EFFECT OF GOVERNMENT SUBSIDIES ON SUPPLY CHAIN DECISION-MAKING AND COORDINATION IN THE CONTEXT OF COVID-19

Considering the two-stage supply chain composed of a leading retailer and a manufacturer under the background of covid-19 epidemic, the retailer determines the anti-epidemic effort level and bears the corresponding costs, and the manufacturer determines the cost sharing rate under the coordination strategy. This paper analyzes the pricing decision, anti-epidemic effort level and cost sharing rate of supply chain under different government subsidy measures and coordination strategies. From the perspective of Stackelberg game, we find that under the background of epidemic situation, government subsidies can improve the level of social welfare; the improvement of marginal income of anti-epidemic efforts is conducive to obtain a higher level of anti-epidemic efforts and social welfare; Within the reasonable marginal revenue range of anti-epidemic efforts, the coordination strategy can improve the retailers’ anti-epidemic efforts and social welfare level, and the government can obtain the maximum anti-epidemic efforts and social welfare level by subsidizing manufacturers with cost sharing. Mathematics Subject Classification. 90B99, 97M40. Received October 21, 2020. Accepted June 1, 2021.

papers in this field analyze the impact of COVID-19 from the overall supply chain, and our research will focus on the analysis of the decision-making of each subject in the supply chain in the context of COVID- 19. In the face of the severe COVID-19 epidemic situation, the government will do its part to make remedial measures [1,12], including emergency adjustments to the national budget and financial assistance to families in need, etc. Usually, the government will also subsidize the supply chain. Reviewing the literature, we find that the government's subsidy measures for the supply chain can be distinguished from the subsidy object, subsidy method, and supply chain type. Chen et al. have studied the optimal subsidy level of the government when the quantity of old products returned to the OEM and to consumers is uncertain, and found that no essential differences exist between providing subsidies directly to the manufacturer and to consumers [11]. Jiang et al. examined three possible subsidy provision strategies depending on which party (the power plant, the villagers' committee or the farmer) the government chooses to subsidise [23]. Wang et al. found that when the e-waste remanufacturing utilization rate is relatively high, the marginal effect of the subsidy to the remanufacturer on economic benefit and the recycle quantity increase as the subsidy increases, and the government should allocate as much support as possible to the remanufacturer [44]. From the way of subsidy, Jian et al. constructed the government cash subsidy strategy model based on the expected profits of supply chain members to analyze the government tax reduction subsidy strategy and cash subsidy strategy [22]. Han et al. found that government carbon subsidies are positive factors in the supply chain operation, which can stimulate the manufacturer to make low-carbon products as expected and choose the high quality-high price development mode [19]. Green consumption subsidy [32] and emission reduction subsidy [26] are also common forms of subsidies. The government will also subsidize different types of supply chains, Ma et al. studied the impact of consumption subsidies on the dual-channel closed-loop supply chain, and considered the impact of consumption subsidies from the perspectives of consumers, the size of the closed-loop supply chain and enterprises [30]. In order to cope with severe environmental problems and reduce the impact on shipping routes downstream of the ocean supply chain, Li et al. used game theory to determine the optimal government subsidy intensity and subsidy reduction point [27]. In the supply chain classification, green supply chain [43], low-carbon supply chain [45] and decentralized Reverse supply chain [20] are common. Social welfare is generally used as the objective function of government subsidies [5]. Our research specifically considers two different types of subsidies: manufacturer's production cost and retailer's anti-epidemic effort cost.
Our research is also closely related to the field of coordination strategy, appropriate coordination strategy can effectively improve supply chain performance [9]. Chen et al. found that using revenue sharing contracts, generate many more supply chain profiles for the PV supply chain comparing to the profiles eared by those supply chain strategies without any subsidy [8]. When Huang et al. studied the financing services of the third party logistics, he concluded that the profit of supply chain can achieve Pareto improvement, and the wholesale price contract may realise profit maximisation and channel coordination in the supply chain system under certain condition [21]. Ghosh et al. studied two models of cost sharing -one in which the retailer offers a cost sharing contract and the other in which the retailer and manufacturer bargain on the cost sharing contract [17]. In particular, we consider a special cost sharing strategy.
In summary, we establish a supply chain decision model in the context of pandemic, and analyze the influence of subsidy factors and coordination factors on optimal supply chain decision by comprehensively considering government subsidy factors and coordination mechanism. Combining mathematical analysis and example analysis, this paper investigates how to obtain the maximum effort level of anti-epidemic, the overall performance level of the supply chain and the level of social welfare, so as to provide some suggestions for the government and members of the supply chain to make decisions under the epidemic situation. This is not only conducive to the stability and sustainable development of the supply chain, but also contributes to the recovery of socio-economic status and the alleviation of the severity of the epidemic.

Model description
We consider a secondary supply chain that is affected by the epidemic and is composed of upstream manufacturers (M ) and downstream retailers (R). Manufacturers produce a specific type (not the products mentioned above that have increased sales during the epidemic). Compared with the normal period (no epidemic situation), manufacturers are in urgent need of local governments to implement subsidy measures due to the lack of human resources of employees, sales channels and the reduction of orders from downstream retailers.
On the other hand, under the impact of COVID-19, brick-and-mortar retailers in severely affected areas have been hit hard and do not have sufficient capacity to fight the epidemic. Therefore, this study mainly discusses online retailers that do not require high store rents (Hereinafter referred to as retailer). During the epidemic period, downstream retailers are responding to the local government's active anti-epidemic businessmen with social responsibility. Compared with the normal period, retailers need to bear greater risks when hiring employees, and the difficulty of logistics and transportation is greatly increased. The freight on the way will also increase with the severe social situation. We regard this retailer's behavior as a highly socially responsible anti-epidemic effort to compensate for the losses of manufacturers in severely affected areas. In order to obtain the marginal effect of the effort, suppose the retailer's anti-epidemic effort cost is quadratic type θ 2 e 2 , where e > 0 is the retailer's anti-epidemic effort level, and θ > 0 is the retailer's anti-epidemic effort cost coefficient, similar to Fan et al. [13], Atasu and Subramanian [3], Arora and Ceccagnoli [2], and Chen et al. [7]. Considering a product that is priced at p, the customer with a net surplus v − p ≥ 0 will buy it [14]. From Figure 1, all the customers with valuations in the interval [p, 1] will buy the product. Therefore, the demand of the product is Q = 1 p dv = 1 − p for 0 ≤ p ≤ 1. While making anti-epidemic efforts, retailers sell products to customers at the sales price p. Active and effective anti-epidemic efforts will reduce the severity of the epidemic and accelerate the progress of social recovery from the epidemic, which coincides with the will of consumers. Therefore, such anti-epidemic efforts meet the value needs of consumers. From Figure 2, all the customers with valuations in the interval [p, 1 + e] will buy the product. Therefore, the demand of the product is Q = 1+e p dv = 1 + e − p for 0 ≤ p ≤ 1.
In order to accelerate the recovery of the economic level, the government will encourage all members of the supply chain to make efforts to control the epidemic, that is, the government will take subsidies for them [42], and government subsidies are collectively referred to as Φ. Considering a variety of subsidy methods, when subsidizing manufacturers, the government subsidizes the cost c of each product produced by the manufacturer, then Φ = sQ, where s > 0 is the government's subsidy coefficient to the manufacturer, then the manufacturer's production cost per-unit of product is reduced to c − s. It should be noted that what we are talking about is the price and demand when the market clears. When subsidizing retailers, the government subsidizes the retailer's anti-epidemic effort level e, then Φ = se, where s > 0 is the government subsidy coefficient to the retailer (the following describes how to distinguish the various subsidy coefficients), which means the greater the anti-epidemic efforts made by retailers, the more government subsidies they will receive. This is also one of the endogenous reasons for retailers' efforts to control the epidemic and develop the socio-economic level.
The government subsidy s is determined by the government's profit function, that is, the social welfare function maximization [10]. Considering the increase in social benefits and the improvement effect of the epidemic EI brought about by the retailer's anti-epidemic efforts, government subsidies Φ, total profits of the supply chain π, and consumer surplus GS as components of the social welfare function, so SW = EI − Φ + π + GS. EI corresponds to the external benefits of retailers' continued anti-epidemic efforts. Different from the marginal benefit set by Raz et al. [37] and Atasu et al. [4], the unit of demand is a constant, i.e. EI = αD, and in order to reflect the continuous benefits of anti-epidemic efforts to social welfare, determine the best level of anti-epidemic efforts, and highlight the status of anti-epidemic efforts in the social welfare function, we define EI = αe, where α(0 < α < 1) is the marginal benefit of the government's anti-epidemic efforts. The government subsidy Φ has been explained in the previous article, and Φ is different under different subsidy types. The third part is the total profit of the supply chain, that is, the sum of the profit of the manufacturer and the retailer. When calculating consumer surplus, we use p 0 to denote the price when demand is zero, i.e. p 0 = 1 + e, then consumer surplus is denoted as (p 0 − p)D/2, and GS = D 2 /2 is calculated.
We analyze the influence of different government subsidy strategies on supply chain decision-making under the background of pneumonia caused by covid-19, considering three scenarios: the government subsidizes the manufacturer separately, the government subsidizes the retailer separately, and the government subsidizes both. Retailers' participation in the anti-epidemic activities will produce a lot of costs. In order to accelerate the progress of social recovery from the epidemic, ensure the stable operation of the supply chain and even improve the performance of the supply chain, manufacturers consider to take cost-sharing coordination measures with retailers. Similar to Cao et al. [5] and Fan et al. [13], we study the situation that manufacturers and retailers share the cost of anti-epidemic efforts of retailers. In particular, it is assumed that the proportion of antiepidemic effort cost borne by retailers is λ, and the remaining 1 − λ is borne by manufacturers. The order of the game can be expressed as Figure 3: The following table summarizes all the symbols that appear in this section:

Model calculation
For ease of exposition, we define System xy, where x ∈ {n, c, e, b} stands for the government does not subsidize, the government subsidizes the manufacturer's cost of the product, the government subsidizes the retailer's level of effort and government subsidizes both, and y ∈ {Y, N } represents whether or not to adopt a coordination strategy. For example, System eN indicates that the government subsidizes the effort of retailers and those manufacturers and retailers do not adopt a coordinated strategy.

System nN
In the case of System nN , the government does not subsidize members of the supply chain, i.e. Φ = 0, and the manufacturer and retailer do not adopt coordination strategy, i.e. λ = 1. In this case, the overall benefits of the manufacturer, retailer and supply chain are: Similarly, we can derive the expression of supply chain members and social welfare in the remaining cases.

System cN
In accordance with the game sequence given in the previous article, we use reverse induction to solve the supply chain decision. In order to avoid the length of the text, we show the specific equilibrium solution solving process in Appendix A. While ensuring the positive constraints of the parameters and the second-order optimal conditions, we will further analyze and discuss the equilibrium solution.

Managerial implications
In this section, we first analyze the endogenous causes of government subsidies in the context of epidemic situation when the national finance is in crisis. Section 4.1 compares the social welfare level with or without government subsidies and the optimal retailer's anti-epidemic effort level. Then we compared some equilibrium results for all five subsidies cN , cY , eN , eY and bN . Sections 4.1 and 4.2, respectively analyzed and calculated the best retailers' anti-epidemic effort level and the best social profit level. Then, considering the impact of the cost coefficient, we discussed the impact of the retailer's anti-epidemic effort cost coefficient on some equilibrium results in various situations, this analysis is shown in Section 4.3. In order to facilitate the analysis, this article takes α = 0.1 and 1 < θ < 3 (satisfy the quadratic conditions in the appendix) for analysis. For the main idea of not overcrowding, our interested readers refer to Appendix B, which contains all the calculations and proofs in this section.

Supply chain decision analysis with or without government subsidy
In this section, we compare and analyze the optimal decision of supply chain with or without government subsidy. To be specific, Proposition 4.1a compared and analyzed the optimal level of anti-epidemic efforts with and without government subsidies, while Proposition 4.1b examined the level of social welfare with and without government subsidies.
Summarily, we write the result as: Proposition 4.1a shows that, regardless of whether coordination strategy is considered, the optimal level of anti-epidemic efforts is always higher with government subsidies than without government subsidies. Obviously, when the government subsidizes the retailers, the increase of income brought by the improvement of antiepidemic efforts is greater than the increase of cost, and the government subsidies have a positive impact on the "anti-epidemic efforts". At the same time, with the improvement of anti-epidemic efforts, the overall performance of the supply chain, consumer surplus and marginal income of anti-epidemic efforts will also increase, so will the level of social welfare. When the government subsidizes the manufacturer, the unit cost of the manufacturer will be reduced, the retailer can get a lower wholesale price in the process of bargaining with the manufacturer, more cost will be used to improve the level of anti-epidemic efforts, the overall performance of the supply chain, consumer surplus and the sustainable income of anti-epidemic efforts will also be increased, so the level of social welfare will also be improved.

Comparison of retailers' anti-epidemic efforts and coordination levels
In this section, we examine the impact of various subsidy measures on the level of anti-epidemic efforts. Specifically, Proposition 4.2a examined the optimal anti-epidemic effort level in several situations without a coordinated strategy; Proposition 4.2b compared the optimal anti-epidemic effort level when a coordinated strategy is adopted and when a coordinated strategy is not adopted; Proposition 4.2c compared the coordination levels under the two coordination strategies.
Summarily, we write the result as: Comparison of the level of anti-epidemic efforts and coordination levels under different subsidies where ψ e1 and ψ e2 are defined in the proof.
Proposition 4.2a analyzes the optimal e without considering the coordination strategy. When the government subsidizes e and c at the same time, the optimal level of anti-epidemic effort is always greater than that when the government only subsidizes e. This shows that under the premise of subsidizing e, when the government subsidizes c again, the unit cost of the manufacturer will be reduced, the retailer can get a lower wholesale price in the process of bargaining with the manufacturer, more costs will be used to improve the anti-epidemic effort level, and the optimal anti-epidemic effort level will be improved accordingly. Figure 4 shows that when ψ e1 > 0 is satisfied, the optimal level of anti-epidemic effort when the government subsidizes e is greater than that when the government subsidizes c, this shows that when the government subsidizes e, the increase of subsidies has a greater incentive effect on retailers' efforts to improve the anti-epidemic level than when the government subsidizes c, the decrease of wholesale price has a greater incentive effect on retailers' efforts to improve the anti-epidemic level. Figure 5 shows that when ψ e2 > 0 is satisfied, the government provides subsidies to e and c at the same time, the optimal anti-epidemic effort level is larger than that of the government subsidies alone.  In other words, when ψ e2 > 0, on the premise of subsidizing c, the government subsidizes e again, the cost increase caused by improving the level of anti-epidemic efforts is less than the subsidy amount, so the level of anti-epidemic efforts is increased. Proposition 4.2b shows that coordination strategy can improve the optimal level of anti-epidemic efforts in supply chain when government subsidies e. A coordinated strategy means that manufacturers bear part of the cost of anti-epidemic efforts, while retailers have a lower cost of anti-epidemic efforts, which can motivate retailers to increase the level of anti-epidemic efforts. Figure 6 shows that in the case of government subsidy c, the coordination strategy can improve the optimal anti-epidemic effort level of the supply chain only when ψ e3 > 0 is satisfied. It can be found from the observation in Figure 6 that, within a reasonable range, the possibility of ψ e3 > 0 (i.e. e cN < e cY ) is high, which indicates that when manufacturers receive government subsidies, they have a strong willingness to share the anti-epidemic effort with retailers, this is because when the government subsidizes c, the manufacturer's unit cost is reduced, and more of the cost can be used to share the cost of anti-epidemic efforts. Finally, we conclude from e eY < e cY that when the government subsidizes c, the coordination strategy adopted by the supply chain subjects has a more stimulating effect on the improvement of anti-epidemic efforts.
Proposition 4.2c shows that the coordination level of System eY is always higher than that of System cY , because when the government subsidizes c, manufacturers are more willing to share the cost of anti-epidemic efforts.

Comparison of social welfare levels
In this section, we examine the effects of various subsidy measures on the level of social welfare. To be specific, Proposition 4.3a compares the optimal level of social welfare in several situations without coordination strategy, while Proposition 4.2b compares the optimal level of social welfare in the case of coordination strategy and without coordination strategy.
Summarily, we write the result as: Comparison of social welfare levels under different subsidy types.
where ψ SW1 and ψ SW2 are defined in the proof.
Proposition 4.3a shows that, without considering the coordination strategy, for any given type of subsidies, the social welfare function presents an absolute relationship, that is, the social welfare function is the largest when the government subsidizes c, followed by the government subsidizes both c and e, and the smallest is the government subsidizes e. This shows that when the government subsidizes the manufacturer, the manufacturer's unit cost will be reduced, and the retailer can get lower wholesale price in the process of bargaining with the manufacturer. More cost of the retailer will be used to improve the anti-epidemic effort level, and then the optimal anti-epidemic effort level will be increased. Then the marginal benefits of anti-epidemic efforts, overall benefits of the supply chain and consumer surplus all increase, and the increase rate is higher than the increase rate of government subsidies (the difference between the two changes is set as ∆ς cN ), so the level of social welfare increases. Similarly, ∆ς cN > ∆ς bN > ∆ς eN can be obtained from Proposition 4.3a.
Proposition 4.3b shows that when ψ SW2 > 0 is established in Figure 7, the coordination strategy can improve the optimal social welfare level of the supply chain when c is subsidized by the government. Adopting coordination strategy means that manufacturers bear part of the cost of anti-epidemic efforts, and retailers' cost  of anti-epidemic efforts is reduced, which can encourage retailers to improve the level of anti-epidemic efforts. With the improvement of anti-epidemic efforts, the cost of government subsidies increases, and the increase rate is higher than the marginal benefit of anti-epidemic efforts, the overall benefit of supply chain and the increase rate of consumer surplus. Therefore, when the government subsidies c, the coordination strategy will reduce the level of social welfare. Similarly, we get ∆ς cY > ∆ς eY . When analyzing System eN and System eY , we find that they are similar to e cN and e cY in Section 4.2. As shown in Figure 8, the optimal social welfare function presents different relationships under different constraints. When ψ SW1 > 0, there is ∆ς eY > ∆ς eN , that is, coordination strategy can improve the optimal level of social welfare when the government subsidizes e. ∂e cN ∂θ < 0, ∂e cY ∂θ < 0, ∂e eN ∂θ < 0, ∂e eY ∂θ < 0, and ∂e bN ∂θ < 0.
Proposition 4.4 shows that when the cost coefficient θ of anti-epidemic efforts of retailers increases, the optimal anti-epidemic effort level under five kinds of subsidies will decrease. Obviously, when θ increases, retailers will incur more costs if they take the same actions to fight the epidemic, which will discourage retailers' enthusiasm for fighting the epidemic, and the optimal level of fighting the epidemic will decrease accordingly. Proposition 4.5 shows that when the cost coefficient θ of anti-epidemic efforts of retailers increases, the optimal level of social welfare under five kinds of subsidies will decrease. Obviously, the expansion of θ will reduce the overall benefit of the supply chain, the marginal income of the anti-epidemic efforts will also decrease, and the level of social welfare will also decline.

Numerical analysis
The above model mainly analyzes the optimal decision when the government has different subsidy strategies. Next, we assign the relevant parameters in the model to further analyze and verify the relevant conclusions. From the above analysis, it can be seen that the marginal benefit α of the government's anti-epidemic efforts is the main variable affecting the decision-making of the supply chain. Therefore, we select c = 0.9, θ = 2.5 within an appropriate range to focus on the analysis of the impact of the marginal benefit of the anti-epidemic efforts.
The relationship between the level of anti-epidemic efforts and the marginal benefits of anti-epidemic efforts is shown in Figure 9. As can be seen from Figure 9, e is in a positive linear relationship with α. When the marginal benefit of anti-epidemic efforts increases, the optimal anti-epidemic effort level of retailers will also increase. When α = 0.1, the anti-epidemic effort level in System cY is the highest, which verifies Proposition 4.2. At the same time, we also found that no matter which entity is subsidized separately by the government, the cost-sharing strategy is always conducive to improving the level of anti-epidemic efforts of retailers. When the marginal benefit of anti-epidemic effort is within the normal level (α ∈ (0, 0.262)), System cY is always the best choice, because the optimal anti-epidemic effort level is the highest at this time. Figure 10 shows that the profit of manufacturers increases with the increase of the marginal benefit of antiepidemic efforts. When the government subsidizes e, adopting a coordinated strategy means that manufacturers share the cost of anti-epidemic efforts of retailers. At this time, the cost of anti-epidemic efforts of retailers will decrease, and retailers will increase the level of anti-epidemic efforts, and the demand for products will  increase accordingly. The increase in orders has increased profits by more than the share of the cost of the anti-epidemic effort, so manufacturers' profits have increased. For manufacturers, when the marginal benefit of anti-epidemic efforts is low (α ∈ (0, 0.147)), the manufacturer's profit under System cN is the highest; when the marginal benefit of anti-epidemic efforts is high (α ∈ (0.147, 0.3)), the manufacturer's profit under System cY is the highest. Figure 11 shows that the retailer's profit increases with the increase of the marginal benefit of anti-epidemic efforts. When the government subsidizes e, adopting a coordinated strategy means that manufacturers share the cost of anti-epidemic efforts of retailers. At this time, the cost of anti-epidemic efforts of retailers decreases, retailers will increase the level of anti-epidemic efforts, and the demand for products will increase accordingly. Higher orders and higher government subsidies have increased profits more than the cost of the anti-epidemic effort, so retailers' profits have increased. For retailers, when the marginal benefit of anti-epidemic efforts is at a normal level (α ∈ (0, 0.244)), the manufacturer's profit under System cN is the highest; when the marginal benefit of anti-epidemic efforts is higher (α ∈ (0.244, 0.3)), the manufacturer's profit under System cY is the highest. The relationship between the level of social welfare and the marginal benefits of anti-epidemic efforts is shown in Figure 12. As can be seen from Figure 12, SW is positively correlated with SW. When the marginal benefit of anti-epidemic efforts increases, the level of social welfare will also increase. When α = 0.1, the social welfare level in the case of System cY is the highest, which verifies Proposition 4.3. At the same time, we also find that no matter which subject is subsidized by the government alone, when the marginal benefit of anti-epidemic efforts is not within the minimum range (α ∈ (0.048, 0.3)), the cost-sharing strategy is always conducive to improving the anti-epidemic efforts of retailers, and the System cY is always the optimal choice, because the social welfare level is the highest at this time.

Managerial insights
In the context of COVID-19, enterprises in the supply chain actively participate in the fight against the epidemic and may share the additional costs arising from the fight against the epidemic, and the government will provide cost subsidies and anti-epidemic efforts subsidies to enterprises in the supply chain. We build a model to analyze the operation of the supply chain, the optimal solution of the supply chain when the social welfare is maximized, and the influence of relevant parameters on the optimal solution. The analysis results are summarized as follows.
In the context of the pandemic, the social and economic benefits decrease and the government will face a certain financial crisis. However, when the government takes appropriate subsidy measures for manufacturers and retailers, the benefits of members of the supply chain, the overall performance level of the supply chain and the level of social welfare will increase. Therefore, the government has an incentive to subsidize members of the supply chain, which can improve socio-economic benefits in the context of the epidemic, contribute to socio-economic recovery, and reduce the socio-economic impact of the epidemic. On the other hand, it is obvious that with the increase of the marginal benefits of the anti-epidemic efforts, the better the effect of government subsidies on the improvement of social welfare level will be. Generally speaking, the measures taken by the government to improve the marginal benefits of anti-epidemic efforts include: actively publicizing the anti-epidemic efforts of members of the supply chain to mobilize consumers' purchase desire; We will actively introduce epidemic prevention and control policies, improve the efficiency of enterprises' resumption of work and production, speed up the efficiency of the circulation of materials and products, and improve the logistics efficiency in the supply chain. Accelerating the breakthrough of technical barriers and the research and development of key logistics technologies are conducive to improving the efficiency of enterprises' efforts to fight against the epidemic, so as to improve the marginal income of efforts to fight against the epidemic.
The government has the motivation to subsidize the members of the supply chain. Under the marginal benefits of different anti-epidemic efforts, different subsidy measures can maximize the anti-epidemic effort level or social welfare level. When the coordination strategy is not taken into account, the social welfare will be the largest when the government chooses to subsidize the manufacturer (α ∈ (0, 0.194)), and the social welfare will be the largest when the marginal benefit of the anti-epidemic efforts exceeds this range (α ∈ (0.194, 0.3)). When we analyze the optimal decision of supply chain, we find that the cost sharing strategy can effectively improve the anti-epidemic effort level of retailers and social welfare level.
By comprehensively considering all subsidy scenarios, it can be concluded that when the marginal benefits of anti-epidemic efforts are within the normal range (α ∈ (0.048, 0.242)), and when the government subsidizes manufacturers in the supply chain that adopt the cost-sharing strategy, the anti-epidemic efforts and social welfare levels are superior to those of other subsidy strategies. Therefore, the following implications can be drawn. Under the epidemic background, member enterprises in the supply chain should actively adopt the costsharing strategy of anti-epidemic efforts, which can improve the profit level of member enterprises in the supply chain. At the same time, coordinated strategies can also improve the anti-epidemic efforts and the level of social welfare, which can alleviate the difficulties brought by the epidemic and effectively speed up the recovery of the whole society from the epidemic. The government should subsidize manufacturers who adopt a coordinated strategy, which can improve anti-epidemic efforts and social welfare. Management implications are not universal when the marginal benefits of anti-epidemic efforts are too small or too high.
On the other hand, the increase of the cost coefficient θ of anti-epidemic efforts will reduce the level of antiepidemic efforts and social welfare. In order to achieve a higher level of anti-epidemic efforts and social welfare, the government should introduce epidemic prevention and control policies, accelerate the process of epidemic normalization, accelerate the progress of enterprises' resumption of work and production, accelerate the research and development of vaccines, and accelerate the research and development of new and high technologies to improve logistics efficiency and reduce the cost coefficient of anti-epidemic efforts.

Future research directions
The first extension of the model proposed in this paper is that the anti-epidemic behavior of manufacturers can be quantified. In fact, when manufacturers are in the epidemic situation, their own production and transportation activities will improve the social economic level, accelerate the progress of social recovery from the epidemic, and will generate additional production costs. Secondly, the coordination strategies adopted by manufacturers and retailers may take various forms. This study mainly focuses on the cost-sharing strategy of anti-epidemic efforts, and subsequent studies can try to explore more coordination methods. Thirdly, this study focuses on the cooperation process initiated by retailers. The supply chain in reality will have a variety of driving modes, so we can try to compare and analyze the supply chain decisions driven by different subjects. Finally, the supply chain in reality is more diversified. In order to make our research more universal, subsequent studies can focus on the operation of the closed-loop supply chain or the supply chain with more participants in the epidemic situation.

System nY
The first step is the same as above nN case, we get Hessian Matrix H R (e, p) = , when 2θλ − 1 > 0 is satisfied, solving the first-order optimality condition for e and p, i.e.

System eY
The first step is the same as above nN case, we get Hessian Matrix H R (e, p) = . Lastly, substituting all of the above optimal decision variables into SW, SW is concave in s when ∂ 2 SW ∂s 2 = − 64θ−39 (16θ−9) 2 < 0, i.e. 64θ − 39 > 0, solving ∂SW ∂s = 0 give the optimal solutions:

Appendix B. Proof of propositions
Proof of Proposition 4.1.