## Abstract

As a novel market-based water-saving mechanism, the Water Saving Management Contract (WSMC) project faces interruption risk caused by emergencies like the coronavirus disease-2019 (COVID-19) pandemic. An interruption risk assessment model of WSMC projects is established through a quantitative evaluation of the impact of emergencies on water users based on input-output theory. First, the concept of the interruption risk index (IRI) is defined as a function of the duration of enterprise shutdown (DES). Second, the DES is divided into two parts: the duration caused by COVID-19 and that under other types of emergencies. Third, the risk tolerance threshold is given to estimate the interruption result, and its different consequences are discussed. Finally, a WSMC project in China is taken as a case study, and its interruption risks are analysed. The results show that the IRIs of this WSMC in both 2020 and 2021 are theoretically greater than the risk tolerance thresholds, and the high pandemic prevention standards and conservative pandemic estimates are the main reasons for the above results. The model established in this study provides a reference for WSMC participants to deal with emergencies and provides the theoretical support for the extension of the WSMC.

## HIGHLIGHTS

Firstly a quantitative interruption risk assessment model was established for WSMC.

The economic loss of water users can be estimated under COVID-19 pandemic.

Based on statistical data, the conditional expectation of shutdown time can be obtained under normal emergencies.

## INTRODUCTION

The Water Saving Management Contract (WSMC) is a market-based water-saving mechanism, by which a water-saving service company (WSSC) invests in water-saving reconstruction and obtains income through the water-saving benefits provided by water users (Liu *et al.* 2021). The WSMC faces many risks when it is implemented in the market because of its imperfect system in all aspects (Hu *et al.* 2021), with one of these risks being its interruption risk. WSMC project interruption usually refers to the failure of WSSCs to receive benefits as stipulated by the contract. The occurrence of an emergency may disturb water users’ normal business activities and increase the interruption risk of the WSMC (Li *et al.* 2020). This phenomenon affects the motivation of participants and impedes the popularity of this water-saving mechanism (Gan 2009). Hence how to assess the above risk is a vital issue that must be addressed for WSMC projects (Ma *et al.* 2021).

Except for some macroscopic qualitative descriptions or trend analyses of the WSMC project risk, there are few studies on quantitative models for assessing the above risk. In similar research domains, Mills *et al.* studied the inherent risks of energy performance contract (EPC) projects and divided them into five categories (Mills *et al.* 2006). Then some scholars added the risk types and tried to identify the key risks on the basis of Mills *et al.*’s work (Jia *et al.* 2021). In addition, other domains have adopted methods that rely on expert data for the prioritization of critical risk influencing factors and estimating risk levels, such as the analytical hierarchy process (Liu *et al.* 2020), fuzzy comprehensive evaluation (Mo *et al.* 2021), and fuzzy analytical hierarchy process (Zhen & Vinnem 2021). In addition, more risk forecast models have been developed based on process-driven aspects such as floods (Marquez *et al.* 2020), droughts (Seyedabadi *et al.* 2020), and project benefits (Nesticò *et al.* 2020). Recently there have been an increasing number of studies investigating data-driven risk assessment models including those dealing with machine learning (Cantos *et al.* 2020), artificial neural networks (Teles *et al.* 2020), and so on (Amin *et al.* 2019). However, WSMC projects obviously belong to mechanisms with long project cycles, low returns and weak liquidity. Therefore, a constant assessed risk level of WSMC projects may be imprecise during the whole contract period (Jokhadze & Schmidt 2020). In addition, since the WSMC is in its infancy, there are few statistical data points for the current data-driven models.

In classical research, risk is defined as a function of probability and consequence (Shi *et al.* 2020). Since the difficulty of determining the probability of an emergency occurring and being a small probability event, the equal probability method and scenario analysis are usually adopted in traditional studies to evaluate this risk (Shen *et al.* 2021). Hence, most studies focus on emergency risk as a function of consequences (Yang *et al.* 2017). The cost recovery and profit gain of the WSSCs in the project are completely dependent on the amount of water saved by water users; hence, the consequences of WSMC risk under emergencies are the losses caused by a reduction in the amount of water saved. Figure 1 shows the chain logic diagram that explains the details of emergency impact transfer. The duration of enterprise shutdown (DES) of water users and the water-saving benefit for WSSCs play bridging roles between emergencies and WSMC interruption risk. Water users’ DESs are affected by both the hardware facilities and economic environment of the cities in which they are located (Fraccascia 2019). Regarding the first aspect, the water supply shortage caused by emergencies leads to a decline in water users’ production, and no water consumption means that no water has been saved. Regarding the other aspect, water users face an economic environmental problem that does not meet the requirements of normal business activities. For example, for water users in the car wash or hotel industries, while COVID-19 does not pose a threat to their own operating facilities or water supply systems, they are affected by reduced travel due to prevention and control policies (Chang *et al.* 2020). Community resilience refers to the sustained ability to resist, absorb and accommodate the effects of natural hazards by utilizing available resources, and it has received great attention across a variety of fields and disciplines (Faulkner *et al.* 2018). Community resilience exactly represents the water users’ risk tolerance ability from two aspects, i.e., hardware facilities and the economic environment. According to its definition, enterprises’ resilience reflects the ability of an enterprise to function effectively and recover successfully in the aftermath of disasters. Under emergencies, the main difference between WSMC water users and ordinary water users is that the former needs to consider the water-saving benefits payment. Hence, the losses of water users consist of economic losses as well as this payment.

Based on the above two impacts of emergencies on the WSMC, this paper establishes an interruption risk assessment model of WSMC projects under emergencies. Due to their low probability and high impact, the consequences caused by emergencies are chosen as the main factor of interruption risk in this paper (Özen & Şahin 2020). To describe the above risk, the interruption risk index (IRI) of WSMC projects under emergencies is defined as a function of both the consequences and the previous production of water users. Considering the state of emergency of the world today, the DES is divided into two parts: that affected by COVID-19 and that affected by other types of emergencies. For the COVID-19 part, this paper adopts epidemic status indices and epidemic prevention standards to calculate the DES under COVID-19. For the other part, the community resilience index (CRI) is applied to evaluate the expected DES of some given types of emergencies. Then economic loss is calculated based on input–output theory, and the water-saving benefits payment is also obtained. Once the risk tolerance threshold is given, the interruption risk of WSMCs can be estimated, and the interruption consequences are discussed. Finally based on a WSMC case, a discussion and conclusions are provided.

## METHOD

In this section, the IRI is defined to describe the interruption risk level of WSMC projects. Then an IRI assessment model is established, considering COVID-19 and other types of emergencies. In addition, the risk tolerance threshold is defined and the interruption results are introduced. Finally, if the WSMC project is interrupted, then the consequences of such an interruption are discussed.

### Interruption risk and the IRI

Figure 2 shows the technology roadmap of the IRI assessment model of the WSMC under emergencies. The following content is the process of establishing the above model according to the technology roadmap.

### Calculation of the DES under different emergencies

DES refers to the duration of an enterprise stopping its normal business activities and its unit in this paper is day(s). Unless otherwise stated, the DES specifically denotes the shutdown days of water users.

During the DES period, an enterprise's production decreases, which leads to losses. Then, the DES consists of two parts. The first is the DES under COVID-19, and the other is the DES under other types of emergencies.

### Calculation of the DES under COVID-19

According to practical experience and a literature review, the impact of COVID-19 on water users mainly concerns policy requirements and customer demand (Renukappa *et al.* 2021). The above problems can be dealt with by considering the current status condition of COVID-19. Several indices are selected to represent the current status of COVID-19, and epidemic prevention standards are set up for each of these indices. Then the DES under COVID-19 can be calculated by the following theorem:

**Theorem 1.**Let denote the DES under COVID-19, and then according to Equation (3):where is the number of shutdown days of water users in month during the period of interest, , and

*M*refers to the total number of months of interest.

** Proof**. According to practical experience and a literature review, four indices are selected to describe the current status of COVID-19, the details of which are shown in Table 1.

Indices . | Name . | Description . |
---|---|---|

Existing confirmed cases | The number of existing confirmed cases across the country by day | |

New confirmed cases | The number of new confirmed cases that are reported across the country by day | |

Percentage of existing indigenous cases | The proportion of the number of existing confirmed cases per day that are reported domestically by day | |

Provincial distribution of existing confirmed cases | The number of provinces in the country where existing confirmed cases are reported by day |

Indices . | Name . | Description . |
---|---|---|

Existing confirmed cases | The number of existing confirmed cases across the country by day | |

New confirmed cases | The number of new confirmed cases that are reported across the country by day | |

Percentage of existing indigenous cases | The proportion of the number of existing confirmed cases per day that are reported domestically by day | |

Provincial distribution of existing confirmed cases | The number of provinces in the country where existing confirmed cases are reported by day |

*Note*: The statistical time of each day is from 0 to 24 hours.

indicates the judgement function of whether the epidemic prevention standard has been met on the day, refers to the collected COVID-19 status data vector on the day, , and is the number of total days in month ,

In Equation (4), is applied to counting the number of days that water users shut down. In particular, in Equation (4) indicates that it is not required that all abovementioned status indices meet the epidemic prevention standards at the same time in a day. For example, the number of existing confirmed cases is high, but most of these cases are imported from abroad; namely, the proportion of domestic cases is low. Then, epidemic prevention standards can also be met on the current day due to the strict prevention and control of imported confirmed cases. Hence, once the pandemic status meets the judgement function in Equation (4), water users can continue their normal business activities on a certain day.

This theorem is proved.

### Calculation of the DES under other types of emergencies

It is assumed that the DES under other types of emergencies is a random variable with a *lognormal* probability distribution function. Since the above distribution function is difficult to fit, first, water users’ resilience, which is a significant factor affecting the DES is considered. Then, according to the definition of the CRI, an uncertain comprehensive evaluation method is applied to calculate the probability distribution function of water users’ resilience. Finally, the expected value of the above DES is calculated by conditioning it on the resilience of water users.

**Theorem 2.**Let denote the DES under other types of emergencies, and let

*R*denote water users’ comprehensive resilience with a known probability distribution function. Then, the expected value of is as in Equation (6):where

*c*is a constant, is a given fixed parameter, is the possible value of random variable

*R*and refers to the water user resilience index under the th emergency, , and

*K*is the number of emergency types of interest.

** Proof**. Water users’ resilience is generally derived from the resilience of the community in which they are located. Community resilience usually contains social, economic and infrastructural elements, in which several quantitative variables are selected to assess the resilience index (Matin & Taylor 2015). Based on statistical data, the variables have been presented in the extant literature (Qin

*et al.*2017).

*m*is the number of years in which data were collected. We assume that water users’ degrees of resilience under different types of emergencies are different, and that these differences depend on the levels of the above three elements under different types of emergencies. Then the experience and knowledge of industry experts are investigated to determine the weights of under some given types of emergencies based on an uncertain comprehensive evaluation method. Water users’ resilience under a given type of emergency can be calculated by Equation (10):where is water users’ resilience index under a given type

*k*emergency, is the weight of element under a type

*k*emergency, , , and

*K*is the number of different types of considered emergencies. Let random variable

*R*denote water users’ resilience index, and let represent the event in which water users’ resilience index equals under a type

*k*emergency. Since emergencies are usually small probability events and are difficult to determine, according to the

*equally liability criterion*, it is assumed that the probability of occurrence of each type of emergency is equal; i.e., as in Equation (11):

*lognormal*distribution. Then, the expectation of is as in Equation (12). involves two parameters, i.e., and , where is a given fixed parameter, and is a function of water users’ resilience index

*R*.

*R*; as in Equation (13):where

*c*is a constant. Since

*R*is a random variable, it is clear that is also a random variable and has the following probability distribution, as in Equation (14).

This theorem is proved.

### Comprehensive DES under emergencies

*T*denote the total DES under emergencies; then, according to Equation (15):and where

*L*is the total days of the period of interest.

### Enterprises’ economic loss under emergencies

Enterprises’ economic loss under emergencies is the loss caused by the decline in enterprises’ production. Theorem 3 presents such economic loss.

** Proof**. According to input–output theory, enterprises’ economic loss is related to the decline in the enterprise production after the occurrence of emergencies, as well as the duration of such a decline (Xu & Liang 2019). To simplify the corresponding calculation, water users’ output curve before the occurrence of an emergency can be represented by its daily mean output value. In addition, it is assumed that after the occurrence of an emergency, the enterprise shuts down with no production, so the output value is zero. Hence, enterprises’ economic loss is defined as the product of the unit time loss and the DES of enterprises, according to the Equations (16) and (17). This theorem is proved.

### Actual water-saving benefit payment for WSSCs in the current year

Due to the stipulation in the WSMC, the water-saving benefit payment afforded to water users is dependent on the duration of their normal business operations in the current year. Theorem 4 shows the calculation of actual water-saving benefit payments.

**Theorem 4**. Let denote the actual water-saving benefit payment, which can be described by the following Equation (18):where is the required water-saving benefit for WSSCs in the current year, as stipulated in the contract, and

*L*is the total days in the current year.

** Proof**. The contract states that, the water-saving benefit payment afforded to water users is dependent on the duration of their normal business operations in the current year. Hence, the water-saving benefit payment is determined by the required water-saving benefit for WSSCs as stipulated in the contract and the duration of water users’ production decline in the current year; i.e., according to the Equation (18), where the second part of the right side of the above equation is the proportion of normal business days. This theorem is proved.

### Risk tolerance threshold and WSMC interruption results

If , then water users cannot bear the loss caused by an emergency, so they do not have the ability to pay the required benefit to the WSSC, and thus, the WSMC project is interrupted.

If , then water users can bear the loss and can pay the corresponding benefit to the WSSC, and thus, the WSMC project can continue.

### Estimation of the consequences of project interruption

*et al.*2021), and thus are not covered in this paper.

## RESULTS

### Background and data

Hebei University of Engineering (HUE) is located in Handan, Hebei Province, China. Handan suffers from a severe water resource shortage, and faces many difficulties in building a water-saving society. In December 2014, HUE signed an agreement with Beijing Cathay WSSC to start the WSMC project. This project adopts the shared mode, which means that HUE should share the water-saving benefit with Cathay WSSC for the purpose of the investment recovery of the latter. As a special type of water user, HUE obtains its financial income from Hebei Provincial Finance. In general, this income is used to pay the water consumption expenditure of its campus to the water supply company. Through the WSMC, a part of the remaining income is used to pay the water-saving benefits to the WSSC.

However, the worldwide outbreak of the COVID-19 pandemic in early 2020 disturbed the normal working plan of HUE. Since online teaching has become necessary, faculty members and students have had the dates they should return to campus postponed. In this case, the actual water consumption on the HUE campus is far less than that of the previous period. A reduction in water consumption means a reduction in water financial income from the central government. In this paper, the reduction in water expenditure of HUE refers to the economic loss of water users, and the duration of the postponement refers to the DES of HUE in terms of COVID-19. Thus far, we know that COVID-19 is a critical emergency worldwide. This paper attempts to evaluate the interruption risk of the WSMC project due to the postponement due to the impact of COVID-19 on HUE in 2020, and this risk level can be used for an assessment basis in a future period.

Currently, the COVID-19 pandemic is still ongoing, and its impact on WSMC projects remain unknown in 2021. In January 2021, there were still small outbreaks of COVID-19 in China. For example, on 6 January 2021, Shijiazhuang city, Hebei Province, began to report a number of new indigenous cases. Moreover, the impact of local conventional natural disasters and other emergencies in Handan have become of great concern to the WSMC implementers and potential participants. This paper considers 14 types of emergencies that may occur in Handan, including earthquakes, hailstorms, rainstorms, waterlogs, high temperatures, droughts, smog and haze, enterprise safety accidents, traffic and transportation accidents, public facilities and equipment accidents, environmental pollution, group diseases, food safety and occupational hazards, and economic security incidents. Considering the above emergencies and COVID-19, the DES of HUE in 2021 needs to be estimated, and the interruption risk of its WSMC project should be discussed.

To establish the IRI assessment model of the WSMC, the following data were collected.

**Data a:** Daily COVID-19 pandemic data from 20 January, 2020, to 20 January, 2021, including the daily number of confirmed cases, daily number of new cases, daily proportion of existing local cases and daily distribution of existing cases in 31 provinces and autonomous regions (excluding Hong Kong, Macao and Taiwan), are collected. Data are from National Health Commission of the PRC and Tencent epidemic real-time data platform.

**Data b:** Since no long-term quantitative prediction research on the COVID-19 pandemic has been found in China, the domestic pandemic is stable in 2021 according to international predictions. Therefore, in this study, the relatively stable data of the domestic epidemic from July to December 2020 were selected to approximately replace the COVID-19 status of 2021, and the data were extended to the whole year of 2021.

**Data c:** CRI data from 2015 to 2019 for Handan are collected. Data are from Handan Statistical Yearbooks from 2016 to 2020.

**Data d:** Experts’ data for the HUE resilience estimation are collected. Twenty industry experts from HUE, Handan City Bureau of statistics, China Institute of Water Resources and Hydropower Research (IWHR), Beijing Cathay WSSC, Tsinghua University and other units are invited to determine the weights for water user resilience based on the uncertain comprehensive evaluation method.

*et al.*2018), the importance degree of each indicator is divided into five levels, and each level is set as an independent

*Zigzag*uncertain variable, i.e. , , , , and . The expectation for each level is calculated below:

### Estimation of interruption risk of the WSMC project in HUE under COVID-19 in 2020

For the WSSC, the project interruption will cause additional long-term losses, namely, the unrecovered investment cost of the project and the opportunity cost of this part of capital.

### Comprehensive assessment of WSMC interruption risk of HUE in 2021

## DISCUSSION

The IRIs of HUE WSMC in both 2020 and 2021 are greater than their tolerance thresholds. These results show the high risk level of WSMC projects since their system is in its infancy. In general, however, the IRI in 2021 is less than that in 2020 and is even less than the risk tolerance threshold in 2020. This finding indicates that the impact of COVID-19 on water users in 2020 is very serious, especially when combined with the comprehensive impact of the other 14 types of emergencies in 2021. From Figure 4, the interruption risk of WSMC in 2021 results from the reduction in the risk tolerance threshold of HUE in 2021. Because of the decrease in the share proportion of water-saving benefits for the WSSC, the required payment of water users decreased, which reflects that their expected payment was less than before. The following sections discuss the interruption risk in 2020 and 2021.

### Analysis of the WSMC interruption risk of HUE in 2020 under COVID-19

#### An explanation that WSMC is theoretically interrupted but practically uninterrupted

According to the above results, the WSMC project of HUE in 2020 is theoretically interrupted. In fact, this project has continued under COVID-19 in 2021, and the following reasons for these results are presented.

- a.
As a public institution, HUE has relatively stable financial income, which does not affect its normal activities in the next year due to the decrease in income in the current year. Therefore, HUE dealt with losses in 2020 due to the impact of COVID-19, and normal financial income is thus supporting activities in 2021.

- b.
As a novel water-saving mechanism, the main purpose of the implementation of WSMC project in HUE is to save water resources and to enhance the awareness of water savings, rather than only to reduce water consumption. Therefore, the continuation of the WSMC project aims to achieve the original purpose of preserving water conservation when students and faculty members return to campus in 2021.

- c.
Beijing Cathay WSSC signed a contract for the WSMC project with HUE to participate in water resource conservation. Moreover, as a new market-based water-saving mechanism, the WSMC needs a display window through which the WSSC can promote the project and attract potential customers. Therefore, during the COVID-19 pandemic in 2020, Cathay WSSC also provided some preferential policies to jointly maintain the progress of the project.

Hence, the role of the experimental unit and the demonstration effect are the main reasons for the survival of the HUE WSMC project under COVID-19.

#### Analysis of the DES of HUE in 2020

The DES of HUE under COVID-19 in 2020 accounted for 67% of the year. This result shows that the water users faced a high risk of stopping business activities and becoming bankrupt. So there was made a selection of the epidemic prevention standard vector and the degree of the epidemic prevention standard every month were the main reasons for this.

The epidemic prevention standards selected in this study mainly refer to the existing literature and the actual average level of various industries. For different types of water users, the above standards are usually different, and their sensitivities to national epidemic prevention and control policies are also different. In this study, the four epidemic prevention standards,, were changed to obtain a comparison of their DESs, as shown in Figure 5. The above radar chart shows that the change in and has a great influence on the shutdown time of water users. When the two values become smaller, it will play a decisive role in the length of time that the water user is out of business. This finding suggests that companies pay more attention to epidemic prevention standards for newly confirmed cases. In addition, Figure 5 indicates the influence of index changes on the annual shutdown time of water users. Due to its slight change, we can observe the change in parameters from the number of days of failing to meet the epidemic prevention standard every month. Below, we look at the change in the number of days of shutdown for water users in each month for each epidemic prevention standard.

From Figure 6, it can be found that the parameter changes in the epidemic prevention standard vector had almost no effect on the number of shutdown days of water users in the early stage of the epidemic outbreak. The epidemic conditions of each day from January 2020 to March 2020 did not meet the epidemic prevention standards. After April 2020, although domestic epidemic prevention and control gradually stabilized, local reported cases and imported cases from abroad also increased the number of shutdown days per month for water users.

When calculating the cumulative number of days of shutdown per month by Equation (5), the standard is high for normal water users, which refers to the epidemic conditions that need to meet the epidemic prevention standards for every day in a month. In fact, after April 2020, with the reopening of most parts of the country, sporadic cases in individual areas arose because nationwide measures to suspend business were not being implemented. Therefore, this study adjusted the monthly epidemic prevention standard conditions, when the monthly was not more than 3, 5, 10 and 15 days to observe the annual number of days of shutdown of water users (Figure 7). This shows that in 2020, the pressure of epidemic prevention and control were still severe in all months, and the outbreak of local epidemics affected the situation of epidemic prevention and control in the whole country. When the standard was relaxed to 15 days per month, the number of DES days was less than the above results, but this standard could not be applied to practical prevention. There are great risks when water users continue their normal business activities under pandemic conditions and do not reach the required daily standards for approximately half of the month. This is irresponsible in terms of its effect on people's health and for the prevention and control of the epidemic nationwide.

According to the model calculation, HUE was in normal operations for 4 months in 2020. However, colleges and universities generally took the semester as the unit to implement teaching activities, and for the health and safety of teachers and students. In May 2020, although the conditions of the epidemic reached the standard, the school still did not open until the end of the semester after receiving guidance from authorities. In fact, HUE officially carried out offline teaching in the autumn of September 2020 until the beginning of January 2020.The model calculated that December 2020 and January 2021 did not meet the epidemic prevention standards. However, due to the particularity of the campus and the fact that local cases were reported at the end of December 2020, the semester ended normally. Therefore, although the shutdown time of HUE calculated by this model is somewhat different from the actual situation, universities, as special water users, were greatly affected by the policy; thus, this model still has some utility in describing their shutdown time.

#### Analysis of the risk tolerance threshold of HUE in 2020

According to Equation (20), water users’ loss bearing coefficient is set to 40% in HUE; nevertheless, when the value of is equal to 30%, normal enterprises have to either take remedial measures or shut down. On the one hand, the abovementioned particularity of HUE provides more space for loss bearing than that of general water users. Moreover, the interruption risk result of the WSMC project in HUE shows that even if the risk tolerance threshold of HUE becomes sufficiently high, its WSMC tends to be interrupted, which indicates that COVID-19 has a strong impact on water users with heavy losses. For general, for water users without other support, it is difficult to survive during this epidemic.

### Analysis of comprehensive assessment of the WSMC interruption risk in 2021

#### Analysis of the interruption risk of the WSMC in 2021

The results show that there is still a theoretical interruption risk of the WSMC project in HUE in 2021. As mentioned above, the IRI has decreased significantly compared with that in 2020. The reason why the interruption risk occurs is that the risk threshold is also reduced due to the reduction in required water-saving benefit payments in 2021. In general, the decreased IRI indicates that the impact of COVID-19 on water users has weakened, which increases confidence among project participants.

#### Analysis of the DES of HUE in 2021 under emergencies

Figure 8 shows the proportions of the DESs of COVID-19 and other emergencies in the total DES in 2021. Obviously, the DES under COVID-19 reflects the main problem in total DES in 2021. However, the above result does not indicate that water users have to stop their normal business activities again, as they did in 2020. The pandemic status of COVID-19 in 2021 is a conservative estimate based on the collected pandemic data from July to December in 2020. Actually, in terms of the four indices in Table 1, the pandemic status in China almost reached the required standards of Equation (23), since the implementation of epidemic prevention measures, such as flight fusing, vaccination, and activity limitations. Hence, the DES under COVID-19 may be shorter than the estimated DES, resulting in a lower interruption risk level of the WSMC project of HUE.

The determination of the water user resilience probability distribution function is derived from the three elements of the CRI based on expert data. Because the ability of an enterprise to sustain itself during an emergency varies according to its type, the possible value of water users’ resilience is the weighted sum of the social, economic and infrastructural resilience index. The above weights are determined by experts’ experiences due to the traits of the considered emergencies. For example, for emergency *floods*, the weight of the infrastructural resilience index is greater than that of the other two indices, but the weight of the economic resilience index is greater than that of the other two indices for emergency *droughts*. Hence, all the possible values of water users’ resilience under each type of emergency are obtained. This method overcomes the barrier that few data points can be collected from enterprises under emergencies. In addition, the traditional method for measuring an enterprise's resilience is in the form of a questionnaire, which directly interviews the person in charge of the enterprise to understand some of its situations. One of the most common indicators is the following: ‘How long does it take your business to recover from an emergency?’ We believe that the answer to the above question is affected by the resilience of enterprises, and for some types of emergencies, owners cannot answer this question at all. Hence, combining the statistical data and experts’ knowledge, the method in this paper can improve the objectivity of all water users.

According to Equation (13), it is assumed that there is a negative linear correlation between parameter and the water users’ resilience, which results in the expected DES under other types of emergencies being sensitive to constant *c*. The above result is derived from the DES being a lognormal random variable, and its expected value is an exponential form in terms of parameter . According to the industry's experience, we selected an appropriate value for *c* to process the calculation. In future research, more objective methods need to be investigated on how to determine the above parameters.

## CONCLUSIONS

The occurrence of emergencies increases water users’ losses and results in the interruption risk of the WSMC project. This paper has established a quantitative assessment model for the above risk and applied it in a WSMC project in China. The work in this paper showed the risk level of the case and analysed the main reasons for these results. The above work in this paper provides theoretical support for WSMC participants in dealing with emergencies and improves the WSMC mechanism system for larger markets. Coupling experts’ experiences with statistical data, water users’ resilience was dealt with as a random variable with equal probability of distribution. Some functions and parameters used in this method are still subjective and limited, and thus, this topic is one direction for future research.

## ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (No. 61873084). We thank the National Health Commission of the PRC (http://www.nhc.gov.cn/) and the Tencent epidemic real-time data platform (https://xw.qq.com/act/qgfeiyan).

## DECLARATION OF COMPETING INTERESTS

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

## DATA AVAILABILITY STATEMENT

All relevant data are available from an online repository or repositories at the National Health Commission of the PRC http://www.nhc.gov.cn/xcs/xxgzbd/gzbd_index.shtml.