Dynamic Adjustment Mechanism and Differential Game Model Construction of Mask Emergency Supply Chain Cooperation Based on COVID-19 Outbreak

Abstract

Incorporating the impact of the COVID-19 pandemic on the mask supply chain into our framework and taking mask output as a state variable, our study introduces the differential game to study the long-term dynamic cooperation of a two-echelon supply chain composed of the supplier and the manufacturer under government subsidies. The study elaborates that government subsidies can provide more effective incentives for supply chain members to cooperate in the production of masks compared with the situation of no government subsidies. A relatively low wholesale price can effectively increase the profits of supply chain members and the supply chain system. The joint contract of two-way cost-sharing contract and transfer payment contract can promote production technology investment efforts of the supply chain members, the optimum trajectory of mask production, and total profit to reach the best state as the centralized decision scenario within a certain range. Meanwhile, it is determined that the profits of supply chain members in the joint contract can be Pareto improvement compared with decentralized decision scenario. With the increase of production technology investment cost coefficients and output self-decay rate, mask outputs have shown a downward trend in the joint contract decision model. On the contrary, mask outputs would rise with growing sensitivity of mask output to production technology investment effort and increasing sensitivity of mask demand to mask output.

1. Introduction

The COVID-19 pandemic has had a massive impact on life safety and the global economy since its outbreak at the end of 2019. According to data released, the WHO (World Health Organization) reported a total of 64,603,428 confirmed cases worldwide on 4 December 2020, and a total of 1,500,614 people succumbed to COVID-19. Although COVID-19 prevention and control have achieved initial success, the black swan event makes medical protection supplies demand to show a blowout situation. In the COVID-19 pandemic, no country can survive alone. It is essential to replenish health emergency supplies such as medical material and personal protective equipment (PPE) for pandemic preparedness and prevention. It is equally important to strengthen emergency supply chain elasticity for rapid expansion and conversion. Therefore, we need to make effective coordination decisions to safeguard a stable, reliable, and rapid supply of health emergency supplies.

The COVID-19 pandemic has largely restricted population mobility around the world. Therefore, medical masks have become significant medical protective products, and we consider the medical masks to be a symbol of the fight against COVID-19. This gives the impression that there has been a huge demand gap in many countries. As the provider of an important medical protection material, the mask industry is not only related to the safety of people’s lives but also related to the public health protection of the entire society. Therefore, after the COVID-19 outbreak, global mask enterprises have accelerated mask expansion and production through technological transformation and shifts over time. Consequently, the total production volume of medical masks has rapidly increased, effectively alleviating the “hard to find a mask” situation.

Research related to the COVID-19 pandemic and economic development has attracted significant attention from many scholars, some of whom have proposed corresponding countermeasures against mask shortages. Morales-Narvaez and Dincer [1] improved personal protective equipment (PPE) to fill the gap in protective materials and maintain the stability of the PPE supply chain. Due to shortages of hand sanitizers, masks, and medical gloves, some experts have proposed various methods to mitigate risk [2,3]. The salt-based solution pretreatment and soaking strategy can boost the filtering capacity of surgical masks [4]. In addition, scholars propose some COVID-19 relevant supply chain issues. A game approach was proposed and procedures such as storage capacities and effective measures were utilized to avoid straining PPE supply chains against the second sweep of the COVID-19 pandemic [5]. Santini indicated that increased inter-regional collaboration and a steadier supply of reagents can increase testing capacity against the COVID-19 pandemic [6].

The mask supply chain is composed primarily of upstream Meltblown cloth suppliers, midstream mask manufacturers, and downstream hospitals/pharmacies. COVID-19 has triggered a significant impact on the upstream, midstream, and downstream supply chain reflected in both supply and price. The first aspect is the supply influence. Firstly, it is reported that the price of Meltblown production line equipment is 10–50 times that of a fully automatic flat mask machine. The core of mask production lies in the production and electret treatment of Meltblown non-woven fabrics. The technical requirement of other production links are not high, so mask supply chain coordination should focus on the supply of Meltblown cloth products, which is the core base material of masks. Secondly, due to the mobility restriction of major labor forces, manufacturers can hardly resume their work activities. Thirdly, global travel restrictions affect the logistics efficiency of Meltblown cloth and reduce the on-time delivery rate of masks. Eventually, due to the pandemic differences in various regions, the production schedule and production volume are not fairly matched. The other aspect is the price influence. The increase in labor costs and logistics costs contributed by the pandemic and special process requirements for the core material of masks have caused substantial changes in the production cost and mask supply, which brings considerable conflicts to the originally coordinated supply chain and disrupts the supply chain balance. Therefore, to actively respond to the impact of COVID-19, close collaboration between upstream and downstream enterprises is required. In addition, to jointly explore effective strategies to increase mask output, mask manufacturers can increase production technology investments. The raw material supplier can adopt technology investments to increase the production of critical raw material. As the emerging leader and resource coordinator for handling major pandemic incidents, the government plays an essential role in sponsoring supply chain coordination and enlightening effectiveness. To promptly resume a healthy supply chain, the relevant administrations must be vigilant to the production coordination issues of member enterprises and coordinate the imbalance of market supply and demand through specific intervention methods such as subsidy policies. We can take these measures to solve material supply interruption, labor shortages, and cost pressure, which improve collaboration between upstream and downstream enterprises. Furthermore, the government can guide supply chain members to make production decisions and formulate a reasonable profit distribution or cost distribution mechanism for supply chain members in the specific implementation process, optimizing the supply chain from an unbalanced state to a balanced state. Therefore, both the manufacturer and the supplier are committed to increasing production technology investment and choosing effective cooperation output strategies and coordination mechanisms. Studying the dynamic coordination of mask emergency supply chain in the significant pandemic scenario has essential theoretical and practical significance.

The differential game entails multiple participants attempting to optimize their independent and incompatible goals and eventually gain optimal Nash equilibrium strategies. More specifically, one or more state variables change over time according to the differential equation. Due to the impact of COVID-19, mask production is a multi-cycle rather than a single-cycle process. From a long-term and dynamic perspective, it is more realistic to employ differential games for research. However, few studies incorporate the impact of COVID-19 into the research on the emergency supply chain and consider the characteristics of dynamic demand. Based on the above analysis, this research aims to solve the following problems.

(1)

How does the trajectory of mask production shift over time?

(2)

How do parameters affect supply chain decisions? These parameters consist of the supplier and manufacturer’s production technology investment cost coefficients, the sensitivity coefficients of production volume to the production technology investment efforts of the supply chain members, the sensitivity coefficient of mask demand to mask production volume, and government subsidies.

(3)

How does the joint contract affect the optimal strategy and coordinate the mask emergency supply chain?

Our study applies the joint contract, combining two-way cost-sharing contract and transfer payment contract to optimize the mask emergency supply chain. Introducing differential game and optimal control theory, we consider the impacts of COVID-19, price, and government subsidies as endogenous variables on the mask demand. Furthermore, our results also highlight that the joint contract realizes the coordination consequence on the mask emergency supply chain, reduces the production risk, and increases the profits of the supply chain members.

The rest of our paper consists of the following sections. Section 2 outlines a literature review. Section 3 presents the parameter descriptions. The optimal output strategy for the supplier-led decentralized supply chain and centralized supply chain is analyzed in Section 4, constructing a differential game model for the joint production of the supplier and manufacturer in a major pandemic and adopting the collective contract for emergency supply chain coordination. Section 5 conducts numerical analysis to gain more management enlightenment. Finally, we summarize the conclusions and propose future directions in Section 6.

2. Literature Review

Three relevant research streams are integrated from the literature, including contract coordination, cooperation between enterprises in emergency scenarios, and the government’s role in supply chain cooperation and decision activities.

Firstly, our paper focuses on contract coordination. The contract mechanism becomes an effective method for supply chain coordination. The main differences lie in various components of supply chain members and different incentive methods, such as revenue sharing contract [7], quantity discount contract [8], option contract [9], wholesale price contract [10], etc. These contracts can efficiently realize the coordination performance of the emergency supply chain. With the gradual complexity of emergencies, two or more joint contracts to coordinate supply chains have also become a research focus. For example, Wang and Choi constructed three contract models of revenue sharing, cost sharing, and two-part tariffs under total carbon control and trade emission constraints [11]. In response to the availability of wind-power energy equipment, Liu et al. put forward three types of contracts to analyze coordination issues [12]. Li et al. designed a capacity retention contract and quantity flexibility contract as a risk-sharing coordination mechanism that incentivizes the enterprise to boost production volume and improve the overall performance [13]. A risk aversion contract or risk-sharing contract was designed to realize the coordination under stochastic returns and asymmetric productivity information [14,15]. Based on the risk-neutral manufacturer and risk-averse retailers, Wang, Guo, and Wang studied coordination contract design issues, including transaction volume, transfer payment, and profit distribution rules [16]. He, Ma, and Pan proposed a risk diversification contract with different risk attitudes of supply chain members [17]. Research indicated that this contract had a larger market than the option contract, capacity reservation contract, etc. Adhikari, Bisi, and Avittathur studied coordination issues of a five-echelon textile supply chain [18].

Secondly, the supply chain is usually composed of two or more enterprises. Emergency events lead to the supply chain changing from a balanced state to an unbalanced state. Therefore, the cooperative relationship research between upstream and downstream enterprises in the emergency scenario is predominantly imperative. For example, Li, Chen, and Ai designed a wholesale price contract and two-part tariffs contract and then determined the Bayes–Nash equilibrium solution of supply chain members [19]. Zhou et al. employed the abatement cost-sharing contract to coordinate the carpet supply chain with fairness concerns [20]. Based on the dual-channel supply chain composed of manufacturer and packaging companies, Jabarzare and Rasti-Barzoki analyzed the best pricing, quality decision, and supply chain members’ profits through three methods [21]. The upstream manufacturer and downstream retailers conjointly bore product liability costs incurred by quality defects [22]. Zhang and Wang employed a duopoly game model to study the influences of enterprise horizontal and vertical fairness [23]. Liu, Chen, and Fang proposed dynamic coordination methods of a temporary static allocation mechanism and a compensation mechanism for the supply chain [24]. Zhou and Ye established a cooperative advertising contract and an emission reduction cost-sharing contract to realize dual-channel supply chain coordination [25]. Sharma, Dwivedi, and Singh constructed a supplier-led game model and coordinated through the options contract, concluding that the fairness sensitivities of retailers and suppliers have opposite effects on the entire supply chain [26]. A two-echelon drug supply chain composed of drug distributors and drug retailers with random demand was studied [27]. Parsaeifar et al. analyzed the horizontal competitive Nash equilibrium and vertical competitive Stackelberg equilibrium strategy based on a three-echelon supply chain and performed some parameters sensitivity analysis [28]. Tian et al. constructed a Stackelberg game model and analyzed the complexity of the multi-channel supply chain system [29]. Additionally, some scholars proposed collaborative issues from other supply chain examples. Iris et al. adopted a heuristic algorithm to solve the flexible ship loading problem regarding the cooperation between the terminal operator and the liner shipping company [30]. Verdonck analyzed the impacts of three collaborative strategy settings on freight transport companies [31]. To solve the Berth Allocation Problem, Venturini et al. implemented the cooperation between shipping lines and terminals and concluded that speed is a crucial factor and can reduce the overall time and fuel emissions [32]. Lai et al. designed an effective mechanism that considers both system optimization and cost allocation to solve the collaborative production–distribution issue [33].

Lastly, the government has also given tremendous support to the cooperation decision-making activities in the supply chain. For example, Sheu and Chen calculated that government fiscal policy can interfere in enterprises’ competition in the green supply chain [34]. The government’s consumer subsidies can help stimulate the decision-making of both manufacturer and retailer, and consumer subsidies may not benefit online retailers conversely [35]. Zhang and Yousaf designed a two-part tariffs contract involving government taxes and subsidies to coordinate the green supply chain [36]. Research has indicated that government intervention and additional demand for customers’ green preferences could affect the green improvement degree. Chen and Ivan Su formulated game theory model types for the photovoltaic supply chain based on government subsidy policies and fairness concerns, elaborating on the influence of coordination approaches on solutions and performance of the supply chain [37]. Heydari, Govindan, and Jafari pointed out that government incentives can coordinate the reverse closed-loop supply chain in terms of tax exemptions and subsidies [38]. Peng and Pang adopted CVaR(conditional value at risk) to describe farmers’ risk aversion behavior [39]. Research has indicated that the impact of government agricultural subsidies on farmers’ profits depends on farmers’ risk aversion. Peng, Pang, and Cong applied the quantity discount contract and the revenue sharing contract with emission reduction subsidies to coordinate carbon emission reduction decisions [40].

To sum up, scholars have conducted in-depth research on coordination issues in the emergency supply chain and have achieved productive results. However, most of the literature focuses on emissions reduction cooperation or retailers’ optimal green advertising investment strategies. Few literature categories employ differential games to study where the supplier and the manufacturer participate in production technology investments to fight against pandemic risk. Besides, as a key factor affecting mask demand, the pandemic risk directly acts upon the enthusiasm of supply chain members for joint production. In conclusion, our paper utilizes the differential game model to elaborate on the dynamic cooperation of mask-related raw materials and production enterprises. This article consists of: (1) considering that both the supplier and the manufacturer are involved in production technology investment, we construct centralized and decentralized differential game models, and the joint contract is employed to solve the coordination imbalances; (2) elaborating on the main impact of government subsidies on product production technology investment, demand, and supply chain member profits (i.e., the effect of government subsidies on corporate decision-making behavior). Especially after the introduction of the joint contract, government subsidies can increase the enthusiasm of each member enterprise for dynamic cooperation and production technology investments.

3. Model Assumptions and Symbol Description

Based on the literature analysis and the pandemic’s impact, we established the framework of the mask emergency supply chain, shown in Figure 1. A two-echelon supply chain was defined, including government, mask supplier (Meltblown cloth), and mask manufacturer. The COVID-19 pandemic that has disrupted mask supply chain balance greatly influences supply factors such as mask labor, logistics distribution efficiencies, and production capacities. On the other side, mask price also fluctuates with demand. To restore the mask supply chain balance, supply chain member enterprises need to work together to increase mask production accordingly, i.e., the supplier provides sufficient Meltblown cloth raw materials, and the manufacturer invests in production capacity technology. Meanwhile, to inspire the members to invest in production capacity technology, the government also provides the supplier and manufacturer with specific capacity cost subsidies.

This article outlines the following assumptions:

Assumption 1.

Similar to [41,42], the production technology investment costs affect the mask production volume. It assumes that production technology investment costs of the supply chain members are related to their efforts and have convex characteristics. Therefore, production technology investment effort cost of the supplier$C_s\left(t\right)$and production technology investment effort cost of the manufacturer$C_m\left(t\right)$can be defined as:

$$C_s\left(t\right)=\frac{{\mu }_s}{2}{E}_s^2\left(t\right),C_m\left(t\right)=\frac{{\mu }_m}{2}{E}_m^2\left(t\right)$$

(1)

where$E_s\left(t\right)$and$E_m\left(t\right)$represent the production technology investment efforts of supplier and manufacturer at time t individually, and${\mu }_s({\mu }_s>0)$and${\mu }_m({\mu }_m>0)$denote the production technology investment cost coefficients of supplier and manufacturer, respectively.

Assumption 2.

Production technology investment effort has a positive impact on mask output. Assuming that the existing production technology and equipment continue to age over time, there is a natural decay rate of mask production [43]. Presuming that mask production volume is dynamically changing, its dynamic change rate is affected by the production technology investment efforts. The differential equation of mask production is expressed as:

$$\dot{G}\left(t\right)=\alpha E_s\left(t\right)+\beta E_m\left(t\right)-\delta G\left(t\right)$$

(2)

where$\dot{G}\left(t\right)$and$G_0\left(G_0\ge 0\right)$represent the mask production volume at time$t$, and initial time, respectively,$\alpha$and$\beta$refer to sensitivity coefficients of production volume to the production technology investment efforts of the supplier and manufacturer, and δ(δ > 0) denotes the natural attenuation coefficient of production.

Assumption 3.

Liu et al. and Zhang et al. [44,45] present that price and non-price factors can influence market demand. Both supplier and manufacturer are committed to increasing production volume. Therefore, the price of masks and production capacity improvement efforts of the mask supplier and manufacturer determine market demand [46]. Consequently, the mask demand$D\left(t\right)$at time$t$is expressed as:

$$D\left(t\right)=\left[a-bp\left(t\right)\right]\left[D_0+\epsilon G\left(t\right)\right]$$

(3)

where$D_0$refers to the initial mask demand,$a$refers to the market capacity,$b$is the elasticity coefficient of demand to mask price,$p\left(t\right)$represents mask market price at time$t$, and$\epsilon$denotes the sensitivity coefficient of mask demand to mask production.

Assumption 4.

This article focuses on the impact analysis of supplier and manufacturer’s production technology investment on product demand. The inventory backlog is not taken into account in simplifying the model, nor the impact of price on product demand. The enterprises make decisions based on complete information.

Assumption 5.

$w\left(t\right)$represents the supplier’s wholesale price and satisfies$0\le w\left(t\right)\le p\left(t\right)$. The supplier and the manufacturer have neutral risk preferences and seek to maximize their profits at all times. The discount factors$\rho$of all members are the same.

Assumption 6.

To encourage the supply chain members to increase capacity investment, the government will provide certain special subsidies to the Meltblown cloth supplier and mask manufacturer, and set${\phi }_s$and${\phi }_m$as the government’s cost subsidy adjustment factor for the increases in production capacity of the supply chain members, where${\phi }_s,{\phi }_m\ge 0,{\phi }_s,{\phi }_m\in \left(0,1\right)$.

The relevant variables of supplier, manufacturer, and supply chain are represented by the subscript s, m, and sc, respectively. The longstanding profits in three circumstances are defined as:

$$J_s={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[\mathrm{w}\left(a-bp\left(t\right)\right)\left(D_0+\epsilon G\left(t\right)\right)-\frac{{\mu }_s}{2}{E}_s^2\left(t\right)+{\phi }_s\frac{{\mu }_s}{2}{E}_s^2\left(t\right)\right]dt,$$

(4)

$$J_m={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[\left(\mathrm{p}-\mathrm{w}\right)\left(a-bp\left(t\right)\right)\left(D_0+\epsilon G\left(t\right)\right)-\frac{{\mu }_m}{2}{E}_m^2\left(t\right)+{\phi }_m\frac{{\mu }_m}{2}{E}_m^2\left(t\right)]dt,$$

(5)

$$J_{SC}={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[\mathrm{p}\left(a-bp\left(t\right)\right)\left(D_0+\epsilon G\left(t\right)\right)-\left(1-{\phi }_s\right)\frac{{\mu }_s}{2}{E}_s^2\left(t\right)-\left(1-{\phi }_m\right)\frac{{\mu }_m}{2}{E}_m^2\left(t\right)]dt.$$

(6)

For the writing convenience, $t$ will not be listed below.

4. Differential Game Model for Joint Production of the Supplier and the Manufacturer Based on COVID-19 Pandemic

4.1. Differential Game Model of Decentralized Decision

The supply chain members focus on maximizing their profits in the decentralized decision scenario (denoted by superscript $D$). In this model, the Meltblown cloth supplier is the frontrunner of the Stackelberg game in the mask emergency supply chain. Here is the sequence of actions. Initially, the government controls the best subsidy ratios for the Meltblown cloth supplier and the mask manufacturer. Secondly, the supplier determines the base material wholesale price and the production capacity effort for Meltblown cloth. Finally, the manufacturer determines the sale price of masks and the mask production capacity effort. The decision issues of supplier and manufacturer are respectively defined as:

$$\left\{\begin{array}{c}\underset{E_s}{\mathit{max}}{J}_s={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[w\left(a-bp\right)\left(D_0+\epsilon G\right)-\frac{{\mu }_s}{2}{E}_s^2+{\phi }_s\frac{{\mu }_s}{2}{E}_s^2]dt\\ \underset{p,E_m}{\mathit{max}}{J}_m={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[\left(p-w\right)\left(a-bp\right)\left(D_0+\epsilon G\right)-\frac{{\mu }_m}{2}{E}_m^2+{\phi }_m\frac{{\mu }_m}{2}{E}_m^2]dt\end{array}\right..$$

(7)

Theorem 1.

The equilibrium results of the decentralized decision model under a major pandemic are as follows.

(1) 

The optimal trajectory of mask production is expressed as:

$$G^{D*}\left(t\right)=G_{\infty }^D+\left(G_0-G_{\infty }^D\right)e^{-\delta t}$$

(8)

where$G_{\infty }^D=\frac{{\alpha }^2\left(a-bw\right)w\epsilon }{2\delta \left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\left(a-bw\right)}^2\epsilon }{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}$.

(2) 

The optimum equilibrium strategies for supply chain members are expressed as:

$$\left\{\begin{array}{c}{E}_s{}^{D*}=\frac{\alpha \left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}\\ E_m{}^{D*}=\frac{\beta {\left(a-bw\right)}^2\epsilon }{4b\left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}\\ p^{D*}=\frac{a+bw}{2b}\end{array}\right..$$

(9)

(3) 

The optimum profit values of supplier and manufacturer are given respectively as:

$$\left\{\begin{array}{c}{J}_s^{D*}=e^{-\rho t}[\frac{\left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)}{G}^{D*}\left(t\right)+\frac{\left(a-bw\right)w{D}_0}{2\rho }+\frac{{\alpha }^2{\left(a-bw\right)}^2{w}^2{\epsilon }^2}{8\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\left(a-bw\right)}^{3}w{\epsilon }^2}{8b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}]\\ J_m^{D*}=e^{-\rho t}[\frac{{\left(a-bw\right)}^2\epsilon }{4b\left(\delta +\rho \right)}{G}^{D*}\left(t\right)+\frac{{\left(a-bw\right)}^2{D}_0}{4b\rho }+\frac{{\beta }^2{\left(a-bw\right)}^4{\epsilon }^2}{32\rho b^2{\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}+\frac{{\alpha }^2{\left(a-bw\right)}^{3}w{\epsilon }^2}{8b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}]\end{array}\right.$$

(10)

Proof. 

The decision situations of supply chain members are Equation (7), which is solved by reverse induction. After the moment $t$, the optimum value function of the manufacturer’s long-term profit is $J_m^{D*}\left(E_m\right)=e^{-\rho t}{V}_m^D\left(G\right)$. Following the optimal control theory, $V_m^D\left(G\right)$ satisfies the Hamilton–Jacobi–Bellman (HJB) function for any given G ≥ 0:

$$\rho V_m^D\left(G\right)=\underset{E_m}{\max }\left\{\left(p-w\right)\left(a-bp\right)\left(D_0+\epsilon G\right)-\frac{{\mu }_m}{2}{E}_m^2+{\phi }_m\frac{{\mu }_m}{2}{E}_m^2+V_{mG}^{D^{\prime }}\left(\alpha E_s+\beta E_m-\delta G\right)\right\}.$$

(11)

Calculating the first-order partial derivatives for Equation (11) with respect to $E_m$ and $p$ and setting them to 0, i.e., $\frac{\partial \rho V_m^D\left(G\right)}{\partial E_m}=0$, $\frac{\partial \rho V_m^D\left(G\right)}{\partial p}=0$, we can obtain:

$$E_m^D=\frac{\beta V_{mG}^{D^{\prime }}}{\left(1-{\phi }_m\right){\mu }_m},p^{D*}=\frac{a+bw}{2b}.$$

(12)

Similarly, after the moment $t$, the optimal value function of supply’s long-term profit is $J_s^{D*}\left(E_s\right)=e^{-\rho t}{V}_s^D\left(G\right)$. $V_s^D\left(G\right)$ satisfies the Hamilton–Jacobi–Bellman (HJB) equation for any given G ≥ 0:

$$\rho V_s^D\left(G\right)=\underset{E_s}{\max }\left\{w\left(a-bp\right)\left(D_0+\epsilon G\right)-\frac{{\mu }_s}{2}{E}_s^2+{\phi }_s\frac{{\mu }_s}{2}{E}_s^2+V_{sG}^{D^{\prime }}\left(\alpha E_s+\beta E_m-\delta G\right)\right\}.$$

(13)

Calculating the first-order partial derivative for Equation (12) with respect to $E_s$ and $p$ and setting them to 0, i.e., $\frac{\partial \rho V_s^D\left(G\right)}{\partial E_s}=0$, we can obtain:

$$E_s^D=\frac{\alpha V_{sG}^{D^{\prime }}}{\left(1-{\phi }_s\right){\mu }_s}.$$

(14)

Setting ${\pi }_s=\frac{\left(a-bw\right)w}{2}$ and ${\pi }_m=\frac{{\left(a-bw\right)}^2}{4b}$ and substituting Equations (12) and (14) into Equations (11) and (13) to sort it out, we can obtain:

$$\rho V_s^D\left(G\right)=\left[\frac{\left(a-bw\right)w}{2}\epsilon -\delta V_{sG}^{D^{\prime }}\right]G+\frac{\left(a-bw\right)w}{2}{D}_0+\frac{{\alpha }^2{(V_{sG}^{D^{\prime }})}^2}{2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{V}_{sG}^{D^{\prime }}{V}_{mG}^{D^{\prime }}}{\left(1-{\phi }_m\right){\mu }_m},$$

(15)

$$\rho V_m^D\left(G\right)=\left[\frac{{\left(a-bw\right)}^2}{4b}\epsilon -\delta V_{mG}^{D^{\prime }}\right]G+\frac{{\left(a-bw\right)}^2}{4b}{D}_0+\frac{{\beta }^2{(V_{mG}^{D^{\prime }})}^2}{2\left(1-{\phi }_m\right){\mu }_m}+\frac{{\alpha }^2{V}_{sG}^{D^{\prime }}{V}_{mG}^{D^{\prime }}}{\left(1-{\phi }_s\right){\mu }_s}.$$

(16)

Following the structure of Equations (15) and (16), it is presumable that $V_s^D\left(G\right)=m_{1}G+m_2$, $V_m^D\left(G\right)=r_{1}G+r_2$, where $m_1$, $m_2$, $r_1$ and $r_2$ are constants. Substituting their partial derivatives for $G$ into Equations (15) and (16), we can obtain

$$\left\{\begin{array}{c}{m}_1=\frac{\left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)}\\ m_2=\frac{\left(a-bw\right)w{D}_0}{2\rho }+\frac{{\alpha }^2{\left(a-bw\right)}^2{w}^2{\epsilon }^2}{4\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\left(a-bw\right)}^{3}w{\epsilon }^2}{4b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\end{array}\right.,$$

(17)

$$\left\{\begin{array}{c}{r}_1=\frac{{\left(a-bw\right)}^2\epsilon }{4\mathrm{b}\left(\delta +\rho \right)}\\ r_2=\frac{{\left(a-bw\right)}^2{D}_0}{4b\rho }+\frac{{\beta }^2{\left(a-bw\right)}^4{\epsilon }^2}{32\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}+\frac{{\alpha }^2{\left(a-bw\right)}^3{\epsilon }^2}{4b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}\end{array}\right..$$

(18)

Substituting Equations (17) and (18) into Equations (12) and (14), respectively, to obtain Equations (9), which are the optimal equilibrium strategies for the supplier and the manufacturer, and substituting Equation (9) into state Equation (2). Then we solve the differential equation to get Equation (8). Theorem 1 is proved.

According to Equations (17) and (18), $J_m^{D*}\left(E_m\right)=e^{-\rho t}{V}_m^D\left(G\right)$, $J_s^{D*}\left(E_s\right)=e^{-\rho t}{V}_s^D\left(G\right)$, $V_s^D\left(G\right)=m_{1}G+m_2$ and $V_m^D\left(G\right)=r_{1}G+r_2$, we can obtain the overall long-term profits in the decentralized decision model. □

4.2. Differential Game Model of Centralized Decision

The supply chain ponders the overall profit as the goal to regulate each member’s production technology investment effort in the centralized decision scenario (denoted by superscript $C$), then the decision goal is given as follows:

$$\underset{E_s}{\max }{J}_{sc}^C={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[p\left(a-bp\right)\left(D_0+\epsilon G\right)-\left(1-{\phi }_s\right)\frac{{\mu }_s}{2}{E}_s^2\left(t\right)-\left(1-{\phi }_m\right)\frac{{\mu }_m}{2}{E}_m^2\left(t\right)]dt.$$

(19)

Theorem 2.

In the centralized decision model, equilibrium results under a major pandemic are given as follows.

(1) 

The optimal trajectory of mask production is expressed as

$$G^{c*}\left(t\right)=G_{\infty }^c+\left(G_0-G_{\infty }^c\right)e^{-\delta t}$$

(20)

where$G_{\infty }^c=\frac{{\alpha }^2{a}^2\epsilon }{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{a}^2\epsilon }{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}$.

(2) 

The optimum equilibrium strategies for both supplier and manufacturer are expressed as

$$\left\{\begin{array}{c}{E}_s{}^{c*}=\frac{\alpha a^2\epsilon }{4b\left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}\\ E_m{}^{c*}=\frac{\beta a^2\epsilon }{4b\left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}\\ p^{C*}=\frac{a}{2b}\end{array}\right..$$

(21)

(3) 

The optimum profit values of supplier and manufacturer are given respectively as

$$J_{sc}^{C*}=e^{-\rho t}[\frac{a^2\epsilon }{4b\left(\delta +\rho \right)}{G}^{c*}\left(t\right)+\frac{a^2{D}_0}{4b\rho }+\frac{{\alpha }^2{a}^4{\epsilon }^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{a}^4{\epsilon }^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}].$$

(22)

Proof. 

The specific solution method is similar to the proof of Theorem 1.

After the moment $t$, the optimum value function of the long-term profit of the supply chain is $J_{sc}^{C*}\left(E_s,E_m\right)=e^{-\rho t}{V}_{sc}^C\left(G\right)$. As per the optimal control theory, $V_{sc}^C\left(G\right)$ satisfies the Hamilton–Jacobi–Bellman (HJB) function for any given G ≥ 0:

$$\rho V_{sc}^C\left(G\right)=\underset{E_s,E_m}{\max }\left\{p\left(a-bp\right)\left(D_0+\epsilon G\right)-\left(1-{\phi }_s\right)\frac{{\mu }_s}{2}{E}_s^2-\left(1-{\phi }_m\right)\frac{{\mu }_m}{2}{E}_m^2+V_{scG}^{C^{\prime }}\left(\alpha E_s+\beta E_m-\delta G\right)\right\}.$$

(23)

Calculating the first-order partial derivatives for Equation (23) with respect to $E_s$ and $E_m$ and setting them to 0, i.e., $\frac{\partial \rho V_{sc}^C\left(G\right)}{\partial E_s}=0,\frac{\partial \rho V_{sc}^C\left(G\right)}{\partial E_m}=0,\frac{\partial \rho V_{sc}^C\left(G\right)}{\partial p}=0$, we can obtain

$$E_s^C=\frac{\alpha V_{scG}^{C^{\prime }}}{\left(1-{\phi }_s\right){\mu }_s},E_m^C=\frac{\beta V_{scG}^{C^{\prime }}}{\left(1-{\phi }_m\right){\mu }_m},p^C=\frac{a}{2b}.$$

(24)

Substituting Equation (24) into Equation (23) to sort out, we can obtain

$$\rho V_{sc}^C\left(G\right)=\left(\frac{a^2}{4b}\epsilon -\delta V_{scG}^{C^{\prime }}\right)G+\frac{a^2}{4b}{D}_0+\frac{{\alpha }^2{(V_{scG}^{C^{\prime }})}^2}{2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{(V_{scG}^{C^{\prime }})}^2}{2\left(1-{\phi }_m\right){\mu }_m}.$$

(25)

Similarly, it can be assumed that $V_{sc}^C\left(G\right)=l_{1}G+l_2$, where $l_1$, $l_2$ are constants. Substituting their partial derivatives for $G$ into Equation (25), we can obtain

$$\left\{\begin{array}{c}{l}_1=\frac{a^2\epsilon }{4b\left(\delta +\rho \right)}\\ l_2=\frac{a^2{D}_0}{4b\rho }+\frac{{\alpha }^2{a}^4{\epsilon }^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{a}^4{\epsilon }^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}.\end{array}\right.$$

(26)

We substitute Equation (26) into Equation (24) to obtain Equations (21), which are the optimal equilibrium strategies for the supplier and the manufacturer, and we substitute Equation (21) into state Equation (2). Then we solve the differential equation to get Equation (20). Theorem 2 is proved.

According to Equation (26), $J_{sc}^{C*}\left(E_s,E_m\right)=e^{-\rho t}{V}_{sc}^C\left(G\right)$, $V_{sc}^C\left(G\right)=l_{1}G+l_2$, we can obtain the overall long-term profits in the centralized decision scenario. □

4.3. Decision Model in Joint Contracts Decision Scenario

According to the above analysis, the production technology investment effort and overall profit of the mask emergency supply chain in the decentralized decision scenario are lower than the optimal strategy values in the centralized decision scenario. As there is a double marginal effect, our study proposes a joint contract to increase the production enthusiasm of supply chain members based on the decentralized decision scenario. Along with joint contract coordination, the supplier and the manufacturer incentivize to bear part of production technology investment costs for each other. Meanwhile, the party with greater profit increase needs to provide a particular transfer payment as compensation to the party with less profit increase (or decrease). Assuming that ${\omega }_1$ is the proportion of production technology investment cost shared by the supplier for the manufacturer, ${\omega }_2$ is the proportion of production technology investment cost shared by the manufacturer for the supplier, and F is the transfer payment between the supplier revenue and manufacturer revenue, the decision model under the joint contract (denoted by superscript H) is concluded as

$$\left\{\begin{array}{c}\underset{E_s}{\max }{J}_s={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[w\left(a-bp\right)\left(D_0+\epsilon G\right)-\left(1-{\phi }_s-{\omega }_2\right)\frac{{\mu }_s}{2}{E}_s^2-{\omega }_1\frac{{\mu }_m}{2}{E}_m^2+F]dt\\ \underset{p,E_m}{\max }{J}_m={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[\left(p-w\right)\left(a-bp\right)\left(D_0+\epsilon G\right)-\left(1-{\phi }_m-{\omega }_1\right)\frac{{\mu }_m}{2}{E}_m^2-{\omega }_2\frac{{\mu }_s}{2}{E}_s^2-F]dt\end{array}\right.$$

(27)

When $F>0$, it means that the supplier’s profit has increased less or even decreased, and the manufacturer will compensate the supplier with a certain amount of revenue. When $F<0$, it means that the manufacturer’s profit has increased less or even decreased, and the supplier will compensate the manufacturer with a certain amount of revenue.

Theorem 3.

In the joint contract decision scenario, the equilibrium results under a major pandemic are as follows.

(1) 

The optimal trajectory of mask production is expressed as

$$G^{H*}\left(t\right)=G_{\infty }^H+\left(G_0-G_{\infty }^H\right)e^{-\delta t}$$

(28)

where$G_{\infty }^H=\frac{{\alpha }^2{a}^2\epsilon }{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{a}^2\epsilon }{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}$.

(2) 

The optimal production technology investment effort cost ratio is shared by the supplier for manufacturer. Meanwhile, the optimum production technology investment effort cost ratio is shared by the manufacturer for the supplier, and the optimal equilibrium strategies for the production technology investment of the supplier and manufacturer are given as

$$\left\{\begin{array}{c}{w}_1=\frac{[a^2-{\left(a-bw\right)}^2]\left(1-{\phi }_m\right)}{a^2}\\ w_2=\frac{\left[a^2-2b\left(a-bw\right)w\right]\left(1-{\phi }_s\right)}{a^2}\\ E_s{}^{H*}=\frac{\alpha a^2\epsilon }{4b\left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}\\ E_m{}^{H*}=\frac{\beta a^2\epsilon }{4b\left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}\\ p^{H*}=\frac{a+bw}{2b}\end{array}\right..$$

(29)

(3) 

The optimal profit values of supplier and manufacturer are expressed respectively as

$$\left\{\begin{array}{c}{J}_s^{H*}=e^{-\rho t}\left\{\frac{\left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)}{G}^{H*}\left(t\right)+\frac{\left(a-bw\right)w{D}_0}{2\rho }+\frac{F}{\rho }+\frac{{\alpha }^2\left(a-bw\right)w{\epsilon }^2{a}^2}{16b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^2{a}^{2}w\left(2a-3bw\right)}{32b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\right\}\\ J_m^{H*}=e^{-\rho t}\left\{\frac{{\left(a-bw\right)}^2\epsilon }{4b\left(\delta +\rho \right)}{G}^{H*}\left(t\right)+\frac{{\left(a-bw\right)}^2}{4b\rho }{D}_0-\frac{F}{\rho }+\frac{{\beta }^2{\left(a-bw\right)}^2{\epsilon }^2{a}^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}+\frac{{\alpha }^2{\epsilon }^2{a}^3\left(a-2bw\right)}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}\right\}\end{array}\right..$$

(30)

Proof. 

The decision problems of supplier and manufacturer are Equations (27), which are solved by reverse induction. After the moment $t$, the optimum value function of the manufacturer’s long-term profit is $J_m^{H*}\left(E_m\right)=e^{-\rho t}{V}_m^H\left(G\right)$. Similar to the previous two models, $V_m^H\left(G\right)$ satisfies the Hamilton–Jacobi–Bellman (HJB) function for any given G ≥ 0:

$$\rho V_m^H\left(G\right)=\underset{E_m}{\max }\left\{\left(p-w\right)\left(a-bp\right)\left(D_0+\epsilon G\right)-\left(1-{\phi }_m-{\omega }_1\right)\frac{{\mu }_m}{2}{E}_m^2-{\omega }_2\frac{{\mu }_s}{2}{E}_s^2-F+V_{mG}^{H^{\prime }}\left(\alpha E_s+\beta E_m-\delta G\right)\right\}.$$

(31)

Calculating the first-order partial derivative for Equation (31) with respect to $E_m$ and setting it to 0, i.e., $\frac{\partial \rho V_m^H\left(G\right)}{\partial E_m}=0$, $\frac{\partial \rho V_m^H\left(G\right)}{\partial p}=0$, we can obtain

$$E_m^H=\frac{\beta V_{mG}^{H^{\prime }}}{\left(1-{\phi }_m-{\omega }_1\right){\mu }_m},p^H=\frac{a+bw}{2b}.$$

(32)

Similarly, after the moment $t$, the optimal value function of the supplier’s long-term profit is $J_s^{*}\left(E_s\right)=e^{-\rho t}{V}_s^H\left(G\right)$. Similar to the previous two models, $V_s^H\left(G\right)$ satisfies the Hamilton–Jacobi–Bellman (HJB) equation for any given G ≥ 0:

$$\rho V_s^H\left(G\right)=\underset{E_s}{\max }\left\{w\left(a-bp\right)\left(D_0+\epsilon G\right)-\left(1-{\phi }_s-{\omega }_2\right)\frac{{\mu }_s}{2}{E}_s^2-{\omega }_1\frac{{\mu }_m}{2}{E}_m^2+F+V_{sG}^{H^{\prime }}\left(\alpha E_s+\beta E_m-\delta G\right)\right\}.$$

(33)

Calculating the first-order partial derivative for Equation (33) with respect to $E_s$ and setting it to 0, $\frac{\partial \rho V_s^H\left(G\right)}{\partial E_s}=0$, we can obtain

$$E_s^H=\frac{\alpha V_{sG}^{H^{\prime }}}{\left(1-{\phi }_s-{\omega }_2\right){\mu }_s}.$$

(34)

Substituting Equations (32) and (34) into Equations (31) and (33) to sort out, we can obtain

$$\rho V_s^H\left(G\right)=\left({\pi }_s\epsilon -\delta V_{sG}^{H^{\prime }}\right)G+{\pi }_s{D}_0+F+\frac{{\alpha }^2{(V_{sG}^{H^{\prime }})}^2}{2\left(1-{\phi }_s-{\omega }_2\right){\mu }_s}+\frac{{\beta }^2{V}_{mG}^{H^{\prime }}[2V_{sG}^{H^{\prime }}\left(1-{\phi }_m-{\omega }_1\right)-V_{mG}^{H^{\prime }}{\omega }_1]}{2{\left(1-{\phi }_m-{\omega }_1\right)}^2{\mu }_m},$$

(35)

$$\rho V_m^H\left(G\right)=\left({\pi }_m\epsilon -\delta V_{mG}^{H^{\prime }}\right)G+{\pi }_m{D}_0-F+\frac{{\beta }^2{(V_{mG}^{H^{\prime }})}^2}{2\left(1-{\phi }_m-{\omega }_1\right){\mu }_m}+\frac{{\alpha }^2{V}_{sG}^{H^{\prime }}[2V_{mG}^{H^{\prime }}\left(1-{\phi }_s-{\omega }_2\right)-V_{sG}^{H^{\prime }}{\omega }_2]}{2{\left(1-{\phi }_s-{\omega }_2\right)}^2{\mu }_s}.$$

(36)

Following the structure of Equations (35) and (36), it is presumable that $V_s^H\left(G\right)=n_{1}G+n_2$, $V_m^H\left(G\right)=x_{1}G+x_2$, where $n_1$, $n_2$, $x_1$ and $x_2$ are constants. Substituting their partial derivatives for $G$ into Equations (35) and (36), we can obtain

$$\left\{\begin{array}{c}{n}_1=\frac{\left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)}\\ n_2=\frac{\left(a-bw\right)w{D}_0}{2\rho }+\frac{F}{\rho }+\frac{{\alpha }^2{\left(a-bw\right)}^2{w}^2{\epsilon }^2}{8\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s-{\omega }_2\right){\mu }_s}+\frac{{\beta }^2{\left(a-bw\right)}^3{\epsilon }^2\left[4bw\left(1-{\phi }_m-{\omega }_1\right)-\left(a-bw\right){\omega }_1\right]}{32b^2\rho {\left(\delta +\rho \right)}^2{\left(1-{\phi }_m-{\omega }_1\right)}^2{\mu }_m}\end{array}\right.,$$

(37)

$$\left\{\begin{array}{c}{x}_1=\frac{{\left(a-bw\right)}^2\epsilon }{4b\left(\delta +\rho \right)}\\ x_2=\frac{{\left(a-bw\right)}^2}{4b\rho }{D}_0-\frac{F}{\rho }+\frac{{\beta }^2{\left(a-bw\right)}^4{\epsilon }^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m-{\omega }_1\right){\mu }_m}+\frac{{\alpha }^2{\left(a-bw\right)}^{2}w{\epsilon }^2\left[\left(a-bw\right)\left(1-{\phi }_s-{\omega }_2\right)-wb{\omega }_2\right]}{8b\rho {\left(\delta +\rho \right)}^2{\left(1-{\phi }_s-{\omega }_2\right)}^2{\mu }_s}\end{array}\right..$$

(38)

Substituting Equations (37) and (38) into Equations (12) and (14), respectively, we can obtain the optimal equilibrium strategies for the supplier and the manufacturer, i.e., $E_s{}^{H*}=\frac{\alpha \left(a-bw\right)w\epsilon }{2\left(1-{\phi }_s-{\omega }_2\right){\mu }_s\left(\delta +\rho \right)}$ and $E_m{}^{H*}=\frac{\beta {\left(a-bw\right)}^2\epsilon }{4b\left(1-{\phi }_m-{\omega }_1\right){\mu }_m\left(\delta +\rho \right)}$.

When the decision situation in the joint contract scenario is consistent with the case in the centralized decision scenario, i.e., $E_s{}^{H*}=E_s{}^{c*}$ and $E_m{}^{H*}=E_m{}^{c*}$ are established simultaneously to realize the mask supply chain coordination, Equation (29) can be obtained. Substituting Equation (29) into state Equation (2), we solve the differential equation to get Equation (28). Theorem 3 is proved.

According to Equations (37) and (38), $J_m^{H*}\left(E_m\right)=e^{-\rho t}{V}_m^H\left(G\right)$, $J_s^{*}\left(E_s\right)=e^{-\rho t}{V}_s^H\left(G\right)$, $V_s^H\left(G\right)=n_{1}G+n_2$ and $V_m^H\left(G\right)=x_{1}G+x_2$, we can obtain the overall long-term profits of supply chain members in the joint contract scenario. □

Comparing the optimal profit and optimal equilibrium strategy in three situations, we can obtain the following inferences.

Corollary 1.

In the decentralized decision scenario, the production technology investment efforts are positively affected by the sensitivity coefficient of production to production technology investment effort ($\alpha$or$\beta$), the sensitivity coefficient$\epsilon$of mask demand to mask production, and the government’s production investment cost subsidy coefficient (${\phi }_s$or${\phi }_m$) to the supplier and manufacturer. Besides, the production technology investment efforts are also negatively affected by their production technology investment cost coefficient (${\mu }_s$or${\mu }_m$), natural attenuation coefficient$\delta$, and discount factor$\rho$. Therefore, the production technology investment effort and the overall profit of the mask emergency supply chain will both increase with the growth of$\alpha$or$\beta$, indicating that the more sensitive production volume is to the change in the production technology investment effort, the better the cooperation effectiveness will be of the supply chain members to increase production volume. With the increase of$\epsilon$, production technology investment efforts and total profit will increase accordingly, indicating that the more sensitive mask demand fluctuations are in mask production, the better the cooperation effectiveness between supplier and manufacturer will be. The production technology investment effort and total profit will decrease with the growth of${\mu }_s$or${\mu }_m$, indicating that the higher the investment in production technology, the more challenges there will be to achieve efficient cooperation. Furthermore, the worse the cooperation effectiveness, the lower the overall profit. With the increase of$\delta$, the production technology investment effort and total profit will progressively decrease. When the value of$\delta$reaches a certain level, the negative effect of the natural attenuation of production will be greater than the positive effect of production technology investment. Therefore, we should take measures to increase government incentives to encourage production technology investment.

Moreover, in the centralized decision model, the influence of relevant parameters on decision variables $E_s$, $E_m$ and state variable $G\left(t\right)$ is similar to the situation in the decentralized decision, which is limited by space and will not be repeated.

Corollary 2.

In the decentralized decision model, the optimal trajectory of mask production volume changes monotonically, i.e., when$G_0<G_{\infty }^D$, the mask production volume increases monotonously with time. When$G_0>G_{\infty }^D$, the mask production volume decreases monotonously with time. When$G_0=G_{\infty }^D$, the mask production volume is constant. The optimal trajectory changes of mask production volume in the centralized decision model and joint contract decision model are consistent with the changes in the decentralized decision model, which will not be recurring.

Corollary 3.

Comparing the decision-making results in the three scenarios:$E_s{}^{D*}<E_s{}^{c*}=E_s{}^{H*}$,$E_m{}^{D*}<E_m{}^{c*}=E_m{}^{H*}$,$G^{C*}\left(t\right)=G^{H*}\left(t\right)\ge G^{D*}\left(t\right)$, i.e., introducing the joint contract, the production technology investment efforts, and the optimal trajectory of production have all improved at the same time. It is indicated that the mask sale price p is constantly greater in the decentralized decision scenario than in the centralized decision scenario regardless of the price w. Besides, the mask sale price is increasing and the production technology investment effort of the manufacturer is decreasing with the increase of$w$, while the production technology investment effort of the supplier is increasing for any$w\in \left[0,\frac{a}{2b}\right]$and decreasing for any$w\in \left[\frac{a}{2b},\frac{a}{b}\right]$.

Proof. 

When $E_s{}^{H*}-E_s{}^{D*}=\frac{\alpha \epsilon \left[a^2-2\left(a-bw\right)wb\right]}{4b\left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}>0$, then $E_s{}^{H*}>E_s{}^{D*}$. In the same way, we can obtain $E_s{}^{c*}=E_s{}^{H*}$.

When $E_m{}^{H*}-E_m{}^{D*}=\frac{\beta \epsilon [a^2-{\left(a-bw\right)}^2]}{4b\left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}>0$, then $E_m{}^{H*}>E_m{}^{D*}$. In the same way, we can obtain $E_m{}^{c*}=E_m{}^{H*}$.

When $G^{C*}\left(t\right)-G^{D*}\left(t\right)=G_{\infty }^C+\left(G_0-G_{\infty }^C\right)e^{-\delta t}-G_{\infty }^D-\left(G_0-G_{\infty }^D\right)e^{-\delta t}$ $=\left(1-e^{-\delta t}\right)\left[\frac{{\alpha }^2\epsilon \left[a^2-2\left(a-bw\right)wb\right]}{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2\epsilon \left[a^2-{\left(a-bw\right)}^2\right]}{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}\right]\ge 0$, then $G^{C*}\left(t\right)\ge G^{D*}\left(t\right)$. In the same way, we can obtain $G^{C*}\left(t\right)=G^{H*}\left(t\right)$, indicating that the optimal trajectory of mask production in the joint contract model is consistent with the optimal state of the centralized decision model. □

Corollary 4.

Under the joint contract, the proportions of supplier and manufacturer individually bearing the other’s production technology investment costs satisfy$w_1=\frac{[a^2-{\left(a-bw\right)}^2]\left(1-{\phi }_m\right)}{a^2}, w_2=\frac{\left[a^2-2b\left(a-bw\right)w\right]\left(1-{\phi }_s\right)}{a^2}$. $\frac{\partial w_1}{\partial w}=\frac{2b\left(a-bw\right)\left(1-{\phi }_m\right)}{a^2}>0$indicates that the proportion of production technology investment cost shared by the supplier for the manufacturer is proportional to the wholesale price.

$\frac{\partial w_2}{\partial w}=\frac{-2b\left(a-2bw\right)\left(1-{\phi }_s\right)}{a^2}<0$indicates that the proportion of production technology investment cost shared by the manufacturer for the supplier is inversely proportional to the wholesale price.

Corollary 5.

Under the three decision models, the production technology investment effort is directly proportional to the government’s subsidy rates to supply chain members. The total profit and mask production volume are all proportional to the government’s respective subsidy rate.

Proof. 

In the case of introducing a joint contract, $\frac{\partial E_s{}^{H*}}{\partial {\phi }_s}=\frac{\alpha a^2\epsilon }{4b\left(\delta +\rho \right){\left(1-{\phi }_s\right)}^2{\mu }_s}>0$ indicates that the production technology investment effort is proportional to the government’s subsidy rates, and the same can be proved. Similarly, $\frac{\partial E_m{}^{H*}}{\partial {\phi }_m}>0$, $\frac{\partial J_s{}^{H*}}{\partial {\phi }_s}>0$, $\frac{\partial J_m{}^{H*}}{\partial {\phi }_m}>0$, $\frac{\partial J_s{}^{H*}}{\partial {\phi }_m}>0$, $\frac{\partial J_m{}^{H*}}{\partial {\phi }_s}>0$, $\frac{\partial G^{H*}\left(t\right)}{\partial {\phi }_s}>0$ and $\frac{\partial G^{H*}\left(t\right)}{\partial {\phi }_m}>0$ can be proved. Therefore, the above deduction is proved. Similarly, it can be proved that in the centralized decision and decentralized decision scenarios, the production technology investment effort is directly proportional to the government’s subsidy rates. The total profit and mask production volume are proportional to the government’s respective subsidy rates. □

Corollary 5 indicates that whether the supplier and the manufacturer intend to cooperate depends not only on their respective margins but also on the sensitivity coefficients of production to the supplier and the manufacturer’s production technology investment efforts, government subsidy rates, and other factors. Especially when the natural attenuation rate of production volume is relatively large, which affects the supplier and the manufacturer’s production enthusiasm, the government can increase the production technology subsidy rate to strengthen the cooperation willingness of enterprises.

Corollary 6.

Compared with the decentralized decision model, the overall profit in the centralized decision scenario has increased. The earnings of supply chain members have increased in the joint contract decision model, and total profit is lower than that of the centralized decision scenario.

Proof. 

$$J_{sc}^{C*}-J_s^{D*}-J_m^{D*}=e^{-\rho t}\left\{\left[\frac{a^2\epsilon }{4b\left(\delta +\rho \right)}{G}^{c*}\left(t\right)-\frac{\left(a-bw\right)\left(a+bw\right)\epsilon }{4b\left(\delta +\rho \right)}{G}^{D*}\left(t\right)\right]+\frac{b^2{w}^2{D}_0}{4b\rho }+\frac{{\alpha }^2{\epsilon }^{2}a\left[a^3-4bw{\left(a-bw\right)}^2\right]}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^2{w}^2\left(6a^2-8abw+3b^2{w}^2\right)}{32\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\right\}.$$

where $G^{C*}\left(t\right)\ge G^{D*}\left(t\right),\frac{{\alpha }^2{\epsilon }^{2}a\left[a^3-4bw{\left(a-bw\right)}^2\right]}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}>0,\frac{{\beta }^2{\epsilon }^2{w}^2\left(6a^2-8abw+3b^2{w}^2\right)}{32\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}>0$, we can obtain $J_{sc}^{C*}>J_s^{D*}+J_m^{D*}$.

$$J_{sc}^{C*}-\left(J_s^{H*}+J_m^{H*}\right)=e^{-\rho t}\left\{\frac{b^2{w}^2\epsilon }{4b\left(\delta +\rho \right)}{G}^{c*}\left(t\right)+\frac{b^2{w}^2{D}_0}{4b\rho }+\frac{2{\alpha }^2{\epsilon }^2{a}^2{b}^2{w}^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{2{\beta }^2{\epsilon }^2{a}^2{b}^2{w}^2}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\right\}.$$

where $G^{C*}\left(t\right)>0$, we can obtain $J_{sc}^{C*}>\left(J_s^{H*}+J_m^{H*}\right)$.

Because $J_s^{H*}-J_s^{D*}=e^{-\rho t}\left\{\frac{\left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)}\left[G^{H*}\left(t\right)-G^{D*}\left(t\right)\right]+\frac{F}{\rho }+\frac{{\alpha }^2\left(a-bw\right)w{\epsilon }^2[{\left(a-bw\right)}^2+b^2{w}^2]}{16b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^{2}w\left[a^2\left(2a-3bw\right)-4{\left(a-bw\right)}^3\right]}{32b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\right\}\ge 0$, we can obtain $J_s^{H*}\ge J_s^{D*}$. Similarly, we can obtain $J_m^{H*}\ge J_m^{D*}$. □

Corollary 7.

The proportion of the production technology investment cost shared by the manufacturer for the supplier is inversely proportional to the government’s subsidy rate to the supplier in the joint contract decision model. The proportion of the production technology investment cost shared by the supplier for the manufacturer is inversely proportional to the government’s subsidy rate to the manufacturer. The government subsidies help to achieve supply chain coordination.

Proof. 

Because $\frac{\partial w_1}{\partial {\phi }_m}=-\frac{[a^2-{\left(a-bw\right)}^2]}{a^2}<0$, then $w_1$ decreases with the increase of ${\phi }_m$ and the same can be proved. Because $\frac{\partial w_2}{\partial {\phi }_s}<0$, then $w_2$ decreases with the increase of ${\phi }_s$. It is evident that the external subsidy rate of the supply chain adversely affects the cost-sharing ratio among internal members.

Before and after the introduction of the joint contract, the supplier’s profit margin is

$$\begin{array}{c}\Delta J_s=J_s^{H*}-J_s^{D*}=e^{-\rho t}\left\{\frac{\left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)}\left[G^{H*}\left(t\right)-G^{D*}\left(t\right)\right]+\frac{F}{\rho }+\frac{{\alpha }^2\left(a-bw\right)w{\epsilon }^2[{\left(a-bw\right)}^2+b^2{w}^2]}{16b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^{2}w\left[a^2\left(2a-3bw\right)-4{\left(a-bw\right)}^3\right]}{32b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\right\}.\\ \frac{\partial \left(J_s^{H*}-J_s^{D*}\right)}{\partial {\phi }_s}=e^{-\rho t}\left\{\frac{{\alpha }^2\left(a-bw\right)w{\epsilon }^2[{\left(a-bw\right)}^2+b^2{w}^2]}{16b\rho {\left(\delta +\rho \right)}^2{\left(1-{\phi }_s\right)}^2{\mu }_s}+\frac{{\beta }^2{\epsilon }^{2}w\left[a^2\left(2a-3bw\right)-4{\left(a-bw\right)}^3\right]}{32b\rho {\left(\delta +\rho \right)}^2{\left(1-{\phi }_m\right)}^2{\mu }_m}\right\}>0\text{.}\end{array}$$

Before and after introducing the joint contract, the mask supplier’s profit margin increases with the growth of the government’s subsidy rate to the supplier. The same can be obtained $\frac{\partial \left(J_m^{H*}-J_m^{D*}\right)}{\partial {\phi }_m}>0$, and the mask manufacturer’s profit margin increases with the growth of the government’s subsidy rate to the mask manufacturer. In summary, the government provides a certain degree of subsidies to supply chain members, which can benefit supply chain coordination. □

Corollary 8.

When$w_1=\frac{[a^2-{\left(a-bw\right)}^2[\left(1-{\phi }_m\right)}{a^2}$, $w_2=\frac{\left[a^2-2b\left(a-bw\right)w\right]\left(1-{\phi }_s\right)}{a^2}$and$F\in \left[F_{min},F_{max}\right]$($F_{min}$and$F_{max}$satisfy Equation (39)), the joint contract not only incentivizes the supplier and the manufacturer to increase production input but also enables the overall profit to reach Pareto improvement.

$$\left\{\begin{array}{c}{F}_{min}=-\frac{\left(a-bw\right)\rho w{\epsilon }^2}{8b\delta {\left(\delta +\rho \right)}^2}\left(1-e^{-\delta t}\right)\left\{\frac{{\alpha }^2\left[a^2-2\left(a-bw\right)wb\right]}{\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2\left[a^2-{\left(a-bw\right)}^2\right]}{\left(1-{\phi }_m\right){\mu }_m}\right\}-\frac{{\alpha }^2\left(a-bw\right)w{\epsilon }^2[{\left(a-bw\right)}^2+b^2{w}^2]}{16b{\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}-\frac{{\beta }^2{\epsilon }^{2}w\left[a^2\left(2a-3bw\right)-4{\left(a-bw\right)}^3\right]}{32b{\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\\ F_{max}=\frac{{\left(a-bw\right)}^2\rho {\epsilon }^2}{16b^2\delta {\left(\delta +\rho \right)}^2}\left(1-e^{-\delta t}\right)\left\{\frac{{\alpha }^2\left[a^2-2\left(a-bw\right)wb\right]}{\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2\left[a^2-{\left(a-bw\right)}^2\right]}{\left(1-{\phi }_m\right){\mu }_m}\right\}+\frac{{\alpha }^2{\epsilon }^2\left[a^3\left(a-2bw\right)-4{\left(a-bw\right)}^{3}bw\right]}{32b^2{\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^2{\left(a-bw\right)}^{2}bw\left(2a-bw\right)}{32b^2{\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\end{array}\right.$$

(39)

Proof. 

The differences between the member’s profit in the joint contract decision and the decentralized decision scenario:

$$\Delta J_{sc}=J_{sc}^{H*}-J_{sc}^{D*}=\left\{\frac{\left(a-bw\right)\left(a+bw\right)\epsilon }{4b\left(\delta +\rho \right)}\left[G^{H*}\left(t\right)-G^{D*}\left(t\right)\right]+\frac{{\alpha }^2{\epsilon }^2\left[a^3\left(a-4bw\right)+2a{b}^2{w}^2\left(3a-2bw\right)\right]}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^2{w}^2\left(2a-3bw\right)\left(2a-bw\right)}{32\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\right\},$$

$$\begin{gathered} \Delta J_s=e^{-\rho t}\{\frac{\left(a-bw\right)w\epsilon }{2\left(\delta +\rho \right)}\left(1-e^{-\delta t}\right)\left[\frac{{\alpha }^2\epsilon \left[a^2-2\left(a-bw\right)wb\right]}{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2\epsilon \left[a^2-{\left(a-bw\right)}^2\right]}{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}\right] \\ +\frac{F}{\rho }+\frac{{\alpha }^2\left(a-bw\right)w{\epsilon }^2[{\left(a-bw\right)}^2+b^2{w}^2]}{16b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^{2}w\left[a^2\left(2a-3bw\right)-4{\left(a-bw\right)}^3\right]}{32b\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\} \end{gathered}$$

$$\begin{gathered} \Delta J_m=e^{-\rho t}\{\frac{{\left(a-bw\right)}^2\epsilon }{4b\left(\delta +\rho \right)}\left(1-e^{-\delta t}\right)\left[\frac{{\alpha }^2\epsilon \left[a^2-2\left(a-bw\right)wb\right]}{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2\epsilon \left[a^2-{\left(a-bw\right)}^2\right]}{4b\delta \left(\delta +\rho \right)\left(1-{\phi }_m\right){\mu }_m}\right] \\ -\frac{F}{\rho }+\frac{{\alpha }^2{\epsilon }^2\left[a^3\left(a-2bw\right)-4{\left(a-bw\right)}^{3}bw\right]}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^2{\left(a-bw\right)}^{2}bw\left(2a-bw\right)}{32b^2\rho {\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}\}s. \end{gathered}$$

Because $G^{H*}\left(t\right)>G^{D*}\left(t\right)$, then $\Delta J_{sc}>0$.

When $w_1=\frac{\left[a^2-{\left(a-bw\right)}^2\right]\left(1-{\phi }_m\right)}{a^2}$ and $w_2=\frac{\left[a^2-2b\left(a-bw\right)w\right]\left(1-{\phi }_s\right)}{a^2}$, the mask supply chain is in a coordinated state, and the joint contract decision model is more advanced than the decentralized decision model in terms of the overall profit. However, the gains of the supply chain members have not yet been determined to increase or not, then it is necessary to further explore the range of transfer payment $F$ to ensure that revenues of the supplier and the manufacturer in the joint contract decision model are higher than the profits in the decentralized decision model. Therefore, the necessary condition for the realization of the joint contract is that the members’ incomes in the joint contract decision model are not lower than their incomes in the decentralized decision model. Then we have: $\Delta J_s\ge 0$ and $\Delta J_m\ge 0$.

When $F\ge -\frac{\left(a-bw\right)\rho w{\epsilon }^2}{8b\delta {\left(\delta +\rho \right)}^2}\left(1-e^{-\delta t}\right)\{\frac{{\alpha }^2\left[a^2-2\left(a-bw\right)wb\right]}{\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2\left[a^2-{\left(a-bw\right)}^2\right]}{\left(1-{\phi }_m\right){\mu }_m}\}-\frac{{\alpha }^2\left(a-bw\right)w{\epsilon }^2[{\left(a-bw\right)}^2+b^2{w}^2]}{16b{\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}-\frac{{\beta }^2{\epsilon }^{2}w\left[a^2\left(2a-3bw\right)-4{\left(a-bw\right)}^3\right]}{32b{\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}$, $\Delta J_s\ge 0$.

When $F\le \frac{{\left(a-bw\right)}^2\rho {\epsilon }^2}{16b^2\delta {\left(\delta +\rho \right)}^2}\left(1-e^{-\delta t}\right)\{\frac{{\alpha }^2\left[a^2-2\left(a-bw\right)wb\right]}{\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2\left[a^2-{\left(a-bw\right)}^2\right]}{\left(1-{\phi }_m\right){\mu }_m}\}+\frac{{\alpha }^2{\epsilon }^2\left[a^3\left(a-2bw\right)-4{\left(a-bw\right)}^{3}bw\right]}{32b^2{\left(\delta +\rho \right)}^2\left(1-{\phi }_s\right){\mu }_s}+\frac{{\beta }^2{\epsilon }^2{\left(a-bw\right)}^{2}bw\left(2a-bw\right)}{32b^2{\left(\delta +\rho \right)}^2\left(1-{\phi }_m\right){\mu }_m}$, $\Delta J_m\ge 0$.

Therefore, when $F\in \left[F_{min},F_{max}\right]$, $\Delta J_s\ge 0$ and $\Delta J_m\ge 0$. Corollary 8 is proved. □

5. Numerical Analysis

To further analyze the above three decision-making results, our paper presents MATLAB to simulate and visually evaluate the related parameters’ influence on the mask supply chain. Assume the relevant parameters are set as benchmarks: $a=$ 70, $b=$ 2, $\alpha =0.6, \beta =0.4, \epsilon =3, \delta =$ 0.3, $\rho =0.8$, ${\mu }_s=$ 16, ${\mu }_m=$ 25, $D_0=20, G_0=0$, ${\phi }_s=0.3, {\phi }_m=0.6, t=$ 1. To meet the actual situation, we suppose $w=$ 5.

5.1. Comparison Results of Numerical Examples

Substituting related parameters into Theorem 1, Theorem 2, and Theorem 3, we can get the equilibrium results in three scenarios as shown in

Table 1. Comparison results of numerical examples in different decision scenarios without government subsidies.

Variables	Centralized Scenario	Decentralized Scenario	Joint Contract Scenario
$G\left(t\right)$	41.7076	14.7380	41.7076
$D\left(t\right)$	5079.30	1926.40	4353.70
$E_s$	62.6420	15.3409	62.6420
$E_m$	26.7273	19.6364	26.7273
$p$	17.5	20	20
$w_1$	-	-	0.2653
$w_2$	-	-	0.7551
$J_S$	-	7256.30	14,739
$J_m$	-	22,234	29,461
$J_{SC}$	60,832	29,490.3	44,200

and

Table 2. Results comparison of numerical examples in different decision scenarios with government subsidies.

Variables	Centralized Scenario	Decentralized Scenario	Joint Contract Scenario
$G\left(t\right)$	69.4784	28.3249	69.4784
$D\left(t\right)$	7995.20	3149.20	6853.10
$E_s$	89.4886	21.9156	89.4886
$E_m$	66.8182	49.0909	66.8182
$p$	17.5	20	20
$w_1$	-	-	0.1061
$w_2$	-	-	0.5286
$J_S$	-	12,914	23,383
$J_m$	-	36,506	47,878
$J_{SC}$	96,756	49,421	71,261

. Table 1 is the equilibrium result in the case of no government subsidies, and Table 2 is the equilibrium result in the case of government subsidies.

According to Table 1 and Table 2, we can obtain the following conclusions.

(1)

The production technology investment efforts, the mask production, and product demand under the joint contract decision scenario have reached the level of the centralized decision scenario.

(2)

Compared with the case of no government subsidies, under the influence of government subsidy policies, the profits of the supplier, the manufacturer, and the supply chain system in these three decision situations have increased. Besides, under the influence of the government subsidy policy, the cost-sharing ratio among supply chain members is inconsistent with that under the case of no government subsidies. It represents that government subsidies can change the cost composition of supply chain members, thereby affecting the profits of supply chain members under the game relationship.

(3)

In the case of no government subsidies, compared with the centralized decision scenario, the production technology investment efforts of suppliers and manufacturers under the decentralized decision scenario have decreased by 75.51% and 26.53%, respectively. The mask production has decreased by 62.07%, product demand has decreased by 64.66%, and the total profit of the supply chain system has been reduced by 51.52%.

In the case of government subsidies, the production technology investment efforts of suppliers and manufacturers under the decentralized decision situation have decreased by 75.51% and 26.53%, respectively. The mask production has decreased by 60.61%, the production demand has decreased by 59.23%, and the total profit of the supply chain system has decreased by 48.92%.

It is indicated that, regardless of whether there are government subsidies, the supplier and the manufacturer are willing to cooperate in production. Moreover, supply chain members are more profitable, and their enthusiasm for cooperation tends to be higher in the case of government subsidies.

(4)

In the case of no government subsidies, compared with the decentralized decision scenario, the production technology investment efforts, mask production, and product demand have all been found to increase after the introduction of the joint contract. The manufacturer’s profit has increased by 32.50%, the total profit of the supply chain has increased by 49.88%, and the supplier’s profit has doubled.

In the case of government subsidies, the production technology investment efforts, mask production, and product demand tend to increase. The profits of suppliers and manufacturers have increased by 81.07% and 31.15%, respectively, and the total profit of the supply chain has increased by 44.1%, achieving the centralized decision situation.

It denotes that there is a double marginal effect in the mask emergency supply chain under the decentralized decision scenario, and the joint contract can achieve supply chain coordination.

5.2. Optimal Trajectory Analysis of State Variables

Due to space limitations, some parameters were selected for analysis.

Substituting the relevant parameter values into three models and supposing $t\in \left[0,10\right]$, we can observe the optimal trajectories of mask production volume in the decentralized decision scenario and centralized decision scenario in Figure 2.

Trajectories of mask production in both the decentralized decision model and centralized decision model have a time-stable trend in Figure 2. Because $G^{D*}\left(t\right)<G^{c*}\left(t\right)$, the optimal trajectory of mask production increases monotonically, and the increase is faster in the centralized decision model.

Figure 3 and Figure 4, respectively, show the impact of the sensitivity coefficient of the supplier’s production technology investment effort $\alpha$ on mask production volume and the influence of the supplier’s production technology investment cost coefficient ${\mu }_s$ on mask production in the decentralized decision scenario and the joint contract decision scenario.

Figure 3 illustrates that the mask production increases with the increase of $\alpha$ in the decentralized decision model and joint contract decision model. It indicates that the more sensitive the mask production is toward the supplier’s production technology investment, the more straightforwardly the supplier’s production technology investment affects mask output. The joint contract decision scenario permanently shows a better outcome than the centralized decision scenario in terms of the mask production volume. Similarly, the joint contract model beats the decentralized decision model in terms of supply chain cooperative production. From Figure 4, it is obvious that the mask production declines where there is a rise in production technology investment cost coefficient ${\mu }_s$ in the decentralized decision model and the joint contract model. This is because the higher the supplier’s production technology investment cost, the greater the difficulty in achieving cooperative production. At this time, the cooperation effect of the supply chain members is relatively weaker.

To better understand the influence of mask demand sensitivity coefficient $\epsilon$ regarding mask production volume and the influence of production volume natural attenuation coefficient $\delta$ regarding mask production volume in the decentralized decision scenario and the joint contract decision scenario, supposing $\epsilon =\left[1,2,3,4\right]$ and $\delta =\left[0.2,0.4,0.6,0.8\right]$, we get the effect of the variation of $\epsilon$ on mask production volume and the effect of the variation of $\delta$ on mask production volume in two scenarios, as shown in Figure 5 and Figure 6.

It can be seen from Figure 5a,b that the influence trend of the change of ε on the mask production is roughly the same in the decentralized decision scenario and the joint contract decision scenario. At the same time, when other conditions remain unchanged and the value of ε is fixed, the output in the joint contract decision scenario is higher than the value in the decentralized decision scenario. It is obvious that the joint contract has certain effects in coordinating the cooperation of supply chain members. Figure 5 illustrates that the optimal production volume trajectory increases over time and eventually stabilizes when $\epsilon$ is constant, indicating that the increased production volume is getting smaller and smaller; the cooperation production process is controllable. When t is fixed, the optimal trajectory slope of production volume increases with the increase in the sensitivity coefficient $\epsilon$ of demand to mask production volume, indicating that as demand increases, supply chain member enterprises have better cooperative production effects.

From Figure 6a,b, the impact trend of the change of δ on the mask production remains the same in the two scenarios. Besides, when the value of δ is the same, the mask production under the joint contract scenario is greater than that of the decentralized decision scenario. It shows that the joint contract can help to coordinate the cooperation of supply chain members and increase mask production. Figure 6 demonstrates that the greater the natural attenuation coefficient $\delta$ of production is, the less obvious the cooperative production effect of supply chain member enterprises is.

To study how the wholesale price affects the optimal strategy of the supply chain, supposing $w\in \left[0,20\right]$, we obtain the effect of the variation of $w$ on supply chain members’ profits and production technology investment efforts in the decentralized decision scenario as shown in Figure 7.

Figure 7a presents that supplier’s profit, manufacturer’s profit, and the profit of the supply chain system increase first and then decrease as the wholesale price increases. Figure 7b illustrates that the supplier’s production technology investment effort is directly proportional to the wholesale price, while the manufacturer’s production technology investment effort is inversely proportional to the wholesale price. Combining Figure 7a,b, the result can be explained that although the higher wholesale price can encourage manufacturers to invest in production technology, it cannot bring optimal profits to the manufacturer and supply chain system. On the contrary, a relatively low wholesale price can maximize the optimal profits of the manufacturer and the supply chain system. This is because, although a relatively low wholesale price can reduce the supplier’s production technology investment effort, it can encourage the manufacturer to invest more in production technology effort. Therefore, when the wholesale price is lower, the profits of the supplier, the manufacturer, and the supply chain system can be improved. This result demonstrates that the wholesale price can play a good incentive role in the supply chain system.

5.3. Influence Analysis of Government Subsidies

To better understand the influence of government subsidies on the optimal strategy of the supply chain, supposing ${\phi }_s\in \left[0,1\right]$ and ${\phi }_m\in \left[0,1\right]$, we can get the influence of the variation of government subsidies on product production and member enterprises’ profits in the decentralized decision model and the centralized decision model as shown in Figure 8.

Figure 8a presents that mask production volume increases with the increases of ${\phi }_s$ and ${\phi }_m$. Figure 8b shows that when ${\phi }_s$ is constant, mask production volume rises over time and tends to be stable. When ${\phi }_m$ is fixed, mask production volume increases with the increase of ${\phi }_s$, indicating that government subsidy incentives can be adopted to motivate supply chain enterprises so that they can cooperate. It comes to light that supplier profits in the decentralized decision model rise with the increases of government subsidies ${\phi }_s$ and ${\phi }_m$, and the profit of the supply chain in the centralized decision model increases with the increases of government subsidies ${\phi }_s$ and ${\phi }_m$ in Figure 8c,d. It is indicated that subsidies to the supplier and the manufacturer can boost their profits and their partners’ profits simultaneously. At the same time, the government can not only boost the overall profits of the supply chain system by increasing subsidies to suppliers, but also boost the overall profits of the supply chain system by increasing subsidies to its partners. Figure 9 demonstrates that the overall profits in the decentralized decision model and the centralized decision model increase with the increase of ${\phi }_s$, and when the government subsidy rate ${\phi }_s$ to the supplier is constant, the centralized decision model is better than the decentralized decision model in terms of the overall profit.

5.4. Coordination Effect Analysis of Joint Contract

According to the benchmark parameter values, we can calculate that $w_1=0.4412$ and $w_2=0.5588$. Supposing $Fϵ\left[-100,000,100,000\right]$, we can obtain the relationship between increased long-term profit and transfer payment, as shown in Figure 10, in the joint contract model $(w_1$, $w_2$, $F$).

It can be seen from Figure 10 that the profit increment of the supply chain after the joint contract coordination remains unchanged, and has no direct relationship with $F$. Meanwhile, the supplier profit increment is proportional to $F$ and the manufacturer’s profit increment is inversely proportional to $F$. It is apparent that when $F=F_{min}=-1.8740\times {10}^4$, the supplier’s profit increase is 0 before and after the joint contract coordination, and the manufacturer obtains supply chain profit increase in the joint contract scenario in Figure 10. When $F=F_{max}=1.9902\times {10}^7$, the manufacturer’s profit increase is 0 before and after joint contract coordination, and the supply chain profit increase is all obtained by the supplier in the joint contract scenario. When $F<-1.8740\times {10}^4$, the supplier’s profit will not increase but decrease in the joint contract scenario, and the supplier will reject the contract. When $F>1.9902\times {10}^7$, the manufacturer will reject the contract. Therefore, when $Fϵ\left[-1.8740\times {10}^4,1.9902\times {10}^7\right]$, compared with the decentralized decision scenario, the supplier profit increment and the manufacturer profit increment after coordination are both greater than 0 (shaded part in Figure 10). It denotes that the profits of both supplier and manufacturer will grow simultaneously in the joint contract scenario, so as to comprehend the effective coordination of the mask emergency supply chain. In the actual production process, to lessen the pandemic impact on mask production, the supplier and the manufacturer mutually negotiate the specific value of transfer payment $F$.

6. Conclusions

As background to the COVID-19 pandemic, our paper presents optimal control theory and differential game theory to study cooperative mask production between Meltblown cloth suppliers (leaders) and mask manufacturers (followers), and to evaluate the impact of government subsidies on cooperative production of supply chain members. Moreover, considering that mask production volume is affected by the sensitivity coefficient of production technology investment effort, and mask demand is affected by market prices and production volume, differential game models are constructed for the centralized decision model, the decentralized decision model, and the joint contract decision model. Meanwhile, the mutual cost-sharing ratio and the transfer payment interval of the joint contract are specified. Ultimately, optimal decisions are compared and analyzed through calculation examples. The following conclusions are drawn:

(1)

The production technology investment exertions of supplier and manufacturer, the simultaneous optimum trajectory of production, and the profits of supply chain members are amplified in the joint contract decision scenario. The overall profit of the joint contract decision model reached the level of the centralized decision scenario. Therefore, our joint contract introduction can maximize supply chain profits and meet mask market demand under the COVID-19 pandemic. At the same time, the balance strategies of production technology investment effort under the three decisions do not change with time.

(2)

With the growth in the impact of the supplier’s production technology investment effort sensitivity coefficient on production volume and the increase in mask demand sensitivity coefficient to mask production volume, mask production volume showed an increasing trend, indicating that the more sensitive production volume is to the supplier’s production technology investment effort, the more apparent cooperative production effects of supply chain members are. With the growth in the impact of the supplier’s production technology investment cost coefficient on mask production and the increase in the self-decay rate of mask production, mask production represents a downward trend, indicating that high input costs will weaken the cooperative enthusiasm of the supplier and the manufacturer. The effect of long-term collaborative output will deteriorate as the natural aging speed of invested production equipment accelerates. At this point, the government can subsidize the production inputs of the supplier and the manufacturer to increase cooperation enthusiasm. Additionally, government subsidies can increase the profits of supply chain members and their partners more than in the case of no government subsidies.

(3)

Compared with the decentralized decision model, production technology investment efforts in the centralized decision scenario, the optimum trajectory of mask production, and the supply chain’s general profit are favorably improved.

(4)

When the joint contract meets certain conditions, the optimal production technology investments of both supplier and manufacturer and overall profit are improved compared with the decentralized decision scenario, reaching a certain level of the centralized decision scenario. At the same time, it also makes the profits of the supplier and the manufacturer achieve double Pareto improvement.

Our paper principally concentrates on the impact analysis of the joint contract on mask production, supply chain coordination effects, price on the profits and production technology investment efforts, and government subsidies on the strategy of cooperative production. However, some limitations exist in our approach and the relationship between the demand and the supply were not considered. At some point, we can ponder the relationship between demands and supplies in the analysis framework. At the same time, our paper only studies the two-echelon supply chain composed of one supplier and one manufacturer. Furthermore, we can elaborate on mask production and demand decision issues in the multi-echelon supply chain. Additionally, considering the particularity of emergency medical supplies, choosing a cold chain transportation mode from an environmental and financial perspective may be a future research direction.