incorrect) impres-sion that the standard approach has to be preferred with respect to the novelproposal. But considering the indicator function F applied to ¯ z , Ψ(¯ z ) and Simulation of the current model Proposal
Number of full hospitals = 5
Proposal emptymore than half of available bedsless than half of available bedsalmost fullfull (a) (b)
Critical points: 10
Critical points: 11
Critical points: 10
Simulation of the current model
Critical points: 11
Proposal
Figure 7.
Model comparison from the case of central areas out-breaks: final configurations Ψ(¯ z ) (left) and Φ(¯ z ) (right). Thecolors attached to each cell/node represent its corresponding rela-tive capacity (0=green, 1/2=yellow, 3/4=magenta, 4/5=red,6/7=black) Φ(¯ z ), we obtain F (cid:0) w , ¯ z (cid:1) = 7 . , F (cid:0) w , Ψ(¯ z ) (cid:1) = 5 . , F (cid:0) w , Φ(¯ z (cid:1) ) = 4 . , as a consequence of the fact that memory of the dynamics –represented by w – is now considered, and again the minimum value is achieved for the SRHstrategy.4.2. Simulation of peripheral outbreaks.
Next, we focus on the caseof outbreaks occurring in the vicinity of a node located at the edge of theCartesian grid. Specifically, we take n = 9 and the same ground state z of the previous example, and an inflow matrix of patients w given by thesubmatrix (cid:18) (cid:19) located at the top-right corner of the null matrix. Then, the initial config-uration ¯ z = z + w is given by¯ z =
43 2 5 2 1 3 2 6 43 3 1 5 6 2 3 1 36 1 5 3 2 5 2 3 41 3 2 1 6 3 4 5 61 4 1 2 2 5 1 2 34 1 3 4 5 6 2 5 34 2 3 5 2 2 6 3 11 6 5 2 4 4 2 1 2 , with the cell above the threshold capacity located at position (1 , OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 19 others, a possible output of the standard strategy is the final configurationgiven by Ψ(¯ z ) = . On the other hand, by employing the SRH approach, we pass through anintermediate state , to reach the final configurationΦ(¯ z ) = . Figure 8 provides a representation of the two configurations Ψ(¯ z ) andΦ(¯ z (cid:1) , by indicating the level of saturation reached in each hospital.Finally, the indicator function F defined at the beginning of Section 4takes the following values for the different configurations, F (cid:0) w , ¯ z (cid:1) = 7 . , F (cid:0) w , Ψ(¯ z ) (cid:1) = 5 . , F (cid:0) w , Φ(¯ z (cid:1) ) = 4 . , and this fact contributes to the advantages of the SHR approach.5. Beyond the Abelian Sandpile paradigm
The mathematical model introduced in this article lays the foundation foran optimisation of the healthcare system management. A notable featureof the novel proposal is its scalability to various levels of description, andalso the possibility of improving the experimental simulation by including
Simulation of the current model Proposal
Number of full hospitals = 5
Proposal emptymore than half of available bedsless than half of available bedsalmost fullfull (a) (b)
Critical points: 10
Critical points: 10
Critical points: 10
Proposal
Critical points: 10
Simulation of the current model
Figure 8.
Model comparison from the case of peripheral out-breaks: final configurations Ψ(¯ z ) (left) and Φ(¯ z ) (right). Thecolors attached to each cell/node represent its corresponding rela-tive capacity (0=green, 1/2=yellow, 3/4=magenta, 4/5=red,6/7=black) more realistic situations and different types of medical facilities (emergencyrooms, external care points, ...) into an integrated dynamical system.5.1. Sandpiles with internal dissipation.
In order to incorporate otherrelevant elements into the SHR model, such as recovery or (unfortunately)death of patients, we postulate an elimination mechanism inherent to thesystem, which is translated in mathematical formalism by considering thepresence of some dissipation during the evolution.For instance, we describe the effect of removing particles from the networkby a simple subtraction of a (randomly chosen, but possibly measurable)distribution ζ = ( ζ , ζ , . . . , ζ p ) with the obvious constraint that0 ≤ ζ i ≤ Φ( z ) i for any index i = 1 , , . . . , p . The modified model workflow is divided into the following steps.1-2.
Initialisation/Inflow . We proceed as in Section 3.1 ;3-4.
Hub/Additional toppling . We proceed as in Section 3.1 ;5.
Internal dissipation . We subtract the distribution ζ to the interme-diateconfiguration z , leading to the final configuration Φ(¯ z ) = z − ζ ;6. Iteration . We reinitialize the ground state with z equal to Φ(¯ z ),and we restart from 2.We refer to this modified model as Sandpile with Internal Dissipation (SID).In the long run, after a large but finite number of iterations, a balancebetween the inflow and outflow steps has also to be incorporated in orderto guarantee conservation of the total number of patients. In particular, it OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 21 could be useful to add the hypothesis that N (cid:88) n =1 p (cid:88) i =1 w ni = N (cid:88) n =1 p (cid:88) i =1 ζ ni , where N is the total number of iterations, with w n and ζ n denoting theinflow and outflow contributions at the time-step n , respectively. Indeed,we notice that if the lefthand side is larger than the righthand side, thewhole system risks to undergo a finite time collapse, by reaching its totalcapability –sum of the capacities of each single structure– in finite time.Other interesting features can be added to provide the model with ahigher level of realism: for instance, the presence of some inertia to thetransfer process [26], giving preference to structures with a certain level ofhospitalized patients [21], or constrain additional bulk dissipation [46].5.2. Conclusion and perspetives.
From the analysis developed in this ar-ticle, we observe that the standard healthcare system management typicallygenerates highly unstable and unbalanced configurations, where specific ge-ographical areas with semi-empty hospitals alternate with others where allmedical facilities are saturated, especially during sudden and unforeseenevents like the spread of epidemics. Instead, following the SRH strategy foroptimized management of connections between hospitals, seems to producemore sparse allocation of patients, which has to be considered as a preferableconfiguration in terms of load-balancing.On the other hand, it is crucial to improve the exchange of informationand to provide decision-making tools to the local structures, in order tooptimise the healthcare response in normal times and to avoid the collapse ofindividual hospitals in times of crisis. There are many relevant consequencesin the socio-economic field: among others, we stress the riveting possibility ofthe automation of health protocols, meaning to build an application capableof learning something from the data autonomously, without receiving explicitinstructions from the outside.Such conceptual experimentation could create a learning environment inwhich policy makers may gain a better understanding of how the systemresponds to their decisions, providing forecasts of potential different choicesand strategies. We are aware that a paradigm shift is required and we hopeto give a contribution to this respect. It is worth stressing that the agreementwith realistic experimental data is presently very limited. However, theconceptual framework we have proposed in this article applies in principleto many different contexts, and these research directions are currently beingexplored.
Acknowledgements
The authors thank the Department of Mathematics G. Castelnuovo, SapienzaUniversity of Rome, for hosting the electronic workshop
COVID-19 calls for Math-ematics – –which has motivated this article.The authors thank prof. Ferdinando Romano (Department of Public Health andInfectious Diseases, Sapienza University of Rome) for the stimulating discussionsabout the logistic issues of the healthcare system organisation. The authors also thank prof. Stefano Finzi Vita (Department of Mathematics G.Castelnuovo, Sapienza University of Rome) for useful references on the mathemat-ical theory of the Abelian Sandpile model.
Authors’ contributions
All authors equally contributed to the conception and design of this article, theacquisition, analysis and interpretation of data, the drafting and revision of themanuscript.
Consent for publication
All authors read and approved the submitted manuscript. All authors haveagreed to be personally accountable for the contributions, and to ensure that ques-tions related to the accuracy or integrity of any part of this article are appropriatelyresolved and documented in the literature.
References [1] Adami C,
Self-organized criticality in living systems , Physics Letters A :1 (1995)29–32[2] Aleksiejuk A, Holyst JA, Kossinets G,
Self-organized criticality in a model of collectivebank bankruptcies , International Journal of Modern Physics C :3, (2002) 333–341[3] Bak P, “How nature works: The science of self-organized criticality”, Springer-Verlag,New York, 1996.[4] Bak P, Tang C, Wiesenfeld K, Self-organized criticality: An explanation of the /f noise , Physical Review Letters :4 (1987) 381–384[5] Bak P, Tang C, Wiesenfeld K, Self-organized criticality , Physical Review A :1(1988) 364–374[6] Barbu V, The steepest descent algorithm in Wasserstein metric for the sandpile modelof self-organized criticality , SIAM Journal on Control and Optimization :1 (2017)413–428[7] Bartolozzi M, Leinweber DB, Thomas AW, Self-organized criticality and stock marketdynamics: An empirical study , Physica A: Statistical Mechanics and its Applications :2-4 (2005) 451–465[8] Basu U, Mohanty PK,
Self-organised criticality in stochastic sandpiles: Connectionto directed percolation , EPL :6 (2014) 60002[9] Biondo AE, Pluchino A, Rapisarda A,
Modeling financial markets by self-organizedcriticality , Physical Review E - Statistical, Nonlinear, and Soft Matter Physics :4(2015) 042814[10] Cajueiro DO, Andrade RFS, Controlling self-organized criticality in complex net-works , European Physical Journal B :2 (2010) 291–296[11] Carlson JM, Swindle GH, Self-organized criticality: Sandpiles, singularities, and scal-ing , Proc. Nat, Acad. Sci. USA :15 (1995) 6712–6719[12] Casartelli M, Zerbini M, Metric features of self-organized criticality states in sandpilemodel , Journal of Physics A: Mathematical and General :5 (2000) 863–872[13] Chen Y, Zhou Y, Scaling laws and indications of self-organized criticality in urbansystems , Chaos, Solitons and Fractals :1 (2008) 85–98[14] Creutz M, Abelian sandpiles , Computer in Physics (1991) 198–203[15] de Arcangelis L, Perrone-Capano C, Herrmann HJ, Self-organized criticality modelfor brain plasticity , Physical Review Letters :2 (2006) 028107[16] Dhar D, Sandpiles and self-organized criticality , Physica A: Statistical Mechanics andits Applications :1-2 (1992) 82–87[17] Dhar D,
Self-organized critical state of sandpile automaton models , Physical ReviewLetters :14 (1990) 1613–1616[18] Dhar D, The Abelian sandpile and related models , Physica A (1999) 4–25.[19] Dhar D,
Theoretical studies of self-organized criticality , Physica A :1 (2006) 29–70[20]
OSPITAL MANAGEMENT IN THE COVID-19 EMERGENCY 23 [21] Falk J, Winkler M, Kinzel W,
On the effect of the drive on self-organized criticality ,Journal of Physics A: Mathematical and Theoretical :40 (2015) 405003[22] Frigg R, Self-organised criticality - What it is and what it isn’t , Studies in Historyand Philosophy of Biological and Biomedical Sciences :3 (2003) 613–632[23] Gaveau B, Schulman LS, Mean-field self-organized criticality , Journal of Physics A:Mathematical and General :9,005 (1991) L475–L480[24] Glinton R, Paruchuri P, Scerri P, Sycara K, Self-organized criticality of belief propa-gation in large heterogeneous teams , Springer Optimization and Its Applications, 40,2010, 165–182[25] Gross J.L., Yellen J., Zhang P., “Handbook of Graph Theory”, 2nd edition. Chapmanand Hall/CRC 2014.[26] Head DA, Rodgers GJ,
Crossover to self-organized criticality in an inertial sandpilemodel , Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Inter-disciplinary Topics :3 (1997) 2573–2579[27] Hergarten S, Landslides, sandpiles, and self-organized criticality , Natural Hazardsand Earth System Science :6 (2003) 505–514[28] Hoffmann H, Payton DW, Optimization by Self-Organized Criticality , Scientific Re-ports :1 (2018) 2358[29] Ivashkevich EV, Priezzhev VB, Introduction to the sandpile model , Physica A (1998) 97–116.[30] J´arai AA,
Sandpile models , Probability surveys (2018) 243–306[31] Kari J. Theory of cellular automata: A survey , Theoretical Computer Science :1-3(2005) 3–33[32] Lavine, J.S., Bjornstad, O.N., Antia, R., Immunological characteristics govern thetransition of COVID-19 to endemicity, Science (2021) – DOI: 10.1126/science.abe6522[33] Leo G, Lodi A, Tubertini P, Di Martino M,
Emergency department management inLazio, Italy , Omega (2016) 128–138[34] Lindenfors P., “For whose benefit? The Biological and Cultural Evolution of HumanCooperation” Springer, 2017[35] Malamud BD, Turcotte DL, Self-organized criticality applied to natural hazards , Nat-ural Hazards :2-3 (1999) 93–116[36] Manna SS, Sandpile models of self-organized criticality , Current Science :3 (1999)388–393[37] Markovi´c D, Gros C, Power laws and self-organized criticality in theory and nature ,Physics Reports :2 (2014) 41–74[38] Mauro JC, Diehl B, Marcellin RF, Vaughn DJ,
Workplace accidents and self-organizedcriticality , Physica A: Statistical Mechanics and its Applications (2018) 284–289[39] Morales L.F., Charbonneau P.,
Self-organized criticality in solar flares: A cellularautomata approach
Nonlinear Processes in Geophysics :4 (2010) 339–344[40] Newman MEJ, Evidence for self-organized criticality in evolution , Physica D: Non-linear Phenomena :2-4 (1997) 293–296[41] Paczuski M, Bassler KE,
Theoretical results for sandpile models of self-organized crit-icality with multiple topplings , Physical Review E - Statistical Physics, Plasmas, Flu-ids, and Related Interdisciplinary Topics :4B (2000) 5347–5352[42] Ramos RT, Sassi RB, Piqueira JRC, Self-organized criticality and the predictabilityof human behavior , New Ideas in Psychology :1 (2011) 38–48[43] Sarkar P., A brief history of cellular automata , ACM Computing Surveys :1 (2000)80–107[44] Tadi´c B, Dankulov MM, Melnik R, Mechanisms of self-organized criticality in socialprocesses of knowledge creation , Physical Review E :3 (2017) 032307[45] Trudeau R.J. “Introduction to Graph Theory”, Dover Books on Mathematics, DoverPublications Inc. 1994.[46] Tsuchiya T, Katori M, Proof of breaking of self-organized criticality in a nonconserva-tive Abelian sandpile model , Physical Review E - Statistical Physics, Plasmas, Fluids,and Related Interdisciplinary Topics :2 (2000) 1183–1188[47] Turcotte DL, Self-organized criticality , Rep Prog Phys :10 (1999) 1377–1429 [48] Wolfram S., “A new kind of science”, Wolfram Media, 2002. [49] https://covid19.who.int/ [50] https://en.wikipedia.org/wiki/Severe_acute_respiratory_syndrome [51] https://en.wikipedia.org/wiki/Influenza_A_virus_subtype_H1N1 [52] Xi N, Ormerod P, Wang Y, Technological innovation, business cycles and self-organized criticality in market economies , EPL :6 (2012) 68005[53] Yuan J, Ren Y, Shan X, Self-organized criticality in a computer network model , Phys-ical Review E - Statistical Physics, Plasmas, Fluids, and Related InterdisciplinaryTopics :2 (2000) 1067–1071(Roberta Martucci) Dipartimento di Matematica G. Castelnuovo, SapienzaUniversit`a di Roma, piazzale Aldo Moro 2 - 00185 Roma (Italy)
Email address : [email protected] (Corrado Mascia) Dipartimento di Matematica G. Castelnuovo, Sapienza Uni-versit`a di Roma, piazzale Aldo Moro 2 - 00185 Roma (Italy)
Email address : [email protected] (Chiara Simeoni) Laboratoire de Math´ematiques J.A. Dieudonn´e CNRS UMR7351, Universit´e Cˆote D’Azur, Parc Valrose - 06108 Nice Cedex 2 (France)
Email address : [email protected] (Filippo Tassi) Dipartimento di Matematica G. Castelnuovo, Sapienza Uni-versit`a di Roma, piazzale Aldo Moro 2 - 00185 Roma (Italy)
Email address ::