Personal protective equipment market coordination using subsidy.

During a pandemic, various resources, including personal protective equipment (PPE), are required to protect people and healthcare workers from getting infected. Due to the high demand and limited supply chain, countries experience a shortage in PPE products. This global crisis imposes a decline in the international trade of PPE supplies. In fact, most governments implement a localization strategy motivating domestic manufacturers to pivot their operations to respond to PPE demands. An oligopolistic market cannot reach the socially optimal coverage without government subsidies. On the other hand, the government subsidy pays the proportion of production costs to reach the socially optimal coverage, while the government's budget is limited. Therefore, the government collaborates with manufacturers via procurement contracts to increase the supply of PPE products. We propose the first supply chain model of PPE products that investigates manufacturer costs and government expenditure. We consider how different behavioral aspects of manufacturers and government can self-organize towards a system optimum. Additionally, we integrate the consumer surplus, producer surplus, and societal surplus into the game model to maximize social benefit. A cost-sharing contract under the system optimum between government and manufacturers is designed to increase the production of PPEs and hence, helps in reducing the number of infected individuals. We conducted our computational study on real data generated from the mask usage during the Covid-19 pandemic in Los Angeles (LA) County to respond to the reported PPE shortage. Under the socially optimal strategy, the PPE coverage increases by up to 33%, and the number of infected individuals reduces by up to 30% compared to other strategies.

the number of nodes

the set of links

the set of links with fixed production capacity

the set of links with added production capacity

time period per week

1 if an element of incidence matrix of link on path , and 0 otherwise

the set of paths until the demand markets

the set of all paths

the total quantity coverage of PPE item per week

the number of PPE production for path per week

the flow of the product on link per week

the flow of the product on link per week which have fixed production capacity

the flow of the product on link per week which have capability to increase the productivity of PPE items

the upper bound on the capability of manufacturers on link under fixed production capacity

the upper bound on the capability of manufacturers on link with added production capacity

the capacity of links per week with added production capacity

the contact rate per week

intensity function per week

the number of infected individuals per week

the number of susceptible individuals per week

the number of infected individuals under low-risk per week

the number of infected individuals under high-risk per week

the number of susceptible individuals who have low risk per week

the number of susceptible individuals who have high risk per week

the demand price function per week for the entire population () and for large facilities ()

consumer price ($)

PPE burn rate for infected individuals per week

PPE burn rate of staff per week in hospitals and medical centers

the number of staff at hospitals and medical centers

the percentage of PPE coverage per week

the infection loss of individuals per week

total population

a coefficient (the burden of new infections on the health care sector)

the speed of learning for the health care service

the damage of pandemic on economic which is corresponding to the percentage of the annual GDP

the percentage of government subsidy

the revenue function per week

the total operational cost related to link per week

the investment capacity function

the burden of infection on healthcare sector due to the PPE shortage

Introduction

A pandemic such as Covid-19 can deeply impact the global trade of medical products. For instance, the U.S. trade deficit (that is the gap between exports and imports) expanded by about 20%, reaching a value of $8.65 billion (Federal Reserve Bank of St. Louis on the Economy Blog, 2020, Peterson Institute for International Economics, 2020). On the other hand, suppliers experience a sharp demand for essential medical items because of the speedy consumption of medical products during the pandemic. One of the most common challenges that people and healthcare workers face was limited access to personal protective equipment (PPE). PPE includes N95 respirators, surgical masks, gloves, eye and foot protection, protective hearing devices, hard hats, respirators and full-body suits, ventilators, test kits, swabs, sanitizer, and disinfectant.

Before the Covid-19 pandemic PPE used to be produced by only a few countries and the rest of the world were dependent on them for PPE production (Loey et al., 2021). During the Covid-19 pandemic, governments issue policies to support domestic production and distribution of PPE by providing economic subsidies to manufacturers for pivoting their operations to produce PPE. In the short time, multiple agencies, including government and private companies attempt to increase PPE supplies and productive capacities. However, in the long time, a balance between trade surplus and budget deficit should be considered.

The localization of critical medical industries builds a resilient supply chain strategy that not only responds to the PPE shortage during a pandemic, but also provides economic development opportunities for domestic manufacturers, specifically for small and medium-sized businesses. The employment growth increases with the economic fundamentals conducive to advanced manufacturing. This localization strategy can help manufacturers to increase the benefits by potentially reducing the production costs of PPE items. This strategy can also develop a skilled workforce that increases capabilities to create product innovations for the PPE market.

It is impossible to encourage manufacturers to produce PPE products when there are high costs to pivot their operations. An oligopolistic market is unable to achieve the socially optimal coverage without government subsidy (Geoffard & Philipson, 1997). Governments look at PPE not in the context of how much fund is needed to support the firms, but in the context of what is the impact if they do not have PPE available in the private and public sectors. However, the government’s budget is limited and needs to be assigned more effectively to reach the socially optimal coverage of PPE items. The government collaborates with manufacturers via procurement contracts to pivot their operations to produce PPE items. Government subsidies help domestic companies pay part of the cost of production. However, an optimal strategy is needed to efficiently allocate subsidies to manufacturers to achieve a social equilibrium coverage. Hence, the lack of a centralized coordination system for the PPE market may make these actions only partially successful.

Manufacturers also need to invest in pivoting their operations based on their business strategy. For example, firms need to have access to the source of prototypes for designing their products. Therefore, the government needs to assign subsidies to firms more effectively in order to reach an equilibrium coverage (Liu et al., 2021). In this study, we consider an equilibrium system between government and manufacturers that reduces the sum of expected operational and health costs for both players that are higher than if each player acts individually. The investment cost of companies on links is also considered to increase the production capacity of PPE products. The government allocates subsidies to firms based on firms’ performance in order to achieve an equilibrium coverage.

The lack of a centralized coordination strategy was experienced during the Covid-19 pandemic. While healthcare workers and people were concerned about the PPE supply, a response strategy to increase the PPE supply was absent. For example, only a few raw materials are needed to manufacture N95 masks. While these materials are readily available, the PPE shortage was reported during the Covid-19 pandemic by healthcare sectors (Shullih & LeMay, 2021). Therefore, a proactive contingency strategy needs to be prepared at the pre-disruption stage of the pandemic and can be implemented at the post-disruption stage (Elavarasan et al., 2021). In this paper, we propose a coordination system to respond to the Covid-19 types of pandemics that provides the socially optimal coverage in the PPE market.

An SIR model is an epidemiological model that estimates the number of infected individuals with an infectious disease in a closed population over time. There are three categories of population during any epidemic; “susceptible”, “infected”, and “removed/recovered” population. Jahedi and Yorke (2020) introduced a simple SIR model to estimate the number of infected people and the number of susceptible people using satellite equations. Most epidemiological models do not address manufacturing concerns. On the other hand, supply chain models often address logistical issues but do not consider the key characteristics of epidemiology. For instance, the number of infected individuals increases due to the shortage of PPE supply. Hence, this study incorporates the SIR model into the supply chain coordination problem to estimate the expected system costs based on epidemiological dynamics.

We use fear-sentiment information to estimate the percentage of individuals who have switching behaviors that is equivalent to the PPE burn rate for the entire population. Then, a supply-chain resilience model for PPE products is proposed that includes different stakeholders in terms of suppliers, manufacturers, consumers, and government. Furthermore, the impact of different strategies on the proposed game model is studied. Our contributions include the following epidemic management features:

We propose a medical supply chain model with the epidemiology of the disease that coordinates manufacturers and the government to respond to medical items shortage.

We investigate an actual demand price function by modeling the elasticity of demand and individuals’ willingness to pay to avoid risks.

We study a system problem between government and manufacturers that considers expected production, purchasing, and health costs due to the shortage of PPE products.

We design a cost-sharing contract to allocate government subsidy to manufacturers that it provides incentives to both parties.

A socially optimal strategy with the objective of maximizing social welfare is proposed to achieve the largest market coverage.

The rest of this paper is organized as follows: The literature is reviewed in Section 2. Section 3 presents a formal problem definition. Section 4 introduces the proposed coordination system for the PPE market during the Covid-19 types of pandemics. The numerical results of a real-world case study are given in Section 5. Finally, the concluding remarks are presented in Section 6.

Literature review

After the Covid-19 pandemic, domestic companies, including small, medium, and large sizes, are motivated by the government for pivoting their operations to produce essential medical items. Government and private companies put in their best efforts to make a balance between the supply and the demand of PPE products. Manufacturers need to have access to actual demands for PPE products during the pandemic. Companies are supported by the emergency response team in the duration of pandemic by estimating the demand projection of PPE items. However, small and medium companies face excess inventory buildup due to trading again with the global supply chain. Therefore, an equilibrium status between the demand and supply can solve the shortage of PPE products.

One way to estimate PPE demand for the entire population during a pandemic is to conduct a survey with a large number of interviews or questionnaires. However, the survey can be conducted at a single time, and it is expensive to conduct the survey again. For instance, the New York Times estimated mask usage by conducting a single survey New York Times Coronavirus Face Mask Map. Additionally, the estimation of PPE usage by multiplying the entire population and the rate of PPE usage per person is not an efficient method because the number of individuals who use PPE over time cannot be estimated by this method. Therefore, such these methods are not effective ways to show PPE usage in society over time. Loey et al. (2021) proposed a deep learning model for medical face mask detection that helps governments to prevent the COVID-19 transmission. Sayarshad (2022) classified the sentiment of individuals by collecting a corpus of text posts on Twitter. The percentage of susceptible individuals with high fear and low fear using the fear-sentiment information of individuals is estimated. This model provided a real-time demand projection for the entire population based on individuals’ propensities that can help manufacturers to respond to the PPE demands.

Several models have been applied in supply chain models to compute the equilibrium coverage of medical and protective products to respond to an outbreak. For instance, Daniele and Sciacca (2021) introduced an optimization model for the PPE supply chain that considers the supplier’s expected cost function in terms of the production cost and investment cost to produce PPE products. Nagurney (2021) proposed a game model for the supply chain during the COVID-19 pandemic under different scenarios of the labor market. Manufacturers pick a production volume that maximizes expected profits or minimizes expected net costs, while the government selects to cover the PPE products needed for a fraction of the population. In this study, we propose an equilibrium model between manufacturers and the government that minimizes the total system costs (in terms of the expected health and production costs). This competitive system can solve the demand issue and excess inventory buildup where small and medium companies can compete to respond to the PPE shortage based on their business strategy.

The government collaborates with manufacturers via procurement contracts to improve the PPE supply. The government allocates subsidies to firms in the proportion of production costs to achieve the most extensive equilibrium coverage. Liu et al. (2021) investigated the role of government incentives in supporting firms’ sustainability strategies. Fu et al. (2018) investigated government subsidy strategies on a supply chain of electric vehicles (EVs) in general that considers different players in terms of suppliers, manufacturers, and consumers where the impact of different subsidy strategies on social welfare and consumer surplus is considered. Zhang et al. (2019) proposed a game model using government subsidy for charging service fees that considers the interests of the main players in the electric vehicle (EV) charging market. Raz and Ovchinnikov (2015) extended a pricing policy into the news-vendor problem for a generic supply chain using government subsidy. Li and Dong (2021) evaluated government regulations such as purchasing regulation and price regulation to control the shortage of life-saving goods during pandemics when manufacturers want to obtain the maximum profit. Taylor and Xiao (2014) studied whether a donor should subsidize the sales or purchases of malaria drugs in Africa. Likewise, a Stackelberg game is proposed by Levi et al. (2017) to increase the consumption of malaria drugs where an uniform subsidy can be optimal for malaria drugs. Bernstein and Federgruen (2005) investigated a competitive decentralized supply chain model to find the equilibrium behavior of retailers under demand uncertainty.

Several models have focused on combining SIR models and logistics models. For instance, Savachkin and Uribe (2012) proposed an SIR model to distribute the resources that minimizes the impact of pandemics on public healthcare sectors. Rachaniotis et al. (2012) investigated a scheduling model using an SIR model to allocate the limited medical resources during the pandemics. A dynamic logistics model for the allocation of medical resources is studied by Liu and Zhang (2016) that can be used to control the pandemics. Li et al. (2021) proposed an optimization formulation that incorporates an epidemiology model to consider the interactive effects of infectious disease dynamics and supply chain disruption. Rozhkov et al. (2022) investigated a simulation model that combines supply chain design, pandemic dynamics, and operational dynamics to consider the impacts of the Covid-19 pandemic on supply chains. In this study, we propose a game model between government and manufacturers that considers expected production cost, purchasing cost, and health cost due to the shortage of PPE items. Moreover, an optimization model based on the social welfare function is considered to achieve the largest coverage of PPE items during the pandemics.

Several studies address the vaccine supply chain; however, the supply chain of medical and protective equipment has different characteristics on the demand and supply sides. For example, Chandra and Vipin (2021) proposed centralized and decentralized systems for the vaccine supply chain where the government is a buyer, and the manufacturer is a supplier. Sinha et al. (2021) predicted demands to reduce vaccine wastage where a cost-effective multi-echelon inventory model is studied to protect the customer service level. Chick et al. (2008) proposed a game theory model to design a contract of influenza vaccination by considering the transmission rate and the rate of infected people. Bauch et al. (2003) considered individuals’ behavior into a game theory to evaluate the rate of individuals who will be vaccinated. Arifoglu et al. (2012) considered the impact of uncertainty on the demand side and supply side in the influenza vaccine supply chain by solving a game model. Mamani et al. (2012) studied the socially efficient coverage of vaccines by incorporating the government subsidy and vaccine producer under an oligopoly market. Alam et al. (2021) considered fifteen challenges to COVID-19 vaccine supply chain using fuzzy set method.

In this research work, for the first time, we propose a coordination system to respond to the PPE shortage that incorporates the special characteristics of epidemiology. We focus on the supply chain network model that considers manufacturer costs and government expenditure. We propose an equilibrium system that minimizes the system cost in terms of the expected production and health costs. Using the proposed model, we design a cost-sharing contract between manufacturers and the government that satisfies the goals of both parties. Hence, a cost-sharing contract under the system optimum condition is studied that increases the production of PPEs and hence, helps in reducing the number of infected individuals. Lastly, we integrate the consumer surplus, producer surplus, and societal surplus (the cost that may accrue to society, such as the burden of infection on the healthcare sector) into the model to maximize social welfare.

Problem definition

We denote our medical supply chain network by where is the set of all the links and is the set of nodes, see Fig. 1 for an illustration. The set is a union of the three sets the set of manufacturing companies, the set of distribution centers, the set of demand markets. A link in the network either connects a manufacturer company to a distribution center, or a distribution center to a storage center, or a storage center to a demand node. A path of links corresponds to the possible shipment of products from a firm to a demand node. Note that the index is dropped for convenience since the proposed model can be solved for each PPE item . In this study, we consider our case study for face masks .

Fig. 1

Medical supply chain network.

The set of links is divided into two sets where is the set of links in which firms have capabilities to increase the production capacity of PPE items by investing in machines, designs, and prototypes. The second set contains links that have fixed production capacity of PPE items. The set of demand nodes can be written as where denotes the PPE demands of the entire population and represents the PPE demands of large facilities and public sectors such as hospitals, police and fire stations.

This study investigates a resilient supply chain model to coordinate the PPE market. We combine supply chain design, disease dynamics, and operational dynamics in the proposed game model. In this model it is assumed manufacturers and the government work together to respond to the PPE shortage. The proposed supply chain of PPE products is described in Fig. 2, which contains several modules. We consider a non-homogeneous susceptible individuals to estimate the consumption vector of PPE products for the entire population. Social media such as Twitter provides fear-sentiment information utilized by the epidemiological model to determine the ratio of people who have switching behaviors during the pandemics (Sayarshad, 2022). Based on the fear-sentiment information, we consider an epidemiological model with a heterogeneous susceptible population. We estimate the number of susceptible individuals who have low-risk behaviors, i.e., the number of individuals who use PPE products to protect themselves from the infection. This is equivalent to the consumption vector of PPE items for the entire population.

Fig. 2

The proposed supply chain model for PPE market coordination.

The proposed demand price function is calculated by multiplying the number of individuals with switching behaviors and the probability that individuals have the willingness to pay for PPEs (or masks) to avoid risks of the disease. Government and manufacturers collaborate during the pandemic to respond to the shortage. We study how different behavioral aspects of manufacturers and government can self-organize towards a system optimum. We investigate the proposed supply chain model to minimize the expected total system cost.

We consider the expected health cost due to the PPE shortage during the pandemics. When an individual gets infection, there are costs borne by the society. During the pandemics, individuals might change their behaviors by getting fear-sentiment information about the disease. We called them the individuals who have low-risk behaviors. This is equivalent to the consumption vector of PPE products for the entire population. Otherwise, if the individuals do not change their behaviors and act as usual, they have high-risk behaviors. Hence, the associated reduction in infections only depends on PPE coverage in the market. Lastly, a social welfare function is established by considering the producer surplus, consumer surplus in terms of the cost of PPE products for individuals and the infection cost of individuals, and the societal surplus in terms of the burden of infection on healthcare sectors. The proposed supply chain model is considered in a way that maximizes social welfare. The proposed model has the following main applications:

Using the proposed model, we can evaluate manufacturers’ capabilities to pivot their operations to reach the most extensive equilibrium coverage of medical items after a pandemic. Thus, the proposed supply-chain resilience reduces the risk of excess inventory buildup where small and medium manufacturers can compete with large manufacturers to produce medical products.

Using the proposed model, we can analyze how different behavioral aspects of manufacturers and government can self-organize towards a system optimum. Hence, the manufacturing emergency response team (MERT) can provide response and recovery plans for pandemics may cause for the society in the future.

Using the proposed model, we can estimate the consumption vector for PPE products that helps manufacturers to predict the actual demands for PPE items over time.

Using the proposed model, we can estimate the net demand for each traded PPE item by determining the total consumption vector and the total production quantity by domestic firms. This can be a great strategy for future pandemics where domestic manufacturers play a key role in the global trade network. Moreover, a balance between trade surplus and budget deficit can be determined by the proposed strategy.

After the Covid-19 pandemic, many national governments have to rethink their offshoring strategies for the PPE products. A coordination system for the supply of PPE items can be developed using the proposed model that combines the domestic supply networks and the global trade networks. Hence, offshoring and onshoring strategies for the PPE market can be evaluated using the proposed model.

The proposed model

We investigate an equilibrium model between manufacturers and the government. The system problem (called Problem 1) is studied by the expected health cost and production cost between the government and manufacturers. A cost-sharing contract between manufacturers and government is considered to allocate government subsidies to manufacturers in order to achieve the socially optimal coverage. Hence, the aim of maximizing social welfare is more efficient. A socially optimal coverage to reach the largest market coverage under the objective of maximizing social benefit (called Problem 2) is also proposed. A connection between supply and demand functions is provided by implementing the proposed supply chain model, which includes an epidemiology model of the disease. Each of these elements is explained in the following sections.

Expected net manufacturer costs

The utility function of manufacturers is to minimize the total expected costs. The manufacturer picks a production quantity that minimizes the expected costs, as shown in Eq. (1). The first expression of Eq. (1) is the operational costs of links in the supply chain network. The second term of Eq. (1) is the total investment cost on the set of links that can increase the production capacity of PPE products. The last expression of Eq. (1) is the total revenue of manufacturer. Therefore, the expected net manufacturer costs is determined by The optimal coverage of PPE products is calculated by minimum value between the production quantity on paths and a fraction of the aggregate demand for PPE products during each time period. We convert Eq. (2) by the following constraints, which is an equivalent linear constraint. Let be a big constant and define a new binary decision variable , which is either 1 or 0. Thus, we have The link flows of each manufacturer is equal to the sum of PPE products on paths that contain that link, thus we have where is equal 1 if link is contained in path , and 0 otherwise.

Manufacturers increase their manufacturing capacity by investing in automation and technical technologies, including designing and testing prototypes. For instance, investment cost of adding capacity on a link can be estimated by the types of machines. A basic machine can produce 50 face masks per minute (or 32400 3-ply masks per day), while it is 400 face masks per minute (or 273600 3-ply masks per day) with a high-speed machine (Allan, 2022, Ilyukhina, 2022). Therefore, the link capacity can be increased as (Daniele & Sciacca, 2021): where is the upper bound on the added capacity by manufacturers on link . We now discuss how production capacity is related to product flow. Specifically, we assume that the production capacity at each link is a linear production function. Hence, we have that (Daniele & Sciacca, 2021): Moreover, manufacturers have fixed existing production capacity on links. We consider that there is a bound on production capacity on the links, such that (Daniele & Sciacca, 2021)

Incorporation of an epidemiology model

This section combines supply chain design, disease dynamics, and operational dynamics to consider the impacts of pandemics on supply chains. The SIR model is one of the most basic models of disease transmission within a population. For instance, Jahedi and Yorke (2020) investigated an SIR model as shown in Eqs. (7a), (7b) to estimate the ratio of individuals who are infectious per week and the ratio of individuals who are susceptible per week . Susceptible individuals on average make infectious contact which is the intensity function of Poisson distribution. According to Poisson distribution, the probability that no new infectious case happens in week is . Hence, the ratio of new infectious cases and the ratio of susceptible individuals in week is estimated by

Model E+ (7) suggests that we can calculate the contact rate in week by The susceptible individuals on average infectious contact will vary over time, so it makes more sense to predict the intensity by an online prediction method based on a non-homogeneous Poisson process (NHPP) that can be considered in the future work (Sayarshad, 2015, Sayarshad and Chow, 2016).

Sayarshad (2022) studied a heterogeneous susceptible individuals where the population is divided into two groups. One group is the number of susceptible individuals with low-risk behaviors , while another group is the number of susceptible individuals with high-risk behaviors . The ratio of PPE (or mask) usage is equivalent of the ratio of susceptible individuals who have switching behaviors during the pandemics, thus we have (Sayarshad, 2022): where is a coefficient to amplify the fear-sentiment since it is between [−1, 0]. During the pandemics, the final sentiment information contains the new sentiment information and the prior sentiment information that is affected by the memory performance. Hence, an individual is able to learn new information by , while the past information is being forgotten by . Lastly, the final sentiment, , is (Sayarshad, 2022):

Let represents the degree of learning and denotes the rate of forgetting of information, then after the longest memory epoch, memory performance can be measured by . More details about the fear-sentiment function can be found in Sayarshad (2022).

An epidemiology model only focuses on the epidemics of the disease and does not address manufacturing concerns. On the other hand, a resilient supply chain model needs to address logistical concerns that considers the key characteristics of epidemiology. For example, the number of infected individuals increases due to the PPE shortage. We incorporate the number of PPE coverage in the epidemiology model of the disease to estimate the infectious population due to the supply shortage. Hence, the number of infected individuals who have low-risk behaviors during the pandemics is formulated by Similarly, the additional constraint that estimates the number of infected individuals who have high-risk behaviors is: Moreover, the total number of infected individuals is the sum of the number of infected individuals with low-risk and high-risk behaviors. Hence, we have

Demand price function

After the Covid-19 pandemic, the demand for medical items such as PPE items goes up, and the PPE market experiences a shortage of supply. Domestic companies, including small, medium, and large sizes, are invited by the government to pivot their operations to produce the medical items. After the shortage of PPE supply, small and medium companies have the demand issue and excess inventory buildup (FreightWaves). Hence, the actual demand for PPE was absent in the supply chain resilience models. This section estimates the demand projection for PPE products under the assumption of elastic demand. The demand price function is determined by multiplying the number of individuals with switching behaviors and the probability that individuals are willing to pay for PPEs (or masks) to avoid risks of the disease. where the probability that individuals are willing to pay and use PPE products is determined by Theorem 1.

The PPE is also a critical item for large facilities such as hospitals, police stations, and fire stations. Hence, PPE demand for health-care sectors can be computed by the PPE burn rate at hospitals as follow: where is the ratio of hospitalization, is the burn rate of PPE items for infected individuals, and is the burn rate of PPE items by staff at healthcare sectors, including hospitals and medical health centers.

We consider the infection loss of individuals to estimate the probability that individuals are willing to pay for PPE products to avoid the risk of disease. For example, we investigate mask usage by individuals in society as shown in Fig. 3. Suppose that a population consisting of individuals is considered. Let denote the fraction of mask coverage in the society. The infection loss of an individual is , and is consumer price of mask (or PPE) for an individual, which is calculated by multiplying the number of mask usage per week and the mask price.

Fig. 3

PPE consumer and infection loss of individuals.

The infection result of every individual can be one of the four categories: (a) individuals who wear a mask and get the infection, (b) individuals who wear a mask and are healthy, (c) individuals who do not wear a mask and get the infection, and (d) individuals who do not wear a mask and are healthy. The ratio of infected individuals who have low-risk behaviors is and the ratio of infected individuals who have high-risk behaviors is . Hence, the expected infection cost for individuals who wear a mask or do not wear a mask is calculated by and , respectively. After simplification, the infection loss of marginal individuals is determined by Eq. (16). where is the infection loss of the marginal individual in period . The following theorem is considered to calculate the probability that individuals want to join the system and use the PPE items.

If the infection loss of marginal individual is higher than the previous time , , the individual selects to wear a mask (or use PPE). Hence, the probability that individuals are willing to use the PPE products to avoid the risk of infection is

Let , be random variables with Poisson distributions where the intensity function of infection is defined by and , respectively. We define and based on Eqs. (16), (7a). Hence, for any , we have Using Markov inequality and independence, we have . is the moment generating function of and the same for  (Ross, 1998). By substituting the forms of the moment general function, we get a bound that is equal to . Thus, we finally have .

Expected government costs

In response to the pandemics, the government allocates subsidies to help manufacturers to pivot their operations. The government expenditure increases due to collaboration with manufacturers. Hence, the goal of the government problem is to minimize the expected production cost and the expected health cost (in terms of the burden of infection on the health-care sector due to PPE shortage).

After a pandemic, the new infection imposes a burden on the health care system. The health care sector is not adequately prepared to deal with it first, but their knowledge will change over time. Hence, the health care stress cost depends on the and two other factors, and . shows the health care cost due to the stress of health care and presents the efficiency level or the speed of learning related to the healthcare sector. Thus, the burden of infections on the healthcare system due to PPE shortage is determined by: Based on Eq. (19) given by Gonzalez-Eiras and Niepelt (2020) and Sayarshad (2022), is calibrated by the damage of the pandemic on the economic of a country. For example, in the USA, it was about 13 trillion dollars which is corresponding to 61% of the annual U.S. GDP (). It is assumed that the health care sector can be prepared to deal with the pandemic during a half of year or in (365/2) 182 days.

The second term of (21) is the expected purchasing cost for PPE products by government, where the PPE coverage per week by manufacturers is computed by Eq. (2). Moreover, manufacturers keep the optimal production level when the expected revenue exceeds the cost per unit, because the PPE production is profitable for firms. The cost revenue ratio is a measure of efficiency that compares a manufacturer’s expenses to its earnings. Hence, we consider constraint (20) which is based on the cost revenue ratio to ensure that manufacturers act optimally (Chick et al., 2008).

Formally, the government problem is

Total system cost

This section considers an equilibrium model between manufacturers and the government. As shown in Eq. (22), the system problem between government and manufacturers is determined by the purchasing cost, the expected health cost, and production cost. The first expression of Eq. (22) is the expected purchasing cost by the government. The second term of Eq. (22) is the burden of infection on the healthcare sector due to the shortage of PPE supply. The third expression of Eq. (22) is the operation costs of links to produce PPE products. The last term of Eq. (22) is the investment cost of links to increase the capacity. Thus, the system problem is:

System Problem

Cost-sharing contracts

An oligopolistic market without government subsidies cannot reach the socially optimal coverage of PPE products. The government can offer subsidy to manufacturers that helps to achieve the demand–supply equilibrium. For instance, the government pays the proportion of production costs under the cost-sharing contract. However, the government budget is limited. In this section, a cost-sharing contract between the government and manufacturers is considered to increase production, leading to a reduction in the number of infected individuals.

A fixed percentage of subsidy reaches the optimal coverage of masks products. The subsidy covers the proportion of total production costs (in terms of the operation costs and the investment costs on links). Hence, the government cost function increases under cost-sharing contracts as follows: where is the percentage of coverage costs, . The proportion of production costs can be shared by the government under the cost-sharing contract. Moreover, manufacturers keep the optimal production level under cost-sharing contracts as Eq. (24). Other constraints of Problem 1 remain unchanged.

Therefore, a mechanism to allocate government subsidies is proposed that satisfies manufacturers and government and it helps to achieve a higher coverage of PPE during pandemics.

Social welfare function

An optimal social coverage of PPE products is considered in this section that integrates producer surplus, consumer surplus, and societal surplus under the objective of maximizing social welfare.

Producer surplus

In order to determine producer surplus, we consider the difference between the operational costs of links and the revenue of firm. Hence, we have Let be the ratio of infected individuals with low-risk behaviors and use PPE products to avoid the risk of infection and the ratio of infected individuals with high-risk behaviors. The ratio of infection depends jointly on the ratio of PPE coverage in the market and the ratio of susceptible individuals who have high-risk and low-risk behaviors (Bauch & Earn, 2004). Under the PPE coverage for the population , the ratio of infected individuals for the entire population is defined by: After simplification, we get The willingness to pay for any consumer product depends on Eq. (26). Substituting these and letting , the equilibrium price of PPE products is Note that the equilibrium price is associated with the optimal market coverage where the ratio of infection depends on PPE coverage . In other words, the equilibrium price of PPE products is the marginal reduction of infection due to use of PPE products by individuals.

Consumer surplus

We consider heterogeneous susceptible individuals for PPE consumers during pandemics. The consumer surplus is defined by the total expected cost of individuals willing to use PPE products. Thus, we have

Government surplus

In addition to the costs incurred by individuals and the manufacturers for producing the PPE items, we also consider the expected health cost that may accrue on the society due to the shortage of PPE supply during the pandemics. The expression in the right-hand side of Eq. (29) is the burden of infection on the health care system where it is proportional to the impact of PPE shortage on infectious population and a factor that shows the health-care performance during the pandemic. where shows the health care cost due to the stress of health care and denotes the efficiency level or the speed of learning related to the healthcare sector. Combining all these together, the social welfare is defined as follows:

Social Welfare Problem

Formulating the proposed model in the form of a stage-by-stage algorithm

The proposed coordination system for PPE market including the epidemiology model can be summarized as the following stage-by-stage algorithm:

Stage 1: Input parameters including , , , , , , and time step should be specified. Detailed information about other input parameters is shown in Table 1.

Table 1

Input parameters of scenario 2.

Problem setup
Set of links (A) {1→2,1→3,1→4,1→5,1→6,1→7,2→8,2→9,3→8,3→9,4→8,
4→9,5→8,5→9,6→8,6→9,7→8,7→9,8→10,9→11,10→12,
10→13,11→12,11→13}
Set of links (A_1) {1→4,1→5,2→9,3→8,3→9,4→8,4→9,5→8,
5→9}
Set of links (A_2) {1→2,1→3,1→6,1→7,2→8,6→8,6→9,7→8,7→9,8→10,9→11,
10→12,10→13,11→12,11→13}.
Set of paths (p): p_1=1→2,2→8,8→10,10→12, p_2=1→2,2→8,8→10,10→13,
p_3=1→2,2→9,9→11,11→12, p_4=1→2,2→9,9→11,11→13,
p_5=1→3,3→8,8→10,10→12, p_6=1→3,3→8,8→10,10→13,
p_7=1→3,3→9,9→11,11→12, p_8=1→3,3→9,9→11,11→13,
p_9=1→4,4→8,8→10,10→12, p_10=1→4,4→8,8→10,10→13,
p_11=1→4,4→9,9→11,11→12, p_12=1→4,4→9,9→11,11→13,
p_13=1→5,5→8,8→10,10→12, p_14=1→5,5→8,8→10,10→13,
p_15=1→5,5→9,9→11,11→12, p_16=1→5,5→9,9→11,11→13,
p_17=1→6,6→8,8→10,10→12, p_18=1→6,6→8,8→10,10→13,
p_19=1→6,6→9,9→11,11→12, p_20=1→6,6→9,9→11,11→13,
p_21=1→7,7→8,8→10,10→12, p_22=1→7,7→8,8→10,10→13,
p_23=1→7,7→9,9→11,11→12, p_24=1→7,7→9,9→11,11→13.
Production capacity with investment (Γ¯_n,α): Γ¯_n,1→4=4000000, Γ¯_n,1→5=3100000,
Γ¯_n,2→9=1750000, Γ¯_n,3→8=1250000, Γ¯_n,3→9=1300000,
Γ¯_n,4→8=1000000, Γ¯_n,4→9=1000000, Γ¯_n,5→8=2500000, Γ¯_n,5→9=1750000
Production capacity with fixed products (Γ¯_n,γ), Γ¯_n,1→2=15200000, Γ¯_n,1→3=15600000,
Γ¯_n,1→6=4500000, Γ¯_n,1→7=4300000, Γ¯_n,2→8=2500000, Γ¯_n,6→8=1250000, Γ¯_n,6→9=1300000,
Γ¯_n,7→8=1000000, Γ¯_n,7→9=1000000, Γ¯_n,8→10=2500000, Γ¯_n,9→11=7700000,
Γ¯_n,10→12=4500000, Γ¯_n,10→13=6000000, Γ¯_n,11→12=7000000, Γ¯_n,11→13=90000000.
Operation cost (c_a), c_1→2=0.002, c_1→3=0.003, c_1→4=0.004, c_1→5=0.005, c_1→6=0.006,
c_1→7=0.004, c_2→8=0.003, c_2→9=0.002, c_3→8=0.003, c_3→9=0.004, c_4→8=0.005, c_4→9=0.006,
c_5→8=0.004, c_5→9=0.003, c_6→8=0.006, c_6→9=0.004, c_7→8=0.003, c_7→9=0.002,
c_8→10=0.003, c_9→11=0.004, c_10→12=0.005, c_10→13=0.006, c_11→12=0.004, c_11→13=0.003.
The cost of investment capacity (π_a) π_1→4=0.001, π_1→5=0.002, π_2→9=0.004, π_3→8=0.003,
π_3→9=0.001, π_4→8=0.002, π_4→9=0.004, π_5→8=0.003, π_5→9=0.003.

Stage 2: We estimate the number of infected individuals and the number of susceptible individuals using Eqs. (7a), (7b), respectively.

Stage 3: We estimate the temporary fear-sentiment information using Twitter data, which its details can be found in Sayarshad (2022).

Stage 4: We compute the final fear-sentiment information using Eq. (10) where memory performance of individuals during the pandemic for learning and forgetting information about the disease is considered.

Stage 5: The number of susceptible individuals who have low-risk behaviors is estimated by Eq. (9a). This is equivalent to the PPE burn rate for the entire population.

Stage 6: We determine the demand price function for the entire population and large facilities such as hospitals and medical centers using Eqs. (14), (15), respectively.

Stage 7: We solve the proposed game model under different strategies, as shown in Section 4.

Stage 8: If has reached the last time step of the study period, go to the next stage. Otherwise, increment and go back to stage 2.

Stage 9: Report results: the link flow of PPE item on link , the equilibrium path flow , the investment cost of adding capacity on a link , the quantity coverage of PPE item , infected individuals with low-risk behaviors , infected individuals with high-risk behaviors , the fraction of coverage .

By implementing the above process for all time periods, we can determine the equilibrium coverage of PPE products based on different strategies. A flowchart of the proposed model is shown in Fig. 4.

Fig. 4

The proposed algorithm.

Experimental results

Experimental tests are performed to study the efficiency of the proposed model. We investigate a case study in LA County under different strategies to show the efficiency of the coordination supply chain model in a real-world case. We used a laptop computer with a 2.50 GHz processor, Intel Core i7-2450, 12 GB of RAM, and 64 bit Windows 10 Professional operating system. The model was coded in Julia 1.7.1 and we solved the mathematical models using the Juniper package (Kröger et al., 2018), Jump package (Lubin & Dunning, 2015), and Ipopt package in Julia.

Data

We carried out an experimental study of the Covid-19 pandemic using the proposed model for mask coordination based on real-world data. The list of our data used for the case study is presented below:

Individuals change their behaviors by following social distancing and wearing a mask. Therefore, actual mask usage in LA County can be estimated by the rate of individuals who have low-risk behavior. Similar to Sayarshad (2022), we apply the fear-sentiment information to calculate the percentage of individuals who have switching behaviors. Fig. 5 shows mask usage during Covid-19 in LA County for 36 weeks.

Fig. 5

Susceptible individuals who have switching behaviors in LA County .

The demand for the mask in large facilities such as the hospitals and medical centers in LA County is calculated by Eq. (15) which is based on the mask burn rate per shift (Centers for Disease Control and Prevention: PPE burn rate) and the number of staff at hospitals and medical centers in LA County (Department of Public Health, 2020, Hospitals and Medical Centers, 2020). The burn rate of mask per week for each individual is assumed to be and the consumer price for face mask is assumed to be $1.5. The burn rate of mask for infected individuals per week is assumed to be . We assume that the burn rate of masks per week for staff in hospitals and medical centers is  (Centers for Disease Control and Prevention: PPE burn rate).

The number of Covid-19 confirmed positive cases in LA County that is used in the SIR model, can also be found in the USAFacts dataset (USAFACTS).

The health care cost due to the stress of health care is , and the speed of learning for the health care service is assumed to be  (Sayarshad, 2022). Coefficient to amplify the final sentiment is assumed to be  (Sayarshad, 2022).

Illustrative example

Our supply chain network consists of one firm, six manufacturers, two distribution centers, two distribution storage centers, and two demands market, see Fig. 6 for an illustration. The input parameters of our case study are shown in Table 1. The information contains the set of links, the set of paths from firm until the demand nodes and the production capacities on links under different capacity. We assume that the cost function for operation cost is given: The cost function for the investment capacity is assumed as: Moreover, the total revenue of production is assumed to be: The parameters related to the cost functions are also presented in Table 1.

Fig. 6

The supply chain network of medical items.

In this study, we proposed a demand price function that considers the elasticity of demand. The demand function is determined by multiplying the number of individuals with switching behaviors and the probability that individuals have the willingness to pay for masks to avoid getting infected. The aggregate demand for masks in hospitals and the whole population is shown in Fig. 7. We investigate our game setting using three strategies as follow:

Fig. 7

The aggregate demand for masks per week in LA County ().

Strategy 1 (MC): minimizing net manufacturer costs.

Strategy 2 (SC): minimizing total system costs (Problem 1).

Strategy 3 (SW): maximizing social welfare (Problem 2).

We solve our case study under the first strategy that minimizes net manufacturer costs. The CPU run-time in seconds of the proposed model for the test case is about 6 s. The equilibrium links flow of mask production for one week is shown in Table 2. We experienced a higher number of infection cases in LA County during the selected week. The equilibrium path flows of mask products is presented in Table 2. According to the results of strategy 1, the production capacity on links that have capabilities to increase the capacity is zero. The results show that a monopoly market is unable to solve the PPE shortage during a pandemic when manufacturers act individually. Hence, the government needs to allocate subsidies to firms to achieve the largest coverage of PPE products.

Table 2

Equilibrium results of strategy 1.

Link flow of mask production f_n,a
f_18,(1→2)=1184176.3,	f_18,(1→6)=2002103.9,	f_18,(1→7)=1613716.5,	f_18,(2→8)=1184176.0,
f_18,(6→8)=702104.9,	f_18,(6→9)=1299999.0,	f_18,(7→8)=613717.5,	f_18,(7→9)=999999.0,
f_18,(8→10)=2499999.0,	f_18,(9→11)=2299999.2,	f_18,(10→12)=1230335.6,	f_18,(10→13)=1269663.4,
f_18,(11→12)=1114669.5,	f_18,(11→13)=1185329.6.
Equilibrium mask product path flows x_n,p
x_18,p_1=579952.1,	x_18,p_2=604223.8,	x_18,p_17=346785.1,	x_18,p_18=355319.8,
x_18,p_19=627807.2,	x_18,p_20=672191.8,	x_18,p_21=303598.1,	x_18,p_22=310119.4,
x_18,p_23=486861.7,	x_18,p_24=513137.3.

We find an equilibrium coverage of mask products where the government and firms collaborate together to respond to mask shortages during the pandemic. The equilibrium links flow of mask production for one week is shown in Table 3. The equilibrium path flows of mask products is presented in Table 3. The production capacity on links is demonstrated in Table 3. The CPU run-time in seconds of the proposed model for the test case is about 10 s.

Table 3

Equilibrium results of strategy 2.

Link flow of mask production f_n,a
f_18,(1→3)=1085450.9,	f_18,(1→6)=2030485.8,	f_18,(1→7)=1679560.3,	f_18,(2→8)=1089953.9,
f_18,(2→9)=1213849.2,	f_18,(3→9)=1085450.9,	f_18,(6→8)=730485.8,	f_18,(6→9)=1300000.0,
f_18,(7→8)=679560.3,	f_18,(7→9)=1000000.0,	f_18,(8→10)=2500000.0,	f_18,(9→11)=4599300.1,
f_18,(10→12)=1215925.4,	f_18,(10→13)=1284074.6,	f_18,(11→12)=1792711.5,	f_18,(11→13)=2806588.5.
Equilibrium mask product path flows x_n,p
x_18p_1=527924.5,	x_18p_2=562029.4,	x_18p_3=427148.3,	x_18p_4=786700.9,
x_18p_7=417728.2,	x_18p_8=667722.6,	x_18p_17=355973.0,	x_18p_18=374512.9,
x_18p_19=523245.7,	x_18p_20=776754.3,	x_18p_21=332027.9,	x_18p_22=347532.4,
x_18p_23=424589.2,	x_18p_24=575410.8.
Equilibrium link for the investment of PPE production capacity Γ_n,a
Γ_18,(2→9)=1213849.2,	Γ_18,(3→9)=1085450.9.

Lastly, the socially optimal coverage for coordinating masks to maximize social welfare is solved. The equilibrium links flow of mask production for one week is shown in Table 4. The equilibrium path flows of mask products are presented in Table 4. The production capacity on links is demonstrated in Table 4. The CPU run-time in seconds of the proposed model for the test case is about 20 s.

Table 4

Equilibrium results of strategy 3.

Link flow of mask production f_n,a
f_18,(1→2)=2190728.4,	f_18,(1→3)=1045077.0,	f_18,(1→4)=190088.9,	f_18,(1→5)=191865.8,
f_18,(1→6)=1930276.4,	f_18,(1→7)=1551263.6,	f_18,(2→8)=1034907.8,	f_18,(2→9)=1155820.6,
f_18,(3→8)=98818.9,	f_18,(3→9)=946258.1,	f_18,(4→8)=53869.8,	f_18,(4→9)=136219.1,
f_18,(5→8)=53914.5,	f_18,(5→9)=137951.3,	f_18,(6→8)=655194.5,	f_18,(6→9)=1275082.0,
f_18,(7→8)=576546.9,	f_18,(7→9)=974716.7,	f_18,(8→10)=2473252.4,	f_18,(9→11)=4626047.7,
f_18,(10→12)=1218345.1,	f_18,(10→13)=1254907.3,	f_18,(11→12)=2216352.9,	f_18,(11→13)=2409694.9.
Equilibrium mask product path flows x_n,p
x_18,p_1=505923.8,	x_18,p_2=528984.0,	x_18,p_3=551212.4,	x_18,p_4=604608.3,
x_18,p_5=49384.3,	x_18,p_6=49434.7,	x_18,p_7=455159.9,	x_18,p_8=491098.1,
x_18,p_9=26934.3,	x_18,p_10=26935.6,	x_18,p_11=67927.1,	x_18,p_12=68291.9,
x_18,p_13=26958.3,	x_18,p_14=26956.2,	x_18,p_15=68824.1,	x_18,p_16=69127.2,
x_18,p_17=323779.2,	x_18,p_18=331415.2,	x_18,p_19=604949.9,	x_18,p_20=670132.1,
x_18,p_21=285365.3,	x_18,p_22=291181.6,	x_18,p_23=468279.5,	x_18,p_24=506437.2.
Equilibrium link for the investment of PPE production capacity Γ_n,a
Γ_18,(1→4)=217203.1,	Γ_18,(1→5)=219057.8,	Γ_18,(2→9)=1178661.0,	Γ_18,(3→8)=128395.7,
Γ_18,(3→9)=968816.6,	Γ_18,(4→8)=88101.6,	Γ_18,(4→9)=164406.4,	Γ_18,(5→8)=87803.7,
Γ_18,(5→9)=165925.8.

Equilibrium behavior evaluation

In this section, we evaluate the proposed game model by comparing three strategies that were mentioned in Section 5.2. We compare the optimal number of coverage when different strategies are investigated. As shown in Fig. 8, the maximum mask coverage is found when social welfare is maximized. On the other hand, the minimum mask coverage is observed when net manufacturing costs is minimized. The number of coverage under system optimum is also presented in Fig. 8 where the government collaborates in mask production ramp up to combat pandemics. It is interesting to note that the manufacturers produce a fixed production quantity per week when manufacturers act individually. Hence, the capacity on links that can be increased through investments is zero, as shown in Table 2. Under socially optimal level, the number of coverage increases by 34% and 2% compare to strategy 1 and strategy 2, respectively. Table 5 shows the equilibrium behavior evaluation of three strategies.

Fig. 8

Optimal PPE coverage using different strategies.

Table 5

Equilibrium behavior of different strategies.

Strategies	σ_n	Relative difference (%)	Υ_n	Relative difference (%)	I_n	Relative difference (%)
MC	0.577	−33%	4799998.2	−34%	8247.1	+30%
SC	0.844	−2%	7175440.8	−2%	6952.6	+10%
SW	0.857	–	7285511.4	–	6325.6	–

Moreover, we compare the fraction of mask coverage where three strategies are studied as shown in Fig. 9. Total demands for the mask are satisfied under social welfare maximization. While the minimum fraction of coverage is observed under the first strategy where manufacturers act individually to produce PPE items. The coverage fraction under the socially optimal level increases by 33% and 2% compared to strategy one and strategy two, respectively. Interestingly, we can respond to mask shortages more efficiently when the government collaborates with firms under the system problem and social welfare maximization.

Fig. 9

Optimal fraction of coverage in the market.

Finally, three strategies are evaluated by comparing the number of infections. Under the socially optimal level, the minimum number of infections is found, as shown in Fig. 10. On the other hand, the number of infection cases increases under the objective of minimizing the net manufacturer costs. Hence, the number of infections reduces under government intervention due to increased mask supply. As shown in Table 5, the number of infected individuals under the goal of maximizing social welfare was reduced by 30% and 10% compared to strategy one and strategy two, respectively.

Fig. 10

The number of infection under different strategies.

Cost-sharing contract and evaluation

In this section, we consider a cost-sharing contract that increases the production of PPEs and hence, helps in reducing the number of infected individuals. The government needs to pay the proportion of production costs to firms. Otherwise, the cost of infections and the burden of infection on the healthcare sector can be increased. Therefore, the government collaborates with manufacturers via a cost-sharing contract to increase the supply of mask (or PPE) products. In order to compare the cost-sharing contract between manufacturers and the government, we evaluate the proposed game model based on the following strategies:

Before contract: manufacturers act individually to produce PPE products during the pandemic.

After contact, 70% costs: manufacturers collaborate with the government and 70% of production costs can be paid by the government.

After contact, 100% costs: manufacturers collaborate with the government and 100% of production costs can be paid by the government.

The results of coverage under different actions between government and manufacturers are shown in Fig. 11. Manufacturers collaborate with the government to produce masks where 70% and 100% of production costs are paid by the government. Note that the system optimum cannot be fully achieved when costs coverage by government is less than 70%. As shown in Table 6, the fraction of mask coverage when manufacturers act individually to produce masks decreases by 32%, while the optimal fraction of coverage increases by 32% and 27% when the government pays 70% and 100% of production costs, respectively. The proposed model can estimate the optimal government subsidy that needs to achieve an optimal equilibrium coverage of PPE products. Moreover, a balance between trade surplus and budget deficit can be determined by the proposed strategy in order to deal with the global supply chain.

Fig. 11

The fraction of mask coverage.

Table 6

An comparison of strategies under different status of contract.

Strategies	σ_n	Relative difference (%)	Υ_n	Relative difference (%)	I_n	Relative difference (%)
Before contract	0.577	−32%	4799998.2	−33%	8247.1	+19%
After contract, 70% costs	0.620	−27%	5096003.3	−29%	7912.2	+14%
After contract, 100% costs	0.844	–	7175440.8	–	6952.6	–

We evaluate the optimal number of mask coverage where manufacturers act individually or collaborate with the government via a cost-sharing contract. The results of actions between both parties are shown in Fig. 12. As shown in Table 6, the optimal number of coverage with government involvement increases by up to 32% compared to a strategy without government intervention. Therefore, without government subsidies, an oligopolistic market cannot reach the socially optimal coverage.

Fig. 12

Mask coverage.

Furthermore, we compare the number of infected individuals that is related to the mask coverage in the market. The number of infected individuals under different actions between government and manufacturers is shown in Fig. 13. As shown in Table 6, when the government intervenes, the number of infected individuals decreases by up to 19% compared to a strategy when manufacturers act individually. The Covid-19 infection wave has caused 13 trillion dollars costs in the U.S. which this amount corresponds to 61% of the annual U.S. GDP (Scherbina, 2020). The economic burden of healthcare associated with new infections that the government pays for is extremely high. Hence, the cost-sharing contract can optimize the coordination between manufacturers and the government.

Fig. 13

The number of infection.

Conclusion

The first coordination model of the PPE supply chain was proposed that investigates the socially optimal coverage using the subsidy. We proposed a medical supply chain model with the epidemiology of the pandemic that considers expected production cost, purchasing cost, and health cost due to the shortage of PPE products. We evaluated different strategies between government and manufacturers where the elasticity of demand and willingness to pay to avoid risks were investigated. A socially optimal coverage model with the objective of maximizing social welfare was also proposed to achieve the largest market coverage. Compared to other strategies, the fraction of coverage under the socially optimal level increased by up to 33%. Additionally, the number of infected individuals under social welfare maximization was reduced by 30%.

An oligopolistic market cannot achieve socially optimal coverage without government subsidies. Therefore, a cost-sharing contract to allocate government subsidies to manufacturers was investigated that increased production, leading to a reduction in infection cases. The results showed that the optimal number of coverage with government involvement increased by up to 32% compared to a strategy without government intervention. Additionally, the number of infected individuals with government intervention decreased by up to 19% compared to when manufacturers acted individually.

The proposed model can be extended in the future research works, as below:

After the Covid-19 pandemic, many countries have to rethink their offshoring strategies for the PPE market. The net demand for each traded PPE item can be estimated by the total consumption vector for the PPE item and the total production quantity of PPE items that should be produced by the domestic manufacturing firms (in terms of private sectors and public sectors). Hence, different response plans in terms of trading with the global supply chain and considering a localization strategy by domestic firms would be evaluated in future research.

An international trade resilience with the epidemiology of the disease can be investigated in future research where countries compete for limited supplies of PPE items to maximize social benefits (Li et al., 2021, Salarpour and Nagurney, 2021). The balance of trade (in terms of trade surplus and trade deficit) with the goal of social benefit could be investigated in the future study in order to find the best strategies for future pandemics (Nader et al., 2022).

A blockchain technology that provides smart contracts and improves the transparency of information between stakeholders, including suppliers, market demand, and government, would be considered in future research (Bhushan et al., 2020, Kumar et al., 2020).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.