Novel infectious diseases pose a serious challenge to policy makers and healthcare systems. Emerging from Wuhan, China, the ongoing Coronavirus Disease 2019 (COVID-19) pandemic, caused by the Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has wreaked havoc globally. This is due to its rapid rate of transmission, its virulence and the inability of most countries to adequately prepare for such a disease [1]. Identified in December 2019, the disease now has a global distribution with over 30 million confirmed cases and almost one million confirmed deaths as of September 2020 according to the World Health Organisation (WHO) [2]. COVID-19 is primarily transmitted through respiratory droplets and the WHO has identified two principal routes through which these are carried between people. The first mode of transmission involves a person being in direct, close contact with someone who carries the virus (within one metre) where they become directly exposed to potentially infectious respiratory droplets. The second mode of transmission involves contact with fomites in the immediate vicinity of the infected person [3]. Nguyen et al. estimated that frontline healthcare professionals (HCPs) had a 3.4 times higher risk than non-healthcare workers of contracting COVID-19, even when adjusting for the probability of being tested [4]. Indeed, approximately 10% of the confirmed cases in China [5] and up to 9% of all cases in Italy have been among healthcare professionals [6] as of the date of publication of these studies. This increased risk does not only pose a problem for the HCPs themselves, but also poses a major threat to the elderly and vulnerable populations they care for, since outbreaks within healthcare settings are important amplifiers of infection [7]. One of the most crucial ways by which transmission of this virus (and other infectious diseases) is reduced is the use of proper Personal Protective Equipment (PPE) [7]: the WHO recommends a surgical mask, goggles, or face shield, gown, and gloves to be worn as PPE in their COVID-19 PPE guidelines. If an aerosol-generating procedure is performed, the surgical mask is replaced with an N95 or FFP2/3 respirator which provides a greater level of filtration than surgical masks [8]. These guidelines are replicated globally with minor differences. In the United Kingdom (the case study reported in this paper), the most recent Public Health England guidelines are essentially the same [9]. Globally, medical resources, particularly PPE have come under unprecedented demand. This has led to shortages in many countries with some rationing their use of PPE and in some cases reusing disposable material [10]. The natural consequence of this has been the disproportionately high rate of infection amongst HCPs which in turn contributes to disease spread. This may not necessarily result from a national shortage of PPE but rather local shortages resulting from inefficient distribution of resources in timely manner [11].

In spite of the limited applicability of resource allocation methods, as choices have to be made, they can aid in making the decisions that achieve the best health outcomes (see [12] for a review on the use of such methods in epidemic control). Single- and multi-objective optimisation methods have been used frequently to address problems of resource allocation and scheduling of purchase orders for medical supplies (see [13] for a comprehensive survey and taxonomy). Such methods are centralised in the sense that a central planner seeks to optimally coordinate supply activities for the entire system, e.g. minimise overall costs for the central planner. In contrast to this, game theory is a tool that allows decentralised decision-making [14, 15]. That means, different entities in the system can make decisions based on their individual preferences. For instance, they can schedule their orders to minimise their own costs. The decentralised approach leaves the actors with more freedom and is the more applicable direction for our scenario.

Game theory has been applied to a variety of subject areas such as biology [16], economics [17], computer science [18] and energy [19, 20]. In general, game theory is used to mathematically model systems of competing agents. Usually these individuals act in a selfish and rational manner. In this context, selfish can be understood as being only interested in their own good, i.e. an agent strives to maximise its outcome irrespective of the outcome of others [14]. Furthermore, rational means that there is a clear logical reasoning behind every decision [15]. The most widely used solution concept for a non-cooperative game is the Nash equilibrium [21]. It is achieved when none of the players has an incentive to change their strategy unilaterally.

Despite its various applications in supply chain management (cf. [22] and the references therein), only few studies have applied game theory to the management of medical supplies. To the best of our knowledge, a 2008 study [23] (an extended version was later published in [24]) was the first to develop a game theoretic approach for stockpiling of critical medical items. In their model, the authors propose a non-cooperative game where hospitals stockpile critical medical resources in preparation for disasters. It is proposed that these hospitals determine their individual stockpile levels strategically in a way that minimises their expected total spend. Their model uses a cost function that consists of the cost of ordering, borrowing, storage and a penalty cost in the case of shortages. Although their model proves to give some breathing space to the medical supply chain at the onset of an epidemic, it is only intended as a preparation scheme, and therefore may suffer from inapplicability if an epidemic lasts longer than the planning period considered. The model assumes a given likelihood of the occurrence and severity of a pandemic, which is, in practice, heavily unpredictable. Their work was further developed in [25] by introducing network constraints, thus giving realistic hospital sharing policies. The authors find that deficits between stockpiles and demand can be reduced through central stockpiling and through increasing penalties for deficits.

Game theory has also been applied to the problem of drug allocation [26], where countries behave selfishly to minimise their expected number of infections. Unlike in [27] where the authors propose a similar resource competition among countries, their model imitates the stochasticity of infection transmission parameters. The resulting comparison between this selfish allocation scheme and a utilitarian division scheme, where a central planner (e.g. WHO) makes all of the allocation decisions, shows that having a central planner reduces the total number of worldwide infections considerably. Although their proposed model can aid in understanding how countries, acting in their own self interest, would behave in an epidemic, it does not address the problem of resource distribution within a given country. [28] compares selfish vaccination coverage, which the authors call the ‘Nash vaccination‘, with group optimal coverage, which they call the ‘utilitarian vaccination’. The authors find that the cost of vaccination would be a pivotal factor in determining the effectiveness of Nash vs. utilitarian coverage. However, this can result in different outcomes depending on the level of disease severity according to age, such as in the case of chickenpox.

Most recently, Nagurney et al. proposed a novel decentralised model for medical supply chain with multiple supply and demand points [29]. In their model, selfish consumers who have stochastic demands make purchasing decisions that minimise their total expenditure. The authors assume that the disutility function of a consumer consists of a linear cost of demand, a quadratic cost of transportation and a penalty for shortages/surpluses. Although the quantification of the shortage/surplus penalty can be debated, the authors conclude by suggesting that shortages in supply can be avoided by redirecting global production efforts to instead adding or increasing local production capabilities.

In this paper, we propose a game theoretic approach for managing PPE supplies during a pandemic. We take inspiration from the electricity storage scheduling game developed by the co-authors in [30], where a decentralised system of individually owned home energy systems served by the same utility company schedule their day-ahead battery usage over a full year. This decentralised system resulted in improved energy efficiency for the utility company and cost savings for the participating users.

This work proposes a centralised-decentralised approach to the PPE supply chain (cf. Section 2). In the proposed architecture, healthcare facilities report their demand to a central entity. This central entity is assumed to have the commitment power to fulfill its orders and set the costs for PPE. Given the actions of all game participants, healthcare facilities optimise their PPE orders by making stockpiling decisions individually. Our approach is centralised in the sense that cost and supply is controlled by a central entity. It is also decentralised because healthcare facilities make their stockpiling decisions independently. By adapting the model developed in [30] and applying it to COVID-19 related PPE demand in England, we study the effects of early stockpiling as well as increasing storage capacities on PPE supply in challenging circumstances. Additionally, we examine the impact of a putative second wave on PPE supply and investigate whether delaying this putative second wave can ease the challenge of fulfilling the PPE demand of healthcare facilities. Finally, insights and conclusions from this study are drawn and suggestions regarding PPE supply in disaster management are discussed. The contributions of this work can be summarised as follows:

The remainder of this paper is organised as follows. In Section 2 we provide an overview of the system architecture, explain the chosen cost function and give details on the game formulation. Section 3 contains information on the experimental setup and Section 4 presents an analysis of the results with simulations of different scenarios. This is followed by a discussion and recommendations in Section 5 and finally, Section 6 concludes the paper and points out future research directions.

During the COVID-19 pandemic, most countries have experienced strained healthcare resources and even shortages. The efficient management of personal protective equipment (PPE) supply has proved vital in limiting the spread of the virus and in keeping the healthcare professionals protected. In this paper, after designing a game theory-based model to enhance stock management of PPE supply, we investigate as a case study PPE provision in hospitals of the English National Health Service (NHS). England was selected as not only has it been one of the first and most severely hit countries, but it has also released COVID-19 data in a transparent and timely manner. The datasets and assumptions that were used in this case study are described in Section 3. Data on occupied hospital beds with illness related to COVID-19 [31] was used as a proxy to generate PPE demand profiles.

In this paper, we propose a game-theoretic approach for the supply of PPE to healthcare facilities. We consider the case study of England as it was one of the countries severely impacted by the COVID-19 pandemic and where shortages in medical resources, especially PPE, were reported [10]. The players in this game structure are assumed to report their demand and are committed to paying their respective invoices. The players are also assumed to have the financial independence that inspires their selfish behaviour. Although this game is intended for independent healthcare providers, such as NHS Trusts in England, only region-level COVID-19 data are publicly available. Therefore, in this experiment, we assume that each of the seven NHS England regions can act selfishly in a way that represents the interests of its accommodated Trusts. We believe this has no effect on the insights drawn from this experiment since a region is merely a group of Trusts.

The model we have constructed to analyse the provision of PPE has important implications for the management of the pandemic hitherto. Firstly, we have shown objectively that early stockpiling and increasing storage capacity would have helped to massively reduce the costs associated with the containment of the pandemic. Secondly, the early, sufficient and cheap provision of PPE would have had a substantial impact on the ability to contain the pandemic, and protect both the most vulnerable patients and the healthcare professionals that care for them [7].

Based on these findings and the mathematics underlying our approach, we have identified the key elements that govern the aggregated demand (and therefore, the cost) of PPE in our scenario. The two key parameters involved in saving costs by reducing the demand for PPE in the context of a pandemic that were identified using this approach are: (i) the storage capacity (for PPE) available to the health system. We have shown that increasing the storage capacity enhances the ability to stockpile which in turn helps flatten the demand curve (making it easier for suppliers to meet this demand in a cost-effective way); and (ii) the date when stockpiling of PPE commences. We have shown that earlier stockpiling would have saved costs and ensured adequate provision of PPE for the first peak of the pandemic. These two factors will also prove crucial when planning for a second wave. Of secondary importance is the ability to predict the date of the second wave. This is because, providing the second wave occurs after the new year, the degree of cost saving that occurs diminishes to negligible levels beyond this point in time. Using our game-theoretic approach, we have found that the NHS would be able to achieve all the cost-saving advantages if the second wave was to occur in 2021. A practical implication of this is that policymakers may need to keep some of the restricting measures that are in place to control the pandemic until beyond this point in time.

In order to highlight the significance of having dedicated and sufficient storage for PPE in preparation for a putative second wave, we show a comparison between different storage capacities in Fig 7. In this figure, the aggregate demand profile for all NHS England regions is shown both with and without storage scheduling for a second wave of PPE demand that peaks in mid November. As shown, there is a substantial improvement in cost savings between the different storage capacities, especially when it is amplified by a factor of five and 10, resulting in an improvement in cost saving of 15.9% and 7.7% respectively. This means that the challenge of securing PPE during a second wave can be eased by a considerable amount when additional storage space is provided for stockpiling PPE. The figure also shows that further storage enhancement beyond this would result in less cost-saving improvements and would therefore have little impact on the challenge of securing sufficient PPE. This is especially visible when storage is expanded from 15 times to 20 times the standard storage space. Indeed, enhancing the PPE storage capacity by 15 times, perhaps by using temporary storage facilities as an interim solution, results in a relatively steady demand profile as shown in Fig 7.

In this paper, we developed a game-theoretic model for scheduling PPE supply for healthcare facilities. In this discrete time dynamic game, healthcare facilities make stockpiling decisions that minimise their PPE ordering cost. Our model adopts a centralised-decentralised approach to the PPE supply chain, where a central entity controls the PPE pricing formula, yet is committed to fulfilling the orders independently placed by healthcare providers. Based on publicly available COVID-19 hospitalisation data for NHS England regions, we performed simulations to investigate the impact of three key factors on PPE security of supply. These factors comprise: (i) the stockpiling start date, (ii) the time of the peak of a putative second wave, and (iii) the amount of storage available for medical PPE. The two most critical parameters were found to be the storage capacity and the stockpiling start date, while the timing of the second wave has only had a moderate impact on the challenge of securing PPE. Within our model we observe that enhancing PPE storage capacities by a factor of 15 is sufficient to considerably lower the peak demand at any given day and effectively minimises the strain on the health care system. (This does not imply that dealing with the pandemic in this scenario is rendered trivial. It rather represents the most favourable way to act during the crisis).

While the shortage of PPE supplies for care home workers during the first peak in the UK was a topic of great concern, since those shortages were blamed for care home outbreaks which led to a large number of lost lives [55], we explicitly excluded this from our study due to the lack of publicly available data. Access to such data would provide further insights into demand patterns which can potentially influence the game. Extensions to our work can include introducing supply constraints where demand cannot always be fulfilled, perhaps by adding a shortage penalty to the cost function. Also, our model assumes that healthcare facilities can predict their demand without forecasting errors. This can be enhanced by adding stochastic elements that capture the uncertainty in demand [30]. The model can also be extended by investigating scenarios where hospitals can share their resources by having mutual agreements or using the selfish sharing approach proposed in [56]. This could be via using a scheme to incentivise matching scheduled regional demand with the production of the supplier. Another possible extension of this research is to propose a model that finds the optimal storage capacity for each game player. Although the above refinements may give greater insight, our model already has the potential to be a useful tool that can help governments and policy makers in decision-making in relation to critical PPE supplies. Finally, if one wishes to extend this work to incorporate the global picture, this could be achieved by introducing another layer of competition modelling the orders of the different nations. Here, a Leader-Follower structure would be required and most suitably captured by a Stackelberg game [14].